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https://mathoverflow.net/questions/323191/fine-tuning-the-growth-rate-of-the-degrees-of-polynomials	# Fine tuning the growth rate of the degrees of polynomials Let $$r$$ be an integer with $$r>1$$. Suppose that if $$k\geq 0$$, then $$p_{k}(x)$$ is a polynomial with nonnegative integer coefficients with $$p_{k}(0)=1$$ but where $$p_{k}\neq 1$$. Suppose that $$\prod_{k=0}^{\infty}p_{k}(x)=\frac{1}{1-rx}.$$ For each $$n>0$$, let $$t_{n}$$ be the number indices $$k$$ where $$\deg(p_{k}(x))=n$$. Then is it possible to select polynomials $$(p_{k})_{k\geq 0}$$ where $$\|t_{n}-\frac{r^{n}}{n}\|=O(\alpha^{n})$$ for each $$\alpha>1$$? How slowly can the function $$n\mapsto\|t_{n}-\frac{r^{n}}{n}\|$$ grow? How slowly can the function $$n\mapsto\max(0,\frac{r^{n}}{n}-t_{n})$$ grow? For example, can we have $$\max(0,\frac{r^{n}}{n}-t_{n})=O(\alpha^{n})$$ for all $$\alpha>1$$? This question is motivated by very large cardinals. • It's not so hard to get $\alpha = \sqrt{r}$, or, for the second version, $r^{1/3}$. – Will Sawin Feb 14 at 3:57	2019-03-25 14:59:12	{"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": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 18, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.916174590587616, "perplexity": 154.40055180239173}, "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-2019-13/segments/1552912203991.44/warc/CC-MAIN-20190325133117-20190325155117-00263.warc.gz"}
https://testbook.com/question-answer/find-the-dynamic-load-carrying-capacity-of-a-rolle--5ff4aa0cce9289888f3b5091	# Find the dynamic load-carrying capacity of a roller bearing if the shaft rotates at 1500 rpm, the radial load acting on the bearing is 6 kN and the expected life for 90% life of the bearing is 8100 hours. Free Practice With Testbook Mock Tests ## Options: 1. 6 kN 2. 44 kN 3. 54000 kN 4. 60000 kN ### Correct Answer: Option 2 (Solution Below) This question was previously asked in UPPSC AE Mechanical 2019 Official Paper I ## Solution: Concept: The approximate rating of the service life of a ball or roller bearing is based on the given fundamental equation. $${\rm{L_{10}}} = {\left( {\frac{{\rm{C}}}{{\rm{P}}}} \right)^{\rm{k}}}$$ L10 = Rated bearing life (in million revolutions) P = Load acting in bearing. k = 3 for ball bearing and k = 10/3 for the roller bearing. The relation between life in million revolutions and life in hours is given by: $$L_{10}=\frac{L_{10h}\;\times \;N \;\times \;60}{10^6}$$ where, L10h = rated bearing life in hours and N = speed of rotation in rpm. Calculation: Given: N = 1500 rpm, P = 6 kN, L10h = 8100 hours, roller bearing k = 10/3 We know that; $${\rm{L_{10}}} = {\left( {\frac{{\rm{C}}}{{\rm{P}}}} \right)^{\rm{k}}} \times {10^6}{\rm\;{revolution}}$$ and; $$L_{10}=\frac{L_{10h}\;\times \;N \;\times \;60}{10^6}$$ Combining both equations: $$\frac{L_{10h}\;\times \;N \;\times \;60}{10^6}= {\left( {\frac{{\rm{C}}}{{\rm{6}}}} \right)^{{\frac{10}{3}}}}$$ $$\frac{8100\;\times \;1500 \;\times \;60}{10^6}= {\left( {\frac{{\rm{C}}}{{\rm{6}}}} \right)^{{\frac{10}{3}}}}$$ C = 43.34 kN	2021-08-03 10:33:11	{"extraction_info": {"found_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": 0.2467358261346817, "perplexity": 6540.866991535612}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1627046154457.66/warc/CC-MAIN-20210803092648-20210803122648-00277.warc.gz"}
http://tex.stackexchange.com/tags/math-mode/info	# Tag Info is about typesetting mathematical content, e.g. tweaking the appearance of spacing and symbols in a formula, or producing specific mathematical constructs. For questions about "wrapper" environments like equation or align, consider using . If you have questions about specific mathematical symbols, add and check out How to look up a symbol? first. Tutorials and documentation Useful packages A selection of good questions Special tags External links and descriptions are taken from the TeXblog maths collection.	2014-11-23 02:15:44	{"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": 0.7959126234054565, "perplexity": 6800.29596421318}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416400378956.26/warc/CC-MAIN-20141119123258-00198-ip-10-235-23-156.ec2.internal.warc.gz"}
http://mathoverflow.net/questions/36260/roots-of-quadratic-forms	This may be a very silly question, but I was wondering what is known about the roots of a quadratic form over variables $x_1,\ldots,x_n,y_1,\ldots,y_m$ in the finite field $\mathbb{F}_p$. I'm not even so much interested in characterizing the solutions as I am in counting them (or showing some relation between them), but any help would be greatly appreciated (as well as a pointer to a good introductory article/text on quadratic and modular forms in general).	2015-05-24 03:39:32	{"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": 0.5040785074234009, "perplexity": 70.37928734798035}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207927824.81/warc/CC-MAIN-20150521113207-00319-ip-10-180-206-219.ec2.internal.warc.gz"}
https://math.stackexchange.com/questions/2958316/interpretations-and-models-in-propositional-logic	# Interpretations and models in propositional logic I am trying to understand model vs. interpretation of sentences vs. theories, with respect to a propositional calculus. Is a model of a wff just a sentence that is true under some interpretation? Or a set of wffs that are all true under some interpretation? Or when an interpretation satisfies a wff (or set of wffs)? Does it make sense to say that an interpretation is a model if all valid propositions are true? Or an interpretation being a model iff all theorems from the theory are still valid under the interpretation? • What books are you reading? There's a certain variety in definitions. – Fabio Somenzi Oct 16 '18 at 18:46 • It's possible that considering propositional logic is making this too simplified, and that, at the cost of some additional complexity, the distinctions would be clearer in predicate logic. For example, you could identify a truth function $\mathsf{Prop}\to\mathbf{2}$ with a subset of $\mathsf{Prop}$, the set of propositions. Doing this for first-order logic, though, doesn't make as much sense. (The equivalent would be making a family of relations, i.e. subsets of a cartesian product, indexed by arity, $n$, of the form $\mathsf{Pred}_n\times D^n$. The propositional case is then $n=0$.) – Derek Elkins left SE Oct 16 '18 at 21:59 An interpretation of a truth-functional [i.e. classic] propositional calculus $$\text P$$ is an assignment to each propositional symbol $$\text P$$ of one or the other (but not both) of the truth values truth ($$\text T$$) and falsity ($$\text F$$), and an assignment to the connective symbols of $$\text P$$ of their usual truth-functional meanings. Example : Let the language of $$\text P$$ made of the following list of prop symbols : $$\text {At} = \{p_0, p_1,\ldots \}$$ and let $$\{ \lor, \lnot \}$$ the set of connectives. An interpretation is an assignment $$v : \text {At} \to \{ \text T, \text F \}$$ such that, e.g. $$v(p_0)= \text T$$ and $$v(p_1)= \text F$$, etc. Using $$v$$ and the truth tables for $$\lor$$ and $$\lnot$$ we can easily compute the truth value of a formula whatever of $$\text P$$, like e.g. $$(p_0 \lor \lnot p_1)$$. If $$\varphi$$ is a formula of $$\text P$$ and we have $$v(\varphi)= \text T$$, we say that the interpretation $$v$$ satisfies formula $$\varphi$$ (and we can write : $$v \vDash \varphi$$). A formula of propositional logic is true under an interpretation iff the interpretation assigns the truth value $$\text T$$ to that formula. If a formula is true under an interpretation, then that interpretation is called a model of that formula. Thus, an interpetation satisfies a formula $$\varphi$$ iff it is a model of the formula. In (classic) propositional logic a formula $$\varphi$$ is a tautology (or valid) iff it is true in every interpretation, i.e. such that : $$v \vDash \varphi$$, for every assignment $$v$$. Examples of tautologies : $$(p_0 \to p_0), (p_0 \lor \lnot p_0)$$. • So an interpretation can be a model of a formula, $v \models \varphi$, but then we are also able to say a set of wffs "models" a formula, e.g. $\Gamma \models \varphi$? Or is this wrong and it's more accurate to say $\Gamma \models \varphi$ means $\varphi$ is satisfied in all models of $\Gamma$? – user525966 Oct 16 '18 at 19:06 • @user525966 - Yes; the symbol $\Gamma \vDash \varphi$, with $\Gamma$ a set of formulas, means that $\varphi$ is a semantic consequence of $\Gamma$. See your previous post How is my understanding of ⊢,⊨,→, etc – Mauro ALLEGRANZA Oct 16 '18 at 19:15 • @user525966 The second one. You may be confused by the command \models. This symbol is used in two totally different ways. $\mathcal M\models \Gamma$ “the structure $\mathcal M$ is a model for the set of sentences $\Gamma$” is the use where “models” is a apt verb. The usage $\Gamma\models \phi$ would never be put into English as “$\Gamma$ is a model for $\phi.$.” (In fact, some model theory books, that don’t need to worry about confusion with the syntactical notion, use $\vdash$ here instead.) – spaceisdarkgreen Oct 16 '18 at 19:21 • @user525966 the second usage would be put into English as “$\phi$ is a semantic consequence of $\Gamma.$” – spaceisdarkgreen Oct 16 '18 at 19:27 • @user525966 no. A model of what? All valid propositions are true in any interpretation whatsoever. That’s what it means to be valid. – spaceisdarkgreen Oct 16 '18 at 19:34	2020-10-25 05:30:53	{"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": 28, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9158371090888977, "perplexity": 351.8182522343551}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107887810.47/warc/CC-MAIN-20201025041701-20201025071701-00507.warc.gz"}
https://math.stackexchange.com/questions/3363654/number-of-non-isomorphic-k-partite-graphs	# Number of non-isomorphic k-partite graphs. There is a set of vertices and it is divided into $$k$$ independent sets. The cardinalities of these sets are following: $$a_{1},a_{2},...,a_{k}$$ How many non-isomorphic graphs determined on these sets are there? I hope that someone will help me. Regards. • Where does this problem come from? The answer is, as far as I know, not known even for bi-partite graphs.. See for example math.stackexchange.com/questions/2321657/… – N. S. Sep 20 at 18:51 • The question is easy if you ask for the number of different labeled such graphs, but "isomorphic" makes it very hard. – N. S. Sep 20 at 18:52 • This MSE link treats the case of non-isomorphic bipartite graphs. – Marko Riedel Sep 21 at 14:44	2019-10-14 20:51:54	{"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": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 2, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8481837511062622, "perplexity": 395.7749627296586}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1570986655310.17/warc/CC-MAIN-20191014200522-20191014224022-00075.warc.gz"}
http://codeforces.com/blog/entry/62218?locale=ru	Автор majk, 2 недели назад, , Hungry for yet another contest? On Sunday, October 7, 2018 at 17:05 UTC the Lyft Level 5 Challenge will start with the Round 1! This is a combined round having 7 problems and lasting 2 hours, and it will be rated. The top 100 participants of this round will win a Lyft Level 5 Challenge t-shirt. The top 30 contestants located in the San Francisco Bay area will be invited to the Final Round. In the Final Round the top three onsite contestants will fight for the cash prizes: • First place: $2000 • Second place:$1000 • Third place: \$500 Interested in an internship or a job at Lyft? Many thanks to: I'll be on the community Discord server shortly after the contest to discuss the problems. UPDATE 1: The scoring distribution will be 500-1000-1500-2250-2750-3250-4000. UPDATE 2: The contest is over and there is an editorial. UPDATE 3: Congratulations to the winners: • • +352 • » 2 недели назад, # \| -78 But, is it rated? • » » 13 дней назад, # ^ \| -22 Maybe you can have a look at the first paragraph :D • » » » 13 дней назад, # ^ \| +69 Maybe you can have a look at the tags :D The last tag is obviously the reason he posted that. • » » » » 13 дней назад, # ^ \| ← Rev. 2 → +33 Well some people didn't get that :p • » » » » » 11 дней назад, # ^ \| -12 IT WON'T BE RATED!!! --------> https://www.youtube.com/watch?v=gehfVhX7Spk • » » » » 13 дней назад, # ^ \| +30 Seems I didn't realize that it's a joke( » 2 недели назад, # \| 0 Clashes with the last 15 minutes of Liverpool vs Man city, sorry but watching and supporting Mo Salah is my choice :D • » » 2 недели назад, # ^ \| ← Rev. 2 → +95 I've spoken with him and he said he would not play football because match clashes with contest ;D • » » » 2 недели назад, # ^ \| +38 Maybe Mo wants to participate and win a T-shirt :) • » » » » 12 дней назад, # ^ \| +48 Guess we could call his solutions... Mo's algorithm. • » » 2 недели назад, # ^ \| -45 While the world cup takes place in Russia. I do not see any comment on football. Not relevant, but I like Messi :) » 13 дней назад, # \| +1 On the other Lyft Level 5 challenge blog, it still says the elimination round has 5 problems. Could you change that please? • » » 13 дней назад, # ^ \| +12 Thank you, fixed » 13 дней назад, # \| -32 Look at the tags!! One of them is "dont ask if rated" LOL XDXDXD!!!! » 13 дней назад, # \| -9 Its time is very unfriendly to Chinese players. • » » 13 дней назад, # ^ \| +117 Blame Trump • » » 12 дней назад, # ^ \| +3 Isn't 1 AM a nice time since it's late enough that it definitely won't conflict with anything but it's also not too late to stay up for? That's how I feel about 1 AM contests, anyway. • » » » 12 дней назад, # ^ \| +6 I see you don't have a job/class you have to be at 9 am on Monday ;) • » » » » 12 дней назад, # ^ \| +3 Oh I have class, I just don't go if I don't wake up in time :P • » » » 12 дней назад, # ^ \| +13 Found the American. » 13 дней назад, # \| +58 So about the internship/jobs will you take look at the cf profiles or just consider the contest results ? • » » 12 дней назад, # ^ \| +68 That moment when you are confused whether to risk international master or not » 13 дней назад, # \| +73 [Many thanks to] Alice and Bob for playing a crucial role in some of the statements How come no one ever thanks Vasya for his help in problem statements :( • » » 12 дней назад, # ^ \| +3 Cuz he is green. » 13 дней назад, # \| +101 is it rated? can it bring inspiration or disappointment to us? » 13 дней назад, # \| +32 majk your last contest also was about alice and bob, do you have any fantasy? » 12 дней назад, # \| -40 Is It RaTeD ? • » » 12 дней назад, # ^ \| +49 swap(Is,It) • » » » 12 дней назад, # ^ \| +22 It Is RaTeD? » 12 дней назад, # \| ← Rev. 2 → 0 It seems that I can register whenever I want until the contest ends.So, is it a ACM-like contest?If yes, is it of extended-ACM style (with hacking phase after the contest ends) ?Sorry for my unnatural English. • » » 12 дней назад, # ^ \| +2 As it has point distribution so i hope it will be like normal codeforcses rounds » 12 дней назад, # \| +18 It is a CuSi Round for Chinese. » 12 дней назад, # \| ← Rev. 2 → -20 Set the start 15 min later plz! :'(( Liverpool vs ManCity! :(( • » » 12 дней назад, # ^ \| -7 Never mind, WreckingBall has solved the problem! ;D • » » » 12 дней назад, # ^ \| -7 The game whitout Mo is perfect to! :( • » » » » 12 дней назад, # ^ \| 0 My last hope crashed ); » 12 дней назад, # \| -25 I wish everyone success ! • » » 12 дней назад, # ^ \| -27 Or maybe I wish myself contribution !? • » » » 12 дней назад, # ^ \| -30 Maybe ! • » » » 12 дней назад, # ^ \| 0 I see some bitter irony in all this downvotes. » 12 дней назад, # \| -56 ?DETAR TI SI » 12 дней назад, # \| ← Rev. 2 → -11 Hope, It will be an awesome round for both div1 and div2 users. :) • » » 12 дней назад, # ^ \| +8 Before getting job, i need some positive rating ! » 12 дней назад, # \| ← Rev. 3 → -31 :( • » » 12 дней назад, # ^ \| 0 And also downvoting whoever say it... » 12 дней назад, # \| 0 Clashes with Liverpool vs Manchester city :-/ » 12 дней назад, # \| +10 I dont wanna die without a tshirt. » 12 дней назад, # \| +1 I just wonder that why there's no explanation about interactive problems in the announcement, some kind of information I mean. » 12 дней назад, # \| 0 Contest over in an hour.never done interaction problem before..It would have been better if there was just one. » 12 дней назад, # \| +3 When tourist is there.. no one has to worry that a question will go unsolved..:) » 12 дней назад, # \| 0 Sorry for silly ques..Can someone share his approach for Problem C.. I used DFS but got TLE in TC 6.. • » » 12 дней назад, # ^ \| +2 Maybe you didn't go from the biggest number to the smallest... • » » » 12 дней назад, # ^ \| 0 I did go from biggest number to smallest. Code • » » » » 12 дней назад, # ^ \| 0 I just can't see you code on Codeforces before the end of the systesting... • » » » » » 12 дней назад, # ^ \| +1 Made redundant recursive calls and hence it caused TLE. Now I get AC! >_< • » » » » » » 12 дней назад, # ^ \| 0 Oh, thanks! I've just find out that the order doesn't matter. » 12 дней назад, # \| +29 How to solve D? Couldn't get AC with Pollard Rho. • » » 12 дней назад, # ^ \| +15 Given a number x there's 4 probabilities for iteither x = p^2 or x = p^3 or x = p^4 or x = pq where both p and q are primesso if x has a root,then you can know the primes easily without any struggle,the only problem is pqyou can just try taking the gcd of x with other numbers in the array,if all gcd == 1 or x then p and q are primes number that appears for the first time • » » » 12 дней назад, # ^ \| 0 same approach, but verdicts WA, any corner case? • » » 12 дней назад, # ^ \| ← Rev. 2 → +3 I have solved it using Shanks's square forms factorization which will give you any divisor of n, x such that 1 • » » 12 дней назад, # ^ \| +8 I solved with Fermat factorization (ofc after eliminating p^2, p^3, p^4 and small prime numbers) » 12 дней назад, # \| +5 what are you supposed to do with D if it is a large semiprime • » » 12 дней назад, # ^ \| ← Rev. 3 → -8 I used this method:http://www.naturalnumbers.org/Qfactor3.htmlUPD : Got TLE :/ Nevermind • » » 12 дней назад, # ^ \| +65 Compute gcd between every pair of elements to get some candidates for primes. If those don't divide some not-prime-power number in the input, assume that it consists of two unique primes. • » » » 12 дней назад, # ^ \| 0 thought about this, didn't notice n was 500. phew! • » » » 12 дней назад, # ^ \| 0 I didn't got your solution. Can you please explain a little bit... • » » » » 12 дней назад, # ^ \| +2 See my code or read the editorial. • » » 12 дней назад, # ^ \| +4 You can find the two prime factors x = p * q by gcd with every other number. If that is not possible (the prime factors don't appear in the other numbers, or it only appears in numbers that are equal to the first one) then the prime factors p and q appear in the product the same number of times as x appears in the array. • » » 12 дней назад, # ^ \| ← Rev. 2 → 0 Assuming you have only semiprimes remaining, you could try factoring some of them by calculating GCD of every pair.EDIT: Too late :P » 12 дней назад, # \| 0 Why doesn't Codeforces has __int128? :'( I had problem D if it wasn't because of it » 12 дней назад, # \| 0 Solution for D? • » » 12 дней назад, # ^ \| ← Rev. 3 → +6 There can be four cases in total: The number has 3 divisors, therefore the number is square of some prime. The number has 5 divisors, therefore the number is 4th power of some prime. The number has 4 divisors, the number might be cube of some prime. The number has 4 divisors, the number might be product of two distinct primes. 1,2,3 can be done easily. 4 can be done by this link: http://www.naturalnumbers.org/Qfactor3.html • » » » 12 дней назад, # ^ \| 0 thanks +1 • » » » 12 дней назад, # ^ \| ← Rev. 3 → 0 You wrote "4. The number has 4 divisors..."I think you meant 2EDIT: Nvm i read incorrectly • » » » » 12 дней назад, # ^ \| +5 Hmm, the number has between 3 and 5 divisors. I think you might be mistaken somewhere. » 12 дней назад, # \| ← Rev. 3 → -12 Can anyone explain me how to take input in D? I mean what the fuck is going on there? I kept getting idleness time exceeded, Can anyone tell me why that problem was interactive? • » » 12 дней назад, # ^ \| -8 A number that has 3, 4 or 5 divisor should be a p1p2, p1^2, p1^3 or p1^4, where p1 and p2 are primes. To know what are the primes that multiply two different numbers, take the gcd of each two different numbers in input that is not a power. • » » » 12 дней назад, # ^ \| ← Rev. 13 → 0 Bro, i am not asking the solution, i am asking about the input format.here's my solution: http://codeforces.com/contest/1033/submission/43971194 Tell me why is this giving idleness time exceeded? • » » » » 12 дней назад, # ^ \| 0 You need to flush the output using fflush(stdout) or cout.flush() in C++;. Anyways you'll fail pretest 1. • » » 12 дней назад, # ^ \| 0 You have to flush the output • » » » 12 дней назад, # ^ \| 0 Yeah, i used endl for that. » 12 дней назад, # \| +5 Does anyone know why I might have gotten TLE on problem C pretest 6? » 12 дней назад, # \| 0 I forgot to input n in problem A and spent 1 hour to find the mistake... I need to sleep more often... » 12 дней назад, # \| +5 what a contest.Mathematics rocks. » 12 дней назад, # \| -14 What if the king is being checked in the initial postion in problem A? • » » 12 дней назад, # ^ \| 0 The statement guarantees it's not. • » » » 12 дней назад, # ^ \| -8 Where? • » » 12 дней назад, # ^ \| 0 It is guaranteed that Bob's king is currently not in check and the target location is not in check either. • » » » 12 дней назад, # ^ \| +6 nvm im stupid » 12 дней назад, # \| ← Rev. 3 → 0 What is wrong with this thinking in D?First if a number has 5 divisors it has to be p^4 formif a number has 3 divisors it has to be p^2 formif a number has 4 divisors, we have 2 cases: p^3 or pqall other but pq can be checked easily. for pq part we can first process all other numbers and have a map of divisors and its count. Then for all numbers not processed (pq form) we can check only the divisors which have occurred in previous numbers. Else we dont care about the divisors and just say the count of some divisor would be 1. then the number of divisors would be multiplication of (number of occurrence + 1) for each prime numberEdit Nevermind. for the pq case I considered that all the numbers cannot share a common prime which is obviously a stupid mistake. • » » 12 дней назад, # ^ \| +3 There might be some p * q number with the same values, or at least share the same prime divisor(s). • » » » 12 дней назад, # ^ \| 0 Thanks. realized it later. • » » 12 дней назад, # ^ \| 0 If I understand this correctly, it fails if your input is p2, pq, because you ignore the p2 when you process pq even though they share a factor in p. • » » 12 дней назад, # ^ \| +3 A number of pq form may have a common factor with another number of pq form, which changes calculations a bit! Thus, we need to look into this too, by pairwise checking of any common factors. Even after this some number goes unprocessed, ( we were unable to factorise this pq form number), it means it consists of 2 entirely different primes. » 12 дней назад, # \| 0 In E my approach was to start with n bipartite graphs, binary search for last graph to find a graph such that there are edges between these 2 and then merge them if there exists a valid way. But i cant think how to find an odd cycle if we get to know that there is no valid merging of the 2 bipartite graphs?? » 12 дней назад, # \| -14 Problem D solution can be found here • » » 12 дней назад, # ^ \| -9 Why copied problems.who knows it will just copy paste • » » » 12 дней назад, # ^ \| +20 I see very little in common between these two problems, it is pretty much like saying "Here is wiki article about problem D". • » » » » 12 дней назад, # ^ \| +2 ohh i just wrote it because link leads to a code • » » » » 12 дней назад, # ^ \| ← Rev. 2 → -19 - » 12 дней назад, # \| 0 How to Solve C?? • » » 12 дней назад, # ^ \| -18 Just simple DFS • » » » 12 дней назад, # ^ \| +3 I am stupid • » » » » 12 дней назад, # ^ \| -54 Yes, you should stop doing CP. • » » 12 дней назад, # ^ \| 0 You start from the greatest element and mark that Bob wins that because Alice cannot go to a greater element for sure. Then go to the next smaller number let it be num, let its position be i. You should look at i-num, i-2num .. i+num, i+2num. if there is any case where Bob wins in any of those numbers that means Alice can win because Alice will go there and it will be Bobs turn and we know that if Alice starts there Bob could win so if Bob starts Alice can win. If there isnt any such position you mark it as Bob wins. You keep doing this. • » » 12 дней назад, # ^ \| 0 For every i-th, check i - a[i], i - a[i] 2, ..., i - a[i] * k > 0, AND i + a[i], i + 2a[i], .... , i + a[i] l <= n. Total: n/a[i] iteration. sum of all iterations: sum(n/a[i],i=1..n) = sum(n/i, i = 1..n) ~ O(nlog(n)) • » » » 12 дней назад, # ^ \| 0 But those values must be pre-calculated to check that right? How to do that? • » » » » 12 дней назад, # ^ \| +2 Recursive dp. • » » » » 12 дней назад, # ^ \| +1 const int bob = 2; const int alice = 1; int dp[1<<18]; ... memset(dp, 0, sizeof(dp)); for( i = 1; i <= n; ++i) dp_rec(i); .... void dp_rec(i){ if (dp[i])return; // already calculated. dp[i] = alice; // we assume alice will win. for(int p = i - a[i]; p > 0; p-= a[i]) if (a[p] > a[i]){ dp_rec(p); if(dp[p]==bob) return; } for(int p = i + a[i]; p <= n; p += a[i]) if(a[p] > a[i]) { dp_rec(p); if(dp[p] == bob) return ; // if on p - BOB win, for i-th position alice win. } dp[i] = bob; } • » » » » » 12 дней назад, # ^ \| +5 yeah! thanks.My bad! I kept thinking there must be some simpler solution than memoized dp during the contest , considering it a C problem :(((( » 12 дней назад, # \| 0 In D, I have maintained a hash map of primes to their frequency. First check if it is in the form of p^4, p^3 or p^2. If so update the prime in the hash map and make it 1 in the array. then iterate through the array and find how many numbers are equal to current number, let it be counter. make those elements 1. Take gcd with next elements and when gcd is > 1. let gcd be g, so the current number will have two primes g and number/g. iterate the rest of the array and find how many times g and number/g occurs as divisors, divide the numbers by g or number/g according to which one divides them. Update hash map with cnt+their frequency. For all primes in the hash map iterate the array and see if they divide anything and make those elements in the array 1 and update the map. Finally iterate the array multiply the ans by 2 if there is any elements > 1.If this is the approach I will be sad because I used int instead of long long at some place and got a wrong answer.. » 12 дней назад, # \| ← Rev. 2 → +1 My solution for D:Notice that to have 3, 4, or 5 divisors, the number must be in the form p2, p3, p4 or pq, for primes p, q. It is easy to check if it's one of the first three cases, in which case we can store a map of primes to the number of times they exist in the total product.If we encounter a number of the form pq, then we gcd with all the previous elements to find a possible prime factor. If we find a prime factor, we then know p and q and can process it as we processed the other numbers. If not, then we count how many previous instances of pq happened (call this k), and multiply our current answer by .EDIT: Oh but I failed systests anyway RIP. • » » 12 дней назад, # ^ \| 0 I instead multiplied it by (k+1)^2 as there is no other occurrence of those p and q. Why are you multiplying with ((k+2)/(K+1))^2 ? » 12 дней назад, # \| 0 Why I am not able to submit the solution after the contest ends • » » 12 дней назад, # ^ \| 0 Because the system test is still running » 12 дней назад, # \| +33 why is D a "technically" iteractive problem? I dont get it • » » 12 дней назад, # ^ \| +75 Because of hacks. When you submit a hack, we need to verify whether it is a valid test case, i.e. whether the number has between 3 and 5 divisors. You can't do that without factorising the input, and that factorisation cannot be done fast enough.For that reason, you submit the factorisation as a hack. The interactor then reads this factorisation, and writes the product on its standard output. From then on, it's just a normal (non-interactive) task.I am sorry for the confusion that it apparently caused. • » » » 12 дней назад, # ^ \| +3 "You can't do that without factorising the input, and that factorisation cannot be done fast enough." So what exactly will go wrong if you'll try to factorize it? Will it be something like "checker TL exceeded", since you have to proceed that input within (let's say) 1 second, or is it "we can do that, but it will load CF servers too much" (sounds questionable due to small number of hacks). I don't think it is "you'll have to wait for the verdict for too long" either, since it isn't instant anyway. Could you please add a few details about what the issue is, for those who don't know how CF works? I'm just curious about what is the technical limitation behind it. • » » » » 12 дней назад, # ^ \| +6 You would get a "Judgement Failed" verdict with a comment that validator exceeded the time limit. Which is what I was getting in Polygon when preparing the problem.Someone with a deeper knowledge of the platform might be able to tell you more details. • » » » » » 12 дней назад, # ^ \| +10 Thank you! Together with comment by KAN, it cleared my doubts. • » » » » 12 дней назад, # ^ \| +5 AFAIK the TL on any judge executable (validator, checker, etc) is 10 or 20 seconds. I suppose it can be raised, but is it needed? Also, we'll have to have the same format for all system tests, so we need to factorize them too, that might take long time. » 12 дней назад, # \| +22 Nice contest,Beautiful problems » 12 дней назад, # \| 0 Hi majk, arsijo,There is something wrong with the checker for task D. For example this submission is getting WA on test 9:I assume that the test case is: 3 30 30 42My program locally returns 24 and the checker says it returns 45. I do not see any reason why there should be any difference. Could you take a look into this?Thanks! • » » 12 дней назад, # ^ \| 0 The input is 3 15 15 21 (the first integer on the line is the number of prime factors of ai). » 12 дней назад, # \| -18 Precision... how I love it. Changing from int(c * (1/3)) to int(c ** (1/3) + 0.0001) changed WA11 on AC • » » 12 дней назад, # ^ \| +77 After running into such stuff once or twice you'll probably start doing things like "let's check +-10 in each direction, just to be sure", even if you don't really need that :) • » » 12 дней назад, # ^ \| +10 And then comes a time you have to do something like this XD long long int sq = sqrt(n); while(sqsq > n) sq--; while(sqsq <= n) sq++; » 12 дней назад, # \| +14 If only the constraints of D was 1e18,I've been a candidate :(( » 12 дней назад, # \| 0 Until this moment I realized how weak the pretests of D was. Still, my bad after all, so no huge complaints. Great problemset overall, to be honest (even I myself took away my very chance to be in top100). » 12 дней назад, # \| +14 Nice Contest , Quick editorial,Fast sys testing. Thanks Codeforces!!!! » 12 дней назад, # \| -15 Wow I barely reached yellow! Purples are half div2 — half div1, so now I'm completely div 1. Also it seems now I'm a nerd: http://codeforces.com/blog/entry/61683#comment-456882 :) » 12 дней назад, # \| +15 Why did you cut off solution from solution? First one is already tedious enough to write... • » » 12 дней назад, # ^ \| +7 In reality, the uses around 17000 queries. » 12 дней назад, # \| 0 How well did we need to do to get an intership? » 12 дней назад, # \| ← Rev. 2 → +7 My C got TLE, it ran in 1450ms with time limit being 1s, can anybody explain why is it so? It is sieve-like implementation so the time complexity should be O(n*log(n)).Hi, majk, could you help?Submission : 43960105 • » » 12 дней назад, # ^ \| +86 The sieving part is correct, the part to blame is actually: rep(i,0,n){ s=s+'0'; } Sadly, this is .Alternatives are s += '0' use std::stringstream std::string s(n, '0') The best one in this particular case is 3. • » » » 12 дней назад, # ^ \| +27 Thanks for your reply, it now runs in 77ms. » 12 дней назад, # \| 0 When i though about C problem as a DP problem on DAG, i defined the state such thatdp[i][j]=1 if the player j can win when the game starts at position i, else it equals 0.I found some submissions only use dp[i] and nothing else changed in the rest of the solution, so what is the idea behind the possibility of removing the second dimension in my defined state?Thanks in advance :) • » » 12 дней назад, # ^ \| ← Rev. 4 → +10 If one of the player wins for a given position, then the other one loses. Because there is always one player who wins and the other one loses. I don't see why would you need a second dimension.Unless your state refers to which player starts playing, but I still don't understand why it would be needed since it is specified in the statement that Alice starts playing. • » » » 12 дней назад, # ^ \| +10 Ok i got it, Thanks brother :) » 12 дней назад, # \| 0 What corner case does test 16 for problem D have? My code passed pretests but gave WA for that one. • » » 12 дней назад, # ^ \| ← Rev. 2 → +26 Test 16 is a random test, where each number is a product of two primes slightly over 109. Pretest 10 is the same random test, but with a different seed. So there should not be anything special about it. » 12 дней назад, # \| +63 How do the organizers know who lives in the Bay Area? • » » 12 дней назад, # ^ \| 0 BTW, should you permanently live in Bay Area or you can just be here for the time of contest? • » » » 11 дней назад, # ^ \| ← Rev. 2 → +18 "We invite top-30 participants who don't need sponsored travel to our location. If you get good score in round-1, want to be invited to onsite, and can manage travel costs, don't hesitate to send a personal message and we'll definitely consider you when processing scoreboard."This was what they said here. So, you don't even really need to live there. You just need to be there at the time of the contest. » 12 дней назад, # \| ← Rev. 2 → 0 In problem C , can any one tell me what's wrong in my approach ? why does it give wrong answer on Test 23 ?http://codeforces.com/contest/1033/submission/43970251 • » » 12 дней назад, # ^ \| 0 I solved it with DP , states are , Current State is to pick the optimal choice depends on transitions (For example :- currentTurn for Alice , Alice must search for any odd moves to Win and Bob must search for even moves to Win ) » 9 дней назад, # \| 0 This algorithm is great! It has helped me alot in this problem!	2018-10-19 12:25:45	{"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": 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": 0.5195930600166321, "perplexity": 4746.228474016757}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583512395.23/warc/CC-MAIN-20181019103957-20181019125457-00352.warc.gz"}
https://gmatclub.com/forum/if-a-2b-17b-2a-1-which-of-the-following-must-be-true-about-274512.html	GMAT Question of the Day - Daily to your Mailbox; hard ones only It is currently 23 Oct 2019, 17:16 GMAT Club Daily Prep Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email. Customized for You we will pick new questions that match your level based on your Timer History Track every week, we’ll send you an estimated GMAT score based on your performance Practice Pays we will pick new questions that match your level based on your Timer History If (a + 2b)/(17b - 2a) = 1, which of the following must be true about Author Message TAGS: Hide Tags Math Expert Joined: 02 Sep 2009 Posts: 58464 If (a + 2b)/(17b - 2a) = 1, which of the following must be true about [#permalink] Show Tags 28 Aug 2018, 06:25 00:00 Difficulty: 15% (low) Question Stats: 86% (00:53) correct 14% (01:27) wrong based on 60 sessions HideShow timer Statistics If $$\frac{(a+2b)}{(17b-2a)}=1$$, which of the following must be true about the relationship between a and b? A. a is 3 more than b. B. b is 3 more than a. C. a is $$\frac{1}{5}$$ of b. D. a is 5 times b. E. a is $$\frac{3}{5}$$ of b. _________________ VP Status: Learning stage Joined: 01 Oct 2017 Posts: 1011 WE: Supply Chain Management (Energy and Utilities) Re: If (a + 2b)/(17b - 2a) = 1, which of the following must be true about [#permalink] Show Tags 28 Aug 2018, 06:40 Bunuel wrote: If $$\frac{(a+2b)}{(17b-2a)}=1$$, which of the following must be true about the relationship between a and b? A. a is 3 more than b. B. b is 3 more than a. C. a is $$\frac{1}{5}$$ of b. D. a is 5 times b. E. a is $$\frac{3}{5}$$ of b. Simplifying the given expression, $$\frac{(a+2b)}{(17b-2a)}=1$$ Or, a+2b=17b-2a Or, 3a=15b Or, a=5*b Or, a is 5 times b. Ans. (D) _________________ Regards, PKN Rise above the storm, you will find the sunshine Board of Directors Status: QA & VA Forum Moderator Joined: 11 Jun 2011 Posts: 4784 Location: India GPA: 3.5 Re: If (a + 2b)/(17b - 2a) = 1, which of the following must be true about [#permalink] Show Tags 28 Aug 2018, 07:06 Bunuel wrote: If $$\frac{(a+2b)}{(17b-2a)}=1$$, which of the following must be true about the relationship between a and b? A. a is 3 more than b. B. b is 3 more than a. C. a is $$\frac{1}{5}$$ of b. D. a is 5 times b. E. a is $$\frac{3}{5}$$ of b. $$\frac{(a+2b)}{(17b-2a)}=1$$ Or, $$a + 2b = 17b - 2a$$ Or, $$3a = 15b$$ Or, $$a = 5b$$, Answer must be (D) _________________ Thanks and Regards Abhishek.... PLEASE FOLLOW THE RULES FOR POSTING IN QA AND VA FORUM AND USE SEARCH FUNCTION BEFORE POSTING NEW QUESTIONS How to use Search Function in GMAT Club \| Rules for Posting in QA forum \| Writing Mathematical Formulas \|Rules for Posting in VA forum \| Request Expert's Reply ( VA Forum Only ) Target Test Prep Representative Affiliations: Target Test Prep Joined: 04 Mar 2011 Posts: 2815 Re: If (a + 2b)/(17b - 2a) = 1, which of the following must be true about [#permalink] Show Tags 30 Aug 2018, 18:04 Bunuel wrote: If $$\frac{(a+2b)}{(17b-2a)}=1$$, which of the following must be true about the relationship between a and b? A. a is 3 more than b. B. b is 3 more than a. C. a is $$\frac{1}{5}$$ of b. D. a is 5 times b. E. a is $$\frac{3}{5}$$ of b. Simplifying, we have: a + 2b = 17b - 2a 3a = 15b a = 5b So a is 5 times b. _________________ Jeffrey Miller Jeff@TargetTestPrep.com 122 Reviews 5-star rated online GMAT quant self study course See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews If you find one of my posts helpful, please take a moment to click on the "Kudos" button. Re: If (a + 2b)/(17b - 2a) = 1, which of the following must be true about [#permalink] 30 Aug 2018, 18:04 Display posts from previous: Sort by	2019-10-24 00:16:15	{"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": 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": 0.7382948398590088, "perplexity": 4020.7578865520554}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987836368.96/warc/CC-MAIN-20191023225038-20191024012538-00426.warc.gz"}
https://nbviewer.org/github/feature-engine/feature-engine-examples/blob/main/preprocessing/MatchVariables.ipynb	# MatchVariables¶ MatchVariables() ensures that the columns in the test set are identical to those in the train set. If the test set contains additional columns, they are dropped. Alternatively, if the test set lacks columns that were present in the train set, they will be added with a value determined by the user, for example np.nan. In [1]: import numpy as np import pandas as pd from feature_engine.preprocessing import MatchVariables In [2]: # Load titanic dataset from OpenML data = data.replace('?', np.nan) data['cabin'] = data['cabin'].astype(str).str[0] data['pclass'] = data['pclass'].astype('O') data['age'] = data['age'].astype('float') data['fare'] = data['fare'].astype('float') data['embarked'].fillna('C', inplace=True) data.drop( labels=['name', 'ticket', 'boat', 'body', 'home.dest'], axis=1, inplace=True, ) return data In [3]: data = load_titanic() Out[3]: pclass survived sex age sibsp parch fare cabin embarked 0 1 1 female 29.0000 0 0 211.3375 B S 1 1 1 male 0.9167 1 2 151.5500 C S 2 1 0 female 2.0000 1 2 151.5500 C S 3 1 0 male 30.0000 1 2 151.5500 C S 4 1 0 female 25.0000 1 2 151.5500 C S In [4]: # separate the dataset into train and test train = data.iloc[0:1000, :] test = data.iloc[1000:, :] train.shape, test.shape Out[4]: ((1000, 9), (309, 9)) In [5]: # set up the transformer match_cols = MatchVariables(missing_values="ignore") # learn the variables in the train set match_cols.fit(train) Out[5]: MatchVariables(missing_values='ignore') In [6]: # the transformer stores the input variables match_cols.input_features_ Out[6]: ['pclass', 'survived', 'sex', 'age', 'sibsp', 'parch', 'fare', 'cabin', 'embarked'] ## 1 - Some columns are missing in the test set¶ In [7]: # Let's drop some columns in the test set for the demo test_t = test.drop(["sex", "age"], axis=1) Out[7]: pclass survived sibsp parch fare cabin embarked 1000 3 1 0 0 7.7500 n Q 1001 3 1 2 0 23.2500 n Q 1002 3 1 2 0 23.2500 n Q 1003 3 1 2 0 23.2500 n Q 1004 3 1 0 0 7.7875 n Q In [8]: # Let's drop some columns in the test set for the demo test_t = test.drop(["sex", "age"], axis=1) Out[8]: pclass survived sibsp parch fare cabin embarked 1000 3 1 0 0 7.7500 n Q 1001 3 1 2 0 23.2500 n Q 1002 3 1 2 0 23.2500 n Q 1003 3 1 2 0 23.2500 n Q 1004 3 1 0 0 7.7875 n Q In [9]: # the transformer adds the columns back test_tt = match_cols.transform(test_t) print() The following variables are added to the DataFrame: ['sex', 'age'] Out[9]: pclass survived sex age sibsp parch fare cabin embarked 1000 3 1 NaN NaN 0 0 7.7500 n Q 1001 3 1 NaN NaN 2 0 23.2500 n Q 1002 3 1 NaN NaN 2 0 23.2500 n Q 1003 3 1 NaN NaN 2 0 23.2500 n Q 1004 3 1 NaN NaN 0 0 7.7875 n Q Note how the missing columns were added back to the transformed test set, with missing values, in the position (i.e., order) in which they were in the train set. Similarly, if the test set contained additional columns, those would be removed: ## Test set contains variables not present in train set¶ In [10]: test_t.loc[:, "new_col1"] = 5 test_t.loc[:, "new_col2"] = "test" Out[10]: pclass survived sibsp parch fare cabin embarked new_col1 new_col2 1000 3 1 0 0 7.7500 n Q 5 test 1001 3 1 2 0 23.2500 n Q 5 test 1002 3 1 2 0 23.2500 n Q 5 test 1003 3 1 2 0 23.2500 n Q 5 test 1004 3 1 0 0 7.7875 n Q 5 test In [11]: # set up the transformer with different # fill value match_cols = MatchVariables( fill_value=0, missing_values="ignore", ) # learn the variables in the train set match_cols.fit(train) Out[11]: MatchVariables(fill_value=0, missing_values='ignore') In [12]: test_tt = match_cols.transform(test_t) print() The following variables are added to the DataFrame: ['sex', 'age'] The following variables are dropped from the DataFrame: ['new_col2', 'new_col1'] Out[12]: pclass survived sex age sibsp parch fare cabin embarked 1000 3 1 0 0 0 0 7.7500 n Q 1001 3 1 0 0 2 0 23.2500 n Q 1002 3 1 0 0 2 0 23.2500 n Q 1003 3 1 0 0 2 0 23.2500 n Q 1004 3 1 0 0 0 0 7.7875 n Q Note how the columns that were present in the test set but not in train set were dropped. And now, the missing variables were added back into the dataset with the value 0.	2022-10-01 04:11:15	{"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": 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": 0.17842069268226624, "perplexity": 4382.4428850458935}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1664030335530.56/warc/CC-MAIN-20221001035148-20221001065148-00486.warc.gz"}
https://ncatlab.org/nlab/show/sifted+colimit	# nLab sifted colimit Contents ### Context #### Limits and colimits limits and colimits # Contents ## Definition A sifted colimit is a colimit of a diagram $D \to C$ where $D$ is a sifted category (in analogy with a filtered colimit, involving diagrams of shape a filtered category). Such colimits commute with finite products in Set, by definition. ## Examples ###### Example A motivating example is a reflexive coequalizer. In fact, sifted colimits can “almost” be characterized as combinations of filtered colimits and reflexive coequalizers. (Adamek-Rosicky-Vitale 10) ###### Example (categories with finite products are cosifted Let $\mathcal{C}$ be a small category which has finite products. Then $\mathcal{C}$ is a cosifted category, equivalently its opposite category $\mathcal{C}^{op}$ is a sifted category, equivalently colimits over $\mathcal{C}^{op}$ with values in Set are sifted colimits, equivalently colimits over $\mathcal{C}^{op}$ with values in Set commute with finite products, as follows: For $\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set]$ to functors on the opposite category of $\mathcal{C}$ (hence two presheaves on $\mathcal{C}$) we have a natural isomorphism $\underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \left( \mathbf{X} \times \mathbf{Y} \right) \;\simeq\; \left( \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \mathbf{X} \right) \times \left( \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \mathbf{Y} \right) \,.$	2021-09-18 19:23:40	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 11, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9231576919555664, "perplexity": 1056.5364407469183}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780056572.96/warc/CC-MAIN-20210918184640-20210918214640-00637.warc.gz"}
https://www.varsitytutors.com/hotmath/hotmath_help/topics/random-variable.html	Random Variable We can define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space . The sum of the probabilities for all values of a random variable is $1$ . Example 1: In an experiment of tossing a coin twice, the sample space is . In this experiment, we can define random variable $X$ as the total number of tails. Then $X$ takes the values $0,1$ and $2$ . The table illustrates the probability distribution for the above experiment. $\begin{array}{\|llll\|}\hline \text{Number}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{Tails}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(X\right)\hfill & 0\hfill & 1\hfill & 2\hfill \\ \text{Probability}\hfill & \frac{1}{4}\hfill & \frac{1}{2}\hfill & \frac{1}{4}\hfill \\ \hline\end{array}$ The notation $P\left(X=x\right)$ is usually used to represent the probability of a random variable, where the $X$ is random variable and $x$ is one of the values of random variable. $P\left(X=0\right)=\frac{1}{4}$ is read as "The probability that $X$ equals $0$ is one-fourth." The above definition and example describe discrete random variables... those that take a finite or countable number of values. A random variable may also be continuous, that is, it may take an infinite number of values within a certain range. Example 2: A dart is thrown at a dartboard of radius $9$ inches. If it misses the dartboard, the throw is discounted. Define a random variable $X$ as the distance in inches from the dart to the center. $0\le X<9$	2018-03-23 09:21:54	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 15, "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": 0.8993348479270935, "perplexity": 142.9687871789395}, "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-2018-13/segments/1521257648205.76/warc/CC-MAIN-20180323083246-20180323103246-00664.warc.gz"}
https://techwhiff.com/learn/subject-soil-mechanics-03-the-old-building-a-is/59301	# Subject: soil mechanics 03:- The old building (A) is to be removed and replaced by a... ###### Question: subject: soil mechanics 03:- The old building (A) is to be removed and replaced by a new building (B), as shown in Figure (Q3). The vertical pressure at the center of the clay layer from the building (A) and building (B) is equal to 13.64 kN/m² and 23.3 kN/mº respectively, Find:- 1. O.C.R after removing building (A). 2. The void ratio (e) of the clay layer: a. Immediately after removing building (A). b. After a very long time from removing building (A). 3. Final consolidation settlement of building (B) if to be constructed: a. Immediately after removing building (A). b. After a very long time from removing building (A). 4. Time required to achieve half the final consolidation settlement of building (B) (14 points) 2 Old building N.G.L 2 m W.T.L Sand, y = 15 kN/m2 Clay, y = 20 W=30% G = 2.7. C= 0.2 C = 0.04, C = 0.5m2 /y 8 m Rock #### Similar Solved Questions ##### The long-term liabilities section of CPS Transportation’s December 31, 2020, balance sheet included the following: (FV... The long-term liabilities section of CPS Transportation’s December 31, 2020, balance sheet included the following: (FV of $1, PV of$1, FVA of $1, PVA of$1, FVAD of $1 and PVAD of$1) (Use appropriate factor(s) from the tables provided.) a. A lease liability with 15 remaining lease payments ... ##### Looking to 2021. Their current sales are $16 million, COGS are 30% of sales, operating expenses... looking to 2021. Their current sales are$16 million, COGS are 30% of sales, operating expenses 40% of sales, interest expense 1% of sales and depreciation expense $100,000. Tax 40%, dividend payout 20%. 1 million shares stock. what is their net profit margin? Show work.... 1 answer ##### 11. The combustion analysis of 0.452 g of the hydrocarbon cubane (molecular mass, 104.16 g/mol) gives... 11. 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Interest payable at maturity$ 237 e Textbook and Media ...	2023-03-26 19:17:38	{"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": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.25833097100257874, "perplexity": 8601.550377442662}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1679296946445.46/warc/CC-MAIN-20230326173112-20230326203112-00698.warc.gz"}
http://ahundredwoods.com/harry-potter-npg/55045f-pseudo-random-number-generator	The 2nd Edition; SEED Lectures. All these techniques, except the method of independent sequences, require arbitrary elements of the sequence to be generated e ciently. How to Generate C# Random Numbers, Pseudo vs Secure Random Numbers. In cryptography, PRNG’s are used to construct session keys and stream ciphers. construct a function $$G:\{0,1\}^t\rightarrow\{0,1\}^T, T \gg t$$. Security analysis for pseudo-random number generators Sylvain Ruhault To cite this version: Sylvain Ruhault. 8 min read. We can generate truly random numbers by measuring random fluctuations, known as noise. All uniform random bit generators meet the UniformRandomBitGenerator requirements.C++20 also defines a uniform_random_bit_generatorconcept. The appearance of randomness is provided by performing modulo arithmetic.. PRNGs are suitable for applications where many random numbers are required and where it is useful that the same sequence can be replayed easily. amzn_assoc_marketplace = "amazon"; To generate good pseudo random numbers, we need to start with something that is random; otherwise, the outcome will be quite predictable. Don’t stop learning now. Random number generation / Random Numbers. How to check if a given number is Fibonacci number? Depending on what you are using random numbers for, you will need to decide which type to use. The random module provides a fast pseudorandom number generator based on the Mersenne Twister algorithm. Generating Random Numbers¶ The random() function returns the next random â¦ By far the most widely used PRNG is the Mersenne Twister algorithm which passed numerous tests for statistical randomness and generally creates random numbers faster than other methods. PRNGs generate a sequence of numbers approximating the properties of random numbers. Pseudo Random Number Generator(PRNG) refers to an algorithm that uses mathematical formulas to produce sequences of random numbers. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready. Recommended Time. PRNGs are not suitable for applications where it is important that the numbers are really unpredictable, such as data encryption and gambling. To generate a sequence of true random numbers is a complex problem and hardware generators are used for this. Instead, pseudo-random numbers are usually used. A pseudo-random number generator is an algorithm for generating a sequence of numbers whose properties approximate the properties of sequences of random numbers. The PRNG-generated sequence is not truly random, because it is completely determined by an initial value, called the PRNG's seed. By using our site, you I have used a HP 5082-7356 display which have BCD input and allows to latch input value. For integers, there is uniform selection from a range. generateur pseudo: Ce générateur 100% gratuit génère tout type de pseudonyme/pseudo, il peut en créer pour des femmes, masculin, gamer, Fantasy, Asiatique (manga japonais) donc des pseudo-originaux pour vous simplement et rapidement sans aucun problème en un seul clic, grace à ça vous vous serrez différent ! If you want a different sequence of numbers each time, you can use the current time as a seed. A pseudorandom number generator, also known as a deterministic random bit generator, is an algorithm for generating a sequence of numbers whose properties approximate the properties of sequences of random numbers. Disclosure: As an Amazon Associate we earn commissions from qualifying purchases from Amazon.com. It is called pseudorandom because the generated numbers are not true random numbers but are generated using a mathematical formula. PRNGs generate a sequence of numbers approximating the properties of random numbers. All rights reserved. amzn_assoc_tracking_id = "ezcalcme-20"; Ecole normale supérieure - ENS PARIS, 2015. Random number generators can be hardware based or pseudo-random number generators. Security analysis for pseudo-random number generators. Code - Pseudo Random Number Generator : Chiffrement étrange... bonjour, impossible de se connecter au ssh sur la box billu box, en esperant que sa soit réglé. This random number generator (RNG) has generated some random numbers for you in the table below. Lab Tasks . Many numbers are generated in a short time and can also be reproduced later, if the starting point in the sequence is known. However, none of them generate a truly random number. Find the smallest number whose digits multiply to a given number n, Find n'th number in a number system with only 3 and 4, Program to find GCD or HCF of two numbers, Modulo Operator (%) in C/C++ with Examples, Program to find sum of elements in a given array. To get started, the algorithm requires an initial Seed, which must be provided by some means. However, surprising as it may seem, it is difficult to get a computer to do something by chance as computer follows the given instructions blindly and is therefore completely predictable. You can use this app to call up students in class, rolling dice in a game, pick lottery numbers, and etc. The .Net Framework base class library (BCL) includes a pseudo-random number generator for non-cryptography use in the form of the System.Random class. Practical seed-recovery for the PCG Pseudo-Random Number Generator Charles Bouillaguet, Florette Martinez, Julia Sauvage To cite this version: Charles Bouillaguet, Florette Martinez, Julia Sauvage. The randomness comes from atmospheric noise, which for many purposes is better than the pseudo-random number algorithms typically used in computer programs. Random number generators have numerous applications in various fields like gambling, computer simulation, statistical sampling, cryptography and other areas where producing unpredictable numeric values is crucial. This generator produces a sequence of 97 different numbers, then it starts over again. take-home project): SEED Labs . It is also loosely known as a cryptographic random number generator. Now the aim is to build a pseudo random number generator from scratch! A PRNG starts from an arbitrary starting state using a seed state. Our PRNG is based on Chance.js – a popular open source (MIT License) random generator helper for JavaScript, built on top of the Mersenne Twister algorithm. It is not possible to generate truly random numbers from deterministic thing like computers so PRNG is a technique developed to generate random numbers using a computer. Disclosure: As an Amazon Associate we earn commissions from qualifying purchases from Amazon.com.Copyright © 2017-2020 ezcalc.me. A version of this algorithm, MT19937, has an impressive period of 2¹⁹⁹³⁷-1. Pseudo-Random Numbers. Click 'More random numbers' to generate some more, click 'customize' to alter the number ranges (and text if required). â¦ Listing 1: âGenerating a 128-bit encryption keyâ #include #include #include In case the precision equals to 0 a set of integer pseudo random numbers is generated. This article is contributed by Yash Singla. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above. a closely-guided lab session): Unsupervised situation (e.g. A PRNG starts from an arbitrary starting state using a seed state. It is not so easy to generate truly random numbers. The generator is defined by the recurrence relation: We generate the next random integer using the previous random integer, the integer constants, and the integer modulus. Hardware based random-number generators can involve the use of a dice, a coin for flipping, or many other devices. Pseudo-Random Number Generators. Python, like any other programming technique, uses a pseudo-random generator. Random numbers are used in a wide variety of software applications. Oleh karena itu pembangkit semacam itu disebut pseudo-random number generator (PRNG). For sequences, there is uniform selection of a random element, a function to generate a random permutation of a list in-place, and a function for random sampling without replacement. RANDOM.ORG offers true random numbers to anyone on the Internet. The drand48(), erand48(), jrand48(), lrand48(), mrand48() and nrand48() functions generate uniformly distributed pseudo-random numbers using a linear congruential algorithm and 48-bit integer arithmetic. Cryptography and Se-curity [cs.CR]. This video explain about random number first, then the algorithm used to generate pseudo random number i.e. Many numbers are generated in a short time and can also be reproduced later, if â¦ If you know this state, you can predict all future outcomes of the random number generators. amzn_assoc_design = "in_content"; Download Pseudo Random Number Generator and enjoy it on your iPhone, iPad, and iPod touch. In software, we generate random numbers by calling a function called a “random number generator”. This video explain about random number first, then the algorithm used to generate pseudo random number i.e. âThis app provides an easy way to generate random numbers. amzn_assoc_linkid = "17b0e46c3b11db89cac629853f1068d5"; Pseudo-random Number Generator. amzn_assoc_ad_mode = "manual"; XOR gate is connected to switch to initiate a number generation because by default all outputs of shift register have LOW value at initial state. Inorder Tree Traversal without recursion and without stack! In other words, you can get it to randomly choose a number between one and ten with the press of a button. A pseudorandom number generator, or PRNG, is any program, or function, which uses math to simulate randomness. We want to be able to take a few "true random bits" (seed) and generate more "random looking bits", i.e. The precision defines the number of digits after the decimal point. For a full explanation of the nature of randomness and random numbers, click the â¦ PRNGs generate a sequence of numbers approximating the properties of random numbers. A pseudo-random number generator uses an algorithm of mathematical formulas that will generate any random number from a range of specific numbers. Inputs of shift register are connected to output from XOR gate. However it is not deemed good enough for cryptographic applications. Supervised situation (e.g. This is determined by a small group of initial values. ï¿¿NNT: 2015ENSU0014ï¿¿. In other words, you can get it to randomly choose a number between one and ten with the press of a button. The following program uses the current time as a seed for the pseudo random number generator. De DigiWiki. Most cryptographic applications require random numbers, for example: key generation nonces salts in â¦ VM version: This lab has been tested on our pre-built SEEDUbuntu16.04 VM. This number is generated by an algorithm that returns a sequence of apparently non-related numbers each time it is called. Although sequences that are closer to truly â¦ A pseudo random number generator (PRNG) is a computer algorithm for generating a sequence of numbers whose properties can only approximate the properties of sequence of truly random numbers, because itâs completely determined by an initial value. It is called pseudorandom because the generated numbers are not true random numbers but are generated using a mathematical formula. People use RANDOM.ORG for holding drawings, lotteries and sweepstakes, to drive online games, for scientific applications and for art and music. Explanation: srand() sets the seed which is used by rand() to generate random numbers.time(NULL) return no. With the advent of computers, programmers recognized the need for a means of introducing randomness into a computer program. Enter the number of random values and the minimum and maximum values for the range of random numbers you want to generate. Home Page; SEED Books. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to [email protected] See your article appearing on the GeeksforGeeks main page and help other Geeks. Linear Congruential Generator (LCG) Merupakan PRNG yang paling sederhana dan lebih mudah & cepat dikomputasi. A random number generator is an algorithm that, based on an initial seed or by means of continuous input, produces a sequence of numbers or respectively bits. Generates a set of pseudo random numbers within a predefined range. amzn_assoc_asins = "0387001786,0674107462,9814327743,1452818363"; Check out our other statistics calculators such as Number Sorter or Odds To Probability Calculator. Based on Beans theme for WordPress. Pseudo Random Number Generator: A pseudo random number generator (PRNG) refers to an algorithm that uses mathematical formulas to produce sequences of random numbers. PRNGs generate a sequence of numbers approximating the properties of random numbers. Update: Since this answer was written, the PCG family of algorithms was published that seems to outperform existing â¦ Originally developed to produce inputs for Monte Carlo simulations, Mersenne Twister generates numbers with nearly uniform distribution and a large period, making it suited for a wide range of applications. Pseudo-random Number Generator by Asuna on Mon 11 Mar 2013, 21:47 Bilangan acak ( random ) adalah bilangan yang tidak dapat diprediksi … Code - Pseudo Random Number Generator : Chiffrement étrange... @Tenma : Sécurité informatique : ethical hacking -> vraiment pas top, Hacking the art of exploitation -> très bien (un must), Most computers have built-in random number generators, and we shall take as our starting point in simulation that we can generate the values of pseudo random numbers; moreover, we will act as if these pseudo random numbers were actually true random numbers. Popular examples of such applications are simulation and modeling applications. IACR Transactions on Symmetric Cryptology, Ruhr Universität Bochum, That is, we will act as ifthe sequence of random numbers were actuallya sequence of values of a sample from the uniform (0, 1) â¦ Matt Watson April 16, 2017 Developer Tips, Tricks & Resources. Attention reader! “Why do I need a random number?” The importance of random numbers is not in the number itself (they are common numbers, if taken individually) but in the way they are generated. The service has â¦ It may also be called a DRNG (digital random number generator) or DRBG (deterministic random bit generator). Cryptographic Pseudorandom Number Generator : This PseudoRandom Number Generator (PRNG) allows you to generate small (minimum 1 byte) to large (maximum 16384 bytes) pseudo-random numbers for cryptographic purposes. - [Voiceover] One, two, three, four-- - [Voiceover] For example, if we measure the electric current of TV static over time, we will generate a truly random sequence. Statistical quality of pseudo random numbers are generally sufficient for most of practical applications (but not in the case of cryptography!). Returns a pseudo-random integral number in the range between 0 and RAND_MAX. Fungsinya adalah sebagai berikut: X n = (aX n â 1 + b) mod m dimana: X n = bilangan acak ke-n dari deretnya a, b = koefisien parameter m = modulus Basis dari fungsi ini â¦ Math.NET Numerics provides a few alternatives with different characteristics in randomness, bias, sequence length, performance and thread-safety. Generate number between and = 98. A cryptographically secure pseudorandom number generator or cryptographic pseudorandom number generator (CPRNG) is a pseudorandom number generator with properties that make it suitable for use in cryptography. acknowledge that you have read and understood our, GATE CS Original Papers and Official Keys, ISRO CS Original Papers and Official Keys, ISRO CS Syllabus for Scientist/Engineer Exam, Generate integer from 1 to 7 with equal probability, Generate 0 and 1 with 25% and 75% probability, Random number generator in arbitrary probability distribution fashion, Linear Congruence method for generating Pseudo Random Numbers, Multiplicative Congruence method for generating Pseudo Random Numbers, Additive Congruence method for generating Pseudo Random Numbers, Printing all solutions in N-Queen Problem, Warnsdorff’s algorithm for Knight’s tour problem, The Knight’s tour problem \| Backtracking-1, Count number of ways to reach destination in a Maze, Count all possible paths from top left to bottom right of a mXn matrix, Print all possible paths from top left to bottom right of a mXn matrix, Unique paths covering every non-obstacle block exactly once in a grid, Tree Traversals (Inorder, Preorder and Postorder). // New returns a pseudorandom number generator Rand with a given seed. A pseudo random number generator (PRNG) is a computer algorithm for generating a sequence of numbers whose properties can only approximate the properties of sequence of truly random numbers, because it’s completely determined by an initial value. Example. The generated bit strings should "look random" to an adversary. Hence, the numbers are deterministic and efficient. Most people chose this as the best definition of pseudo-random-number-generator: A random number generator... See the dictionary meaning, pronunciation, and sentence examples. English. It has a humongously large period, but also a relatively humongous state (2.5 kB). amzn_assoc_placement = "adunit0"; Seedable PRNG Based On Multiply/ Add/ Overflow . When we measure this noise, known as sampling, we can obtain numbers. I've tested it for a while and it looks random to me, about the same as llFrand(). Pseudo-Random Numbers. Pseudo Random Number Generator: A pseudo random number generator (PRNG) refers to an algorithm that uses mathematical formulas to produce sequences of random numbers. 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For something like a lottery or â¦ Please use ide.geeksforgeeks.org, generate link and share the link here. of second from JAN 1, 1971 i.e every time we run program we have difference of few seconds which gives the program new seed. Such functions have hidden states, so that repeated calls to the function generate new numbers that appear random. Several pseudo-random number generator parallelizing schemes exist. In my article âHow to get an unbiased RNG from an unbalanced oneâ I showed how to â¦ ï¿¿tel-01236602v2ï¿¿ Aller à : Navigation, rechercher. There really is no limit to how many numbers you are able to choose (i.e: 1 to 100, 100 to 200, etc.). A uniform random bit generatoris a function object returning unsigned integer values such that each value in the range of possible results has (ideally) equal probability of being returned. 2×2 System of Linear Equations Calculator, 3×3 System of Linear Equations Calculator, Linear Least Squares Regression Line Calculator. Writing code in comment? Computer based random number generators are almost always pseudo-random number generators. Linear Congruential Generator is most common and oldest algorithm for generating pseudo-randomized numbers. This clock drives 74HC164 shift register which is used to generate pseudo random number. You can be able to randomly generate â¦ In view that the Mersenne Twister PRNG provides long period, high order of dimensional equidistribution, good speed and reliability it is used within many software applications and packages, including PHP and Python. Pseudo Random Number Generator(PRNG) refers to an algorithm that uses mathematical formulas to produce sequences of random numbers. RAND_MAX is a constant defined in . We demand that this sequence appears \random" to any observer. Features: â¢ Option to set a range â¢ Option to allow duplicates â¢ Store a list of random numbers â¢ Big and easy-to-touch UI â¢ FREE The Mersenne Twister is a strong pseudo-random number generator in terms of that it has a long period (the length of sequence of random values it generates before repeating itself) and a statistically uniform distribution of values. Math.NET Numerics provides a few alternatives with different characteristics in randomness, bias, sequence length, performance and thread-safety. The seed decides at what number the sequence will start. We use cookies to ensure you have the best browsing experience on our website. Widely used PRNG algorithms : Lagged Fibonacci generators, linear feedback shift registers, Blum Blum Shub. They will further learn a standard way to generate pseudo random numbers that are good for security purposes. Cryptographic Pseudorandom Number Generator : This PseudoRandom Number Generator (PRNG) allows you to generate small (minimum 1 byte) to large (maximum 16384 bytes) pseudo-random numbers for cryptographic purposes. You can also choose comma or space as separators between the generated numbers. The Mersenne Twister algorithm is a popular, fairly fast pseudo-random number generator that produces quite good results. For a music jukebox, the accuracy is not very critical. amzn_assoc_region = "US"; Pythonâs random generation is based upon Mersenne Twister algorithm that produces 53-bit precision floats. ). amzn_assoc_ad_type = "smart"; Statistical quality of pseudo random numbers are generally sufficient for most of practical applications (but not in the case of cryptography! This is determined by a small group of initial values. The .Net Framework base class library (BCL) includes a pseudo-random number generator for non-cryptography use in the form of the System.Random class. A pseudo-random number generator uses an algorithm of mathematical formulas that will generate any random number from a range of specific numbers. This module implements pseudo-random number generators for various distributions. The math can sometimes be complex, but in general, using a PRNG requires only two steps: Provide the PRNG with an arbitrary seed. The technique is fast and thread-safe but not suitable from cryptographic purpose. The well known ones being leap frog, sequence splitting, independent sequences and shuf-ï¬ing leap frog [2,12]. Practical seed-recovery for the PCG Pseudo-Random Number Generator. In my article “How to get an unbiased RNG from an unbalanced one” I showed how to extract randomness from any kind of source. This algorithm uses a seed to generate the series, which should be initialized to some distinctive value using function srand. This topic leads us to the question: What is random? 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Produces a sequence of numbers approximating the properties of sequences of random values and minimum!	2021-07-29 11:04:56	{"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": 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": 0.4056612551212311, "perplexity": 1598.9787739710544}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1627046153857.70/warc/CC-MAIN-20210729105515-20210729135515-00616.warc.gz"}
http://physics.stackexchange.com/questions/108640/what-is-an-intuitive-explanation-using-forces-for-the-equatorial-bulge	# What is an intuitive explanation using forces for the equatorial bulge? The earth is not a sphere, because it bulges at the equator. I tried fiddling with centripetal force equations and gravity, but I couldn't derive why this bulge occurs. Is there (a) a mathematical explanation using forces (not energies) and (b) a simple intuitive explanation to explain to others why the bulge occurs? - Essentially a duplicate of physics.stackexchange.com/q/8074/2451 and links therein. – Qmechanic Apr 16 at 12:31 @Qmechanic, those answers seem 10x more complicated than what my question is asking for. Do they really provide a "simple intuitive explanation to explain to others"? – Mew Apr 16 at 12:33 Well, what-if.xkcd.com/92 would seem appropriate here. – Carl Witthoft Apr 16 at 13:03 @Qmechanic, I have also edited to ask for a "forces" approach rather than an "energy" approach which was used in the question you have linked. – Mew Apr 16 at 13:13 @Mew: I don't understand what you're confused about. You say you want an intuitive explanation, but then you don't accept the XKCD explanation, which does exactly that. When a person spins, their arms will tend to fling out; similarly, when the Earth spins, it's Equator tends to fling out. It has nothing to do with "the centrifugal force will exceed gravity", as the force due to the rotation is far smaller than the force due to gravity (as evidenced by the near sphericity of the Earth). – DumpsterDoofus Apr 16 at 14:11 Equatorial bulging of a planet is caused by the combination of gravity and centrifugal force. To show this I will first make a few assumptions: • The planets is assumed to be made up of a liquid of uniform density. • All liquid is at rest relative to itself, which means that there are no shear stresses within the liquid, since this would induce a flow. • The equatorial bulging is small such that the acceleration due to gravity, $\vec{a}_g$, at the surface can be approximated with: $\vec{a}_g=-G\frac{M}{\\|\vec{x}\\|^3}\vec{x}$, where $G$ is the gravitational constant, $M$ the mass of the planet and $\vec{x}$ the position on the surface relative to the center of mass of the planet. • The planet is asymmetric ans rotates around this axis with a constant angular velocity $\omega$. A small volume, $dV$, experiences two volumetric accelerations, namely gravitational and centrifugal, and normal forces by the neighboring liquid in therms pressure. The sum of all accelerations on $dV$ should add up to zero to comply with the second assumption (the centrifugal acceleration already accounts for the fact that the reference frame is rotating). At any point on the surface there is a constant pressure, because above it there would be the vacuum of space. This means that the neighboring liquid, also at the surface, has the same pressure and therefore can not exert any force on each other in the plane op the surface. The only direction that liquid can exert force on each other is in the normal direction to the surface. However the sum of all accelerations still should add up to zero and therefore the sum of the gravitational and centrifugal acceleration should also point in the normal direction of the surface. The magnitude of the gravitational acceleration, $a_g$, is defined by assumption three and its direction is always radially inwards. The magnitude of the centrifugal acceleration, $a_c$, is equal to: $$a_c = \omega^2\sin(\phi)\\|\vec{x}\\|,$$ where $\phi$ is equal to the longitude; its direction is always parallel to the plane of the equator and its line of action always goes through the axis of rotation. These accelerations are illustrated in the figure below. For the next part I will define local unit vectors $\vec{e}_r$ and $\vec{e}_t$, where $\vec{e}_r$ points into the local radial outwards direction and $\vec{e}_t$ is perpendicular to it, lies in the plane spanned by the axis of rotation and $\vec{x}$ and faces the direction closest to the equator. The direction of vectors also correspond with the grey vectors in the figure above. Using these unit vectors, the vector sum of the gravitational and centrifugal acceleration can be written as: $$\vec{a}_g+\vec{a}_c = \vec{e}_r \left(\omega^2\sin(\phi)^2\\|\vec{x}\\| - G\frac{M}{\\|\vec{x}\\|^2}\right) + \vec{e}_t \omega^2\sin(\phi)\cos(\phi)\\|\vec{x}\\|.$$ If there would be no bulging then the normal vector should always point radial in/outwards. However the normal vector has to point in the same direction as the equation above, which means that for $\omega>0$ it will not point in the same direction as $\vec{e}_r$ for all values of $\phi$. This means that the surface will have a small slope, $\alpha$, relative to $\vec{e}_t$ equal to: $$\alpha = \tan^{-1}\left(\frac{\omega^2\sin(\phi)\cos(\phi)\\|\vec{x}\\|}{\omega^2\sin(\phi)^2\\|\vec{x}\\| - G\frac{M}{\\|\vec{x}\\|^2}}\right).$$ As slope means a change of height, and thus radius, when displacing horizontally. To simplify the expression, $r$ will substitute $\\|\vec{x}\\|$. For a slope $\alpha$ the change of the radius, $dr$, for a small change in longitude, $d\phi$, will be equal to: $$dr = \tan(\alpha)rd\phi.$$ By substituting in the equation for $\alpha$ the following differential equation can be obtained: $$\frac{dr}{d\phi} = \frac{\omega^2\sin(\phi)\cos(\phi)r^2}{\omega^2\sin(\phi)^2r - G\frac{M}{r^2}}$$ For $\phi$ is equal to $0$ or $\frac{\pi}{2}$, the poles and the equator respectively, this equation will be zero, however for any value in between it will be positive, since when denominator would become negative it would mean that the centrifugal force will be bigger than gravity and the liquid would be flung into space. So this planet would have the smallest radius near the poles after which the radius will increase with longitude until you reach the equator. - There is a wikipedia article which describes the effect http://en.wikipedia.org/wiki/Equatorial_bulge Basically the bulge is caused by the rotation of the Earth. The centripetal force is given by $F=m\omega^2 r$. Therefore the poles feel a lesser force than the equator which wants to spin out into a disc. This is balanced by gravity which wants the Earth to be spherical. Mathematically the flattening of the Earth is $$f = \frac{5}{4} \frac{\omega^2r_a^3}{GM}$$ where $r_a$ is the average radius and $f$ is the ratio of the major and minor radii. - the centripetal force at the poles is indeed less by your equation, however the component of gravity in this direction is also less at the equator and it exactly balances. This is why I want to see a complete derivation of why it should bulge, because the centripetal argument just doesn't seem to hold up. – Mew Apr 16 at 12:30 Having now actually read the wikipedia I cited I think a better explanation may be in terms of energy. I will try and work out fuller explanation. – nivag Apr 16 at 12:38 I particularly want the explanation using a force approach. I have edited my question accordingly. – Mew Apr 16 at 13:09 If you have ever seen a pizza being made by hand, you will know that when the baker throws the disk of dough in the air, he makes it spin. As he does so, the pizza "disk" gets bigger because the dough on the outside experiences a larger centrifugal force (in the rotating frame of reference of the pizza. Don't start on "there is no such thing", you asked for an intuitive answer). Now think of the earth as that pizza. The bits of the earth near the equator (a larger distance from the axis of rotation) feel a greater force, and are therefore trying to move outward. The force of gravity tries to pull it back. The balance is a slightly distorted sphere: the "bulge". Simple enough, I hope. - gee, thanks, now you made me hungry! – Carl Witthoft Apr 16 at 16:52	2014-11-27 23:19:39	{"extraction_info": {"found_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": 0.8860645890235901, "perplexity": 287.98364256885657}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931009292.37/warc/CC-MAIN-20141125155649-00158-ip-10-235-23-156.ec2.internal.warc.gz"}
http://commons.apache.org/proper/commons-math/apidocs/org/apache/commons/math3/analysis/solvers/UnivariateSolverUtils.html	org.apache.commons.math3.analysis.solvers ## Class UnivariateSolverUtils • java.lang.Object • org.apache.commons.math3.analysis.solvers.UnivariateSolverUtils • ### Method Summary Methods Modifier and Type Method and Description static double[] bracket(UnivariateFunction function, double initial, double lowerBound, double upperBound) This method simply calls bracket(function, initial, lowerBound, upperBound, q, r, maximumIterations) with q and r set to 1.0 and maximumIterations set to Integer.MAX_VALUE. static double[] bracket(UnivariateFunction function, double initial, double lowerBound, double upperBound, double q, double r, int maximumIterations) This method attempts to find two values a and b satisfying lowerBound <= a < initial < b <= upperBound f(a) * f(b) <= 0 If f is continuous on [a,b], this means that a and b bracket a root of f. static double[] bracket(UnivariateFunction function, double initial, double lowerBound, double upperBound, int maximumIterations) This method simply calls bracket(function, initial, lowerBound, upperBound, q, r, maximumIterations) with q and r set to 1.0. static double forceSide(int maxEval, UnivariateFunction f, BracketedUnivariateSolver<UnivariateFunction> bracketing, double baseRoot, double min, double max, AllowedSolution allowedSolution) Force a root found by a non-bracketing solver to lie on a specified side, as if the solver were a bracketing one. static boolean isBracketing(UnivariateFunction function, double lower, double upper) Check whether the interval bounds bracket a root. static boolean isSequence(double start, double mid, double end) Check whether the arguments form a (strictly) increasing sequence. static double midpoint(double a, double b) Compute the midpoint of two values. static double solve(UnivariateFunction function, double x0, double x1) Convenience method to find a zero of a univariate real function. static double solve(UnivariateFunction function, double x0, double x1, double absoluteAccuracy) Convenience method to find a zero of a univariate real function. static void verifyBracketing(UnivariateFunction function, double lower, double upper) Check that the endpoints specify an interval and the end points bracket a root. static void verifyInterval(double lower, double upper) Check that the endpoints specify an interval. static void verifySequence(double lower, double initial, double upper) Check that lower < initial < upper. • ### Methods inherited from class java.lang.Object clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait • ### Method Detail • #### solve public static double solve(UnivariateFunction function, double x0, double x1) throws NullArgumentException, NoBracketingException Convenience method to find a zero of a univariate real function. A default solver is used. Parameters: function - Function. x0 - Lower bound for the interval. x1 - Upper bound for the interval. Returns: a value where the function is zero. Throws: NoBracketingException - if the function has the same sign at the endpoints. NullArgumentException - if function is null. • #### solve public static double solve(UnivariateFunction function, double x0, double x1, double absoluteAccuracy) throws NullArgumentException, NoBracketingException Convenience method to find a zero of a univariate real function. A default solver is used. Parameters: function - Function. x0 - Lower bound for the interval. x1 - Upper bound for the interval. absoluteAccuracy - Accuracy to be used by the solver. Returns: a value where the function is zero. Throws: NoBracketingException - if the function has the same sign at the endpoints. NullArgumentException - if function is null. • #### forceSide public static double forceSide(int maxEval, UnivariateFunction f, BracketedUnivariateSolver<UnivariateFunction> bracketing, double baseRoot, double min, double max, AllowedSolution allowedSolution) throws NoBracketingException Force a root found by a non-bracketing solver to lie on a specified side, as if the solver were a bracketing one. Parameters: maxEval - maximal number of new evaluations of the function (evaluations already done for finding the root should have already been subtracted from this number) f - function to solve bracketing - bracketing solver to use for shifting the root baseRoot - original root found by a previous non-bracketing solver min - minimal bound of the search interval max - maximal bound of the search interval allowedSolution - the kind of solutions that the root-finding algorithm may accept as solutions. Returns: a root approximation, on the specified side of the exact root Throws: NoBracketingException - if the function has the same sign at the endpoints. • #### bracket public static double[] bracket(UnivariateFunction function, double initial, double lowerBound, double upperBound) throws NullArgumentException, NotStrictlyPositiveException, NoBracketingException This method simply calls bracket(function, initial, lowerBound, upperBound, q, r, maximumIterations) with q and r set to 1.0 and maximumIterations set to Integer.MAX_VALUE. Note: this method can take Integer.MAX_VALUE iterations to throw a ConvergenceException. Unless you are confident that there is a root between lowerBound and upperBound near initial, it is better to use bracket(function, initial, lowerBound, upperBound, q, r, maximumIterations), explicitly specifying the maximum number of iterations. Parameters: function - Function. initial - Initial midpoint of interval being expanded to bracket a root. lowerBound - Lower bound (a is never lower than this value) upperBound - Upper bound (b never is greater than this value). Returns: a two-element array holding a and b. Throws: NoBracketingException - if a root cannot be bracketted. NotStrictlyPositiveException - if maximumIterations <= 0. NullArgumentException - if function is null. • #### bracket public static double[] bracket(UnivariateFunction function, double initial, double lowerBound, double upperBound, int maximumIterations) throws NullArgumentException, NotStrictlyPositiveException, NoBracketingException This method simply calls bracket(function, initial, lowerBound, upperBound, q, r, maximumIterations) with q and r set to 1.0. Parameters: function - Function. initial - Initial midpoint of interval being expanded to bracket a root. lowerBound - Lower bound (a is never lower than this value). upperBound - Upper bound (b never is greater than this value). maximumIterations - Maximum number of iterations to perform Returns: a two element array holding a and b. Throws: NoBracketingException - if the algorithm fails to find a and b satisfying the desired conditions. NotStrictlyPositiveException - if maximumIterations <= 0. NullArgumentException - if function is null. • #### bracket public static double[] bracket(UnivariateFunction function, double initial, double lowerBound, double upperBound, double q, double r, int maximumIterations) throws NoBracketingException This method attempts to find two values a and b satisfying • lowerBound <= a < initial < b <= upperBound • f(a) * f(b) <= 0 If f is continuous on [a,b], this means that a and b bracket a root of f. The algorithm checks the sign of $$f(l_k)$$ and $$f(u_k)$$ for increasing values of k, where $$l_k = max(lower, initial - \delta_k)$$, $$u_k = min(upper, initial + \delta_k)$$, using recurrence $$\delta_{k+1} = r \delta_k + q, \delta_0 = 0$$ and starting search with $$k=1$$. The algorithm stops when one of the following happens: • at least one positive and one negative value have been found -- success! • both endpoints have reached their respective limits -- NoBracketingException • maximumIterations iterations elapse -- NoBracketingException If different signs are found at first iteration (k=1), then the returned interval will be $$[a, b] = [l_1, u_1]$$. If different signs are found at a later iteration k>1, then the returned interval will be either $$[a, b] = [l_{k+1}, l_{k}]$$ or $$[a, b] = [u_{k}, u_{k+1}]$$. A root solver called with these parameters will therefore start with the smallest bracketing interval known at this step. Interval expansion rate is tuned by changing the recurrence parameters r and q. When the multiplicative factor r is set to 1, the sequence is a simple arithmetic sequence with linear increase. When the multiplicative factor r is larger than 1, the sequence has an asymptotically exponential rate. Note than the additive parameter q should never be set to zero, otherwise the interval would degenerate to the single initial point for all values of k. As a rule of thumb, when the location of the root is expected to be approximately known within some error margin, r should be set to 1 and q should be set to the order of magnitude of the error margin. When the location of the root is really a wild guess, then r should be set to a value larger than 1 (typically 2 to double the interval length at each iteration) and q should be set according to half the initial search interval length. As an example, if we consider the trivial function f(x) = 1 - x and use initial = 4, r = 1, q = 2, the algorithm will compute f(4-2) = f(2) = -1 and f(4+2) = f(6) = -5 for k = 1, then f(4-4) = f(0) = +1 and f(4+4) = f(8) = -7 for k = 2. Then it will return the interval [0, 2] as the smallest one known to be bracketing the root. As shown by this example, the initial value (here 4) may lie outside of the returned bracketing interval. Parameters: function - function to check initial - Initial midpoint of interval being expanded to bracket a root. lowerBound - Lower bound (a is never lower than this value). upperBound - Upper bound (b never is greater than this value). q - additive offset used to compute bounds sequence (must be strictly positive) r - multiplicative factor used to compute bounds sequence maximumIterations - Maximum number of iterations to perform Returns: a two element array holding the bracketing values. Throws: NoBracketingException - if function cannot be bracketed in the search interval • #### midpoint public static double midpoint(double a, double b) Compute the midpoint of two values. Parameters: a - first value. b - second value. Returns: the midpoint. • #### isBracketing public static boolean isBracketing(UnivariateFunction function, double lower, double upper) throws NullArgumentException Check whether the interval bounds bracket a root. That is, if the values at the endpoints are not equal to zero, then the function takes opposite signs at the endpoints. Parameters: function - Function. lower - Lower endpoint. upper - Upper endpoint. Returns: true if the function values have opposite signs at the given points. Throws: NullArgumentException - if function is null. • #### isSequence public static boolean isSequence(double start, double mid, double end) Check whether the arguments form a (strictly) increasing sequence. Parameters: start - First number. mid - Second number. end - Third number. Returns: true if the arguments form an increasing sequence. • #### verifyInterval public static void verifyInterval(double lower, double upper) throws NumberIsTooLargeException Check that the endpoints specify an interval. Parameters: lower - Lower endpoint. upper - Upper endpoint. Throws: NumberIsTooLargeException - if lower >= upper. • #### verifySequence public static void verifySequence(double lower, double initial, double upper) throws NumberIsTooLargeException Check that lower < initial < upper. Parameters: lower - Lower endpoint. initial - Initial value. upper - Upper endpoint. Throws: NumberIsTooLargeException - if lower >= initial or initial >= upper. • #### verifyBracketing public static void verifyBracketing(UnivariateFunction function, double lower, double upper) throws NullArgumentException, NoBracketingException Check that the endpoints specify an interval and the end points bracket a root. Parameters: function - Function. lower - Lower endpoint. upper - Upper endpoint. Throws: NoBracketingException - if the function has the same sign at the endpoints. NullArgumentException - if function is null. Copyright © 2003–2016 The Apache Software Foundation. All rights reserved.	2016-05-30 20:30:06	{"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": 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": 0.538337767124176, "perplexity": 1668.7693986221543}, "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-2016-22/segments/1464051113990.53/warc/CC-MAIN-20160524005153-00199-ip-10-185-217-139.ec2.internal.warc.gz"}
http://blog.mouans-immobilier.com/q1oxifs3/equivalence-relation-properties-023a3f	The relationship between a partition of a set and an equivalence relation on a set is detailed. If A is an inﬁnite set and R is an equivalence relation on A, then A/R may be ﬁnite, as in the example above, or it may be inﬁnite. Equivalence Relations. Using equivalence relations to deﬁne rational numbers Consider the set S = {(x,y) ∈ Z × Z: y 6= 0 }. 1. Equivalence Relations 183 THEOREM 18.31. Properties of Equivalence Relation Compared with Equality. It is of course enormously important, but is not a very interesting example, since no two distinct objects are related by equality. . For any x ∈ ℤ, x has the same parity as itself, so (x,x) ∈ R. 2. 1. Another example would be the modulus of integers. In a sense, if you know one member within an equivalence class, you also know all the other elements in the equivalence class because they are all related according to $$R$$. Definition: Transitive Property; Definition: Equivalence Relation. We deﬁne a rational number to be an equivalence classes of elements of S, under the equivalence relation (a,b) ’ (c,d) ⇐⇒ ad = bc. Examples: Let S = ℤ and define R = {(x,y) \| x and y have the same parity} i.e., x and y are either both even or both odd. Let R be the equivalence relation … 1. As the following exercise shows, the set of equivalences classes may be very large indeed. Equalities are an example of an equivalence relation. Suppose ∼ is an equivalence relation on a set A. An equivalence class is a complete set of equivalent elements. 1. Math Properties . 1. The parity relation is an equivalence relation. First, we prove the following lemma that states that if two elements are equivalent, then their equivalence classes are equal. Assume (without proof) that T is an equivalence relation on C. Find the equivalence class of each element of C. The following theorem presents some very important properties of equivalence classes: 18. For example, in a given set of triangles, ‘is similar to’ denotes equivalence relations. . 0. Proving reflexivity from transivity and symmetry. Explained and Illustrated . Equivalence relation - Equilavence classes explanation. . reflexive; symmetric, and; transitive. Example 5.1.1 Equality ($=$) is an equivalence relation. Exercise 3.6.2. Equivalent Objects are in the Same Class. Example $$\PageIndex{8}$$ Congruence Modulo 5; Summary and Review; Exercises; Note: If we say $$R$$ is a relation "on set $$A$$" this means $$R$$ is a relation from $$A$$ to $$A$$; in other words, $$R\subseteq A\times A$$. Equivalence Properties . . Remark 3.6.1. Algebraic Equivalence Relations . We will define three properties which a relation might have. Definition of an Equivalence Relation. . Let $$R$$ be an equivalence relation on $$S\text{,}$$ and let $$a, b … The relation \(R$$ determines the membership in each equivalence class, and every element in the equivalence class can be used to represent that equivalence class. Basic question about equivalence relation on a set. . We discuss the reflexive, symmetric, and transitive properties and their closures. A binary relation on a non-empty set $$A$$ is said to be an equivalence relation if and only if the relation is. An equivalence relation on a set S, is a relation on S which is reflexive, symmetric and transitive. Lemma 4.1.9. Then: 1) For all a ∈ A, we have a ∈ [a]. Note the extra care in using the equivalence relation properties. An equivalence relation is a collection of the ordered pair of the components of A and satisfies the following properties - Equivalence Relations fixed on A with specific properties. We then give the two most important examples of equivalence relations. 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Properties and their closures S, is a complete set of triangles, ‘ is similar ’! Which a relation on a set is detailed … Definition: transitive Property ; Definition transitive. Their equivalence classes are equal equivalence relations no two distinct objects are by! The two most important examples of equivalence relations important, but is not a very interesting,... 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http://bsdupdates.com/error-propagation/propagate-error-addition.php	Home > Error Propagation > Propagate Error Addition ## Contents Do this for the indeterminate error rule and the determinate error rule. You can easily work out the case where the result is calculated from the difference of two quantities. This step should only be done after the determinate error equation, Eq. 3-6 or 3-7, has been fully derived in standard form. Uncertainty never decreases with calculations, only with better measurements. useful reference This example will be continued below, after the derivation (see Example Calculation). Solution: Use your electronic calculator. Your cache administrator is webmaster. For example, repeated multiplication, assuming no correlation gives, f = A B C ; ( σ f f ) 2 ≈ ( σ A A ) 2 + ( σ B http://lectureonline.cl.msu.edu/~mmp/labs/error/e2.htm ## Error Propagation Formula Physics For example, the fractional error in the average of four measurements is one half that of a single measurement. Then we'll modify and extend the rules to other error measures and also to indeterminate errors. For such inverse distributions and for ratio distributions, there can be defined probabilities for intervals, which can be computed either by Monte Carlo simulation or, in some cases, by using the Anytime a calculation requires more than one variable to solve, propagation of error is necessary to properly determine the uncertainty. 1. Let's say we measure the radius of a very small object. 2. We know the value of uncertainty for∆r/r to be 5%, or 0.05. 3. are inherently positive. 4. What is the error in R? 5. Under what conditions does this generate very large errors in the results? (3.4) Show by use of the rules that the maximum error in the average of several quantities is the 6. doi:10.1016/j.jsv.2012.12.009. ^ Lecomte, Christophe (May 2013). "Exact statistics of systems with uncertainties: an analytical theory of rank-one stochastic dynamic systems". 7. as follows: The standard deviation equation can be rewritten as the variance ($$\sigma_x^2$$) of $$x$$: $\dfrac{\sum{(dx_i)^2}}{N-1}=\dfrac{\sum{(x_i-\bar{x})^2}}{N-1}=\sigma^2_x\tag{8}$ Rewriting Equation 7 using the statistical relationship created yields the Exact Formula for Propagation of 8. If R is a function of X and Y, written as R(X,Y), then the uncertainty in R is obtained by taking the partial derivatives of R with repsect to each variable, 9. Setting xo to be zero, v= x/t = 50.0 cm / 1.32 s = 37.8787 cm/s. So the modification of the rule is not appropriate here and the original rule stands: Power Rule: The fractional indeterminate error in the quantity An is given by n times the The error propagation methods presented in this guide are a set of general rules that will be consistently used for all levels of physics classes in this department. The problem might state that there is a 5% uncertainty when measuring this radius. Error Propagation Inverse Let's say we measure the radius of an artery and find that the uncertainty is 5%. X = 38.2 ± 0.3 and Y = 12.1 ± 0.2. H.; Chen, W. (2009). "A comparative study of uncertainty propagation methods for black-box-type problems". EngineerItProgram 11.543 προβολές 6:39 Calculating Uncertainty (Error Values) in a Division Problem - Διάρκεια: 5:29. https://en.wikipedia.org/wiki/Propagation_of_uncertainty Example: An angle is measured to be 30°: ±0.5°. Therefore the fractional error in the numerator is 1.0/36 = 0.028. Error Propagation Chemistry Laboratory experiments often take the form of verifying a physical law by measuring each quantity in the law. However, in complicated scenarios, they may differ because of: unsuspected covariances errors in which reported value of a measurement is altered, rather than the measurements themselves (usually a result of mis-specification Since uncertainties are used to indicate ranges in your final answer, when in doubt round up and use only one significant figure. ## Error Propagation Calculator When mathematical operations are combined, the rules may be successively applied to each operation. my site Example: Suppose we have measured the starting position as x1 = 9.3+-0.2 m and the finishing position as x2 = 14.4+-0.3 m. Error Propagation Formula Physics For instance, in lab you might measure an object's position at different times in order to find the object's average velocity. Error Propagation Average Practically speaking, covariance terms should be included in the computation only if they have been estimated from sufficient data. Error Propagation in Trig Functions Rules have been given for addition, subtraction, multiplication, and division. see here Contributors http://www.itl.nist.gov/div898/handb...ion5/mpc55.htm Jarred Caldwell (UC Davis), Alex Vahidsafa (UC Davis) Back to top Significant Digits Significant Figures Recommended articles There are no recommended articles. National Bureau of Standards. 70C (4): 262. A one half degree error in an angle of 90° would give an error of only 0.00004 in the sine. Error Propagation Square Root If you measure the length of a pencil, the ratio will be very high. Generally, reported values of test items from calibration designs have non-zero covariances that must be taken into account if b is a summation such as the mass of two weights, or Advisors For Incoming Students Undergraduate Programs Pre-Engineering Program Dual-Degree Programs REU Program Scholarships and Awards Student Resources Departmental Honors Honors College Contact Mail Address:Department of Physics and AstronomyASU Box 32106Boone, NC this page The trick lies in the application of the general principle implicit in all of the previous discussion, and specifically used earlier in this chapter to establish the rules for addition and If da, db, and dc represent random and independent uncertainties, about half of the cross terms will be negative and half positive (this is primarily due to the fact that the Error Propagation Excel In statistics, propagation of uncertainty (or propagation of error) is the effect of variables' uncertainties (or errors, more specifically random errors) on the uncertainty of a function based on them. Calculus for Biology and Medicine; 3rd Ed. ## When a quantity Q is raised to a power, P, the relative error in the result is P times the relative error in Q. See Ku (1966) for guidance on what constitutes sufficient data2. Error Propagation Contents: Addition of measured quantities Multiplication of measured quantities Multiplication with a constant Polynomial functions General functions Very often we are facing the situation that we need to measure This, however, is a minor correction, of little importance in our work in this course. Error Propagation Definition Generated Mon, 24 Oct 2016 17:42:50 GMT by s_wx1206 (squid/3.5.20) ERROR ANALYSIS: 1) How errors add: Independent and correlated errors affect the resultant error in a calculation differently. For example, you made one measurement of one side of a square metal Uncertainty in measurement comes about in a variety of ways: instrument variability, different observers, sample differences, time of day, etc. This is desired, because it creates a statistical relationship between the variable $$x$$, and the other variables $$a$$, $$b$$, $$c$$, etc... Get More Info Learn more You're viewing YouTube in Greek. There is no error in n (counting is one of the few measurements we can do perfectly.) So the fractional error in the quotient is the same size as the fractional Starting with a simple equation: $x = a \times \dfrac{b}{c} \tag{15}$ where $$x$$ is the desired results with a given standard deviation, and $$a$$, $$b$$, and $$c$$ are experimental variables, each The fractional error may be assumed to be nearly the same for all of these measurements. If this error equation is derived from the indeterminate error rules, the error measures Δx, Δy, etc. General functions And finally, we can express the uncertainty in R for general functions of one or mor eobservables. How would you determine the uncertainty in your calculated values? Since the uncertainty has only one decimal place, then the velocity must now be expressed with one decimal place as well. And again please note that for the purpose of error calculation there is no difference between multiplication and division. Using the equations above, delta v is the absolute value of the derivative times the delta time, or: Uncertainties are often written to one significant figure, however smaller values can allow The coefficients may also have + or - signs, so the terms themselves may have + or - signs. doi:10.2307/2281592. Notes on the Use of Propagation of Error Formulas, J Research of National Bureau of Standards-C. The system returned: (22) Invalid argument The remote host or network may be down.	2018-01-20 15:07:31	{"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": 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": 0.7726188898086548, "perplexity": 868.9722898916198}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084889660.55/warc/CC-MAIN-20180120142458-20180120162458-00587.warc.gz"}
https://zbmath.org/?q=an:1252.46009	# zbMATH — the first resource for mathematics On super-weakly compact sets and uniformly convexifiable sets. (English) Zbl 1252.46009 A famous result of P. Enflo states that a Banach space is superreflexive if and only if it is uniformly convexifiable. R. C. James characterized superreflexivity in terms of the so-called finite tree property. Recall that a Banach space is said to be superreflexive whenever every Banach space finitely representable in it is reflexive. Here, the authors present a localized setting for the previous results. The main theorem reads as follows. A closed convex bounded subset of a Banach space is uniformly convexifiable if and only if it is super-weakly compact. Those notions are the natural counterpart of the corresponding global notions. More precisely, the authors introduce the following definitions. (i) A convex subset $$A$$ of a Banach space is said to be uniformly convex if, for every $$x_0\in A$$, the function $$x\mapsto\\|x-x_0\\|^2$$ is uniformly convex on $$A$$, and $$A$$ is said to be uniformly convexifiable if there exist an equivalent norm on $$X$$ such that $$A$$ is uniformly convex with respect to this norm. (ii) Given two convex subsets $$A$$ and $$B$$ of a Banach space $$X$$, $$A$$ is said to be finitely representable in $$B$$ if, for every $$\varepsilon>0$$, each $$n$$-simplex in $$A$$ can be $$(1+\varepsilon)$$-affinely embedded into $$B$$. (iii) A bounded closed convex subset $$C$$ of $$X$$ is said to be super-weakly compact if, every convex set which is finitely representable in $$C$$, is relatively weakly compact. As another example of how “global” theorems appear in this setting, the authors prove that, for a closed bounded convex subset $$C$$ of a Banach space, the following statements are equivalent. (i) $$C$$ is not superreflexive. (ii) $$C$$ has the finite tree property. (iii) There exists $$\theta>0$$ such that, for every $$n\in\mathbb N$$, there are $$x_1,\ldots,x_n$$ in $$C$$ such that, for all $$1\leq k\leq n-1$$, $$\text{dist}(\text{co}\{x_1,\ldots,x_{k}\},\text{co}\{x_{k+1},\ldots,x_n\})>\theta$$. The main ingredients in the proofs are the classical James’ finite tree theorem, Enflo’s renorming technique, Grothendieck’s lemma and the Davis-Figiel-Johnson-Pełczyński lemma. ##### MSC: 46B20 Geometry and structure of normed linear spaces 46B03 Isomorphic theory (including renorming) of Banach spaces 46B50 Compactness in Banach (or normed) spaces Full Text:	2021-09-20 07:16:37	{"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": 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": 0.9389793276786804, "perplexity": 232.61321855189155}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1631780057033.33/warc/CC-MAIN-20210920070754-20210920100754-00482.warc.gz"}
https://raisingthebar.nl/2018/07/28/solvability-of-an-equation-with-a-square-matrix/?shared=email&msg=fail	28 Jul 18 ## Solvability of an equation with a square matrix Everywhere in this section we assume that $A$ is square of size $k\times k.$ The main problem is about solvability of the equation $Ax=y$. ### Dissecting the problem Exercise 1. If $A$ is invertible, then $N(A)=\{0\}$ and $\text{Img}(A)=R^k.$ Proof. If $Ax=0,$ then $x=A^{-1}Ax=0,$ so $N(A)=\{0\}$ (you could also say that $A$ is one-to-one). From $AA^{-1}=I$ we have $AA^{-1}x=x.$ This tells us that $x=Ay$ with $y=A^{-1}x.$ We see that any $x$ belongs to the image of $A,$ so $\text{Img}(A)=R^k.$ Exercise 2. If $N(A)=\{0\},$ then $A$ is invertible. Proof. Since $A$ is one-to-one, we can use the inverse $f^{-1}$ of $f(x)=Ax$ in the general sense: $f^{-1}(Ax)=x$. It is defined on the image of $A.$ We know that $f^{-1}$ is given by some matrix $B$: $BAx=x$ for all $x.$ Hence, $BAe_i=e_i$ for $i=1,...,k.$ Putting these equations side by side we get (1) $BA=BAI=I.$ (In detail: the identity matrix is partitioned as $I=(e_1,...,e_k),$ so $BAI$ $=(BAe_1,...,BAe_k)=(e_1,...,e_k)=I$). Thus, $B$ is the left inverse of $A,$ and we've seen before that for square matrices this implies existence of the right inverse, invertibility of $A$ and the equation $B=A^{-1}$. Exercise 3. If $\text{Img}(A)=R^k,$ then $A$ is invertible. Proof. If $\text{Img}(A)=R^k,$ then by the second characterization of matrix image $N(A^T)=(\text{Img}(A))^{\perp}=\{0\}.$ Applying Exercise 2 to $A^T,$ we see that the transpose is invertible and hence $\det A=\det A^{T}\neq 0.$ This implies invertibility of $A.$ ### Collecting the pieces together Exercise 4. The following conditions are equivalent: a) $N(A)=\{0\},$ b) $\text{Img}(A)=R^k,$ c) $\det A\neq 0,$ d) $A$ is invertible. Proof. The equivalence c) $\Longleftrightarrow$ d) has been established before. For the implication d) $\Longrightarrow$ a)+b) see Exercise 1. By Exercise 2 we have a) $\Longrightarrow$ d). By Exercise 3, b) $\Longrightarrow$ d. Conclusion. The condition $\det A\neq 0$ is the easiest to check. If it holds, then the equation $Ax=y$ has a unique solution for any $y\in R^k.$ Remark 1. By negating Exercise 4, we see that the following conditions are equivalent: $a^\prime) N(A)\neq \{0\},$ $b^\prime) \text{Img}(A)\neq R^k,$ $c^\prime) \det A=0,$ $d^\prime) A^{-1}$ does not exists. Remark 2. By going through the proof one can check that the result of Exercise 4 holds if $R^k$ is replaced by $C^k$ (the set of vectors with complex coordinates).	2020-02-21 00:44:49	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 124, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.954775333404541, "perplexity": 202.47645716148267}, "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-2020-10/segments/1581875145316.8/warc/CC-MAIN-20200220224059-20200221014059-00077.warc.gz"}
https://en.wikipedia.org/wiki/Cambridge_equation	# Cambridge equation The Cambridge equation formally represents the Cambridge cash-balance theory, an alternative approach to the classical quantity theory of money. Both quantity theories, Cambridge and classical, attempt to express a relationship among the amount of goods produced, the price level, amounts of money, and how money moves. The Cambridge equation focuses on money demand instead of money supply. The theories also differ in explaining the movement of money: In the classical version, associated with Irving Fisher, money moves at a fixed rate and serves only as a medium of exchange while in the Cambridge approach money acts as a store of value and its movement depends on the desirability of holding cash. Economists associated with Cambridge University, including Alfred Marshall, A.C. Pigou, and John Maynard Keynes (before he developed his own, eponymous school of thought) contributed to a quantity theory of money that paid more attention to money demand than the supply-oriented classical version. The Cambridge economists argued that a certain portion of the money supply will not be used for transactions; instead, it will be held for the convenience and security of having cash on hand. This portion of cash is commonly represented as k, a portion of nominal income (the product of the price level and real income), ${\displaystyle P\cdot Y}$). The Cambridge economists also thought wealth would play a role, but wealth is often omitted from the equation for simplicity. The Cambridge equation is thus: ${\displaystyle M^{\textit {d}}={\textit {k}}\cdot P\cdot Y}$ Assuming that the economy is at equilibrium (${\displaystyle M^{\textit {d}}=M}$), ${\displaystyle Y}$ is exogenous, and k is fixed in the short run, the Cambridge equation is equivalent to the equation of exchange with velocity equal to the inverse of k: ${\displaystyle M\cdot {\frac {1}{k}}=P\cdot Y}$ ## History and significance The Cambridge equation first appeared in print in 1917 in Pigou's "Value of Money".[1]Keynes contributed to the theory with his 1923 Tract on Monetary Reform. The Cambridge version of the quantity theory led to both Keynes's attack on the quantity theory and the Monetarist revival of the theory.[2] Marshall recognized that k would be determined in part by an individual's desire to hold liquid cash. In his General Theory of Employment, Interest and Money, Keynes expanded on this concept to develop the idea of liquidity preference,[3] a central Keynesian concept. ## References 1. ^ Patinkin, Don (1 November 1984). Anticipations of the General Theory?: And Other Essays on Keynes. University of Chicago Press. p. 171. ISBN 978-0-226-64874-3. 2. ^ Froyen, Richard T. Macroeconomics: Theories and Policies. 3rd Edition. Macmillan Publishing Company: New York, 1990. p. 70–71. 3. ^ Skidelsky, Robert. John Maynard Keynes: 1883–1946. Penguin: 2003. p. 131.	2019-06-17 09:12:20	{"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": 0.7109718918800354, "perplexity": 2497.5518203064344}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998462.80/warc/CC-MAIN-20190617083027-20190617105027-00491.warc.gz"}
http://mathematica.stackexchange.com/questions/31917/strange-response-with-dynamic-texture-and-mouseposition	# Strange response with dynamic texture and MousePosition I built a cubic, then use MousePosition together with GUIScreenShot to take a floating region with fixed size (200200) on the screen, and use it as dynamic texture on the cubic. My code is Needs["GUIKit"]; $HistoryLength = 1; DynamicModule[{mpp}, Dynamic@{ Refresh[ ClearSystemCache[]; g = GUIScreenShot[With[{mp = MousePosition[]}, mpp = {mp - 100, mp + 100}]]; Graphics3D[{Texture[g], Polygon[coords, VertexTextureCoordinates -> Table[vtc, {6}]]}, Lighting -> "Neutral"], UpdateInterval -> .5], mpp, g}, Initialization :> ( vtc = {{0, 0}, {0, 1}, {1, 1}, {1, 0}}; coords = {{{0, 0, 0}, {0, 1, 0}, {1, 1, 0}, {1, 0, 0}}, {{0, 0, 0}, {1, 0, 0}, {1, 0, 1}, {0, 0, 1}}, {{1, 0, 0}, {1, 1, 0}, {1, 1, 1}, {1, 0, 1}}, {{1, 1, 0}, {0, 1, 0}, {0, 1, 1}, {1, 1, 1}}, {{0, 1, 0}, {0, 0, 0}, {0, 0, 1}, {0, 1, 1}}, {{0, 0, 1}, {1, 0, 1}, {1, 1, 1}, {0, 1, 1}}};)] You can see the result by executing the code then moving your mouse around. Like this, But there are two strange problems, 1) the size of the floating area taken should be always 200200, however it will actually change. 2) the memory usage will constantly grow as we move our mouse, but I have already used $HistoryLength=1 and ClearSystemCache[]. Any suggestion? - ## 1 Answer Look very carefully at the documentation. This had me puzzled for a while too: When the argument consists of screen coordinates of the form {{xmin,xmax},{ymin,ymax}}, a rectangle from the current screen is captured. The key part is {min,max} like PlotRange. Your code produces {{min,min},{max,max}}. I think the memory usage problem is a memory leak in the front end Java code. I don't think there's anything you can do about it except report it to WRI. If you really want to investigate, VisualVM might be helpful, though I have never tried it with Mathematica. - Yes I was too careless, should have written mpp = {mp[[1]] + {-100, 100}, mp[[2]] + {-100, 100}}` instead. I will wait for some more time, if the memory problem cannot be solved, I will report it. Thanks! – saturasl Sep 9 '13 at 21:19	2015-10-09 20:18:34	{"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": 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": 0.22025330364704132, "perplexity": 2495.5170155044093}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443737935292.75/warc/CC-MAIN-20151001221855-00222-ip-10-137-6-227.ec2.internal.warc.gz"}
http://tex.stackexchange.com/questions/94553/longtable-does-not-work-on-ieeeconf-but-works-on-document-class-article	# longtable does not work on ieeeconf, but works on Document class article I am trying to add a long table of figures into latex. I am able to successfully add in \documentclass[12pt]{article}. But when I try adding it in the following document class, I get error: \documentclass[letterpaper, 10 pt, conference]{ieeeconf} Error: Package longtable error: longtable not in 1-column mode \begin{longtable} The code for table I created is: \begin{center} \begin{longtable}{ccp{5cm}\|} \caption{results}\label{tab:support}\\ %Row1 __________________________________________________________________________ Image & result &My comments \\ \raisebox{-.5\height}{\includegraphics[width=140px,height=120px] {Images/support/rgb_000938.eps}} &\raisebox{-.5\height}{\includegraphics[width=200px,height=150px] {Images/support/region_000938_1.eps}} & xxxx \\ \newline \end{longtable} \end{center} - In two column longtable doesn't work as of now. You may use supertabular instead. – Harish Kumar Jan 20 '13 at 7:08 ok...I will try. thanks :) – Swagatika Jan 20 '13 at 10:19 Please have mercy to your readers and don't use tables on a conference paper that lasts more than one column or a page. – percusse Jan 20 '13 at 10:56 thanks for the feedback :) – Swagatika Jan 20 '13 at 11:04 longtable works in two columns if you use the code from multi-page two column table in a single column document Note I used \documentclass[letterpaper, 10pt,]{IEEEconf} as that appears to be the class name with uppercase IEEE and 10pt should have no space, and the version I have in TeXLive 2012 complained about the conference option. \documentclass[letterpaper, 10pt,]{IEEEconf} \usepackage{longtable} \makeatletter \let\saved@longtable\longtable \long\def\foo#1\LT@err#2#3#4!!{\def\longtable{#1#4}} \expandafter\foo\longtable!! \long\def\foo#1\@outputpage#2\@outputpage#3!!{% \def\LT@output{#1\@opcol#2\@opcol#3}} \expandafter\foo\LT@output!! \makeatother \begin{document} \twocolumn[\section{List}] \begin{longtable}{\| l \|\| c \|} \caption{Some square numbers\label{tab:xxx}}\\ $n$&$n$-squared\\ \hline $n$&$n$-squared\\	2014-10-25 09:09:21	{"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": 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": 0.682461142539978, "perplexity": 5152.769440043268}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1414119647914.3/warc/CC-MAIN-20141024030047-00043-ip-10-16-133-185.ec2.internal.warc.gz"}
https://www.shaalaa.com/question-bank-solutions/arc-chord-properties-angle-that-arc-circle-subtends-center-double-that-which-it-subtends-any-point-remaining-part-circle-in-regular-pentagon-abcde-inscribed-circle-find-ratio-between-angle-eda-angle-adc_38526	Share Books Shortlist # In a Regular Pentagon Abcde, Inscribed in a Circle; Find Ratio Between Angle Eda and Angle Adc. - ICSE Class 10 - Mathematics ConceptArc and Chord Properties - the Angle that an Arc of a Circle Subtends at the Center is Double that Which It Subtends at Any Point on the Remaining Part of the Circle #### Question In a regular pentagon ABCDE, Inscribed in a circle; find ratio between angle EDA and angle ADC. #### Solution Arc AE subtends ∠AOE at the centre and ∠ADE at the remaining part of the circle. ∴ ∠ADE = 1/2 xx 72° = 36 [central angle is a regular pentagon at O]	2019-07-23 14:31:17	{"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": 0.6465965509414673, "perplexity": 3208.915807138978}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195529406.97/warc/CC-MAIN-20190723130306-20190723152306-00465.warc.gz"}
http://math.stackexchange.com/questions/44241/study-program-for-algebra-for-an-math-phobic-person?answertab=active	# Study program for algebra for an math-phobic person I have a female friend (Mid 20's) who wants to get an associates degree, who has an absolute phobia of math, and algebra in particular. I'm trying to find a good program that can help her to study algebra to take the entrance exam to the community college, so that she can take as few classes as possible. Does anyone have a suggestion of a great resource, be it an online study course, book, etc that can help her learn enough algebra to at least qualify for an intermediate algebra level? - Stewart Calculus has a review of algebra that looks useful. I tend to take basic algebra for granted in everything I do so I have a hard time understanding how to teach it. Also, why would it matter that your friend is female... – anon Jun 9 '11 at 2:27 @anon: Just that there are some resources around that might be geared more towards females, that's all. – PearsonArtPhoto Jun 9 '11 at 4:03 Last time I checked, there was nothing about math that was gender-specific. – ItsNotObvious Jun 9 '11 at 12:27 Have her take a look at the algebra videos at the Khan Academy. The videos there cover all of basic arithmetic, Algebra I, and Algebra II. - Please correct the name, it is "Khan" – kuch nahi Jun 9 '11 at 4:08 @yayu: OK, thanks. – Jim Belk Jun 9 '11 at 5:14 There are some books by the actress Danica McKellar, who played Winnie on The Wonder Years television series. She got a mathematics Bachelor's at UCLA and is one of three authors on a paper about percolation theory. She has written at least three books at about the level of algebra or even pre-algebra. I understand that the tone is set for teenage girls, that might be off-putting or it might be hilarious. I'm afraid I have not seen the books, but it is clearly a good idea to have such a resource available. • Danica McKellar (2007). Math Doesn't Suck: How to Survive Middle-School Math Without Losing Your Mind or Breaking a Nail. Hudson Street Press. ISBN 978-1594630392. • Danica McKellar (2008). Kiss My Math: Showing Pre-Algebra Who's Boss. Hudson Street Press. ISBN 978-1-59463-049-1. • Danica McKellar (2010). Hot X: Algebra Exposed. Hudson Street Press. ISBN 978-1-59463-070-5. - I edited your answer a bit, since for some reason pound signs caused the text to become very large and bold. – Zev Chonoles Jun 9 '11 at 3:39 @Zev: are you sure this formatting was unintended? It conveyed a subtle message... :) Seriously: the # signs are first level header prefixes for markdown – t.b. Jun 9 '11 at 3:43 Thanks for the edit. I just copied and pasted from the bottom of the Wikipedia page, by the time it got here it had pound signs beginning each line. I took one off, I realized I liked the bold characters, but I was not thinking about the standards of the website (it has been an annoying day for me). Anyway, this is good, keep it. If anyone has seen the books, are they any good? I guess we want the opinion of a middle-school girl on that. – Will Jagy Jun 9 '11 at 4:11 There's a blog post by Terence Tao the first few paragraphs of which are about the first book (and link to other reviews, etc; there's also some discussion in the comments). – ShreevatsaR Jun 9 '11 at 6:10 That's nice, thanks for the blog link. – Will Jagy Jun 9 '11 at 22:24	2015-05-29 13:08:37	{"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": 0.5758358240127563, "perplexity": 2106.9868117556994}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207930109.71/warc/CC-MAIN-20150521113210-00294-ip-10-180-206-219.ec2.internal.warc.gz"}
https://forum.dynare.org/t/dynare-notation/5649	# Dynare notation Hello everyone, I have found a replication of a labour market model that uses a notation of this kind: model; (1/c) = beta(1/c(+1))(1+r(+1)-delta); However, I want to integrate it with another NK model that uses a notation of this kind: model; (1/exp(c(+1)))(1+exp(r(+1))-delta); I thought I could solve by re-writing the first piece of code in in exp-logs (as the second one) but when I do, dynare says it cannot find the steady state anymore. Does anyone understand why that might happen? Cheers Then you must have introduced a mistake when doing this. You must consistently replace all variables that have been redefined. I have tried very hard to debug it, but there must be something that I systematically rewrite wrong thinking it’s right. I wasn’t sure if, for instance c^ALPHA became exp©^ALPHA or (exp©)^ALPHA, so I tried both ways and neither worked. I still think the second one is correct. Except for this, it’s really just 9 short equations. Do you or anyone else spot any consistent mistake? From: ``````var U V M S u e v t z; varexo zi; parameters A a B x l K b c rho sigma; l = 0.039; % obsolescence separation rate a = 0.5; % matching function exponent A = 0.636; % matching function scale parameter B = 0.9967; % discount factor x = 0.5; % worker bargaining weight rho = 0.975; % persistence of productivity process sigma = 0.0076; % variance of productivity process b = 0.9; % value of leisure c = 0.17; % vacancy posting cost K = 26.94; % creation cost model; % Ausiliary definitions for matching function and flow probabilities e = 1-u; t = v/u; % Unemployment value function (eq 16) U = b + B( A t^(1-a)xS(+1) + U(+1) ); % Vacancy value function (eq 17) V = -c + B( A t^(-a)(1-x)S(+1) + V(+1) ); % Output value function (eq 18) M = z + B( (1-l)S(+1) + U(+1) + V(+1) ); % Matching surplus definition (S=M-U-V) (eq 19) S = z - b + c + B((1-l) - A t^(1-a)x - A t^(-a)(1-x))S(+1); % FONC: V=Kn (eq 21) c = B( A * t^(-a)(1-x)S(+1)); % Law of motion for (eq 22) u = u(-1) + l(1-u(-1)) - A(t(-1)^(1-a)) * u(-1); % Log productivity follows an exogenous AR(1) process log(z) = rholog(z(-1)) + zi; end; initval; U = 294.865; V = 0.543609; M = 295.738; S = 0.32959; u = 0.0798989; e = 0.920101; v = 0.04076282; t = 0.510245; z = 1; end; shocks; var zi = sigma^2; end; stoch_simul( nograph, order = 1, periods = 30300, drop = 300, IRF = 90 );`````` To: ``````var U V M S u e v t z; varexo zi; parameters A a B x l K b c rho sigma; l = 0.039; % obsolescence separation rate a = 0.5; % matching function exponent A = 0.636; % matching function scale parameter B = 0.9967; % discount factor x = 0.5; % worker bargaining weight rho = 0.975; % persistence of productivity process sigma = 0.0076; % variance of productivity process b = 0.9; % value of leisure c = 0.17; % vacancy posting cost K = 26.94; % creation cost model; % Auxiliary definitions for matching function and flow probabilities exp(e) = 1-exp(u); exp(t) = exp(v-u); % Unemployment value function (eq 16) exp(U) = b + B( A((exp(t))^(1-a))xexp(S(+1)) + exp(U(+1)) ); % Vacancy value function (eq 17) exp(V) = -c + B( A((exp(t))^(-a))(1-x)exp(S(+1)) + exp(V(+1)) ); % Output value function (eq 18) exp(M) = exp(z) + B( (1-l)exp(S(+1)) + exp(U(+1)) + exp(V(+1)) ); % Matching surplus definition (S=M-U-V) (eq 19) exp(S) = exp(z) - b + c + B((1-l) - A((exp(t))^(1-a))x - A((exp(t))^(-a))(1-x))exp(S(+1)); % FONC: V=Kn (eq 21) c = B( A((exp(t))^(-a))(1-x)exp(S(+1))); % Law of motion for (eq 22) exp(u) = exp(u(-1)) + l(1-exp(u(-1))) - A((exp(t(-1)))^(1-a)) * exp(u(-1)); % Log productivity follows an exogenous AR(1) process z = rhoz(-1) + zi; end; initval; U = 294.865; V = 0.543609; M = 295.738; S = 0.32959; u = 0.0798989; e = 0.920101; v = 0.04076282; t = 0.510245; z = 1; end; shocks; var zi = sigma^2; end; stoch_simul( order = 1, periods = 30300, drop = 300, IRF = 90 );`````` The original code’s source is michael-droste.com/replications/ Cheers When doing an exp()-transformation, you need to adjust the initial values as well. When using [code] var U V M S u e v t z; varexo zi; ``````parameters A a B x l K b c rho sigma; l = 0.039; % obsolescence separation rate a = 0.5; % matching function exponent A = 0.636; % matching function scale parameter B = 0.9967; % discount factor x = 0.5; % worker bargaining weight rho = 0.975; % persistence of productivity process sigma = 0.0076; % variance of productivity process b = 0.9; % value of leisure c = 0.17; % vacancy posting cost K = 26.94; % creation cost model; % Auxiliary definitions for matching function and flow probabilities exp(e) = 1-exp(u); exp(t) = exp(v-u); % Unemployment value function (eq 16) exp(U) = b + B( A((exp(t))^(1-a))xexp(S(+1)) + exp(U(+1)) ); % Vacancy value function (eq 17) exp(V) = -c + B( A((exp(t))^(-a))(1-x)exp(S(+1)) + exp(V(+1)) ); % Output value function (eq 18) exp(M) = exp(z) + B( (1-l)exp(S(+1)) + exp(U(+1)) + exp(V(+1)) ); % Matching surplus definition (S=M-U-V) (eq 19) exp(S) = exp(z) - b + c + B((1-l) - A((exp(t))^(1-a))x - A((exp(t))^(-a))(1-x))exp(S(+1)); % FONC: V=Kn (eq 21) c = B( A((exp(t))^(-a))(1-x)exp(S(+1))); % Law of motion for (eq 22) exp(u) = exp(u(-1)) + l(1-exp(u(-1))) - A((exp(t(-1)))^(1-a)) * exp(u(-1)); % Log productivity follows an exogenous AR(1) process z = rho*z(-1) + zi; end; initval; U = log(294.865); V = log(0.543609); M = log(295.738); S = log(0.32959); u = log(0.0798989); e = log(0.920101); v = log(0.04076282); t = log(0.510245); z = log(1); end; shocks; var zi = sigma^2; end; resid; stoch_simul( order = 1, periods = 30300, drop = 300, IRF = 90 ); `````` [/code] it works. Dear Professor, Thank you so much, I was stuck I couldn’t get why and thus could not proceed with my work. Your work here in this blog is precious. Best, Fabio Dear Prof., One thing remains obscure about the explog version of the model. The steady-state value of unemployment becomes log(0.0798989) = -2.5270, how should this be interpreted? In the normal model 0.08 should be the equilibrium fraction of the household that remains unemployed, but I don’t really know how to interpret the transformed value. Best, Fabio I am not following. After substitution, the new variables are actually the logarithms of the old variables. Steady state unemployment is therefore still 0.0798989.	2022-05-17 01:05:54	{"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": 0.861919105052948, "perplexity": 13106.715187973457}, "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/1652662515466.5/warc/CC-MAIN-20220516235937-20220517025937-00125.warc.gz"}
https://tex.stackexchange.com/questions/605202/why-arent-dageshes-aligned-in-hebrew-in-lualatex	# Why aren't dageshes aligned in Hebrew in LuaLaTex? I'm trying to typeset a single line of Biblical Hebrew, in which some letters have a dagesh diacritic, which is no problem in my text editor in my chosen font (designed for Biblical Hebrew), but for some reason LuaLaTeX is refusing to put the dagesh in the proper place. MWE below, along with example of what should be produced: \documentclass{standalone} \usepackage[nil,bidi=default]{babel} \babelprovide[import=he]{hebrew} \usepackage{fontspec} \setmainfont{KeterYG-Medium} \begin{document} כָּל-עֲמַל הָאָדָם, לְפִיהוּ; וְגַם-הַנֶּפֶשׁ, לֹא תִמָּלֵא. \end{document} Whereas this is what should be produced: • No time to investigate, but it looks like all the nikkudim are misaligned. I notice you haven’t actually specified Hebrew as the main language vía Babel or otherwise identified the text as Hebrew Jul 16 at 5:34 • Did you try with [Renderer=Harfbuzz]? Jul 16 at 6:11 • @UlrikeFischer Thank you! Jul 16 at 16:56 • @DonHosek Good job spotting that—embarassingly, I had uploaded the wrong code (the screenshot was, of course, produced by the correct specifying Hebrew because otherwise TeX would render it backwards!)) Fixed now Jul 16 at 16:56 Combining the comment advice together (language selection, plus font rendering) gives: with various fonts. MWE \documentclass{article}%[border=6pt]{standalone} \usepackage{xcolor} \usepackage[nil,bidi=default]{babel} \babelprovide[import=he]{hebrew} \babelfont{rm}{CMU Serif} \babelfont[hebrew]{rm}[Path=C:/Users/Public/fonts/1/, Extension=.ttf, UprightFont=-Medium, Renderer=HarfBuzz, ]{KeterYG} % \babelfont[hebrew]{ivrtshfr}[Path=C:/Users/Public/fonts/1/, Extension=.ttf, UprightFont=Regular, Renderer=HarfBuzz, ]{Shofar} % \babelfont[hebrew]{ivrtdvd}[Path=C:/Users/Public/fonts/1/, Extension=.ttf, UprightFont=-Medium, Renderer=HarfBuzz, ]{TaameyDavidCLM} % \babelfont[hebrew]{ivrtash}[Path=C:/Users/Public/fonts/1/, Extension=.ttf, UprightFont=-Medium, Renderer=HarfBuzz, ]{TaameyAshkenaz} % \babelfont[hebrew]{ivrttfc}[Path=C:/Users/Public/fonts/1/, Extension=.ttf, UprightFont=*-Medium, Renderer=HarfBuzz, ]{TaameyFrankCLM} % \babeltags{hebrew=hebrew} % \begin{document} { \selectlanguage{hebrew} \colorbox{blue!8}{\texthebrew{כָּל-עֲמַל הָאָדָם, לְפִיהוּ; וְגַם-הַנֶּפֶשׁ, לֹא תִמָּלֵא.}} } { \selectlanguage{hebrew} \textivrtshfr{כָּל-עֲמַל הָאָדָם, לְפִיהוּ; וְגַם-הַנֶּפֶשׁ, לֹא תִמָּלֵא.} } { \selectlanguage{hebrew} \textivrtash{כָּל-עֲמַל הָאָדָם, לְפִיהוּ; וְגַם-הַנֶּפֶשׁ, לֹא תִמָּלֵא.} } { \selectlanguage{hebrew} \textivrtdvd{כָּל-עֲמַל הָאָדָם, לְפִיהוּ; וְגַם-הַנֶּפֶשׁ, לֹא תִמָּלֵא.} } \end{document} • Thank you very much! Jul 16 at 16:57	2021-09-21 17:13:41	{"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": 0.8465918302536011, "perplexity": 13636.094055535055}, "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-2021-39/segments/1631780057225.57/warc/CC-MAIN-20210921161350-20210921191350-00024.warc.gz"}
https://pos.sissa.it/414/514/	Volume 414 - 41st International Conference on High Energy physics (ICHEP2022) - Higgs Physics Strange quark as a probe for new physics in the Higgs sector V.M. Cairo Full text: pdf Pre-published on: December 22, 2022 Published on: Abstract One of the most interesting yet-to-be answered questions in Particle Physics is the nature of the Higgs Yukawa couplings and their universality. Key information in our understanding of this question arises from studying the coupling of the Higgs boson to second generation quarks. Some puzzles in the flavor sector and potential additional sources of CP violation could also have their origins in an extended Higgs sector. Rare Higgs decay modes to charm or strange quarks are very challenging or nearly impossible to detect with the current experiments at the Large Hadron Collider, where the large multi-jet backgrounds makes it difficult to study light quark couplings with inclusive $h\rightarrow q\bar{q}$ decays. Future $e^{+}e^{-}$ machines are thus the perfect avenue to study such phenomena. This contribution presents the development of a novel algorithm for tagging jets originating from the hadronisation of strange quarks (strange-tagging) and the first application of such a strange-tagger to a direct Higgs to strange ($h\rightarrow s\bar{s}$) analysis. The work is performed with the International Large Detector (ILD) concept at the International Linear Collider (ILC), but it is easily applicable to other Higgs factories. The study includes as well a preliminary investigation of a Compact Ring Imaging Cerenkov system (RICH) capable of maximising strange-tagging performance in future Higgs factory detectors. DOI: https://doi.org/10.22323/1.414.0514 How to cite Metadata are provided both in "article" format (very similar to INSPIRE) as this helps creating very compact bibliographies which can be beneficial to authors and readers, and in "proceeding" format which is more detailed and complete. Open Access	2023-01-28 00:36:38	{"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": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4774620234966278, "perplexity": 1589.0052430698818}, "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-2023-06/segments/1674764499468.22/warc/CC-MAIN-20230127231443-20230128021443-00486.warc.gz"}
http://mathhelpforum.com/differential-geometry/112632-help-proving-some-facts.html	# Math Help - Help with proving some facts 1. ## Help with proving some facts Hi, I have to prove the following facts, but I am not sure if I figured them out correctly: f : X --> Y and F : P(X) --> P(Y ) 1) If A1 is a subset of A2, then F(A1) is a subset of F(A2) For this one, I have the following: For all x that are elements of A1 and A2 (there is a y that is an element of F(A1) s.t. F(x)=y) and (y is an element of F(A2) s.t. F(X)=y) Therefore, the fact that x is an element of A1 implies that there is a y that is an element of F(A2) s.t. F(x) = y Hence, F(A1) is a subset of F(A2) (Here I feel that, I missed a step constructing the proof with quantifiers.) Also, 2) For every A that is an element of P(X), A is a subset of the inverse function of F(A) For this one, I have: (For all x that are elements of A, there is a y that is an element of F(A) s.t. F(x)=y s.t. the inverse of F(y)=x) implies that (for all x belonging to A, x is an element of the inverse of F(F(A))) implies that (A belongs to the inverse of F(F(A))). I am also not sure about my logic here, do you guys agree with me? 2. Originally Posted by kaka87 Hi, I have to prove the following facts, but I am not sure if I figured them out correctly: f : X --> Y and F : P(X) --> P(Y ) 1) If A1 is a subset of A2, then F(A1) is a subset of F(A2) For this one, I have the following: For all x that are elements of A1 and A2 (there is a y that is an element of F(A1) s.t. F(x)=y) and (y is an element of F(A2) s.t. F(X)=y) Therefore, the fact that x is an element of A1 implies that there is a y that is an element of F(A2) s.t. F(x) = y Hence, F(A1) is a subset of F(A2) (Here I feel that, I missed a step constructing the proof with quantifiers.) Also, 2) For every A that is an element of P(X), A is a subset of the inverse function of F(A) For this one, I have: (For all x that are elements of A, there is a y that is an element of F(A) s.t. F(x)=y s.t. the inverse of F(y)=x) implies that (for all x belonging to A, x is an element of the inverse of F(F(A))) implies that (A belongs to the inverse of F(F(A))). I am also not sure about my logic here, do you guys agree with me? 1. I think you have it, or if not, you're on the right track. Let $y\in F(A_1)$. Then $\exists x\in A_1$ such that $F(x)=y$. Because $A_1\subset A_2$, $x\in A_2$ also. Thus $y=F(x)\in F(A_2)$, so $F(A_1)\subset F(A_2)$. 2. I'm not sure I understand the second question. It can't be asking you to prove that $A\subset F^{-1}(F(A))$, because that's trivial, but I don't know what else it could be asking.	2014-11-26 15:42:40	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9111445546150208, "perplexity": 466.3287235617806}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931007056.0/warc/CC-MAIN-20141125155647-00145-ip-10-235-23-156.ec2.internal.warc.gz"}
https://juliabase.org/programming/class-based_views.html	# Class-based views¶ Class-based views are highly practical for the add/edit view of physical processes because they keep code duplication at a minimum. In some cases, you get away with only a few lines of code. Mixin classes reduce the redundancy further. Although it is still possible to have ordinary view functions for physical processes, we do not recommend this. If you follow the convention of calling your view class “EditView” and place it in a module called class_name.py, the PatternGenerator will detect it and create the URL dispatch for it. ## The API¶ The API of JuliaBase’s class-based view classes is best described by discussing the attributes and methods of the common base class ProcessWithoutSamplesView. Not only if you derive your views, but also if you need to define your own abstract view class, you should derive it from one of the concrete classes presented in the next section, though, because you probably want to re-use part of their functionality. This class is found in the module samples.utils.views.class_views. class ProcessWithoutSamplesView(kwargs) Abstract base class for the classed-based views. It deals only with the process per se, and in partuclar, with no samples associated with this process. This is done in the concrete derived classes ProcessView (one sample) and ProcessMultipleSamplesView (multiple samples). So, you should never instantiate this one. The methods that you most likely want to redefine in you own concrete class are, with decreasing probability: Note that for is_referentially_valid(), save_to_database(), build_forms(), and get_context_data(), it is necessary to call the inherited method. Since you connect forms with the view class, the view class expects certain constructor signatures of the forms. As for the process model form, it must accept the logged-in user as the first argument. This is the case for ProcessForm and DepositionForm, so this should not be a problem. The derived class (see below) may impose constrains on their external forms either. Variables: form_class – The model form class of the process of this view. model – The model class of the view. If not given, it is derived from the process form class. class_name – The name of the model class, e.g. "clustertooldeposition". process – The process instance this view is about. If we are about to add a new process, this is None until the respective form is saved. forms – A dictionary mapping template context names to forms, or lists of forms. Mandatory keys in this dictionary are "process" and "edit_description". (Derived classes add "sample", "samples", "remove_from_my_samples", "layers", etc.) data – The POST data if we have a POST request, or None otherwise. id – The ID of the process to edit, or None if we are about to add a new one. This is the recommended way to distinguish between editing and adding. preset_sample – The sample with which the process should be connected by default. May be None. request – The current request object. This is inherited from Django’s view classes. template_name – The file name of the rendering template, with the same path syntax as in the render() function. identifying_field – The name of the field in the process which is the poor man’s primary key for this process, e.g. the deposition number. It is taken from the model class. build_forms() Fills the forms dictionary with the forms, or lists of them. In this base class, we only add "process" itself and "edit_description". Note that the dictionary key is later used in the template as context variable. This method has no parameters and no return values, self is modified in-situ. It is good habit to check for a key before setting it, allowing derived methods to set it themselves without doing double work. get_context_data(kwargs) Generates the template context. In particular, we inject the forms and the title here into the context. This method is part of the official Django API. Return: the context dict dict get_next_id() Gets the next identifying value for the process class. In its default implementation, it just takes the maximal existing value and adds 1. This needs to be overridden if the identifying field is non-numeric. Return: The next untaken of the identifying field, e.g. the next free deposition number. object get_title() Creates the title of the response. This is used in the <title> tag and the <h1> tag at the top of the page. Return: the title of the response str is_all_valid() Checks whether all forms are valid. Moreover, this method guarantees that the is_valid() method of every form is called in order to collect all error messages. You may mark any unbound form as valid for this method by setting its attribute dont_check_validity to True. If it is not present, it is assumed to be False. This is helpful for marking unbound forms that should not let the request fail during a POST request. An example is a form in which the user enters the number of to-be-added sub-processes. It is reset (emptied) after each POST request by setting it to a pristine unbound form. However, this must not prevent the view from succeeding. Return: whether all forms are valid bool is_referentially_valid() Checks whether the data of all forms is consistent with each other and with the database. This is the partner of is_all_valid() but checks the inter-relations of data. This method is frequently overriden in concrete view classes. Note that a True here does not imply a True from is_all_valid(). Both methods are independent of each other. In particular, you must check the validity of froms that you use here. Return: whether the data submitted to the view is valid bool save_to_database() Saves the data to the database. Return: the saved process instance samples.models.PhysicalProcess startup() Fetch the process to-be-edited from the database and check permissions. This method has no parameters and no return values, self is modified in-situ. ## Main classes¶ The following names are found in the module samples.utils.views. class ProcessView(kwargs) View class for physical processes with one sample each. The HTML form for the sample is called sample in the template. Typical usage can be very short: from samples.utils.views import ProcessForm, ProcessView class LayerThicknessForm(ProcessForm): class Meta: model = LayerThicknessMeasurement fields = "__all__" class EditView(ProcessView): form_class = LayerThicknessForm class ProcessMultipleSamplesView(kwargs) View class for physical processes with one or more samples each. The HTML form for the sample list is called samples in the template. The usage is analogous to ProcessView. class DepositionView(kwargs) View class for views for depositions with layers. The layers of the process must always be of the same type. If they are not, you must use DepositionMultipleTypeView instead. Additionally to form_class, you must set the step_form_class class variable to the form class to be used for the layers. The layer form should be a subclass of SubprocessForm. class DepositionMultipleTypeView(kwargs) View class for depositions the layers of which are of different types (i.e., different models). You can see it in action in the module institute.views.samples.cluster_tool_deposition. Additionally to the class variable form_class, you must set: Variables: step_form_classes – This is a tuple of the form classes for the layers short_labels – (optional) This is a dict mapping a layer form class to a concise name of that layer type. It is used in the selection widget of the add-step form. class SubprocessForm(view, args, kwargs) Model form class for subprocesses. Its only purpose is to eat up the view parameter to the constructor so that you need not redefine the constructor every time. class SubprocessMultipleTypesForm(view, data=None, kwargs) Abstract model form for all step types in a process. It is to be used in conjunction with MultipleStepTypesMixin. See the views of th cluster-tool deposition in the INM “institute” app for an example for how to use this class. ## Mixins¶ class RemoveFromMySamplesMixin(kwargs) Mixin for views that like to offer a “Remove from my samples” button. In the template, they may add the following code: {{ remove_from_my_samples.as_p }} This mixin must come before the main view class in the list of parents. class SamplePositionsMixin(kwargs) Mixin for views that need to store the positions the samples used to have during the processing. The respective process class must inherit from ProcessWithSamplePositions. In the edit template, you must add the following code: {% include "samples/edit_sample_positions.html" %} This mixin must come before the main view class in the list of parents. class SubprocessesMixin(kwargs) Mixing for views that represent processes with subprocesses. Have a look at institute.views.samples.solarsimulator_measurement for an example. In a way, it is a light-weight variant of the MultipleStepsMixin below. In contrast to that, this mixin doesn’t order its subprocesses (you still may enforce ordering in the show template). For this to work, you must define the following additional class variables: • subform_class: the model form class for the subprocesses • process_field: the name of the field of the parent process in the subprocess model • subprocess_field: the related_name parameter in the field of the parent process in the subprocess model You should derive the model form class of the subprocess from SubprocessForm. This is not mandatory but convenient, see there. In the template, the forms of the subprocesses are available in a list called subprocesses. Furthermore, you should include {{ number.as_p }} in the template so that the user can set the number of subprocesses. This mixin must come before the main view class in the list of parents. class MultipleStepsMixin(kwargs) Mixin class for views for processes with steps. The steps of the process must always be of the same type. If they are not, you must use MultipleTypeStepsMixin instead. The step model must include a field called “number”, which should be ordered. This mixin must come before the main view class in the list of parents. You can see it in action in the module institute.views.samples.five_chamber_deposition. In the associated edit template, you can also see the usage of the three additional template variables steps, change_steps (well, at least their combination steps_and_change_steps), and add_steps. Additionally to form_class, you must set the following class variables: Variables: step_form_class – the form class to be used for the steps. process_field – to the name of the field of the parent process in the step model. class MultipleStepTypesMixin(*kwargs) Mixin class for processes the steps of which are of different types (i.e., different models). The step model must include a field called “number”, which should be ordered. This mixin must come before the main view class in the list of parents. You can see it in action in the module institute.views.samples.cluster_tool_deposition. In the associated edit template, you can also see the usage of the three additional template variables steps, change_steps (well, at least their combination steps_and_change_steps), and add_steps. Moreover, note the use of the step_type and type fields of each layer (= step). Additionally to the class variable form_class, you must set: Variables: step_form_classes – This is a tuple of the form classes for the steps process_field – to the name of the field of the parent process in the step model. short_labels – (optional) This is a dict mapping a step form class to a concise name of that step type. It is used in the selection widget of the add-step form. ## Sub-processes¶ Quite often, there is the need to divide a process further into sub-processes. JuliaBase realises this by special mixin classes. In this section, I discuss the bigger picture of it. Look at the reference above for the details and further information. There are three mixins that deal with sub-processes: ### SubprocessesMixin¶ This is a lightweight solution if you just want to have the forms for sub-processes auto-generated. It lets you create an edit/add view which allows the user to enter the number of sub-processes, and to enter the sub-processes’ data. This mixin does not enforce any ordering of the sub-processes – you may, howvever, enforce an ordering in the show view yourself, possibly by a model setting. Because of the lack of user convenience, this mixin is useful particularly for edit/add views which are primarily used by programs (e.g. crawlers) rather than by human beings. The solar simulator of the INM institute app demonstrates how to use this mixin. ### MultipleStepsMixin¶ This mixin realises the JuliaBase concept of a step. Steps are ordered sub-processes. On the model layer, they are models not derived from Process that contain an interger field number. This field is used to define the ordering, and helps JuliaBase to provide some convenience functionality: Re-ordering steps, duplicating them, deleting them. The parent model, for example the deposition process class, must define a method steps() that returns a query set of all steps, as in: def get_steps(self): return self.layers The MultipleStepsMixin is the main ingredient for the class DepositionView. You can see the latter in action in the 5-chamber deposition views of the INM institute app. #### “My steps”¶ Moreover with steps, your users can use something called “my steps”. It is a list of favourite steps that occur frequently. Every step in this list has a nickname, chosen by the user. When composing a new process, the user can select from this list instead of entering the step data manually. JuliaBase stores the “my steps” list for each user, however, you must add a view that lets the user set this list for each process class(es) that should be supported. In JuliaBase’s example app “institute”, a “my layers” view is included which realises this functionality for deposition layers. ### MultipleStepTypesMixin¶ This mixin is the same as above, but each step may be of a different model class. Using this mixin is slightly more complicated but also more powerful for obvious reasons. The steps must be of a common base model class, of which the concrete model classes are derived. Consequently, the base model class must inherit from jb_common.models.PolymorphicModel. The steps() method may return instances of the base model class, because the class-based view already takes care of finding the actual instance. In the view model, you should derive the forms classes for your step types from SubprocessMultipleTypesForm. This takes care of an extra field step_type that helps the view class to identify the step type from he HTTP POST data. This forms the basis of DepositionMultipleTypeView. You can see the latter in action in the cluster-tool deposition views of the INM institute app.	2020-02-19 23:46:13	{"extraction_info": {"found_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": 0.1917126327753067, "perplexity": 1697.7400383637018}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1581875144429.5/warc/CC-MAIN-20200219214816-20200220004816-00305.warc.gz"}
https://www.strathweb.com/2017/06/using-iactionconstraints-in-asp-net-core-mvc/	Using IActionConstraints in ASP.NET Core MVC · 1585 words · 8 minutes to read ASP.NET Core provides a way to constraint parameter values when matching routes via an IRouteConstraint (read more here) interface. This can be very useful, if you want to disambiguate certain routes from one another. This functionality is built into the routing package and is independent from the MVC framework. However, aside from that, the MVC framework itself also provides an interesting constraint-mechanism - IActionConstraints. Let’s have a look at them today. IActionContraints in the framework 🔗 When you think about filtering and disambiguating the mapping between of a request URL to an actual handler that will run the code to serve a response, action constraints provide an additional layer of configuration that you can use when building your applications. MVC goes through an action selection process to pick the correct action (typically a method on controller, but not necessarily - perhaps that’s a topic for another blog post). Action constraints let you apply additional rules to that action matching mechanism. Internally, the framework relies on IActionConstraints quite a bit. For example, there is a built-in HttpMethodActionConstraint which is used to restrict specific actions to handle specific types of HTTP request methods only. This is the reason why, when you use the [Route] attribute on an action, the action handles all types of requests (as long as the URL matches), but when you use [HttpGet] attribute on an action, the action handles GET requests only. In the latter case, the presence of [HttpGet] will internally result in an application of a HttpMethodActionConstraint configured to allow GET requests only. Authoring custom IActionContraints 🔗 You could very easily create your own custom IActionConstraints. The interface is shown below, and is rather self explanatory (the base IActionConstraintMetadata is just an empty marker interface). public interface IActionConstraint : IActionConstraintMetadata { /// <summary> /// The constraint order. /// </summary> /// <remarks> /// Constraints are grouped into stages by the value of <see cref="Order"/>. See remarks on /// <see cref="IActionConstraint"/>. /// </remarks> int Order { get; } /// <summary> /// Determines whether an action is a valid candidate for selection. /// </summary> /// /// <returns>True if the action is valid for selection, otherwise false.</returns> bool Accept(ActionConstraintContext context); } The whole logic is contained in the bool Accept(ActionConstraintContext context) method, which lets you inspect various contextual information, primarily HttpContext. Based on that you can decide whether the constraint should handle (return true) or miss (return false) the request. Note that returning false here doesn’t immediately create any response to the caller, it simply means that other candidate actions selected in the action selection process in the MVC framework can still get a chance to handle the request. Only if none of the candidate actions is deemed to be valid to handle request, that’s when the caller gets issued a 404 response as the MVC framework couldn’t process the request. A sample custom constraint is shown below: public class MandatoryHeaderActionConstraint : IActionConstraint, IActionConstraintMetadata { { } public int Order => 0; public bool Accept(ActionConstraintContext context) { // only allow route to be hit if the predefined header is present { return true; } return false; } } This constraint requires a presence of a specific header on the request in order to hand over the request processing to an action. So how would you go ahead and apply this action constraint to a specific action? There are two primary ways to do it - via a so called “MVC convention” (for example IActionModelConvention) or via a routing attribute. Applying an action constraint via action convetion 🔗 Conventions allow you to apply some extra information to the actions or to the controllers, that the MVC framework will then use for its various operations - such as controller or action selection. Actions constraints are a great example here. Let’s continue with our example of the mandatory header. Such a mandatory header could for example be a CorrelationId which should be passed by the caller for traceability purposes. Let’s create the following attribute first: public class RequireHeaderAttribute : Attribute { public string RequiredHeader { get; set; } { } } We will use it decorate the actions on which we would like our constraint to be applied. So a potential endpoint might look like this: [Route("api/values")] public class ValuesController : Controller { [HttpGet] public IEnumerable<string> Get() { return new string[] { "value1", "value2" }; } } The final piece needed, is the convention that will scan for the RequireHeaderAttribute and inject the constraint whenever needed. This is shown below: public class MandatoryHeaderConvention : IActionModelConvention { public void Apply(ActionModel action) { foreach (var selector in action.Selectors) { } } } In order to register the convention with MVC, it has to be added when you call AddMvc() at application startup. services.AddMvc(opt => { }); All of that means that hitting /api/values without a CorrelationId header will result in 404. One final thought. Instead of relying on a custom attribute like we did, there could be other ways of discovering whether an action should have a specific constraint applied to it. For example, the action name itself could be conventionally used to indicate that. In such cases, you’d need to inspect the actionName in the IActionModelConvention and make the decision based on that. Applying an action constraint via attribute route 🔗 It is also possible to apply action constraints using attribute routing. However, the default attributes (such as RouteAttribute or HttpGetAttribute) do not support that out of the box. This means, you will need to subclass them. Let’s illustrate that by using a different constraint than before - albeit, a similar one. public class AcceptLanguageActionConstraint : IActionConstraint, IActionConstraintMetadata { private string _locale; public AcceptLanguageActionConstraint(string locale) { _locale = locale; } public int Order => 0; public bool Accept(ActionConstraintContext context) { // only allow route to be hit if the predefined header is present { return true; } return false; } } The action constraint above restricts actions based on the Accept-Language header. This means, you could have multiple actions handling the same route (which normally would be illegal in MVC), and the disambiguation would happen based on the Accept-Language header. The custom route attribute, subclassing the default RouteAttribute is shown next. public class LanguageSpecificRouteAttribute : RouteAttribute, IActionConstraintFactory { public bool IsReusable => true; public LanguageSpecificRouteAttribute(string template, string locale) : base(template) { Order = -10; _constraint = new AcceptLanguageActionConstraint(locale); } public IActionConstraint CreateInstance(IServiceProvider services) { return _constraint; } } Interestingly, the route attribute itself can also act as IActionConstraintFactory, which means it can surface a specific IActionConstraint to the framework. This is a very nice and clean approach. The reason why the Order value is -10, is that we’d like this language-specific route to be processed before other routes. We’ll explain this in a moment. In our case we will force the user of our LanguageSpecificRouteAttribute to supply not just the route template, but also the locale, which will then be used with AcceptLanguageActionConstraint to match the action only when the predefined Accept-Language header value is found. We could now create two separate controllers, one responsible for handling a different locale. [LanguageSpecificRoute("api/values", "de-CH")] public class SwissValuesController : Controller { [HttpGet] public IEnumerable<string> Get() { return new string[] { "value1 for Switzerland", "value2 for Switzerland" }; } } [Route("api/values")] public class ValuesController : Controller { [HttpGet] public IEnumerable<string> Get() { return new string[] { "value1", "value2" }; } } With this set up, we have two controllers that are both responding to the same route - /api/values. Because we gave LanguageSpecificRouteAttribute an order of -10, that route will be given a priority over a route defined with RouteAttribute. This means we will rely on its constraint to either provide an action match (and letting SwissValuesController process the request) or an action miss. In the latter case, ValuesController will get a chance to respond and since it has no action constraints attach to it, that will be our fallback - in case the caller didn’t provide Accept-Language header, or provided one that didn’t resolve to Switzerland. Finally, you could generalize this logic, and avoid having a specialized attribute such as our LanguageSpecificRouteAttribute. Instead, you could create some sort of a reusable ConstrainedRouteAttribute. public class ConstrainedRouteAttribute : RouteAttribute, IActionConstraintFactory { public bool IsReusable => true; public ConstrainedRouteAttribute(string template, Type constraint, params object[] constructorParameters) : base(template) { Order = -10; _constraint = Activator.CreateInstance(constraint, constructorParameters) as IActionConstraint; } public IActionConstraint CreateInstance(IServiceProvider services) { return _constraint; } } This allows you to pass in any type of constraint and tries to apply it to the current action. The usage would then look like this: [ConstrainedRoute("api/values", typeof(AcceptLanguageActionConstraint), "de-CH")] public class SwissValuesController : Controller { [HttpGet] public IEnumerable<string> Get() { return new string[] { "value1 for Switzerland", "value2 for Switzerland" }; } } Applying an action constraint in Strathweb.TypedRouting.AspNetCore 🔗 I also wanted to mention that if you are using the typed routing library that I built, you can also apply action constraints there. The syntax would be as follows: opt.Get("api/values", c => c.Action<VaulesController>(x => Get())). WithConstraints(new MandatoryHeaderConstraint("CustomHeader")); Summary 🔗 And this is it - hopefully this can be useful when you are building your applications. I personally think that in large code bases, being able to redistribute request handling for identical route between different controllers/actions can be quite beneficial.	2023-03-29 01:02:52	{"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": 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": 0.21230152249336243, "perplexity": 4009.9683678807887}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948900.50/warc/CC-MAIN-20230328232645-20230329022645-00184.warc.gz"}
https://onlinetesting.americanbookcompany.com/sample/1550	TCAP Success Grade 5 MATH Chapter 6 ### TCAP Success Grade 5 MATH Chapter 6 Sample 1 pt 2. Which fraction problem has the answer modeled below? 1 pt 5. Cancel or simplify where possible in the following problem. Then multiply. $\frac{6}{8}×\frac{4}{12}$ 1 pt 7. Describe an answer for $3×\frac{1}{2}$. 1 pt 8. Describe an answer for $\frac{3}{4}×\frac{1}{2}$ 1 pt 9. Faye took $\frac{1}{2}$ of the new package of gum. Her friend, Beth, took $\frac{1}{3}$ of the remainder of the package. How much of the package remains?	2020-10-27 20:51:46	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 17, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8051889538764954, "perplexity": 8141.055517071908}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1603107894759.37/warc/CC-MAIN-20201027195832-20201027225832-00395.warc.gz"}
https://www.gamedev.net/forums/topic/114027-downloading-the-platform-sdk--a-horror-story/	#### Archived This topic is now archived and is closed to further replies. This topic is 5726 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic. ## Recommended Posts ##### Share on other sites Howard, I think you would be better suited by simply saving up your lunch money and ordering the CD. ##### Share on other sites What version of the SDK were you downloading. I think I still have the entire set from when I downloaded them a few month back. I could give you a URL to them, if they match. EDIT: I have the November 2001 one [edited by - CWizard on September 11, 2002 7:33:34 PM] ##### Share on other sites I''m with LessBread on this one: 1) Just e-mailing you the headers and libs which you copied on top of the official ones would end up with a horribly messed up build environment - the PSDK installer registers new environment variables, updates all the tools etc. 2) I installed the CD version of the PSDK on this machine, so I just checked the size of the e-mail attachment you''re asking for: 273Mb!!!. That''s including the Win64 and Prerelease stuff. Without those it''s still ~66Mb which isn''t going to be much different to downloading 100Mb from the MS site. In fact when you consider that sending it by e-mail will BASE64/UU encode the data, the size of the e-mail you''re asking someone to send would be 132Mb. Which would most likely bounce too! 3) Where do you live? I''m in Europe (UK), and the CD only cost me $15.95 including international shipping. Going by UK exchange rate''s thats the equivilent of roughly: a) two monthly PC magazines. b) entry fee to a nightclub for one night. c) four bottles of beer in a bar. d) a box of branded CDRs not too much really - especially if you put a cost on the stress and your time downloading ~100Mb from MS or ~132Mb from your e-mail provider. And you get the full thing including all the samples etc, which you can install on any machine as many times as you like (i.e. no repeat 100Mb download if you have to reformat your dev machine etc). -- Simon O''Connor Creative Asylum Ltd www.creative-asylum.com #### Share this post ##### Link to post ##### Share on other sites I think you just summed it up: why would he pay for a lousy programming CD when he could be drinking 4 beers with that money? Mmmm...beer... #### Share this post ##### Link to post ##### Share on other sites If you want I could try bzip'ing the latest PSDK CD ISO image (I have an MSDN subscription) and you could download it in several 1 MB chunks. I'd include MD5 sums for each chunk so you can verify the download was good for each section. You could then re-assemble it and then burn directly to a CD. Please e-mail me if you're interested (place GAMEDEV somewhere in the subject line), I'd be able to start uploading over my cable connection tonight. bzip compression should be able to shave almost 100 mb off of a 600 mb file, even if it is compressed via cab's or whatever... [edited by - SilentReaper on September 11, 2002 9:07:12 PM] #### Share this post ##### Link to post ##### Share on other sites Thanks for all the responses ! CWizard : Thanks for the files, but i don''t think they will work. I was downloading Build 5.2.3663.0 which is the July 2002 release. I am missing files 3,6,8 and 15.I need the core sdk, in files 3,6 & 8. 15 contains the MDAC SDK S1CA : quote: 1) Just e-mailing you the headers and libs which you copied on top of the official ones would end up with a horribly messed up build environment - the PSDK installer registers new environment variables, updates all the tools etc. I dont plan to overwrite my old files. I would extract the new files in a seperate folder, and include the path in VC++ Options>Directories . And AFAIK the enviroment variables are only required by the files in the bin dir.The headers and libs dont need enviroment vars.I will be missing the tools & samples, but its still better than nothing. quote: 2) I installed the CD version of the PSDK on this machine, so I just checked the size of the e-mail attachment you''re asking for: 273Mb!!!. That''s including the Win64 and Prerelease stuff. Without those it''s still ~66Mb which isn''t going to be much different to downloading 100Mb from the MS site. In fact when you consider that sending it by e-mail will BASE64/UU encode the data, the size of the e-mail you''re asking someone to send would be 132Mb. Which would most likely bounce too! i just want the headers and lib, which according to you are ~66MB. If u rar them, you will find that it shrinks to 10% ie 7-8 MB, which you have to admit is a LOT better than a 100MB. quote: the CD only cost me$15.95 including international shipping. Going by UK exchange rate''s thats the equivilent of roughly: a) two monthly PC magazines. Well, it would cost me more than 10 monthly PC magazines.But, the point is I dont have the money right now, however I do have ~200 hrs left in my net account. So this would just be simpler. SilentReaper : Appreciate the offer, but redownloading 500 mb would mean that I get the final thing a few months hence, and I have already been waiting since july. Still, if nothing better comes up, I will mail you. To sum up : The best solution would be if someone could post the missing files from build 3663.0 for download. Failing that, could someone just rar and mail me the headers & lib dirs? Otherwise, I guess i will get started on the new August ver of the SDK thats up for download, and hope to get it completed by oct. --------------------------------------------------------- Murphy''s Law : Anything that can possibly go wrong, will. ##### Share on other sites where are you downloading this thing from? what''s the exact url ''cos i can''t find it... :confused: ##### Share on other sites To bad they didn''t match. I didn''t even know they had a July 2002 build. They are a bit irregular, aren''t they? Nov 2002-July 2002-Aug 2002. • 19 • 10 • 19 • 14 • 20	2018-05-22 08:39:59	{"extraction_info": {"found_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": 0.24666020274162292, "perplexity": 5139.7638525678285}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794864648.30/warc/CC-MAIN-20180522073245-20180522093245-00111.warc.gz"}
https://cstheory.stackexchange.com/questions/40685/does-p-bpp-say-anything-about-space-complexity	# Does $P=BPP$ say anything about space complexity? There are many streaming algorithms with sublinear randomized space but linear deterministic space. Does $P=BPP$ have anything to do with derandomizing space and more importantly but not related to streaming does derandomizing time have anything to do with $L=RL$? • Here's a tenuous connection. $\mathbf{P} = \mathbf{BPP}$ implies some circuit lower bounds. Suitable circuit lower bounds imply $\mathbf{L} = \mathbf{RL}$. (The two circuit lower bounds are not the same.) – William Hoza Apr 30 '18 at 21:40	2019-09-19 12:22:38	{"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": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.692071795463562, "perplexity": 898.4607928437626}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514573519.72/warc/CC-MAIN-20190919122032-20190919144032-00262.warc.gz"}
https://math.stackexchange.com/questions/2826197/find-the-acute-angle-between-the-surfaces-xy2z-3x-z2-and-3x2-y22z-1	# Find the acute angle between the surfaces $xy^2z = 3x + z^2$ and $3x^2-y^2+2z=1$ at the point $(1,-2,1)$ Find the acute angle between the surfaces $xy^2z = 3x + z^2$ and $3x^2-y^2+2z=1$ at the point $(1,-2,1)$ Angle between curves at a point is given by the angle between their tangent planes at the point. $$f(x,y,z):= 3x +z^2 -xy^2z$$ $\nabla f(1,-2,1) = \langle 3-y^2z,-2xyz,2z-xy^2\rangle_{(1,-2,1)} = \langle -1,4,-2 \rangle$ Equation of tangent plane to $xy^2z = 3x + z^2$ will be $-x + 4y -2z + d=0$ Putting $(1,-2,1)$ and solving for $d$ we have $$x-4y+2z=11 \; \; \; (1)$$ Also, a nice way to write equation of tangent plane to curve $ax^2 + by^2 + cz^2 + 2ux + 2vy + 2wz + d =0$ at $P(x_0,y_0,z_0)$ would be: $$ax\cdot x_0 + by\cdot y_0 + cz\cdot z_0 + u(x+x_0) + v(y+y_0) + w(z+z_0) +d=0$$ Hence tangent plane to $3x^2-y^2+2z=1$ will be $3x(1) -y(-2)+(z+1)=1$ $$\Rightarrow 3x +2y+z=0 \; \; \; (2)$$ One of the reason for posting this is, how'd you follow this method to write equation of tangent plane to the first curve $xy^2z = 3x + z^2$? Basically, is it possible to extend this method to equations where degree is greater than $2$ and contains terms such as $xyz$? Now, angle between tangent planes is angle between their normals, Direction ratios of normal to $(1)$ and $(2)$ respectively are $a=\langle 1,-4,2\rangle$ and $b=\langle 3,2,1\rangle$ $\Rightarrow \theta= \arccos(\frac{a\cdot b}{\|a\|\|b\|}) = \arccos(\frac{-3}{7\sqrt{6}})$ Problem here is, how'd I know if this is acute or not, i.e, if this is the answer that I was looking for ? • If the cosine is negative, the angle can't be acute, right? Recall that $\arccos$ takes values in $[0,\pi]$ – saulspatz Jun 20 '18 at 15:20 • @saulspatz So I need to subtract it from 180 degrees? what if the angle lies between 180 to 270 degrees? – So Lo Jun 20 '18 at 15:31 • It won't. $\arccos$ will always give you an angle between $0$ and $180$ degrees. – saulspatz Jun 20 '18 at 15:36 • @saulspatz if I put $a=\langle -1,4,-2\rangle$ then $cos(\theta )$ is positive. But that will not match the answer I got previously. I am getting confused here – So Lo Jun 20 '18 at 16:08 • You'll get $\arccos\left(\frac{3}{7\sqrt{6}}\right)$ which is the supplement of what you had before. What is confusing you? (I haven't checked your arithmetic.) – saulspatz Jun 20 '18 at 16:22 We have $$\vec n_1 = \frac{\nabla (x y^2 z - 3 x + z^2)}{\|\|\nabla(x y^2 z - 3 x + z^2)\|\|} \\ \vec n_2 = \frac{\nabla (3 x^2 - y^2 + 2 z - 1)}{\|\|\nabla(3 x^2 - y^2 + 2 z - 1)\|\|}$$ and the sought angle is $$\varphi = \min(\|\arccos(\pm<\vec n_1,\vec n_2>)\|) = \arccos(\frac{1}{\sqrt{742}})\approx 88^{\circ}$$ • This is not an acute angle. – saulspatz Jun 20 '18 at 16:25 • Also, I think you must have made an arithmetic error. I get the same numerical answer as the OP. – saulspatz Jun 20 '18 at 16:33 • @saulspatz Really the answer is $\min(\|\arccos(\pm<\vec n_1,\vec n_2>)\|) = \arccos(\frac{1}{\sqrt{742}})\approx 88^{\circ}$ – Cesareo Jun 20 '18 at 18:02 Did you go ahead and take arccos$\left(\frac{-3}{7\sqrt{6}}\right)$? I get about 100$^o$. So your answer is no. The acute angle is 180 - 100 or about 80$^o$. • if I put $a=\langle -1,4,-2\rangle$ then $cos(\theta )$ is positive. But that will not match the answer I got previously. I am getting confused here – So Lo Jun 20 '18 at 16:08 The angle between the tangent planes is the angle between normals. Note that if the scalar product between the normals is positive, the angle is acute. If the scalar product is negative, the angle is obtuse. In this case, just take the opposite direction for one of the normals, ($b\rightarrow -b$), or equivalent $$\theta= \arccos\left(\frac{\|a\cdot b\|}{\|a\|\|b\|}\right)$$ to get the acute angle	2021-06-15 02:35:07	{"extraction_info": {"found_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": 0.7001870274543762, "perplexity": 201.6182231936281}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1623487616657.20/warc/CC-MAIN-20210615022806-20210615052806-00294.warc.gz"}
https://socratic.org/questions/ar-4s-2-3d-10-4p-4-is-the-noble-gas-configuration-of-which-element	# [Ar]4s^2 3d^10 4p^4 is the noble gas configuration of which element? Dec 31, 2016 That's not a noble gas configuration. That is, however, a noble gas shorthand for the electron configuration of selenium. Notice how the number of electrons corresponds to the number of protons in a neutral element. Thus, if you start from the $\left[A r\right]$ portion of the electron configuration, you should have accounted for $18$ electrons. In fact, the noble gas portion of the configuration represents the core electrons of the unidentified element. Therefore, everything after it is non-core (the $4 s$ and $4 p$ electrons are your valence electrons here, for the post-transition elements on the fourth row). Again, since the number of electrons equals the number of protons in a neutral element, and there are two $4 s$, ten $3 d$, and four $4 p$ electrons, there must be $18 + 2 + 10 + 4 = 34$ total electrons in the element, which correspond to $34$ total protons in the element. When you look it up, you should locate selenium and find that it has atomic number $34$.	2020-01-29 11:27:36	{"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": 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": 0.5872921943664551, "perplexity": 563.7987727109114}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1579251796127.92/warc/CC-MAIN-20200129102701-20200129132701-00043.warc.gz"}
https://santafe.edu/research/results/papers/7442-topology-controlled-potts-coarseningarxiv-pre	#### Denholm, J. and S. Redner We investigate the long-time relaxation of the q-state kinetic Potts ferromagnet on the triangular lattice that is quenched to zero temperature from a random or an antiferromagnetic initial state. For q = 3, the final state is either the ground state (probability ≈ 0.75), a two-stripe state (probability ≈ 0.09), or a three-hexagon state (probability ≈ 0.16). The relaxation to the hexagonal state is governed by a time that scales as L2 ln L. We provide a heuristic argument for this anomalous scaling and present additional new features of Potts coarsening on the triangular lattice for q = 3 and for q > 3.	2020-08-10 05:11:45	{"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": 0.8252571821212769, "perplexity": 1586.7820790299916}, "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-2020-34/segments/1596439738609.73/warc/CC-MAIN-20200810042140-20200810072140-00369.warc.gz"}
https://www.nature.com/articles/s41396-017-0014-8?WT.ec_id=ISMEJ-201803&spMailingID=56181717&spUserID=OTI4MDAwOTE4MAS2&spJobID=1361993448&spReportId=MTM2MTk5MzQ0OAS2&error=cookies_not_supported&code=8313aef8-e1a4-4299-b86b-dadb09594684	## Introduction Despite decades of focus on the local scale, ecology now recognizes the influence of regional processes of evolution and dispersal on community diversity and composition [1,2,3]. During all stages of community assembly, colonization can influence community diversity and composition by contributing new species and adding individuals to established populations [4,5,6]. Determining how many and which colonists influence local communities is important to understand community structure because the number and identity of species of colonists can regulate richness [7,8], influence species composition [9], trait distribution [9], and ecosystem function [10] at the local scale. In plant and animal ecology, the traditional concept of the regional species pool hypothesizes that species available for colonization depend on large-scale evolutionary and biogeographical processes [11,12]. Due to the large spatial and long temporal scales of these processes, the regional species pool is often assumed to be stable at ecological time scales [11,13,14] and this view has permeated much of the understanding of local and regional processes [15], interspecific interactions [16], and neutral theory [17]. However, the pool of colonists interacting with local scale habitats is often a subset of this large-regional pool, subject to more proximate habitat factors, particularly in microbial systems [18,19,20]. The evolutionary regional species pool can be divided into subsets of colonists that share similar dispersal mechanisms and ecological traits. For example, of all bacteria in alpine glaciers, Rime and collaborators [21] found that microorganisms from glacier sediment, and not from precipitation or streams, colonized newly deglaciated soil. For clarity, we refer to each of these subsets of colonists as a colonist pool, the group of species that arrives to local patches and interacts with local habitats. Previous studies have identified the most important colonist pool among a series of potential pools in some systems (e.g. [20,22,23,24]), but have rarely directly considered the possibility that multiple colonist pools can coexist with several local habitats (but see [25]). For example, although plant communities are often exposed to wind dispersed colonists (seed rain) and colonists from the soil (seed bank), which colonist pool is actually the most influential can depend on soil properties of local habitats [26,27,28]. If a particular local habitat receives colonists from only one colonist pool, comparing the colonist pool with local community composition is informative of the habitat filter [29]. These filters occur when habitat conditions are harsh or stressful for non-tolerant species and local communities become less diverse, less heterogeneous and less similar to the colonist pool [30]. However, there are other possible scenarios when multiple colonist pools influence different habitats (Fig. 1). If neighboring local sites with different habitats receive colonists from the same colonist pool (Fig. 1b), this provides an ideal opportunity to compare their habitat filters. If local sites receive colonists from multiple colonist pools (Fig. 1c), then comparing the local habitat to all contributing colonist pools is necessary to assess the nature of the habitat filters. Here, we examined the influence of multiple colonist pools of sediment bacteria on fiddler crab-associated bacterial communities and ask whether considering multiple colonist pool influences improves our understanding of this system. When fiddler crabs molt, newly formed chitin surfaces of the carapace and the gut are sterile and receive colonists from sources of bacteria in the surrounding environments [31,32]. Colonists are likely to come from the sediment because fiddler crabs feed on organic matter, microalgae and bacteria that they scrape off the surface sediment [33] and burrow to mate, escape predation and avoid extreme temperatures [34]. Sediment bacteria can be subdivided in three dominant habitats in salt marshes based on physicochemical conditions: sediment surface, subsurface, and crab burrows [35,36,37]. Bacteria in the surface sediment experience high oxygen and light availability [36,38] and should be dominated by aerobic heterotrophs, autotrophs, and nitrogen fixers [39,40,41]. Subsurface bacteria experience low oxygen levels and utilize NO, Fe and SO4 as electron acceptors [35,42] and should thus be dominated by taxa capable of anaerobic metabolism [36]. The sediment on the burrow walls is aerated and mixed as fiddler crabs burrow, developing a diverse bacterial assemblages that differ markedly in composition from surface and subsurface bacteria [43,44,45,46,47]. Although microbial communities in these sediments have been studied separately, their combined influences on the infauna have not been assessed yet. Host-associated microbial communities exposed to colonization from sediment bacteria could reveal alternative multiple colonist pool scenarios (Fig. 1). Regardless of how many colonist pools influence a site, all propagules must pass through local habitat filters before recruiting to local communities. In the crab’s carapace, bacteria able to attach and compete for space should be able to maintain membership in the community [48,49]. In contrast, to colonize the gut, bacteria must tolerate the physical scrapping and sediment selection of the crab’s oral appendages [50] and the chemical stress of hepatopancreas secretions [51,52]. In addition to local habitat characteristics, host factors like genotype [53,54,55] or species can act as a filter and determine the variation in the composition of associated microbial communities [56,57]. Empirical studies have however found contradictory evidence about these relationships [58,59] vs. [60], which could result from our lack of understanding of the mechanisms through which host genotype or species regulate its habitats. Fiddler crab species have been found to differ in some physiological factors [61] but not others [62] and whether these factors have an effect on their associated microbial communities is unknown. In contrast, the crab’s sex, has been linked with increased richness in male gut bacteria (Cuellar-Gempeler and Munguia [63]). This diversity effect of sex can be mediated by physiological [64,65,66], or behavioral [67,68] differences between fiddler crab males and females. Importantly, to decipher the role of host factors and habitat filters on bacterial community diveristy and composition, we must first investigate the influence of colonization. The major goal of this study is to investigate the influence of multiple colonist pools on fiddler crab associated bacteria and to compare its effect on composition to that of host-associated factors. Specifically, we asked three questions. First, we ask how different are the colonist pools. We hypothesize that surface, subsurface and burrow bacteria differ based the physicochemical characteristics of each sediment type. Second, we questioned how many and which sediment colonist pool influences the carapace’s and the gut’s bacterial communities. Based on the crab’s interaction with the sediment, we hypothesize that bacteria colonize the gut when the crab feeds on surface sediment and burrow sediment bacteria colonize the carapace as the crab takes refuge. In contrast, we do not expect the subsurface sediment to be an important contributor of colonists. Third, we ask whether host factors like crab sex or species generate variation in community diversity and composition. We hypothesize that sex is more important in determining variation in bacterial composition than host species, based on previous findings that suggest bacterial communities associated with males are more diverse. ## Methods We conducted our study in a 150 × 150 m2 marsh area located on Stedman Island, near Aransas Pass, Texas (27° 53’ 13.56” N, −97° 7’ 0.07 W). The sediment is influenced by inflow from high salinity seawater from Redfish Bay (between 25 and 35 ppt during the summer [69]) and vegetated by scarce black mangrove (Avicenia germinans), saltmarsh cordgrass (Spartina alterniflora) and woody glasswort (Salicornia bigelovii). This salt marsh was chosen because it is co-inhabited by two fiddler crab species (Uca panacea and Uca rapax) that exhibit similar behavior, habitat, and diet [70]. We focused on the bacterial community associated with these crab’s carapace and gut, which represent different habitats for bacteria. While the crab’s carapace is open to colonization [48,49], colonizing the gut requires surpassing physical [50] and chemical filters [51,52]. ### Sampling procedure Sample collection was divided evenly in four sampling dates (July 21, 29 and 31, and August 5, 2014). Adult U. panacea and U. rapax (18 males and 18 females) were collected by hand and stored in individual sterile containers. Using sterile spatulas, we collected a total of 10 samples of each sediment type. Approximately 20 g of surface, subsurface and burrow sediment were collected on sterile 50 mL falcon tubes. Replicates of surface, subsurface and burrows were collected approximately 10 m apart from each other at sites with active burrows. Active burrows were identified based on crab activity (leaving and entering) and signs of recent burrowing (freshly disturbed sediment and crab tracks). Surface sediment was scrapped from the top layer (0 to 2 cm deep). For subsurface samples, we mixed sediment from 7 to 15 cm in depth. Although microbial community composition can vary significantly with depth [71], crabs are exposed to all bacteria over this depth range when burrowing. Likewise, burrow samples were obtained from this depth range but along the surface of fiddler crab burrows. We selected only burrows that extended beyond 15 cm in depth and were wider than 2 cm. Crab and sediment samples were transported to the Marine Science Institute from the University of Texas at Austin for further processing. ### Sample preparation and DNA extraction Upon arrival to the lab, samples were prepared for DNA extraction. Each sample from the sediment was homogenized by vortexing at high speed for 30 s and 2 g were separated, in sterile conditions, for DNA extraction. Although this procedure may not completely homogenize a heterogeneous sediment sample, it allowed mixing of the sediment obtained from different depths in subsurface and burrow samples. Surface samples were treated the same for consistency. Crabs were rinsed with sterile deionized water to remove debris and unattached microorganisms. Then, carapaces were swabbed and scraped to profile the surface community. Crabs were sacrificed by freezing and dissected to obtain gut samples. Then, guts were rinsed with sterile deionized water to avoid food bolus interference and to remove unattached bacteria. All samples were kept in MoBio PowerSoil bead tubes at −80 °C and were processed within 2 months of collection. DNA was extracted using the PowerSoil DNA extraction kit (MoBio) as instructed by the manufacturer. We included samples taken of all sterile materials to control for potential contamination. DNA concentration was quantified with Qubit fluorometer (Qubit 2.0, Invitrogen) using high sensitivity assay reagents. Only samples with more than 0.1 ng/μl DNA yield were used for the study to avoid sample bias. ### 16S rRNA gene library preparation and sequence analysis Individual samples were prepared for Illumina sequencing using a two-step, gene-specific PCR. To avoid host DNA amplification, we targeted the V4 hypervariable region of bacterial 16S using the 515F/806R primer pair (Ong et al [72], Wang and Qian [73]). We used the MiSeq Illumina platform to obtain pair-end 250 bp nucleotide reads. Library preparation and sequencing was done at the University of Texas Genome Sequencing and Analysis Facility (GSAF). The resulting sequences were processed using custom bash scripts and QIIME [74] with Greengenes as reference databank [75]. OTUs (Operationa Taxonomical Units) were defined at the 97% sequence similarity and were picked with an open frame. Please refer to our Supplementary material for details on library preparation and sequence processing. ### Data analysis The following analyses were conducted in the R environment (version 3.3.2, [76]). We removed OTUs assigned to Archaea or unassigned and those found in less than 3 times in less than 1% of the samples. First, we evaluated differences in diversity and community composition among sediment types. To determine whether these results reflect similar sampling effort across samples, we calculated species accumulation curves. Since the species accumulation curves from sediment reached asymptotes (Fig S1), we considered only raw richness for further analyses. We assessed the difference in richness across sediment type by using a GLM fitted with a negative binomial distribution. Then, we calculated Pielou’s evenness as J = H’/lnS [77] and used an ANOVA to determine differences across sediment types. We assessed normality and homoceidacity of residuals using the Shapiro-Wilks test and the Bartlett test, respectively [78,79]. Evenness was inverse square root transformed to meet parametric assumptions. We calculated distance-to-centroid as a measure of community similarity and evaluated differences in similarity across sediment types. We used an analysis of multivariate homogeneity of group dispersions with 999 permutations. This method is multivariate analogue of Levene’s test for homogeneity of variances where non-euclidean distances between objects and group centroids are reduced to principal coordinates to assess beta diversity [80]. We assessed the role of sampling date by adding it as a random factor in these tests (Table S1) but, since we did not find any effects, we removed it from the statistical models presented here. We used a Cannonical Correspondance Analysis (CCA) with standardization by row and column weights to assess compositional differences between the three types of sediment bacteria. CCA is a direct ordination technique that can identify correlations between multivariate data on OTU abundances and the sediment type coded as dummy variables. Significance of the differences in composition was assessed with a permutational MANOVA on the CCA scores [81]. We compare the results from the CCA with non metric multidimensional scaling (NMDS) based on Bray-Curtis and a weighted UniFrac (Fig S4). Since we obtained similar results, we felt confident to proceed with the CCA ordination. We chose this specific ordination tool because it can incorporate the function predict, which allows us to find corresponding scores for crab samples in an unbiased way. Second, we determined the contributions of each sediment pool to each crab-associated community by comparing community composition in each crab sample and the centroid of the various colonist pools. To incorporate the crab sample scores to the sediment CCA, we used the function predict from the package ‘vegan’ (version 2.4.4, [82]). Then, we assigned each crab-associated community to a species pool based on the distance to nearest pool centroid in multivariate CCA space. This approach is useful to estimate similarity between standard and test data without the bias of including all samples in the calculation of the original ordination axes [83]. We assessed the sensitivity of this method by comparing the results from all OTUs with the results from the 100 most abundant taxa (Fig S2) and to the outcome of Source Tracker ([84], Fig S3). Third, we wanted to determine the effect of host-factors on community diversity and composition. We assessed the effect of host-factors on richness and evenness using a three-way ANOVA including habitat (carapace or gut), sex and species. We tested for normality and homoceidacity as described above for sediment samples. Evenness data was square root transformed to meet parametric assumtions. Then, we used a perMANOVA on CCA scores obtained from predict to determine how host factors, such as the type of habitat, or the crab’s sex or species influence bacterial community diversity and composition. We were also interested in assessing the significance of these groupings independently of the assignment methodology from the predict function. Thus, we used cluster analysis to identify groups of samples based on community composition and asked whether these clusters correspond to host-factors or pool assignments. We used a hierarchical clustering analysis with the “ward” method (hclust function, [85]) and estimated the significant number of clusters using the gap statistic [86]. Lastly, we calculated distance-to-centroid as a measure of community similarity and evaluated whether similarity was determined by the host’s sex or species within each habitat. We used an analysis of multivariate homogeneity of group dispersions as described above and tested the role of habitat type, host sex and species. We looked at three taxonomical levels to identify the bacteria that contribute to the above diversity patterns. First, we illustrated bacterial phyla that characterize each sediment type and crab habitat these patterns. We built a heatmap detailing the abundance and distribution of bacterial Phyla using heatmap.2 from the package ‘gplots’ (version 3.0.1, [87]). Then, we focused on the family taxonomic level to identify bacterial abundance patterns that underlie colonist pool relationships. To identify specific families contributing to differences and similarities between sediment and crab substrates, we used SIMPER analysis [88]. SIMPER analysis compares normalized relative abundance between samples measuring differences as Bray-Curtis dissimilarity. The contribution of individual OTU i to compositional differences is estimated with the formula $${\mathrm{d}}_{ijk}{\mathrm{ = }}\|\frac{{{\mathrm{x}}_{ij} - {\mathrm{x}}_{ik}}}{{{\mathrm{\Sigma x}}_{ij} - {\mathrm{x}}_{ik}}}\|$$ where x represents the abundance of OTU iin simple j and k. Although this method can confound the overall abundance with variation between samples [89], we used it to identify bacterial families explaining 80% of the variation in community composition between carapace, gut, surface, and burrow bacteria. We tested for significant differences in abundance of bacterial families identified by SIMPER across sediment and crab substrates using a permutation test (999 permutations). We selected informative families to further tested differences in abundances among sediment types and crabs. We used ANOVAs unless parametric assumptions were not met after transformations. In these cases Kruskal-Wallis test was used instead. To identify specific differences in abundance across substrates, we used post hoc Tukey or Nemeyeni tests, depending on whether data was parametric or not. To correct for false discoveries over tests for each taxa, we used Bonferroni corrections [90]. We used additional heatmaps to illustrate OTU abundance and occupancy patterns of selected families. ## Results Of the original samples, 65 samples were suitable for analysis. After quality-filtering, we had 21 carapace, 15 gut, and 30 sediment samples with a median of 6605 reads per sample and a range between 3000 and 14346 reads. We identified 517 individual bacterial OTUs distributed across 17 phyla and 80 families. ### Colonist pools in the sediment Sediments contained a total of 459 OTUs. Surface and subsurface had comparable richness levels, while burrow sediments had slightly lower richness, albeit not significantly different (Table 1, χ22, 27 = 4.380, p = 0.996). Evenness followed rather the opposite trend, with burrow having a higher value than subsurface and surface having the lowest value (Table 1, F2, 27=16.83, p < 0.001). Surprisingly, the surface did not contain any unique OTUs, while subsurface and burrow sediments had 14 and 36 unique OTUs, respectively (Table 1). These unique OTUs were rare within each sediment type and did not represent more than 2% of the community (Table 1). The CCA revealed significant differences in bacterial community composition among sediment types (Fig. 2, perMANOVA, F2, 27 = 108.73, R2 = 0.889, p < 0.001). While there were clear differences in composition, sediment types also differed in community similarity, with burrow sediment containing the least similar assemblages (permutation test, F value = 45.898, p < 0.001, Fig. 2) In the surface sediment, we found high abundance of Firmicutes, and Spirochaetes (Fig. 3). Particularly, surface sediment bacteria were dominated by taxa from Halanaerobacteriaceae and Balneolaceae (relative contribution ranging from 9.2 to 24.8%, Fig. 4, Table S2). Cyanobacteria were consistently found in burrow and surface sediment but never reached high relative abundances (average relative abundance of 0.8% in surface and burrow sediment), and it was extremely rare in subsurface sediment. Subsurface sediments were instead represented by high abundances of OTUs from Caldithrix, and SAR406 (also known as MGA, [91]), while burrow sediments contained high abundance of Acidobacteria and Nitrospirae (Fig. 3). Subsurface and burrow sediments shared high abundances of Rhodobacteriaceae and Desulfobacteriaceae (Fig. 4). Importantly, sediment types differed in OTUs composition within bacterial families such as Desulfobacteriaceae (Fig. 5, Table S3). ### Colonist pool influences on crab-associated bacteria Based on community similarity, most crab-associated communities were assigned to burrow, few to surface sediment, and none to subsurface sediment pool. While all carapace bacterial communities were (100%) assigned to the burrow, 10% gut communities were assigned to the surface and 90% to the burrow (Fig. 2). Crab gut bacterial communities assigned to the surface sediment corresponded to males from both species. A perMANOVA based on predicted CCA scores of crab samples revealed significant differences in bacterial community composition among carapaces and guts, but also showed an effect due to the interaction between host species and sex (Table 2). Crab bacterial communities were characterized by high abundance of Proteobacteria, Bacteroidetes and Actinobacteria and Verrucomicrobia. While most phyla were more abundant in the carapace, Tenericutes and Planctomycetes were more abundant in the gut. The influences of sediment bacteria were better understood by looking at similarities at the family and OTU taxonomical level (Figs. 4, 5). Carapace and gut bacterial communities held high abundances of Rhodobacteraceae (mostly Paracoccus sp, Rhodobacter sp), Flavobacteriaceae and Saprospiraceae, which are also found, at lower abundance, in burrow and subsurface sediment (Fig. 4). Dominant taxa from the surface (Halanaerobiaceae and Balneolaceae) were absent or present at very low abundances within crab habitats (Fig. 4). Vibrionaceae was an important representative of crab bacterial communities and was found, at lower abundance, in the burrow sediment. At the OTU level, we found that taxa within Desulfobacteraceae found in crab samples reflect those found in burrow sediment (Fig. 5). ### Host factors Habitat within the host was the main explanatory factors for the bacterial communities’ richness and evenness. Carapace bacterial communities had higher richness than gut bacterial communities, with a marginal effect of species (Tables 3, 4). This pattern was probably driven by U. panacea, which had the lowest values of richness in the gut (Table 3). Gut bacterial communities had higher evenness than carapace bacterial communities (Tables 3, 4). There were no significant differences in community similarity between gut and carapace bacterial communities (permutation test, F value = 3.050, p = 0.081, Fig. 2), crab species (F value = 0.616, p = 0.46) or male and female communities (F value = 0.133, p = 0.87). The cluster analysis showed a contrasting effect of host factors on carapace and gut bacterial communities. Gap analysis indicated that carapace bacterial communities clustered in 6 significant groups (Fig. 6a, b). These groups loosely correspond to host factors (Fig. 6a). For example, the first cluster on the left contains only U. rapax samples and the two clusters on the far right are composed of U. panacea. In contrast, gap analysis indicated only one cluster comprising all gut bacterial communities (Fig. 6c, d). ## Discussion Whether a unique regional species pool or multiple sources of colonists influence local communities could have consequences for local community dynamics. Here, we identified three potential colonist pools of bacteria in the salt marsh sediments and their influences in crab-associated bacterial communities. Patterns of similarity in community composition suggested that carapace bacterial communities receive most colonists from burrow sediment (Fig. 2). In contrast, guts received colonists from burrow and, at least in part, from surface bacteria (Fig. 2). This result suggests that multiple colonist pools can influence local patches (Fig. 1b) and individual colonist pool can influence distinct local habitats (Fig. 1c). We found that the three potential colonist pools within the salt marsh sediment differed in OTU composition (Fig. 2). Our findings coincide with previous studies showing strong physicochemical differences driven sediment depth [47] and bioturbation [37,71,92] and suggest that sediment bacteria are under distinct ecological selection pressures. For example, surface sediment was strongly dominated by salt tolerant taxa (Halanaerobiaceae, Balneolaceae) while other OTUs were rare (Fig. 4, Table S2, [93,94,95]. Even with this strong environmental stress, surface bacteria were more diverse than other sediment types, possibly because oxygen, light availability or immigration from other habitats sustain a long tail of rare OTUs (Fig. 3, Table S2). These results coincide with studies highlighting the strong effects of salinity on autotrophic [40] and nitrogen fixers [41] commonly found in the surface. Interestingly, many of these halophilic taxa are also anaerobic and their abundance in the surface sediment suggests fine scale changes in oxygen availability. Subsurface and burrow communities were dominated by taxa consistent with an environment limited in oxygen but abundant in sulfates and nitrates (Figs. 3, 4, [39,42,96]), but the differences between these two types of sediment were less clear. By looking at the OTU taxonomical level, we found groups of OTU from selected families like Desulfobacteraceae that clearly characterize sediment types (Fig. 5, Table S3). This finding suggests that each group of OTU within this family specialize on each sediment type, partitioning the coastal marsh habitat and reducing competition [97]. Physicochemical differences among sediment type appear to be a stronger factor in driving bacterial composition than physical separation between the sediments driving ecological drift. Based on similarities in OTU relative abundance and occupancy between crab-associated habitats and the sediment pools, burrow and surface sediment bacteria colonized fiddler crabs’ guts and carapaces (Fig. 2). Evidence for influence of the burrow on both crab habitats is better seen at two taxonomic levels. At the family level, Rhodobacteraceae and Flavobacteriaceae were important components of burrow sediment and crab-associated bacterial communities (Fig. 4). These taxa are common to the marine environment and have been previously found in association with crustacean exoskeleton and gut [98,99,100]. At the genus level, different taxa within Desulfobacteriaceae characterized each sediment type, but only burrow taxa matched taxa found associated with crabs (Fig. 5, Table S3). These findings support the idea that both gut and carapace habitats receive colonists from the burrow sediment (Fig. 1b). In this study, we emphasize the role of the burrow sediment as a source of colonists, yet these results also suggest a two-way colonization feedback between burrow sediments and crab-associated bacteria. There are two reasons why we suggest that burrow to crab colonization is predominant. First, dominant taxa are shared between the subsurface and the burrow, suggesting that these are sediment and not crustacean adapted taxa. Second, Desulfobacteriaceae are known environmental bacteria requiring resources like sulfates and acetates to fulfill metabolic requirements [101]. Desulfobacteriaceae associated with crabs are likely to undergo constant colonization from the burrow, suggesting a source-sink relationship, at least for this group of taxa (also known as mass effects in metacommunity ecology, [102]). However, we want to highlight that the crab’s activity can also have an important effect on the diversity and composition of burrow sediment bacteria. Previous work showed that areas of increased bioturbation are enriched in Vibrio spp. [103,104] and other Proteobacteria [105]. In our study, Vibrionaceae from crab bacterial communities is likely to colonize the burrow sediment. We propose a complex feedback mechanism between the effects of the crab’s activity on burrow sediment physicochemistry, bacterial migration from the crab to the sediment and colonization of crab habitats from burrow sediment bacteria. We found that crab-associated bacterial communities significantly cluster in groups based on community composition, but only in the carapace (Fig. 6). Contrary to our expectations, the clusters were associated with host species. There may be physiological or behavioral differences between these species that we have not considered. For example, fiddler crab reproductive behavior and physiology varies across species and can influence the associated bacterial communities [65]. In contrast to the carapace bacterial community, there were no significant clusters in gut bacteria (Fig. 6). One possible explanation is that strong filtering in the gut homogenizes these bacterial communities. If this is true, then their composition should remain unchanged and similarity in community composition should be higher [30], yet we failed to find differences in community similarity between gut and carapace bacterial communities. Alternatively, lack of clustering can be explained by considering a gradient in the contribution of multiple colonist pools. On one extreme, guts receive most or all colonists from surface sediment; while towards the other extreme, surface contribution diminishes and burrow contribution increases. Although there are no records of fiddler crab feeding from the burrows, they may be using burrow water when stressed by high temperatures and low water availability [106,107]. In addition, these patterns may be associated with molting stage, or other defining events in the life of each individual crab. Our results suggest that surface sediment bacteria can influence some gut bacterial communities. While this concurs with our expectations based on crab feeding behavior [67], we found that dominant taxa from the surface sediments were not representative of gut bacterial communities (Figs. 3, 4). Similarly, dominant taxa in the gut were not represented in the surface (Figs. 3, 4). A possible explanation is that rare OTUs from the surface sediment influence the distribution of rare OTUs in the crab’s gut. However, when analyzed the patterns from the 100 most abundant species, we found the same relationship between the gut and the surface sediment bacteria (Fig S2). Influential taxa could thus be intermediate OTUs that are rare within the 100 most abundant taxa. A possible explanation for these patterns is that bacteria able to colonize the crab’s gut survive by remaining inactive and dormant within surface sediment. The method we use to assign colonist pools to local communities relies on assignments based on similarity to determine species pool influences. This method is limited by the absence of a direct measure of dispersal, which results in two main issues. First, other sources of colonists may influence fiddler crab-associated communities. For example, bacteria in tidal seawater, plants and other crabs could influence fiddler crab associated bacterial communities. However, marsh sediment contains richer and more abundant bacterial assemblages [108] and has a stronger relationship with fiddler crabs based on their activity budgets [67,68,109], when compared to other sources. Second, our supposition that similarity indicates major contribution of a colonist pool assumes that we have identified all the species pools. Even with these limitations, the method used here allows us to identify likely sources of colonists from the host’s surrounding environment with no previous knowledge and it provides results that are comparable to other methods ([84], Fig S3). Our study highlights colonization from distinct species pools and its consequences for local diversity. Importantly, it suggests that local communities may be influenced by more than one colonist pool and colonist pools can influence more than one local habitat. Grouping species according to their dispersal regimes and large-scale ecological filters (such as sediment type) may result in a more realistic approach to the influence of multiple colonist pools on local communities. This view could aid at bridging our understanding of systems that operate at distinct scales but depend on this balance between colonization and local habitats, such as host-associated bacteria and plant communities. Integrating multiple colonist pools with the traditional evolutionary and biogeographical concept of the species pool may lead to a better understanding of the relationship between colonization, local filters and patterns of diversity across multiple scales.	2023-04-01 15:24:26	{"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": 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": 0.37452301383018494, "perplexity": 6701.497732812741}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1679296950030.57/warc/CC-MAIN-20230401125552-20230401155552-00750.warc.gz"}
https://zbmath.org/?q=an%3A0725.34049	# zbMATH — the first resource for mathematics Compound matrices and ordinary differential equations. (English) Zbl 0725.34049 A survey is given of a connection between compound matrices and ordinary differential equations. A typical result for linear systems is the following. If the n-th order differential equation $$x'=A(t)x$$ is uniformly stable, then a necessary and sufficient condition that the equation has an $$(n-k+1)$$-dimensional set of solutions satisfying $$\lim_{t\to \infty}x(t)=0$$ is that $$y'=A^{[k]}(t)y$$ should be asymptotically stable. For nonlinear autonomous systems, a criterion for orbital asymptotic stability of a closed trajectory given by Poincaré in two dimensions is extended to systems of any finite dimension. A criterion of Bendixson for the nonexistence of periodic solutions of a two dimensional system is also extended to higher dimensions. Reviewer: P.Smith (Keele) ##### MSC: 34D05 Asymptotic properties of solutions to ordinary differential equations 34C25 Periodic solutions to ordinary differential equations Full Text: ##### References: [1] A. C. Aitken, Determinants and Matrices, Oliver and Boyd, Edinburgh, 1956. [2] R. Bellman, Introduction to Matrix Analysis, McGraw-Hill, New York, 1960. · Zbl 0124.01001 [3] W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, Health, Boston, 1965. · Zbl 0154.09301 [4] J. Douglas, Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931), 263-321. JSTOR: · Zbl 0001.14102 [5] M Fiedler, Additive compound matrices and inequality for eigenvalues of stochastic matrices, Czechoslovak Math. J. 24 (99) (1974), 392-402. · Zbl 0345.15013 [6] F. R. Gantmacher, The Theory of Matrices, Chelsea Publ. Co., New York, 1959. · Zbl 0085.01001 [7] W. Hahn, Stability of Motion, Springer-Verlag, New York, 1967. · Zbl 0189.38503 [8] J. K. Hale, Ordinary Differential Equations, Wiley-Interscience, New York, 1969. · Zbl 0186.40901 [9] P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964; Birkhäuser, Boston, 1982. · Zbl 0125.32102 [10] A. S. Householder, The Theory of Matrices in Numerical Analysis, Blaisdell, New York, 1964. · Zbl 0161.12101 [11] P. Lancaster and M. Tismenetsky, The Theory of Matrices, Second Edition with Applications, Academic Press, Orlando, 1985. · Zbl 0558.15001 [12] D. London, On derivations arising in differential equations, Linear and Multilinear Algebra 4 (1976), 179 -189. · Zbl 0358.15011 [13] M. Marcus and H. Minc, A survey of matrix theory and matrix inequalities, Allyn and Bacon, Boston, 1964. · Zbl 0126.02404 [14] A. W. Marshall and I. Olkin, Inequalities: theory of majorization and its applications, Academic Press, New York, 1979. · Zbl 0437.26007 [15] J. Mikusiński, Sur l’equation $$x^(n) +A(t)x=0$$, Ann. Polon. Math. 1 (1955), 207-221. · Zbl 0064.33104 [16] T. Muir, The theory of determinants in the historical order of development, Macmillan, London, 1906. · JFM 37.0181.02 [17] J. S. Muldowney, On the dimension of the zero or infinity tending sets for linear differential equations, Proc. Amer. Math. Soc. 83 (1981), 705-709. JSTOR: · Zbl 0484.34003 [18] ——–, Dichotomies and asymptotic behaviour for linear differential systems, Trans. Amer. Math. Soc. 283 (1984), 465-484. JSTOR: · Zbl 0559.34049 [19] Z. Nehari, Disconjugate linear differential operators, Trans. Amer. Math. Soc. 129 (1967), 500-516. JSTOR: · Zbl 0183.09101 [20] G. B. Price, Some identities in the theory of determinants, Amer. Math. Monthly 54 (1947), 75-90. JSTOR: · Zbl 0029.00210 [21] B. Schwarz, Totally positive differential systems, Pacific J. Math. 32 (1970), 203-229. · Zbl 0193.04501 [22] R. A. Smith, An index theorem and Bendixson’s negative criterion for certain differential equations of higher dimension, Proc. Roy. Soc. Edinburgh 91A (1981), 63-77. · Zbl 0499.34026 [23] ——–, Some applications of Hausdorff dimension inequalities for ordinary differential equations, Proc. Roy. Soc. Edinburgh 104A (1986), 235-259. · Zbl 0622.34040 [24] J. H. M. Wedderburn, Lectures on matrices, Amer. Math. Soc., New York, 1934. · Zbl 0121.26101 [25] H. Wielandt, Topics in the analytic theory of matrices, Lecture notes prepared by R.R. Meyer, University of Wisconsin, Madison, 1967. · Zbl 0178.02104 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.	2022-01-19 07:24:55	{"extraction_info": {"found_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": 0.7342028617858887, "perplexity": 1079.878747750615}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1642320301264.36/warc/CC-MAIN-20220119064554-20220119094554-00084.warc.gz"}
https://meta.stackexchange.com/questions/111110/why-does-this-question-appear-twice-in-the-hot-list	# Why does this question appear twice in the hot list? Allow me to refer you to my freehand circles: I knew Jeff really does like us folks over at Gaming! • You typed "Allow me to refer you to my freehand circles", but my brain saw "Give me infinite upvotes" Nov 5, 2011 at 4:57 • You mean the 40ish votes I got on my answer within a few hours weren't truly earned? :P – user154510 Nov 7, 2011 at 16:01 • @Matthew It appears not. I think we'll let you keep them though. :P – John Nov 7, 2011 at 16:11 • Thank you for this question. Gimli's gender question was a nice read. Dec 9, 2011 at 21:22 I'm not sure how to answer this question without a lot of hand waving and speculation based on what was happening at the time this occurred so I'm going to go with a default answer of c-c-c-caching. We need to catch this one in the act so thanks for bring it to our attention. I will be monitoring the hot list from Stack Exchange for dupes more diligently. Closing this bug after waiting a full month for this issue to come up again. • Well then do you know why it has 2 different hotness values for the same question? – John Jan 11, 2012 at 0:33 Today we welcome another question so hot which needs to be listed twice! Both links point to the same question Need to reduce: $\frac{(3-h) (3+h)}{h - 3}$. Again, the two questions have different hotnesses.	2022-07-02 02:07:07	{"extraction_info": {"found_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": 0.2587614059448242, "perplexity": 1464.4660176216535}, "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/1656103983398.56/warc/CC-MAIN-20220702010252-20220702040252-00552.warc.gz"}
http://www.researchgate.net/publication/224578616_Fundamental_limits_of_almost_lossless_analog_compression	Conference Paper # Fundamental limits of almost lossless analog compression • ##### S. Verdu Dept. of Electr. Eng., Princeton Univ., Princeton, NJ, USA DOI: 10.1109/ISIT.2009.5205734 Conference: Information Theory, 2009. ISIT 2009. IEEE International Symposium on Source: IEEE Xplore ABSTRACT In Shannon theory, lossless source coding deals with the optimal compression of discrete sources. Compressed sensing is a lossless coding strategy for analog sources by means of multiplication by real-valued matrices. In this paper we study almost lossless analog compression for analog memoryless sources in an information-theoretic framework, in which the compressor is not constrained to linear transformations but it satisfies various regularity conditions such as Lipschitz continuity. The fundamental limit is shown to be the information dimension proposed by Renyi in 1959. 0 Bookmarks · 75 Views • Source ##### Conference Paper: Compressive sensing over graphs [Hide abstract] ABSTRACT: In this paper, motivated by network inference and tomography applications, we study the problem of compressive sensing for sparse signal vectors over graphs. In particular, we are interested in recovering sparse vectors representing the properties of the edges from a graph. Unlike existing compressive sensing results, the collective additive measurements we are allowed to take must follow connected paths over the underlying graph. For a sufficiently connected graph with n nodes, it is shown that, using O(k log(n)) path measurements, we are able to recover any k-sparse link vector (with no more than k nonzero elements), even though the measurements have to follow the graph path constraints. We mainly show that the computationally efficient ℓ<sub>1</sub> minimization can provide theoretical guarantees for inferring such k-sparse vectors with O(k log(n)) path measurements from the graph. INFOCOM, 2011 Proceedings IEEE; 05/2011 • Source ##### Article: Density Evolution Analysis of Node-Based Verification-Based Algorithms in Compressive Sensing [Hide abstract] ABSTRACT: In this paper, we present a new approach for the analysis of iterative node-based verification-based (NB-VB) recovery algorithms in the context of compressive sensing. These algorithms are particularly interesting due to their low complexity (linear in the signal dimension $n$). The asymptotic analysis predicts the fraction of unverified signal elements at each iteration $\ell$ in the asymptotic regime where $n \rightarrow \infty$. The analysis is similar in nature to the well-known density evolution technique commonly used to analyze iterative decoding algorithms. To perform the analysis, a message-passing interpretation of NB-VB algorithms is provided. This interpretation lacks the extrinsic nature of standard message-passing algorithms to which density evolution is usually applied. This requires a number of non-trivial modifications in the analysis. The analysis tracks the average performance of the recovery algorithms over the ensembles of input signals and sensing matrices as a function of $\ell$. Concentration results are devised to demonstrate that the performance of the recovery algorithms applied to any choice of the input signal over any realization of the sensing matrix follows the deterministic results of the analysis closely. Simulation results are also provided which demonstrate that the proposed asymptotic analysis matches the performance of recovery algorithms for large but finite values of $n$. Compared to the existing technique for the analysis of NB-VB algorithms, which is based on numerically solving a large system of coupled differential equations, the proposed method is much simpler and more accurate. 04/2011; • Source ##### Conference Paper: Density evolution analysis of node-based verification-based algorithms in compressed sensing [Hide abstract] ABSTRACT: In this paper, we present a new approach for the analysis of iterative node-based verification-based (NB-VB) recovery algorithms in the context of compressive sensing. These algorithms are particularly interesting due to their low complexity (linear in the signal dimension n). The asymptotic analysis predicts the fraction of unverified signal elements at each iteration ℓ in the asymptotic regime where n → ∞. The analysis is similar in nature to the well-known density evolution technique commonly used to analyze iterative decoding algorithms. To perform the analysis, a message-passing interpretation of NB-VB algorithms is provided. This interpretation lacks the extrinsic nature of standard message-passing algorithms to which density evolution is usually applied. This requires a number of non-trivial modifications in the analysis. The analysis tracks the average performance of the recovery algorithms over the ensembles of input signals and sensing matrices as a function of ℓ. Concentration results are devised to demonstrate that the performance of the recovery algorithms applied to any choice of the input signal over any realization of the sensing matrix follows the deterministic results of the analysis closely. Simulation results are also provided which demonstrate that the proposed asymptotic analysis matches the performance of recovery algorithms for large but finite values of n. Compared to the existing technique for the analysis of NB-VB algorithms, which is based on numerically solving a large system of coupled differential equations, the proposed method is much simpler and more accurate. Information Theory Proceedings (ISIT), 2011 IEEE International Symposium on; 09/2011	2014-09-17 16:07:29	{"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": 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": 0.8018829822540283, "perplexity": 560.1148910855277}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657123996.28/warc/CC-MAIN-20140914011203-00208-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"}
http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.rmi/1312906781	### Nonnegative solutions of the heat equation on rotationally symmetric Riemannian manifolds and semismall perturbations Minoru Murata Source: Rev. Mat. Iberoamericana Volume 27, Number 3 (2011), 885-907. #### Abstract Let $M$ be a rotationally symmetric Riemannian manifold, and $\Delta$ be the Laplace-Beltrami operator on $M$. We establish a necessary and sufficient condition for the constant function 1 to be a semismall perturbation of $-\Delta +1$ on $M$, and give optimal sufficient conditions for uniqueness of nonnegative solutions of the Cauchy problem to the heat equation. As an application, we determine the structure of all nonnegative solutions to the heat equation on $M\times(0,T)$. First Page: Primary Subjects: 35K05, 58J99, 35C15, 35B20, 31C35, 31C12, 35J99, 35K15	2013-05-23 19:06:36	{"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": 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": 0.719254195690155, "perplexity": 148.65712774485402}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368703682988/warc/CC-MAIN-20130516112802-00040-ip-10-60-113-184.ec2.internal.warc.gz"}
http://www.rayvaquero.com/	Jul 21 # England: Multi-storey carpark in Liverpool gutted by fire, 1,300 vehicles destroyed Uncategorized Thursday, January 4, 2018 A fire on Sunday night in the seven-storey carpark for the Echo Arena in Liverpool, England destroyed almost all the vehicles parked inside and led to cancellation of the final evening of the Liverpool International Horse Show and evacuation of nearby blocks of flats. The blaze reportedly started with a parked Range Rover Discovery. Investigators with the fire brigade stated that they believe the fire began with an accidental engine fire in the Range Rover at about 4.30 pm. The first call was made at 4.42 and firefighters arrived eight minutes after that. Ultimately twelve engines and 85 firefighters were involved in combatting the blaze. Aerial appliances were used and also three high-volume pumps. Fed by the fuel in vehicles parked inside, the temperature of the fire in the carpark is believed to have reached as high as 1,000°C. It was too hot to be extinguished with water from hydrants, so a high-volume pump was used to draw water from the River Mersey, and two more were brought from other fire brigades in the region. The carpark has seven storeys and a capacity of 1,600 vehicles, and approximately 1,300 were parked in it when the fire broke out. According to Dan Stephens, chief fire officer for Merseyside Fire and Rescue Service, almost all of them were destroyed, with the exception of a few parked on the top level and at corners. “With these very high temperatures, you were never going to put the fire out without the whole building taking hold. The speed at which the fire spreads means you simply aren’t going to put it out,” said Stephens. The carpark itself was severely damaged; according to Joe Anderson, the mayor of Liverpool. It is not in danger of collapsing but will have to be demolished, which will be difficult with the many burned-out cars still inside it, Anderson told the BBC. According to Stephens, there were no serious injuries: one woman injured her hand, and two people were treated for smoke inhalation. A spokesman for the Echo Arena also stated that all animals were safe. All horses were successfully evacuated from the carpark and then removed from the stables after smoke spread to them. Six dogs were also rescued unharmed, two on a lower level in the early stages of the fire and four that had been left in a car on the top level, freed by firefighters on Monday after the fire was put out. The final evening of the four-day Liverpool International Horse Show had been scheduled to begin at 7.30, and had to be cancelled. Many attendees were stranded in the city on New Year’s Eve night. Merseyside police directed people to the Pullman Hotel, where Red Cross assistance was available, and the Liverpool City Council set up an assistance centre at the Lifestyles Gym. A spokesman for the Association of British Insurers has said that insurance companies will “move very quickly” to reimburse owners whose vehicles were destroyed. Nearby blocks of flats were evacuated because of the smoke. Eyewitnesses reported hearing what they at first thought were firecrackers, then “multiple explosions”, “bangs and popping”, “the bangs of car windows exploding”. People reported leaving everything in their cars, including their cellphones, and running for their lives. Mayor Anderson tweeted that cuts to fire services over the last two years made it significantly harder to fight the fire and might have caused it not to be controllable. He also suggested that fire safety in multi-storey carparks had not been sufficiently considered and that installing sprinklers in them might help stop future fires before they become unmanageable, in a letter to Nick Hurd, a member of Parliament. Jul 21 # Gunmen, police clash in south Russian republic of Kabardino-Balkaria Uncategorized Thursday, October 13, 2005Clashes between gunmen and police have broken out in the capital of the southern Russian republic of Kabardino-Balkaria where Chechen rebels claimed responsibility for the insurgency. The fighting in Nalchik started after a coordinated set of attacks were carried out against city security forces and police stations. The airport and government buildings were also targeted. The strike is believed to have come from anywhere between 60 to 500 insurgents. The insurgents used heavy arms fire, coupled with explosions at the outset, that lasted five hours. A police station has been captured and hostages taken. According to reports from Russian press sources, 61 of the gunmen have been killed and 17 have been captured, and there may be dozens of civilian casualties. Seven policemen were also killed. Several government buildings throughout the city are on fire and some schools have been involved and evacuated accordingly. Russian president Vladimir Putin has ordered the army to seal off the city and shoot any insurgent who tries to escape. Officials in the city are saying the situation is slowly returning to normal. Jul 21 # British computer scientist’s new “nullity” idea provokes reaction from mathematicians Uncategorized Monday, December 11, 2006 On December 7, BBC News reported a story about Dr James Anderson, a teacher in the Computer Science department at the University of Reading in the United Kingdom. In the report it was stated that Anderson had “solved a very important problem” that was 1200 years old, the problem of division by zero. According to the BBC, Anderson had created a new number, that he had named “nullity”, that lay outside of the real number line. Anderson terms this number a “transreal number”, and denotes it with the Greek letter ? {\displaystyle \Phi } . He had taught this number to pupils at Highdown School, in Emmer Green, Reading. The BBC report provoked many reactions from mathematicians and others. In reaction to the story, Mark C. Chu-Carroll, a computer scientist and researcher, posted a web log entry describing Anderson as an “idiot math teacher”, and describing the BBC’s story as “absolutely infuriating” and a story that “does an excellent job of demonstrating what total innumerate idiots reporters are”. Chu-Carroll stated that there was, in fact, no actual problem to be solved in the first place. “There is no number that meaningfully expresses the concept of what it means to divide by zero.”, he wrote, stating that all that Anderson had done was “assign a name to the concept of ‘not a number'”, something which was “not new” in that the IEEE floating-point standard, which describes how computers represent floating-point numbers, had included a concept of “not a number”, termed “NaN“, since 1985. Chu-Carroll further continued: “Basically, he’s defined a non-solution to a non-problem. And by teaching it to his students, he’s doing them a great disservice. They’re going to leave his class believing that he’s a great genius who’s solved a supposed fundamental problem of math, and believing in this silly nullity thing as a valid mathematical concept. “It’s not like there isn’t already enough stuff in basic math for kids to learn; there’s no excuse for taking advantage of a passive audience to shove this nonsense down their throats as an exercise in self-aggrandizement. “To make matters worse, this idiot is a computer science professor! No one who’s studied CS should be able to get away with believing that re-inventing the concept of NaN is something noteworthy or profound; and no one who’s studied CS should think that defining meaningless values can somehow magically make invalid computations produce meaningful results. I’m ashamed for my field.” There have been a wide range of other reactions from other people to the BBC news story. Comments range from the humorous and the ironic, such as the B1FF-style observation that “DIVIDION[sic] BY ZERO IS IMPOSSIBLE BECAUSE MY CALCULATOR SAYS SO AND IT IS THE TRUTH” and the Chuck Norris Fact that “Only Chuck Norris can divide by zero.” (to which another reader replied “Chuck Norris just looks at zero, and it divides itself.”); through vigourous defences of Dr Anderson, with several people quoting the lyrics to Ira Gershwin‘s song “They All Laughed (At Christopher Columbus)”; to detailed mathematical discussions of Anderson’s proposed axioms of transfinite numbers. Several readers have commented that they consider this to have damaged the reputation of the Computer Science department, and even the reputation of the University of Reading as a whole. “By publishing his childish nonsense the BBC actively harms the reputation of Reading University.” wrote one reader. “Looking forward to seeing Reading University maths application plummit.” wrote another. “Ignore all research papers from the University of Reading.” wrote a third. “I’m not sure why you refer to Reading as a ‘university’. This is a place the BBC reports as closing down its physics department because it’s too hard. Lecturers at Reading should stick to folk dancing and knitting, leaving academic subjects to grown ups.” wrote a fourth. Steve Kramarsky lamented that Dr Anderson is not from the “University of ‘Rithmetic“. Several readers criticised the journalists at the BBC who ran the story for not apparently contacting any mathematicians about Dr Anderson’s idea. “Journalists are meant to check facts, not just accept whatever they are told by a self-interested third party and publish it without question.” wrote one reader on the BBC’s web site. However, on Slashdot another reader countered “The report is from Berkshire local news. Berkshire! Do you really expect a local news team to have a maths specialist? Finding a newsworthy story in Berkshire probably isn’t that easy, so local journalists have to cover any piece of fluff that comes up. Your attitude to the journalist should be sympathy, not scorn.” Ben Goldacre, author of the Bad Science column in The Guardian, wrote on his web log that “what is odd is a reporter, editor, producer, newsroom, team, cameraman, soundman, TV channel, web editor, web copy writer, and so on, all thinking it’s a good idea to cover a brilliant new scientific breakthrough whilst clearly knowing nothing about the context. Maths isn’t that hard, you could even make a call to a mathematician about it.”, continuing that “it’s all very well for the BBC to think they’re being balanced and clever getting Dr Anderson back in to answer queries about his theory on Tuesday, but that rather skips the issue, and shines the spotlight quite unfairly on him (he looks like a very alright bloke to me).”. From reading comments on his own web log as well as elsewhere, Goldacre concluded that he thought that “a lot of people might feel it’s reporter Ben Moore, and the rest of his doubtless extensive team, the people who drove the story, who we’d want to see answering the questions from the mathematicians.”. Andrej Bauer, a professional mathematician from Slovenia writing on the Bad Science web log, stated that “whoever reported on this failed to call a university professor to check whether it was really new. Any university professor would have told this reporter that there are many ways of dealing with division by zero, and that Mr. Anderson’s was just one of known ones.” Ollie Williams, one of the BBC Radio Berkshire reporters who wrote the BBC story, initially stated that “It seems odd to me that his theory would get as far as television if it’s so easily blown out of the water by visitors to our site, so there must be something more to it.” and directly responded to criticisms of BBC journalism on several points on his web log. He pointed out that people should remember that his target audience was local people in Berkshire with no mathematical knowledge, and that he was “not writing for a global audience of mathematicians”. “Some people have had a go at Dr Anderson for using simplified terminology too,” he continued, “but he knows we’re playing to a mainstream audience, and at the time we filmed him, he was showing his theory to a class of schoolchildren. Those circumstances were never going to breed an in-depth half-hour scientific discussion, and none of our regular readers would want that.”. On the matter of fact checking, he replied that “if you only want us to report scientific news once it’s appeared, peer-reviewed, in a recognised journal, it’s going to be very dry, and it probably won’t be news.”, adding that “It’s not for the BBC to become a journal of mathematics — that’s the job of journals of mathematics. It’s for the BBC to provide lively science reporting that engages and involves people. And if you look at the original page, you’ll find a list as long as your arm of engaged and involved people.”. Williams pointed out that “We did not present Dr Anderson’s theory as gospel, although with hindsight it could have been made clearer that this is very much a theory and by no means universally accepted. But we certainly weren’t shouting a mathematical revolution from the rooftops. Dr Anderson has, in one or two places, been chastised for coming to the media with his theory instead of his peers — a sure sign of a quack, boffin and/or crank according to one blogger. Actually, one of our reporters happened to meet him during a demonstration against the closure of the university’s physics department a couple of weeks ago, got chatting, and discovered Dr Anderson reckoned he was onto something. He certainly didn’t break the door down looking for media coverage.”. Some commentators, at the BBC web page and at Slashdot, have attempted serious mathematical descriptions of what Anderson has done, and subjected it to analysis. One description was that Anderson has taken the field of real numbers and given it complete closure so that all six of the common arithmetic operators were surjective functions, resulting in “an object which is barely a commutative ring (with operators with tons of funky corner cases)” and no actual gain “in terms of new theorems or strong relation statements from the extra axioms he has to tack on”. Jamie Sawyer, a mathematics undergraduate at the University of Warwick writing in the Warwick Maths Society discussion forum, describes what Anderson has done as deciding that R ? { ? ? , + ? } {\displaystyle \mathbb {R} \cup \lbrace -\infty ,+\infty \rbrace } , the so-called extended real number line, is “not good enough […] because of the wonderful issue of what 0 0 {\displaystyle {\frac {0}{0}}} is equal to” and therefore creating a number system R ? { ? ? , ? , + ? } {\displaystyle \mathbb {R} \cup \lbrace -\infty ,\Phi ,+\infty \rbrace } . Andrej Bauer stated that Anderson’s axioms of transreal arithmetic “are far from being original. First, you can adjoin + ? {\displaystyle +\infty } and ? ? {\displaystyle -\infty } to obtain something called the extended real line. Then you can adjoin a bottom element to represent an undefined value. This is all standard and quite old. In fact, it is well known in domain theory, which deals with how to represent things we compute with, that adjoining just bottom to the reals is not a good idea. It is better to adjoin many so-called partial elements, which denote approximations to reals. Bottom is then just the trivial approximation which means something like ‘any real’ or ‘undefined real’.” Commentators have pointed out that in the field of mathematical analysis, 0 0 {\displaystyle {\frac {0}{0}}} (which Anderson has defined axiomatically to be ? {\displaystyle \Phi } ) is the limit of several functions, each of which tends to a different value at its limit: • lim x ? 0 x 0 {\displaystyle \lim _{x\to 0}{\frac {x}{0}}} has two different limits, depending from whether x {\displaystyle x} approaches zero from a positive or from a negative direction. • lim x ? 0 0 x {\displaystyle \lim _{x\to 0}{\frac {0}{x}}} also has two different limits. (This is the argument that commentators gave. In fact, 0 x {\displaystyle {\frac {0}{x}}} has the value 0 {\displaystyle 0} for all x ? 0 {\displaystyle x\neq 0} , and thus only one limit. It is simply discontinuous for x = 0 {\displaystyle x=0} . However, that limit is different to the two limits for lim x ? 0 x 0 {\displaystyle \lim _{x\to 0}{\frac {x}{0}}} , supporting the commentators’ main point that the values of the various limits are all different.) • Whilst sin ? 0 = 0 {\displaystyle \sin 0=0} , the limit lim x ? 0 sin ? x x {\displaystyle \lim _{x\to 0}{\frac {\sin x}{x}}} can be shown to be 1, by expanding the sine function as an infinite Taylor series, dividing the series by x {\displaystyle x} , and then taking the limit of the result, which is 1. • Whilst 1 ? cos ? 0 = 0 {\displaystyle 1-\cos 0=0} , the limit lim x ? 0 1 ? cos ? x x {\displaystyle \lim _{x\to 0}{\frac {1-\cos x}{x}}} can be shown to be 0, by expanding the cosine function as an infinite Taylor series, dividing the series subtracted from 1 by x {\displaystyle x} , and then taking the limit of the result, which is 0. Commentators have also noted l’Hôpital’s rule. It has been pointed out that Anderson’s set of transreal numbers is not, unlike the set of real numbers, a mathematical field. Simon Tatham, author of PuTTY, stated that Anderson’s system “doesn’t even think about the field axioms: addition is no longer invertible, multiplication isn’t invertible on nullity or infinity (or zero, but that’s expected!). So if you’re working in the transreals or transrationals, you can’t do simple algebraic transformations such as cancelling x {\displaystyle x} and ? x {\displaystyle -x} when both occur in the same expression, because that transformation becomes invalid if x {\displaystyle x} is nullity or infinity. So even the simplest exercises of ordinary algebra spew off a constant stream of ‘unless x is nullity’ special cases which you have to deal with separately — in much the same way that the occasional division spews off an ‘unless x is zero’ special case, only much more often.” Tatham stated that “It’s telling that this monstrosity has been dreamed up by a computer scientist: persistent error indicators and universal absorbing states can often be good computer science, but he’s stepped way outside his field of competence if he thinks that that also makes them good maths.”, continuing that Anderson has “also totally missed the point when he tries to compute things like 0 0 {\displaystyle 0^{0}} using his arithmetic. The reason why things like that are generally considered to be ill-defined is not because of a lack of facile ‘proofs’ showing them to have one value or another; it’s because of a surfeit of such ‘proofs’ all of which disagree! Adding another one does not (as he appears to believe) solve any problem at all.” (In other words: 0 0 {\displaystyle 0^{0}} is what is known in mathematical analysis as an indeterminate form.) To many observers, it appears that Anderson has done nothing more than re-invent the idea of “NaN“, a special value that computers have been using in floating-point calculations to represent undefined results for over two decades. In the various international standards for computing, including the IEEE floating-point standard and IBM’s standard for decimal arithmetic, a division of any non-zero number by zero results in one of two special infinity values, “+Inf” or “-Inf”, the sign of the infinity determined by the signs of the two operands (Negative zero exists in floating-point representations.); and a division of zero by zero results in NaN. Anderson himself denies that he has re-invented NaN, and in fact claims that there are problems with NaN that are not shared by nullity. According to Anderson, “mathematical arithmetic is sociologically invalid” and IEEE floating-point arithmetic, with NaN, is also faulty. In one of his papers on a “perspex machine” dealing with “The Axioms of Transreal Arithmetic” (Jamie Sawyer writes that he has “worries about something which appears to be named after a plastic” — “Perspex” being a trade name for polymethyl methacrylate in the U.K..) Anderson writes: We cannot accept an arithmetic in which a number is not equal to itself (NaN != NaN), or in which there are three kinds of numbers: plain numbers, silent numbers, and signalling numbers; because, on writing such a number down, in daily discourse, we can not always distinguish which kind of number it is and, even if we adopt some notational convention to make the distinction clear, we cannot know how the signalling numbers are to be used in the absence of having the whole program and computer that computed them available. So whilst IEEE floating-point arithmetic is an improvement on real arithmetic, in so far as it is total, not partial, both arithmetics are invalid models of arithmetic. In fact, the standard convention for distinguishing the two types of NaNs when writing them down can be seen in ISO/IEC 10967, another international standard for how computers deal with numbers, which uses “qNaN” for non-signalling (“quiet”) NaNs and “sNaN” for signalling NaNs. Anderson continues: [NaN’s] semantics are not defined, except by a long list of special cases in the IEEE standard. “In other words,” writes Scott Lamb, a BSc. in Computer Science from the University of Idaho, “they are defined, but he doesn’t like the definition.”. The main difference between nullity and NaN, according to both Anderson and commentators, is that nullity compares equal to nullity, whereas NaN does not compare equal to NaN. Commentators have pointed out that in very short order this difference leads to contradictory results. They stated that it requires only a few lines of proof, for example, to demonstrate that in Anderson’s system of “transreal arithmetic” both 1 = 2 {\displaystyle 1=2} and 1 ? 2 {\displaystyle 1\neq 2} , after which, in one commentator’s words, one can “prove anything that you like”. In aiming to provide a complete system of arithmetic, by adding extra axioms defining the results of the division of zero by zero and of the consequent operations on that result, half as many again as the number of axioms of real-number arithmetic, Anderson has produced a self-contradictory system of arithmetic, in accordance with Gödel’s incompleteness theorems. One reader-submitted comment appended to the BBC news article read “Step 1. Create solution 2. Create problem 3. PROFIT!”, an allusion to the business plan employed by the underpants gnomes of the comedy television series South Park. In fact, Anderson does plan to profit from nullity, having registered on the 27th of July, 2006 a private limited company named Transreal Computing Ltd, whose mission statement is “to develop hardware and software to bring you fast and safe computation that does not fail on division by zero” and to “promote education and training in transreal computing”. The company is currently “in the research and development phase prior to trading in hardware and software”. In a presentation given to potential investors in his company at the ANGLE plc showcase on the 28th of November, 2006, held at the University of Reading, Anderson stated his aims for the company as being: To investors, Anderson makes the following promises: • “I will help you develop a curriculum for transreal arithmetic if you want me to.” • “I will help you unify QED and gravitation if you want me to.” • “I will build a transreal supercomputer.” • “How much would you pay to know that the engine in your ship, car, aeroplane, or heart pacemaker won’t just stop dead?” • “How much would you pay to know that your Government’s computer controlled military hardware won’t just stop or misfire?” The current models of computer arithmetic are, in fact, already designed to allow programmers to write programs that will continue in the event of a division by zero. The IEEE’s Frequently Asked Questions document for the floating-point standard gives this reply to the question “Why doesn’t division by zero (or overflow, or underflow) stop the program or trigger an error?”: “The [IEEE] 754 model encourages robust programs. It is intended not only for numerical analysts but also for spreadsheet users, database systems, or even coffee pots. The propagation rules for NaNs and infinities allow inconsequential exceptions to vanish. Similarly, gradual underflow maintains error properties over a precision’s range. “When exceptional situations need attention, they can be examined immediately via traps or at a convenient time via status flags. Traps can be used to stop a program, but unrecoverable situations are extremely rare. Simply stopping a program is not an option for embedded systems or network agents. More often, traps log diagnostic information or substitute valid results.” Simon Tatham stated that there is a basic problem with Anderson’s ideas, and thus with the idea of building a transreal supercomputer: “It’s a category error. The Anderson transrationals and transreals are theoretical algebraic structures, capable of representing arbitrarily big and arbitrarily precise numbers. So the question of their error-propagation semantics is totally meaningless: you don’t use them for down-and-dirty error-prone real computation, you use them for proving theorems. If you want to use this sort of thing in a computer, you have to think up some concrete representation of Anderson transfoos in bits and bytes, which will (if only by the limits of available memory) be unable to encompass the entire range of the structure. And the point at which you make this transition from theoretical abstract algebra to concrete bits and bytes is precisely where you should also be putting in error handling, because it’s where errors start to become possible. We define our theoretical algebraic structures to obey lots of axioms (like the field axioms, and total ordering) which make it possible to reason about them efficiently in the proving of theorems. We define our practical number representations in a computer to make it easy to detect errors. The Anderson transfoos are a consequence of fundamentally confusing the one with the other, and that by itself ought to be sufficient reason to hurl them aside with great force.” Geomerics, a start-up company specializing in simulation software for physics and lighting and funded by ANGLE plc, had been asked to look into Anderson’s work by an unnamed client. Rich Wareham, a Senior Research and Development Engineer at Geomerics and a MEng. from the University of Cambridge, stated that Anderson’s system “might be a more interesting set of axioms for dealing with arithmetic exceptions but it isn’t the first attempt at just defining away the problem. Indeed it doesn’t fundamentally change anything. The reason computer programs crash when they divide by zero is not that the hardware can produce no result, merely that the programmer has not dealt with NaNs as they propagate through. Not dealing with nullities will similarly lead to program crashes.” “Do the Anderson transrational semantics give any advantage over the IEEE ones?”, Wareham asked, answering “Well one assumes they have been thought out to be useful in themselves rather than to just propagate errors but I’m not sure that seeing a nullity pop out of your code would lead you to do anything other than what would happen if a NaN or Inf popped out, namely signal an error.”. Jul 21 # Boy dies after falling into river in North Yorkshire, England Uncategorized Monday, June 7, 2010 Last weekend, an eight-year-old boy died after falling into the River Wharfe in North Yorkshire, England. The river is known for its depth and strong undercurrent. The accident took place on Aaron Page’s birthday, following celebrations with over twenty family members. Page made his way to the river near Skipton around 3pm BST (1400 UTC) on Saturday; his brother, aged thirteen, was the first to call for assistance after spotting Aaron in the water. Emergency services subsequently assisted in searching for Aaron; the body was recovered at approximately 6:25 pm the same day. Members of the ambulance service failed to resuscitate him, pronouncing him dead at the scene. No matter how safe a river seems sometimes there are inherent dangers. “This was a happy family day out to celebrate a birthday at a local beauty spot […] and unfortunately this young boy has got into difficulty in an area of water,” stated Inspector Kevin Franks of North Yorkshire Police. “Despite efforts from members of the public and some quite sterling efforts from the emergency services and everyone else there, sadly we’ve been unable to bring him out alive.” Inspector Mike Woodhall of Harrogate police cautioned; “[n]o matter how safe a river seems sometimes there are inherent dangers.” Jul 21 # Locally designed, low emissions car launched in Qatar Uncategorized Friday, November 30, 2012 Qatari non-profit organization Gulf Organization for Research and Development (GORD) launched a low emissions car at the 2012 United Nations Climate Change Conference (COP 18) in Doha. The car was designed and developed in Qatar. Revealed during a press conference at the Qatar National Convention Centre, the car in addition to an internal combustion engine, includes an automotive thermoelectric generator designed to capture waste heat to produce hydrogen. GORD expects the heat waste collecting system to be compatible with any gasoline or compressed natural gas car. GORD chairman Dr Al-Horr summarised the key concepts of the invention in a statement saying, “Our car produces electricity at no cost by capturing thermal waste energy, reducing costs and eliminating the need for an external source of electricity. Also, bulky compressed-hydrogen cylinders are a thing of the past, as our concept accomplishes the production of hydrogen by using water through fuel cells integrated within the car.” Most of the energy in Qatari vehicle comes from the the car’s gasoline tank, supplemented by a thin film photo-voltaic panel on the roof. Normally in a combustion engine, chemical energy stored in a fuel, such as gasoline, is converted into heat energy through combustion. This heat energy is then converted into mechanical energy, manifested as an increase in pressure in the combustion chamber due to the kinetic energy of the combustion gases. The kinetic energy of these combustion gases are then converted into work; because of the inefficiencies in converting chemical energy into useful work, internal combustion engines have a theoretical maximum effiecincy of 37% (with what is achievable in day to day applications being about half of this). Of the chemical energy in the consumed fuel used by an internal combustion engine 40% is dissipated as waste heat. However, the Qatari vehicle uses a thermoelectric generator to convert this waste heat into electricity. Such generators are used in space vehicles, and produce electricity when thermoelectric materials are subjected to a temperature gradient, the greater the gradient the greater the amount of electrcity produced. In the GORD vehicle the electricity produced is used to electrolyse potable water to produce hydrogen which can be introduced into the vehicle’s existing fuel system. The researchers showed that the heat waste collection engine caused a decrease in the car’s emissions, including a decrease of carbon monoxide and nitrous oxide emissions by more than 50%, the fuel efficiency increasing by 20%. On its website, GORD said that the heat waste collector engine is universal, “Any car can be adapted to accommodate the system as it doesn’t alter any electro-mechanical systems”. Jul 21 # At least 55 killed by Hurricane Katrina; serious flooding across affected region Uncategorized Tuesday, August 30, 2005File:Katrina damage.jpg Thirty people died in the Gulf of Mexico resort of Biloxi when Hurricane Katrina demolished a water-side apartment block, Harrison County emergency operations center spokesman Jim Pollard told AP. However, the Mississippi Emergency Management Agency has yet to confirm the news. At least another twenty have died across the rest of Harrison County; this number is expected to rise. The Governor of the state of Mississippi has asked people to stay away from the area for several days. The towns of Biloxi and Gulfport took the brunt of the hurricane’s 140mph+ winds after it veered away from New Orleans as it made landfall. Three more people have been killed by falling trees and at least two died in traffic accidents resulting from the hurricane. Hundreds of people have been rescued by boat and helicopter from the roofs of houses cut off by flood water. The hurricane has caused a storm-surge — the force of the winds has piled the waters of the Gulf of Mexico against the coastline, causing widespread flooding reaching at least a mile inland in places. A levee on the Lake Pontchartrain canal has broken in two places, causing massive flooding. Some parts of New Orleans are now under 20 feet of water. Flooding seems to have reduced to increasing at a rate of one inch an hour and Army Corps of Engineers are on the scene of the breaks. At aproximately 10:00 PM Central city officials confirmed that a major floodpump has failed, which could result in an additional nine feet of floodwaters. The western part of New Orleans has been flooded after a two-block long stretch of the 17th Street Levee gave way on Monday afternoon. Much of the city lies below sea level and depends on flood defences to keep it safe. One hospital — that has 1,000 patients inside — has been surrounded by the water. The vice-president of the center has described seeing whitecaps on the waves of the water in a street outside. A water main pipe has failed, meaning that tap water has been contaminated with flood water, and is no longer safe to drink. The mayor of the city has described seeing “bodies floating in the water.” Some 10,000 people remain in the Superdome stadium, despite the electricity supply failing leaving the indoor stadium in darkness. The coverings of the concrete roof of the structure have been stripped away by the wind. Electricity supplies to 1.3 million people across the south eastern parts of the US have been damaged, and it could be months before power is restored to all affected. Two oil rigs have broken free of their moorings in the Gulf of Mexico, and a third drifted into a bridge in Mobile Bay, Alabama. Katrina has now been downgraded to tropical storm status as it moves northwards across the US, and wind speeds have dropped to 60mph. The current death toll does not include the 11 killed in Florida when Katrina struck there last week. Jul 21 # Ontario Votes 2007: Interview with NDP candidate Rick Morelli, Vaughan Uncategorized Thursday, October 4, 2007 Rick Morelli is running for the NDP in the Ontario provincial election, in the Vaughan riding. Wikinews’ Nick Moreau interviewed him regarding his values, his experience, and his campaign. Stay tuned for further interviews; every candidate from every party is eligible, and will be contacted. Expect interviews from Liberals, Progressive Conservatives, New Democratic Party members, Ontario Greens, as well as members from the Family Coalition, Freedom, Communist, Libertarian, and Confederation of Regions parties, as well as independents. Jul 21 # UN summit results in pledge to mitigate food crisis Uncategorized Friday, June 6, 2008 A three-day United Nations (UN) summit, bringing together leaders from 181 countries, has wrapped up with a pledge by all attending countries to address the global food shortage crisis. Key actions cited include doubling the world’s food production by 2030, providing resources for farmers in poor countries and increasing humanitarian aid in times of crisis. Protests and violent riots have resulted in parts of the world in recent months due to increasing unaffordability, and sometimes unavailability, of food. It is estimated that 862 million people, or just over one eighth of the world’s population, are malnourished. Jul 21 # Small plane crashes in central Berlin Uncategorized Friday, July 22, 2005 A small single-engine ultralight aircraft has crashed on a lawn between the Reichstag building, the seat of the German parliament, and the Bundeskanzleramt, the seat of the chancellor, in Berlin. The aircraft crashed at approximately 2000CET. Amateur footage aired by the ARD shows the plane rapidly descending before crashing on the lawn and bursting into flames. Witnesses at the scene were inconclusive about the heading of the plane. Some stated that it was targeting the Reichstag while others said that the pilot tried to steer away from it. He was killed in the crash. People near the scene attempted to fight the fire with hand-held extinguishers, however the aircraft was mostly destroyed by the time the emergency services arrived. The German authorities have ruled out terrorism. Police are saying that the pilot’s wife has been missing since Monday and they had suspected that he might have something to do with the disappearance. It is possible that he might have committed suicide to avoid prosecution.	2018-07-21 05:58:34	{"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": 0.4214146137237549, "perplexity": 2619.160793826907}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676592387.80/warc/CC-MAIN-20180721051500-20180721071500-00043.warc.gz"}
https://quant.stackexchange.com/tags/ratio/hot	# Tag Info 18 Sharpe ratio is defined as $\frac{(x - r)}{\sigma}$ where $x$ is return, $r$ is the risk free rate and $\sigma$ is volatility. Now levering up $n$ times multiplies both the return and volatility by $n$. But shouldn't the ratio change since $r$ stays the same? Ah, but remember, leverage isn't free. You have to fund leverage, and that cuts out of your return. ... 8 The textbook academic answer is that Sharpe ratio is not impacted by leverage as explained by other answers. However, reality tells a different tale entirely: Imagine you lever up your investments by such amount that your future performance will critically hinge on the following conditions: That those who extended credit to you will not re-call their ... 6 Generally no. Sharpe ratio should vary linearly. Use leverage: the return increases, but so does volatility. De-lever" the return decreases but, so does volatility. 5 "It's compliated" because the trading strategy performance will depend on the data which is most likely serially correlated. So you want to look into bootstrap approaches for time series such as the block bootstrap, or the wild bootstrap. Another approach would be to look into 'random portfolios' or an approximation thereof. The basic idea is to test how ... 4 As a short summary and adaption of the question: You better redefine $\hat{r}_i= \frac{S_{i-1}}{S_1}-1$ and $\hat{S}_i = (1+\hat{r}_i)S_0$. The above definition of $\hat{S}_i$ yields a sample of potential values for $S$ for the future day. This approach is usually applied in historical simulation. The aim here is to use information of the past about the ... 4 Hint: If these 2 stocks have perfect negative correlation (correlation: -1), then you can construct a risk free portfolio. What would the return on that risk free portfolio be? 3 You can look at the contents of the CFA institute : https://blogs.cfainstitute.org/insideinvesting/2013/01/23/how-much-does-apple-make-a-dupont-analysis/ . As there are more and more candidates and CFA charteholders, we could say that their views are becoming or are already the mainstream views. I would add that accounting and corporate finance is not a "... 3 TL;DR: the test statistic's distribution is $N(0,1)$ A bit more information about the Automatic Variance Ratio Test: $H_0$: ${\Delta}r_t$ is serially uncorrelated (where ${\Delta}r_t=r_t-r_{t-1}$) $H_1$: ${\Delta}r_t$ is serially correlated The test statistic is $VR=\sqrt{T/l}[\hat{VR}(l)-1]/\sqrt{2} \quad {\xrightarrow{d}} \quad N(0,1)$ The $d$ over ... 3 Most of the papers concern CDS spreads which you will need to convert to a PD. Paper using country specific fundamentals: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2517018 This paper uses leverage: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2361872 Another one that decomposes them against peer groups: http://papers.ssrn.com/sol3/papers.... 2 It might help to think of the two as special cases of $$S_{i+1}-S_i = \sigma (c+S_i)^\beta \epsilon$$ which looks like a Constant Elasticity of Variance extension. Taking squares of both sides and then logs will (nearly) linearise it, allowing you to carry some basic estimation using OLS. The parameter $c$ will control the lower bound and can impose some ... 2 I am also not aware of any papers in this area. But having developed many such models, I can list the important steps: Decide on the target variable: usual choices are historical default data, agency ratings and expert rankings Create a sample containing the possible predictors Reduce the list with the help of some expert, e.g. exclude all the predictors ... 2 David Addison's discussion of log versus simple(arithmetic) returns in his answer is correct, but this particular calculation has nothing to do with arithmetic versus log returns. Up Capture is defined by Bacon(2004), p. 47 as: $$Up Capture = \frac{\bar{r+}}{\bar{b+}}$$ (mean of the asset returns over mean of the benchmark return) So simple versus log ... 2 I am not familiar with that R package, but I've written a few performance tracking libraries in my past life, so I might be able to add some insight. While it is indeed true that logarithmic returns may be added and subtracted, all non-quant investors and hedge funds present their performances in percent returns. The reason one can't simply add and subtract ... 2 Sure! Sharpe ratio must be defined as the return per unit risk on a zero-cost position. The notion you are referring to achieves this by assuming borrowing at a risk-free rate before investing, so refinancing risks should matter. On a side note, the Sharpe ratio of any ForEx strategy would implicitly have the stuff you mention accounted for. 2 You can define information ratio on ex-ante basis, so you will be using the expected values, and this definition is called alpha omega: $IR=\frac{\alpha}{\omega}$ Let’s represent the risk reversion by $\lambda$ then the value add is: $VA=\alpha-\lambda \omega^2$ Substituting for alpha: $VA=IR \omega -\lambda \omega^2$ Now the value add is maximised ... 2 Having reviewed the documentation sent by Noob2 and rechecking everything, I came to the following conclusion: ((6044−2002)/2002)^1/20.38=5.57% is absolutely wrong. If one does the calculations for this you get 1.035 (I have no idea how I managed do come up with 5.57% in the first place). Thus this resolves the question where I am confused about the ... 1 For anyone looking for this, I ended up calculating average_win and average_loss and then calculating the ratio as: R = average_win / average_loss 1 You should use average monthly return over stddev. 1 There is no such thing as number of independent bets when one is betting on a common random factor as we quants usually do. Grinold & Kahn’s formula is only relevant when the factor payoff is a constant over time. This is not interesting in practice. When the factor payoff is random, then Ding and Martin The Fundamental Law of Active Management: Redux (... 1 1 I have automated downloading of various price data with simple C# console applications that I write with Visual Studio Express (which is free). You are looking for a very specialized task, so I don't think you can get away from programming. I recommend Microsoft .Net for programming. Just create a new console application and use .Net's WebRequest and ... 1 Wouldn't you just weight the p/l equation the same way your position is weighted? So: (P(StockA)0.8)/(P(StockB)0.2) = Net % change in the position 1 It is a measure of profitability commonly used in the Banking industry. We can exclude the other 3 answers: it does not take into account the bank's leverage (i.e. it is a measure of pre-leverage profitability), it has nothing to do with asset composition (since it is based on total assets, irrespective of their composition), and it has nothing to do with ... 1 Bloomberg has a Default Risk model, which is similar to what you are querying. You can see a screenshot in this PDF. There you can also see the kind of variables they use. You can access it by typing DRSK at the CDS screen is Bloomberg. (If the screenshot in the PDF is not clear enough, let me know and I can post one with better resolution from Bbg) This ... Only top voted, non community-wiki answers of a minimum length are eligible	2020-10-31 05:31:11	{"extraction_info": {"found_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": 0.6841325163841248, "perplexity": 1082.6743393384493}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1603107912807.78/warc/CC-MAIN-20201031032847-20201031062847-00431.warc.gz"}
https://zbmath.org/authors/?q=ai%3Aiosifescu.marius	# zbMATH — the first resource for mathematics ## Iosifescu, Marius Compute Distance To: Author ID: iosifescu.marius Published as: Iosifescu, M.; Iosifescu, Marius Homepage: http://www.acad.ro/pag_cv/cv_miosifescu.htm External Links: MGP · Math-Net.Ru · Wikidata · GND Documents Indexed: 134 Publications since 1956, including 22 Books Reviewing Activity: 684 Reviews Biographic References: 1 Publication all top 5 #### Co-Authors 80 single-authored 16 Theodorescu, Radu 5 Herkenrath, Ulrich 5 Kraaikamp, Cor 4 Grigorescu, Serban 4 Tautu, Petre 3 Bereanu, Bernard 3 Beznea, Lucian 3 Brînzănescu, Vasile 3 Marcus, Solomon 3 Rudolph, Andreas 3 Sebe, Gabriela Ileana 3 Timotin, Dan 2 Ciucu, George 2 Limnios, Nikolaos 2 Oprişan, Gheorghe 2 Postelnicu, Tiberiu 2 Purice, Radu 1 Calude, Cristian S. 1 Cioranescu, Doina 1 Demetrescu, M. C. 1 Deshouillers, Jean-Marc 1 Duhamel, Christian 1 Ene, Horia I. 1 Iordache, Octavian 1 Kalpazidou, Sofia 1 Mandl, Petr 1 Marinoschi, Gabriela 1 Mihaileanu, N. N. 1 Mihoc, Gheorghe 1 Onicescu, Octav 1 Popescu, Gheorghe 1 Rădulescu, Vicenţiu D. 1 Sofonea, Mircea 1 Spătaru, Aurel 1 Theordorescu, R. 1 Todor, I. all top 5 #### Serials 19 Revue Roumaine de Mathématiques Pures et Appliquées 8 Comunicările Academiei Republicii Populare Române 7 Académie de la République Populaire Roumaine, Revue de Mathématiques Pures et Appliquées 5 Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences, Paris 4 Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 4 Mathematical Reports 4 Proceedings of the Romanian Academy. Series A: Mathematics, Physics, Technical Sciences, Information Science 3 Studii şi Cercetări Matematice 2 Teoriya Veroyatnosteĭ i eë Primeneniya 2 Indagationes Mathematicae. New Series 2 Analele Universităţii din Craiova. Seria Matematică Informatică 2 Gazeta Matematica şi Fizica. Seria A 2 Cambridge Tracts in Mathematics 2 Annals of Mathematical Statistics 1 Ukraïns’kyĭ Matematychnyĭ Zhurnal 1 Theory of Probability and its Applications 1 Bulletin Mathématique de la Société des Sciences Mathématiques de la République Socialiste de Roumanie. Nouvelle Série 1 Canadian Mathematical Bulletin 1 Journal of Applied Probability 1 Journal of Statistical Planning and Inference 1 Osaka Journal of Mathematics 1 Tokyo Journal of Mathematics 1 Gazeta Matematica. Perfectionare Metodica si Metodologica in Matematica si Informatica 1 Atti della Accademia Nazionale dei Lincei. Serie Ottava. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali 1 Doklady Bolgarskoĭ Akademii Nauk 1 Gazeta Matematică 1 Revue d’Analyse Numérique et de Théorie de l’Approximation 1 Journal of the European Mathematical Society (JEMS) 1 Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences, Série A 1 Bulletin Mathématique de la Société des Sciences Mathématiques et Physiques de la République Populaire Roumaine. Nouvelle Série 1 Bulletin Société Mathématique de Grèce. Nouvelle Série 1 Grundlehren der Mathematischen Wissenschaften 1 Academia Republicii Populare Romine, Filiala Iaşi, Studii şi Cercetări Ştiinţifice, Matematică 1 Mathematica 1 Wiley Series in Probability and Mathematical Statistics 1 Collection Méthodes Stochastiques Appliquées all top 5 #### Fields 63 Probability theory and stochastic processes (60-XX) 31 Number theory (11-XX) 9 General and overarching topics; collections (00-XX) 6 Measure and integration (28-XX) 6 Dynamical systems and ergodic theory (37-XX) 5 History and biography (01-XX) 4 Sequences, series, summability (40-XX) 3 Statistics (62-XX) 2 Operator theory (47-XX) 2 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 2 Biology and other natural sciences (92-XX) 1 Computer science (68-XX) 1 Mechanics of particles and systems (70-XX) 1 Mechanics of deformable solids (74-XX) 1 Fluid mechanics (76-XX) #### Citations contained in zbMATH Open 50 Publications have been cited 459 times in 386 Documents Cited by Year Random processes and learning. Zbl 0194.51101 Iosifescu, Marius; Theodorescu, Radu 1969 Finite Markov processes and their applications. Rev. and exp. ed. Translated from the Roumanian. Zbl 0436.60001 Iosifescu, Marius 1980 Metrical theory of continued fractions. Zbl 1122.11047 Iosifescu, Marius; Kraaikamp, Cor 2002 Dependence with complete connections and its applications. Transl. from the Romanian. Rev., updated and exp. version. Zbl 0749.60067 Iosifescu, Marius; Grigorescu, Serban 1990 Stochastic processes and applications in biology and medicine. Part II: Models. Zbl 0262.92002 Iosifescu, M.; Tautu, P. 1973 On two recent papers on ergodicity in nonhomogeneous Markov chains. Zbl 0249.60031 Iosifescu, Marius 1972 Stochastic processes and applications in biology and medicine. Part I: Theory. Zbl 0262.92001 Iosifescu, M.; Tautu, P. 1973 Dependence with complete connections and its applications. Reprint of the 1990 hardback ed. Zbl 1154.60060 Iosifescu, Marius; Grigorescu, Serban 2009 On finite tail $$\sigma$$-algebras. Zbl 0245.60031 Iosifescu, Marius 1972 An exact convergence rate in a Gauss-Kuzmin-Lévy problem for some continued fraction expansion. Zbl 1366.11091 Iosifescu, Marius; Sebe, Gabriela Ileana 2006 A very simple proof of a generalization of the Gauss-Kuzmin-Lévy theorem on continued fractions, and related questions. Zbl 0774.11040 Iosifescu, Marius 1992 Dependence with complete connections and its applications. Zbl 0584.60064 Grigorescu, Şerban; Iosifescu, Marius 1982 A Poisson law for $$\Psi$$-mixing sequences establishing the truth of a Doeblin’s statement. Zbl 0378.60013 Iosifescu, Marius 1977 On multiple Markovian dependence. Zbl 0279.60052 Iosifescu, Marius 1973 The law of the iterated logarithm for a class of dependent random variables. Zbl 0169.20903 Iosifescu, M. 1968 A note on invariance principles for iterated random functions. (Letter to the editor). Zbl 1042.60017 Herkenrath, Ulrich; Iosifescu, Marius; Rudolph, Andreas 2003 On the Gauss-Kuzmin-Lévy theorem. III. Zbl 1013.11045 Iosifescu, Marius 1997 On denumerable chains of infinite order. Zbl 0266.60064 Iosifescu, M.; Spataru, A. 1973 Iterated function systems. A critical survey. Zbl 1212.60056 Iosifescu, Marius 2009 Sums of $$s+1$$ pseudo $$s$$th powers. Zbl 0987.11060 Deshouillers, Jean-Marc; Iosifescu, Marius 2000 The tail structure of nonhomogeneous finite state Markov chains: A survey. Zbl 0415.60061 Iosifescu, Marius 1979 Limit theorems for $$\varphi$$-mixing sequences. A survey. Zbl 0376.60025 Iosifescu, Marius 1977 On Strassen’s version of the loglog law for some classes of dependent random variables. Zbl 0232.60019 Iosifescu, Marius 1972 On Gauss problem for the Lüroth expansion. Zbl 1286.11125 Iosifescu, Marius; Sebe, Gabriela Ileana 2013 Random systems with complete connections and iterated function systems. Zbl 1047.60102 Herkenrath, Ulrich; Iosifescu, Marius; Rudolph, Andreas 2003 On Denjoy’s canonical continued fraction expansion. Zbl 1046.11052 Iosifescu, M.; Kraaikamp, C. 2003 On a 1936 paper of Arnaud Denjoy on the metrical theory of the continued fraction expansion. Zbl 1061.11502 Iosifescu, Marius 1999 A basic tool in mathematical chaos theory: Doeblin and Fortet’s ergodic theorem and Ionescu Tulcea and Marinescu’s generalization. Zbl 0801.47003 Iosifescu, Marius 1993 A coupling method in the theory of dependence with complete connections according to Doeblin. Zbl 0757.60110 Iosifescu, Marius 1992 A survey of the metric theory of continued fractions, fifty years after Doeblin’s 1940 paper. Zbl 0734.11041 Iosifescu, M. 1990 On mixing coefficients for the continued fraction expansion. Zbl 0689.10054 Iosifescu, Marius 1989 La loi du logarithme iteré pour une classe de variables aléatoires dependantes. Zbl 0159.47302 Iosifescu, M. 1968 Application des systèmes à liaisons completes à un problème de reglage. Zbl 0154.43001 Iosifescu, M.; Mandl, P. 1966 Sur la programmation linéaire. Zbl 0118.35301 Iosifescu, M.; Theordorescu, R. 1963 Spectral analysis for the Gauss problem on continued fractions. Zbl 1304.11079 Iosifescu, Marius 2014 Introduction to stochastic models. Zbl 1207.60001 Iosifescu, Marius; Limnios, Nikolaos; Oprişan, Gheorghe 2010 Metric properties of Denjoy’s canonical continued fraction expansion. Zbl 1205.37019 Iosifescu, Marius; Kraaikamp, Cor 2008 Finite Markov processes and their applications. Revised and expanded edition. Translated from the Romanian. Reprint of the 1980 ed. Zbl 1136.60300 Iosifescu, Marius 2007 A three-dimensional probability distribution in the metrical theory of continued fractions. Zbl 1138.11032 Iosifescu, M.; Kraaikamp, C. 2004 On the Gauss-Kuzmin-Lévy theorem. I. Zbl 0820.11047 Iosifescu, Marius 1994 Doeblin and the metric theory of continued fractions: A functional theoretic solution to Gauss’ 1812 problem. Zbl 0788.11029 Iosifescu, Marius 1993 On $$U$$-statistics and von Mises statistics for a special class of Markov chains. Zbl 0763.60011 Iosifescu, Marius 1992 Recent advances in mixing sequences of random variables. Zbl 0435.60024 Iosifescu, Marius 1980 Recent advances in the metric theory of continued fractions. Zbl 0406.10041 Iosifescu, Marius 1978 On the application of random systems with complete connections to the theory of f-expansions. Zbl 0299.60077 Iosifescu, M. 1974 An extension of the renewal equation. Zbl 0226.60105 Iosifescu, Marius 1972 Processus aléatoires à liaisons completes purement discontinus. Zbl 0241.60039 Iosifescu, Marius 1968 Probability theory and mathematical statistics. Zbl 0166.13601 Iosifescu, Marius; Mihoc, Gheorghe; Theodorescu, Radu 1966 On some models for learning. I - IV. Zbl 0104.13502 Iosifescu, M.; Theodorescu, R. 1961 Über die differenzialen Eigenschaften der reellen Funktionen einer reellen Veränderlichen. Zbl 0103.04003 Iosifescu, Marius 1959 Spectral analysis for the Gauss problem on continued fractions. Zbl 1304.11079 Iosifescu, Marius 2014 On Gauss problem for the Lüroth expansion. Zbl 1286.11125 Iosifescu, Marius; Sebe, Gabriela Ileana 2013 Introduction to stochastic models. Zbl 1207.60001 Iosifescu, Marius; Limnios, Nikolaos; Oprişan, Gheorghe 2010 Dependence with complete connections and its applications. Reprint of the 1990 hardback ed. Zbl 1154.60060 Iosifescu, Marius; Grigorescu, Serban 2009 Iterated function systems. A critical survey. Zbl 1212.60056 Iosifescu, Marius 2009 Metric properties of Denjoy’s canonical continued fraction expansion. Zbl 1205.37019 Iosifescu, Marius; Kraaikamp, Cor 2008 Finite Markov processes and their applications. Revised and expanded edition. Translated from the Romanian. Reprint of the 1980 ed. Zbl 1136.60300 Iosifescu, Marius 2007 An exact convergence rate in a Gauss-Kuzmin-Lévy problem for some continued fraction expansion. Zbl 1366.11091 Iosifescu, Marius; Sebe, Gabriela Ileana 2006 A three-dimensional probability distribution in the metrical theory of continued fractions. Zbl 1138.11032 Iosifescu, M.; Kraaikamp, C. 2004 A note on invariance principles for iterated random functions. (Letter to the editor). Zbl 1042.60017 Herkenrath, Ulrich; Iosifescu, Marius; Rudolph, Andreas 2003 Random systems with complete connections and iterated function systems. Zbl 1047.60102 Herkenrath, Ulrich; Iosifescu, Marius; Rudolph, Andreas 2003 On Denjoy’s canonical continued fraction expansion. Zbl 1046.11052 Iosifescu, M.; Kraaikamp, C. 2003 Metrical theory of continued fractions. Zbl 1122.11047 Iosifescu, Marius; Kraaikamp, Cor 2002 Sums of $$s+1$$ pseudo $$s$$th powers. Zbl 0987.11060 Deshouillers, Jean-Marc; Iosifescu, Marius 2000 On a 1936 paper of Arnaud Denjoy on the metrical theory of the continued fraction expansion. Zbl 1061.11502 Iosifescu, Marius 1999 On the Gauss-Kuzmin-Lévy theorem. III. Zbl 1013.11045 Iosifescu, Marius 1997 On the Gauss-Kuzmin-Lévy theorem. I. Zbl 0820.11047 Iosifescu, Marius 1994 A basic tool in mathematical chaos theory: Doeblin and Fortet’s ergodic theorem and Ionescu Tulcea and Marinescu’s generalization. Zbl 0801.47003 Iosifescu, Marius 1993 Doeblin and the metric theory of continued fractions: A functional theoretic solution to Gauss’ 1812 problem. Zbl 0788.11029 Iosifescu, Marius 1993 A very simple proof of a generalization of the Gauss-Kuzmin-Lévy theorem on continued fractions, and related questions. Zbl 0774.11040 Iosifescu, Marius 1992 A coupling method in the theory of dependence with complete connections according to Doeblin. Zbl 0757.60110 Iosifescu, Marius 1992 On $$U$$-statistics and von Mises statistics for a special class of Markov chains. Zbl 0763.60011 Iosifescu, Marius 1992 Dependence with complete connections and its applications. Transl. from the Romanian. Rev., updated and exp. version. Zbl 0749.60067 Iosifescu, Marius; Grigorescu, Serban 1990 A survey of the metric theory of continued fractions, fifty years after Doeblin’s 1940 paper. Zbl 0734.11041 Iosifescu, M. 1990 On mixing coefficients for the continued fraction expansion. Zbl 0689.10054 Iosifescu, Marius 1989 Dependence with complete connections and its applications. Zbl 0584.60064 Grigorescu, Şerban; Iosifescu, Marius 1982 Finite Markov processes and their applications. Rev. and exp. ed. Translated from the Roumanian. Zbl 0436.60001 Iosifescu, Marius 1980 Recent advances in mixing sequences of random variables. Zbl 0435.60024 Iosifescu, Marius 1980 The tail structure of nonhomogeneous finite state Markov chains: A survey. Zbl 0415.60061 Iosifescu, Marius 1979 Recent advances in the metric theory of continued fractions. Zbl 0406.10041 Iosifescu, Marius 1978 A Poisson law for $$\Psi$$-mixing sequences establishing the truth of a Doeblin’s statement. Zbl 0378.60013 Iosifescu, Marius 1977 Limit theorems for $$\varphi$$-mixing sequences. A survey. Zbl 0376.60025 Iosifescu, Marius 1977 On the application of random systems with complete connections to the theory of f-expansions. Zbl 0299.60077 Iosifescu, M. 1974 Stochastic processes and applications in biology and medicine. Part II: Models. Zbl 0262.92002 Iosifescu, M.; Tautu, P. 1973 Stochastic processes and applications in biology and medicine. Part I: Theory. Zbl 0262.92001 Iosifescu, M.; Tautu, P. 1973 On multiple Markovian dependence. Zbl 0279.60052 Iosifescu, Marius 1973 On denumerable chains of infinite order. Zbl 0266.60064 Iosifescu, M.; Spataru, A. 1973 On two recent papers on ergodicity in nonhomogeneous Markov chains. Zbl 0249.60031 Iosifescu, Marius 1972 On finite tail $$\sigma$$-algebras. Zbl 0245.60031 Iosifescu, Marius 1972 On Strassen’s version of the loglog law for some classes of dependent random variables. Zbl 0232.60019 Iosifescu, Marius 1972 An extension of the renewal equation. Zbl 0226.60105 Iosifescu, Marius 1972 Random processes and learning. Zbl 0194.51101 Iosifescu, Marius; Theodorescu, Radu 1969 The law of the iterated logarithm for a class of dependent random variables. Zbl 0169.20903 Iosifescu, M. 1968 La loi du logarithme iteré pour une classe de variables aléatoires dependantes. Zbl 0159.47302 Iosifescu, M. 1968 Processus aléatoires à liaisons completes purement discontinus. Zbl 0241.60039 Iosifescu, Marius 1968 Application des systèmes à liaisons completes à un problème de reglage. Zbl 0154.43001 Iosifescu, M.; Mandl, P. 1966 Probability theory and mathematical statistics. Zbl 0166.13601 Iosifescu, Marius; Mihoc, Gheorghe; Theodorescu, Radu 1966 Sur la programmation linéaire. Zbl 0118.35301 Iosifescu, M.; Theordorescu, R. 1963 On some models for learning. I - IV. Zbl 0104.13502 Iosifescu, M.; Theodorescu, R. 1961 Über die differenzialen Eigenschaften der reellen Funktionen einer reellen Veränderlichen. Zbl 0103.04003 Iosifescu, Marius 1959 all top 5 #### Cited by 509 Authors 11 Kraaikamp, Cor 9 Kalpazidou, Sofia 9 Peligrad, Magda 8 Iosifescu, Marius 8 Sebe, Gabriela Ileana 8 Theodorescu, Radu 8 Yin, Gang George 7 Lascu, Dan 7 Vassiliou, Panagiotis C. G. 7 Wu, Jun 6 Herkenrath, Ulrich 6 Szewczak, Zbigniew Stanisław 6 Wang, Baowei 5 Dajani, Karma 5 Zhang, Qing 4 Hernández-Lerma, Onésimo 4 Horbacz, Katarzyna 4 Takashima, Keizo 3 Chen, Ting-Li 3 Chu, Jian 3 Kalpazidou, Sophia L. 3 Limnios, Nikolaos 3 Pemantle, Robin 3 Philipp, Walter 3 Pruscha, Helmut 3 Roitershtein, Alexander 3 Saloff-Coste, Laurent 3 Shimaru, Naoto 3 Sokal, Alan D. 3 Stefanov, Valeri T. 3 Su, Hongye 3 Tan, Bo 3 Wickwire, Kenneth H. 3 Wu, Zhengguang 3 Zuniga, Jessica V. 2 Aberkane, Samir 2 Ayral, Hakan 2 Badowski, Grazyna 2 Berkes, István 2 Campi, Marco Claudio 2 Cavazos-Cadena, Rolando 2 Chen, Haibo 2 Cohen, Joel E. 2 Cohn, Harry 2 Faivre, Christian 2 Fan, Ai Hua 2 Feng, Yan 2 Ganatsiou, Chrysoula 2 Georgiou, Andreas C. 2 Gerontidis, Ioannis I. 2 Girstmair, Kurt 2 Goldwurm, Massimiliano 2 He, Jun 2 Heinrich, Lothar 2 Huang, Lingling 2 Hwang, Chii-Ruey 2 Kaijser, Thomas 2 Kingman, John Frank Charles 2 Kirsebom, Maxim Sølund 2 Lakshmivarahan, Sivaramakrishnan 2 Langeveld, Niels Daniël Simon 2 Lapinskas, R. 2 Lasserre, Jean-Bernard 2 Lertchoosakul, Poj 2 Leśniak, Krzysztof 2 Mukhamedov, Farruh Maksutovich 2 Mukherjea, Arunava 2 Nair, Radhakrishnan B. 2 Najarian, Kayvan 2 Nakada, Hitoshi 2 Nicolaie, Alina 2 O’Brien, George L. 2 Park, Juhyun (Jessie) 2 Peng, Li 2 Popescu, Ileana N. 2 Pułka, Małgorzata 2 Rhodius, Adolf 2 Samur, Jorge D. 2 Sanghvi, Arun P. 2 Schratzberger, Bernhard R. 2 Seneta, Eugene 2 Smeets, Ionica 2 Trapani, Lorenzo 2 Truquet, Lionel 2 Văduva, Ion 2 Vandehey, Joseph 2 Wen, Zhixiong 2 Yao, Xin 2 Yin, Baojian 2 Yoshihara, Ken-ichi 2 Zhang, Lixian 2 Zhang, Zhenliang 1 Aaronson, Jon 1 Adke, S. R. 1 Adomian, George 1 Aguilar, Enrique Guerra 1 Aldous, David John 1 Allen, Benjamin L. 1 Allen, Beth 1 Altenberg, Lee ...and 409 more Authors all top 5 #### Cited in 140 Serials 25 Stochastic Processes and their Applications 22 Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 14 Journal of Number Theory 11 Journal of Mathematical Analysis and Applications 11 Statistics & Probability Letters 10 Linear Algebra and its Applications 9 Lithuanian Mathematical Journal 9 Journal of Multivariate Analysis 7 Journal of Statistical Physics 7 Mathematical Biosciences 7 Automatica 7 Monatshefte für Mathematik 6 Acta Mathematica Hungarica 5 Chaos, Solitons and Fractals 5 Acta Arithmetica 5 Journal of Statistical Planning and Inference 5 Theoretical Population Biology 5 Transactions of the American Mathematical Society 5 Ergodic Theory and Dynamical Systems 5 Indagationes Mathematicae. 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Theoretical Informatics and Applications 2 Methodology and Computing in Applied Probability 2 Stochastic Models 2 Bulletin of the Malaysian Mathematical Sciences Society. Second Series 2 Science China. Mathematics 1 International Journal of Modern Physics B 1 Biological Cybernetics 1 Discrete Applied Mathematics 1 Nuclear Physics. B 1 Physics Reports 1 Rocky Mountain Journal of Mathematics 1 Bulletin of Mathematical Biology 1 Annales de l’Institut Fourier 1 Annals of the Institute of Statistical Mathematics 1 The Annals of Statistics 1 Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV 1 Bulletin of the London Mathematical Society 1 Compositio Mathematica 1 Computing 1 Dissertationes Mathematicae 1 International Journal of Mathematics and Mathematical Sciences 1 International Journal of Game Theory 1 Inventiones Mathematicae 1 Journal of Computational and Applied Mathematics 1 Journal of Economic Theory 1 Journal of Optimization Theory and Applications 1 Kodai Mathematical Journal 1 Mathematika 1 Proceedings of the Edinburgh Mathematical Society. Series II 1 RAIRO, Informatique Théorique 1 Rendiconti del Circolo Matemàtico di Palermo. Serie II 1 SIAM Journal on Control and Optimization 1 Siberian Mathematical Journal ...and 40 more Serials all top 5 #### Cited in 39 Fields 230 Probability theory and stochastic processes (60-XX) 93 Number theory (11-XX) 56 Dynamical systems and ergodic theory (37-XX) 33 Measure and integration (28-XX) 32 Systems theory; control (93-XX) 29 Statistics (62-XX) 23 Biology and other natural sciences (92-XX) 22 Computer science (68-XX) 21 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 20 Operations research, mathematical programming (90-XX) 17 Numerical analysis (65-XX) 13 Linear and multilinear algebra; matrix theory (15-XX) 12 Operator theory (47-XX) 10 Statistical mechanics, structure of matter (82-XX) 7 Real functions (26-XX) 6 Ordinary differential equations (34-XX) 6 Information and communication theory, circuits (94-XX) 5 Partial differential equations (35-XX) 5 Quantum theory (81-XX) 4 Combinatorics (05-XX) 4 Harmonic analysis on Euclidean spaces (42-XX) 3 History and biography (01-XX) 3 Difference and functional equations (39-XX) 3 Sequences, series, summability (40-XX) 3 Calculus of variations and optimal control; optimization (49-XX) 2 Functions of a complex variable (30-XX) 2 Functional analysis (46-XX) 1 Algebraic geometry (14-XX) 1 Topological groups, Lie groups (22-XX) 1 Potential theory (31-XX) 1 Special functions (33-XX) 1 Approximations and expansions (41-XX) 1 Integral equations (45-XX) 1 Differential geometry (53-XX) 1 General topology (54-XX) 1 Manifolds and cell complexes (57-XX) 1 Global analysis, analysis on manifolds (58-XX) 1 Mechanics of particles and systems (70-XX) 1 Fluid mechanics (76-XX) #### Wikidata Timeline The data are displayed as stored in Wikidata under a Creative Commons CC0 License. Updates and corrections should be made in Wikidata.	2021-05-14 01:06:18	{"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": 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": 0.3692152500152588, "perplexity": 6876.737671179729}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989616.38/warc/CC-MAIN-20210513234920-20210514024920-00397.warc.gz"}
https://mathematica.stackexchange.com/questions/67293/get-rational-and-irrational-parts	# Get rational and irrational parts Consider an expression of the form $a + b \sqrt{2}$, where $a,b \in \mathbb{Q}$. How can I extract $b$ (or equivalently $a$) from this expression? One can define the conjugate and use it to construct the rational and radical coefficients (rat and rad resp.). Just as PowerExpand assumes bases are positive reals, conj[x] will be correct only if the symbolic variables and functions in an expression x represent rational numbers. conj[x_] := x /. Sqrt[2] -> -Sqrt[2]; rat[x_] := (x + conj[x])/2; rad[x_] := (x - conj[x])/(2 Sqrt[2]); (* {a, b} ) % // Simplify ( { 1/2 ((a - Sqrt[2] b)^2 + (a + Sqrt[2] b)^2), (-(a - Sqrt[2] b)^2 + (a + Sqrt[2] b)^2)/(2 Sqrt[2]) } {a^2 + 2 b^2, 2 a b} ) ( {3/10, -(1/5)} *) More generally, one can extend the definitions to numbers over an arbitrary quadratic extension of the rationals. conj[x_, sqroot_: Sqrt[2]] := x /. sqroot -> -sqroot; rat[x_, sqroot_: Sqrt[2]] := (x + conj[x, sqroot])/2; rad[x_, sqroot_: Sqrt[2]] := (x - conj[x, sqroot])/(2 sqroot); • Yes, of course. Very clever solution. – Tyson Williams Dec 4 '14 at 17:09 You can use ToNumberField: 2/3 + 1/4 Sqrt[2] ToNumberField[%, Sqrt[2]] which produces AlgebraicNumber[Sqrt[2], {2/3, 1/4}] • Very nice. Thanks! – Tyson Williams Dec 4 '14 at 15:39 • What if $a$ and $b$ are symbolic? – Tyson Williams Dec 4 '14 at 15:41 • @TysonWilliams: I'm not sure how to handle that case, as I rarely use the abstract algebra functions. However, Daniel Lichtblau might know how to do it, so you could try asking him (or you could ask it as another question). – DumpsterDoofus Dec 4 '14 at 15:48 • As a mathematician / theoretical computer scientist, I think of $a$ and $b$ as "symbolically rational" (i.e. some rational numbers by assumption). Of course I also know, but often forget, that (1) symbolic, (2) symbolically rational, and (3) and rational are three different "data types" that can all behave differently in Mathematica. – Tyson Williams Dec 4 '14 at 16:18 • BTW: One can use ToNumberField[num][[2, 1]] to extract the rational part ("a") of any quadratic field. No need to even specify the generator! – kirma Feb 1 '16 at 21:42	2019-08-20 08:32:01	{"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": 0.7391484975814819, "perplexity": 3058.6540560028266}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027315258.34/warc/CC-MAIN-20190820070415-20190820092415-00087.warc.gz"}
http://tex.stackexchange.com/questions/111460/how-do-i-use-titledquestion-in-the-exam-class-with-or-without-points	# How do I use \titledquestion in the exam class with or without points? When using the exam class, it is possible to have commands like \question[5] or simply \question (i.e. with no points). The former includes the points within parentheses by default, and the latter omits the points and the surrounding parentheses. To use titled questions within the exam, where the points are surrounded by parentheses, I have used the following command: \qformat{\bf \thequestiontitle ~(\thepoints) \hfill} Now, when I use the \titledquestion{my title}[5] and \titledquestion{my title} commands, they do not behave in the same way that the \question[5] and \question commands do. The difference is that the titled versions always end up including parentheses, whether or not points have been specified. What I want to know is: what is a proper way to modify the argument of \qformat so that the parentheses are only present when a point value has been specified? I was unable to find this in the manual for exam.cls (I assumed it would be somewhere in the section dealing with titled questions). - You can use \if@placepoints to conditionally include the parentheses and the value: \documentclass{exam} \makeatletter \qformat{\bfseries\thequestiontitle\if@placepoints~(\thepoints)\else\fi\hfill} \makeatother \begin{document} \begin{questions} \titledquestion{my title}[5] \titledquestion{my title} \end{questions} \end{document} - Perfect! Thanks very much -- I wonder why this isn't included in the documentation somewhere. – Mark Apr 30 '13 at 1:33 It's weird; the @ suggests the author doesn't want you to use this conditional. Yet it is not any less useful than \ifprintanswers, which is described. – Ryan Reich Apr 30 '13 at 5:44	2015-01-31 18:12:11	{"extraction_info": {"found_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": 0.8027399182319641, "perplexity": 1141.2343487097946}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422120928902.90/warc/CC-MAIN-20150124173528-00000-ip-10-180-212-252.ec2.internal.warc.gz"}
https://math.stackexchange.com/questions/2745825/determine-the-constant-alpha-in-mathbbr-and-the-vector-w-in-mathbbr/2746416	Determine the constant $\alpha \in \mathbb{R}$ and the vector $w \in \mathbb{R}^3$ that is perpendicular to the vector u such that v = α u + w. Determine the constant $\alpha \in \mathbb{R}$ and the vector $w \in \mathbb{R}^3$ that is perpendicular to the vector $u$ such that $v = \alpha u + w$. Need a little help with this question from my linear algebra course, not sure how to find the solution. I know that that the dot product of w and u is equal to zero. the vectors give are $u = (2, -2, 1) v = (3, 1, -1)$ • So use the dot product on both sides of the $v$ equation... – N74 Apr 20 '18 at 10:21 • Sorry I am still a little confused. use the dot with what on the left side of the equation? – Darragh O'Flaherty Apr 20 '18 at 10:27 • On the left side you will have $v \cdot u$ – N74 Apr 20 '18 at 10:30 Let $w=(w_1,w_2,w_3)$. If $w$ is perpendicular to $u$, the equation $w\cdot u=0$ must be satisfied, that is, $2w_1-2w_2+w_3=0$. As you see there are lots of solutions. If you want $v=\alpha u+w$, then you have another equation, in this cases a system of equations indeed, namely $3=2\alpha +w_1$ $1=2\alpha+w_2$ $-1=\alpha+w_3$ Adding the equation $2w_1-2w_2+w_3=0$ from above you've got a linear system of 4 equations and 4 unknowns, so there is precisely one solution that I'm sure you're able to find. • Perfect thank you very much, just the guidance I needed – Darragh O'Flaherty Apr 20 '18 at 11:28 You have $v= \alpha u + w$. Now, dot product by $u$ gives $$v \cdot u = \alpha u \cdot u + w \cdot u$$ We know $w \cdot u=0$ so the equation gives $$3 =\alpha 9$$ so $$\alpha={ 1 \over 3}$$. Knowing $\alpha$ it is straightforward that $$w=\left ({7 \over 3}, {5 \over 3}, -{4 \over 3} \right )$$	2021-04-22 17:06:35	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9191478490829468, "perplexity": 119.83180657045094}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039594341.91/warc/CC-MAIN-20210422160833-20210422190833-00103.warc.gz"}
https://physics.stackexchange.com/questions/484634/can-we-ignore-the-scalar-field-dilaton-term-in-the-polyakov-sigma-model-action/485127	# Can we ignore the scalar field (dilaton) term in the Polyakov sigma-model action when deriving the classical equations of motion? I have the full Polyakov sigma model action: $$$$\begin{split} &S=S_P + S_B + S_\Phi = \\ &- {1 \over 4 \pi \alpha'} \Big[ \int_\Sigma d^2\sigma \sqrt{-g} g^{ab} \partial_a X^\mu \partial_b X^\nu G_{\mu\nu}(X)\, + \\ &+\epsilon^{ab} B_{\mu\nu}(X) \partial_a X^\mu \partial_b X^\nu \, +\alpha'\Phi(X) R^{(2)}(\sigma) \Big] \,. \end{split}$$$$ and I want to derive the classical equations of motion by varying $$X \mapsto X + \delta X$$. I am confused as to what to do with the last term. It is of a higher power of $$\alpha '$$, so I am thinking it can just be ignored, as it's variation will be of a higher order. Is this thinking correct? Does this question even make sense, as I'm trying to derive classical equations from a sigma-model, which as far as I have seen, is used when quantizing the string? • Under $X \to X + \delta X$, we have $\Phi(X) \to \Phi(X+\delta X) = \Phi(X) + \delta X^\mu \partial_\mu \Phi(X)$. – Prahar Jun 6 at 15:10 "For the remainder of this section we consider a purely classical string. Since the dilaton term is multiplied by $$\alpha'$$, it is a quantum correction and does not directly affect the motion of a classical string."	2019-10-19 22:05:11	{"extraction_info": {"found_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": 4, "wp-katex-eq": 0, "align": 0, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.9920988082885742, "perplexity": 154.64249250697108}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1570986700435.69/warc/CC-MAIN-20191019214624-20191020002124-00042.warc.gz"}
https://undergroundmathematics.org/quadratics/quad-solving-sorter	Fluency exercise ## Problem Sort these quadratic equations into groups. • How will you choose to define the groups? • Are there some equations which belong to more than one group? Once you have sorted the equations into groups, write a further equation for each group. (There are blank cards provided for this purpose.) How else could you have sorted the equations? (For a suggestion as to one possible way of grouping the equations, you could use the possible grouping cards.) You might want to download cards with the equations on them, so that you can sort them into groups more easily. $3x^2-17=0$ $x^2-6x-10=0$ $4x^2-7x=0$ $3x^2-5x-11=0$ $x^2-3x+2=0$ $x^2+3x=0$ $(2x-5)^2-9=0$ $(2x-1)(x+3)=0$ $(x-3)^2-4=0$ $x^2-9=0$ $2x^2+5x+2=0$ $2x^2+7x+4=0$ $x^2+6x-72=0$ $(x+1)^2=0$ $x^2-16x=0$ $2x^2+6x+4=0$ $7x^2+14x-21=0$ $3(x+4)^2-8=0$	2018-03-21 14:29:06	{"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": 0.42151349782943726, "perplexity": 1028.8986688379812}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647660.83/warc/CC-MAIN-20180321141313-20180321161313-00544.warc.gz"}
http://atheistfoundation.org.au/forums/showthread.php?s=9a303f5895bd8d8e54c4f4ba7efed6be&p=609435	AFA Forums Language Peeves Register FAQ Calendar Mark Forums Read Off Topic For anything that doesn't fit anywhere above. #141 9th February 2018, 12:20 PM The Irreverent Mr Black Beetles Reunion Join Date: Jan 2009 Location: Toontown Posts: 6,611 Re: Language Peeves The entrance is only available to owners of pistols and foils, or their seconds. Attached Images Untitled.jpg (71.3 KB, 38 views) __________________ EJB Ludwig Boltzman, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics. Perhaps it will be wise to approach the subject cautiously. (Opening lines of "States of Matter", by D.L. Goodstein). #142 10th February 2018, 08:41 AM The Irreverent Mr Black Beetles Reunion Join Date: Jan 2009 Location: Toontown Posts: 6,611 Re: Language Peeves Toontown Crumble, come on down for a little of that beyond-the-grave action: Attached Images Untitled.png (39.2 KB, 37 views) __________________ EJB Ludwig Boltzman, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics. Perhaps it will be wise to approach the subject cautiously. (Opening lines of "States of Matter", by D.L. Goodstein). two dogs liked this post #143 12th February 2018, 09:12 AM The Irreverent Mr Black Beetles Reunion Join Date: Jan 2009 Location: Toontown Posts: 6,611 Re: Language Peeves ABC News, (phweet!) - Here, boy! Attached Images Untitled.jpg (27.0 KB, 30 views) __________________ EJB Ludwig Boltzman, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics. Perhaps it will be wise to approach the subject cautiously. (Opening lines of "States of Matter", by D.L. Goodstein). two dogs laughed at this post #144 13th February 2018, 08:55 AM The Irreverent Mr Black Beetles Reunion Join Date: Jan 2009 Location: Toontown Posts: 6,611 Re: Language Peeves Fairfax Press, come on down! Quote: At the same time the Catholic Church was fighting a family who refused to accept $55,000 compensation for each of their abused daughters, it purchased a Kew mansion for Archbishop Denis Hart now valued at$2.25 million. Personally, I wouldn't give you two bob and a roadkill cane-toad for him. __________________ EJB Ludwig Boltzman, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics. Perhaps it will be wise to approach the subject cautiously. (Opening lines of "States of Matter", by D.L. Goodstein). two dogs liked this post two dogs laughed at this post #145 16th February 2018, 08:30 AM The Irreverent Mr Black Beetles Reunion Join Date: Jan 2009 Location: Toontown Posts: 6,611 Re: Language Peeves "There's a dead Prime Minister on the phone for me? Now that is news!" Good ol' Toontown Crumble. Attached Images Untitled.png (62.6 KB, 29 views) __________________ EJB Ludwig Boltzman, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics. Perhaps it will be wise to approach the subject cautiously. (Opening lines of "States of Matter", by D.L. Goodstein). two dogs laughed at this post #146 27th February 2018, 10:31 AM The Irreverent Mr Black Beetles Reunion Join Date: Jan 2009 Location: Toontown Posts: 6,611 Re: Language Peeves Some regrettably meta copypasta from the Toontown Crumble. Attached Images Untitled.png (50.9 KB, 22 views) __________________ EJB Ludwig Boltzman, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics. Perhaps it will be wise to approach the subject cautiously. (Opening lines of "States of Matter", by D.L. Goodstein). two dogs, Strato laughed at this post #147 2nd March 2018, 12:10 AM The Irreverent Mr Black Beetles Reunion Join Date: Jan 2009 Location: Toontown Posts: 6,611 Re: Language Peeves Gender-bend it like Beckham? Toontown Crumble, yet again. Attached Images Untitled.jpg (46.9 KB, 17 views) __________________ EJB Ludwig Boltzman, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics. Perhaps it will be wise to approach the subject cautiously. (Opening lines of "States of Matter", by D.L. Goodstein). Strato laughed at this post #148 2nd March 2018, 01:48 AM Strato What Me Deluded? Join Date: Jul 2012 Location: The Bellarine, Geelong. Posts: 5,890 Re: Language Peeves Instead of employing proof readers as done in the past, they must just rely on a spell check algorithm thingy. Cost-cutting. After all, the Toontown consumers aren't going to desist from reading The Chronic because of such trivial normal major fuck ups. __________________ Everything admits of a naturalistic explanation. #149 7th March 2018, 02:04 PM The Irreverent Mr Black Beetles Reunion Join Date: Jan 2009 Location: Toontown Posts: 6,611 Re: Language Peeves Er, I guess we'll use the surplus to assist with cloning experiments... Attached Images Untitled.jpg (20.4 KB, 13 views) __________________ EJB Ludwig Boltzman, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics. Perhaps it will be wise to approach the subject cautiously. (Opening lines of "States of Matter", by D.L. Goodstein). two dogs laughed at this post #150 17th March 2018, 08:29 PM The Irreverent Mr Black Beetles Reunion Join Date: Jan 2009 Location: Toontown Posts: 6,611 Re: Language Peeves ABC News QLD, just now: a guy interviewed in a vox-pop talks about "four-month-year-old kids". __________________ EJB Ludwig Boltzman, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics. Perhaps it will be wise to approach the subject cautiously. (Opening lines of "States of Matter", by D.L. Goodstein). two dogs liked this post Bookmarks	2018-03-20 17:46:52	{"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": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.212354838848114, "perplexity": 14273.579680372368}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647519.62/warc/CC-MAIN-20180320170119-20180320190119-00255.warc.gz"}
https://www.physicsforums.com/threads/question-about-newtons-method.397500/	1. Apr 22, 2010 ### math8 Let $$x_{m}$$ and $$x_{m+1}$$ be two successive iterates when Newton's method is applied to a polynomial $$p(z)$$ of degree n. We prove that there is a zero of $$p(z)$$ in the disk $${z \in \textbf{C}: \|z - x_{m}\| \leq n\|x_{m+1} - x_{m}\| }$$. I suppose we may use $$p'(z)/ p(z) = \sum ^{n}_{j=1} 1/ \left( z-r_{j} \right)$$ where $$r_{1}, r_{2}, \cdots , r_{n}$$ are roots of p. We need to show there is a root $$\alpha$$ of $$p(z)$$ for which $$\|\alpha - x_{m}\| \leq n\|x_{m+1} - x_{m}\|}$$ i.e. for which $$\frac{n}{\|\alpha - x_{m}\|} \geq \left\| \frac{p'(x_{m})}{p(x_{m})} \right\| = \sum ^{n}_{j=1} 1/ \|x_{m}-r_{j} \|$$ But I am not sure how to proceed from there.	2018-01-20 23:34:02	{"extraction_info": {"found_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": 0.8801466822624207, "perplexity": 80.37864985478393}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084889736.54/warc/CC-MAIN-20180120221621-20180121001621-00194.warc.gz"}
https://robotics.stackexchange.com/tags/rangefinder/hot	# Tag Info 9 I still think this is off-topic, but it seems I need more space than a comment to show (answer?) why that is so. You are starting from some performance specifications and are looking to get to a set of features you need in your camera. Here is a post from NI about stereo vision that gives a formula for depth resolution: \Delta z = \frac{z^2}{fb}\... 7 Animals and robots both need to understand something about the 3D structure of the world in order to thrive. Because it's so important, animals have evolved a huge number of strategies to estimate depth based on camera-like projective sensors (eyes). Many make use of binocular disparity -- the fact that the distance between the same scene point in two ... 7 FWIW, I once needed to create a plexiglass window/shell for a near-IR camera. Most CCD and CMOS sensors are sensitive in the near IR range (e.g. around 850nm), which is where @user3095849 suggested your sensor sits. I went to a local plexiglass supplier and asked for samples of various sheets (they often have lots of leftover pieces) and simply tried them ... 6 First you need to know the wavelength of the sensor. According to the sharp sensors datasheet for GP2Y0A21YK, the cover should efficiently transmit light throughout the wavelength range of the LED(λ = 850 nm ± 70 nm). Second you need to find the exact material which can transmits the wavelength of the sensor. Some transparent plastics such as acrylic glass ... 4 Imagine someone put you in a wheelchair and blindfolded you, then let you reach your arm out and touch a wall. You could tell how far away the wall was, but as long as you were pushed parallel to the wall, how would you know how far you had gone? You can't count steps or see the end of the hall, so you do not have a way to index your samples of where the ... 3 I'll try to explain this in terms of software. Firstly, it is just next to impossible to have 100 metres (practically, 3 metres is like the best depth accuracy) and we need to be sure that the cameras are capturing images properly with all practical features (vibration-free and even under sunlight or rain)! (Else features will be lost and we end up with ... 3 Vision is actually one kind of rangefinder, although passive. Earlier camera use parallax to find the distance and focus. Nowadays, active rangefinders are developed such as ultrasonic sensors, lidar sensors, which emit a signal and checks the returned time, the time of flight. This is more accurate and could adapt to low light conditions. 3 Yes, there is such a system available today, ScenSor from DecaWave: These tags can measure their distance from base stations using the time of flight of radio packets. They have an precision of about 10cm, I.E. successive samples are randomly distributed in a 10cm diameter cloud around the true location. Also, the radio signal needs a clear line of sight ... 2 Another sensor that you should investigate is ultra-short baseline (USBL). I have used the MicronNav with success in the past. You mentioned accelerometers, but what I think you really meant is an inertial navigation system. These fuse data from accelerometers, gyros, compass, and optionally GPS for a more accurate dead-reckoning. I have used the ... 2 You could use AR markers attached to each robot or in known locations around the room. In fact, this is how it is typically done in research for indoor, multi-robot systems. The libraries for recognizing and extracting range / angle information from AR markers are well developed, and included in the Robot Operating System. Now, you won't need to actually use ... 2 I did casually search for something like this a year or two ago. "Sparse sensing" or "sensing limited" were the sort of phrases that cropped up. Kris Beevers has some interesting publications in this sort of area, such as SLAM With Sparse Sensing. His general approach was to maintain previous sensor readings while changing the direction of the robot, to ... 2 One way you can use SLAM in your setup is to stop the robot every 30 cm or so, and perform a sweep with your lidar. You can then use e.g. one of the 2D SLAM packages from openslam. The problem with sweeping while you are moving is that you cannot get any correspondence information from a single range reading. SLAM works by associating features in one ... 2 Both 1550nm and 900nm ranges are used in different lidars. Velodyne and majority others use 905nm lasers because these diodes are really cheap. 1550nm are used when there is a need for very long distance measurement range, such as Luminar, as the eye safety threshould is several orders of magnitude higher for this wavelength than 905nm. In principle, ... 2 You are correct. Most Velodyne LIDARs (very commonly used on autonomous vehicles), use 905 nm diode lasers, and so do many Hokuyo laser scanners and SICK LIDARs. A new company, Ouster, uses 850 nm lasers, claiming that their choice of lower wavelength is because the "atmospheric water absorption is orders of magnitude lower than at 905nm". Some more info: ... 2 I wouldn't recommend using sensors with different laser frequencies either, because that likely means using an entirely different product, with different accuracy and other properties. don't tilt, shift I think the idea of tilting a laser scanner is very problematic, especially with a height constraint like you have. But what about shifting the scanner up or ... 1 Short answer: They don't need. Cameras can be used as range finders, and an easy example is the Xbox Kinect. Computer vision is an area that is being studied a lot. You can find a lot on authonomous cars, and the problem of SLAM. 1 Are you sure roscore is running? "Failed to contact master" is usually an issue with roscore not initializing. If roscore indeed is running but you are still getting the message, check if ROS_MASTER_URI is set to something other than the default. Or run "rosnode list" and paste the output here. 1 So according to the datasheet from the link provided, the Data interface is indeed UART (LVTTL 3.3V). The datasheet also says it has 2 Modes of operation: Single measurement, continuous ranging 1Hz to 25Hz. The datasheet does not go into the details of the application layer (ie. the commands and structure of the data in those commands), let alone the baud ... 1 Check this, starting from the last part: I think they are using the same technology as the several of their sensors: ( the one you mentioned) In the 100 dollar price range I think you will not find anything that meets your demands. I have no experience with golf laser range finders. 1 I think 0.3m noise is a bit exaggerated for a scanning laser rangefinder. As you saw with the Hokuyo (which is one of the cheapest LIDARs you can get) they say that it is 0.03m range "error" (they do not explicitly state this is 2$\sigma$, but I have tested the noise profile myself and it is consistent with 2$\sigma$). My experience with laser scanners is ... 1 Am I exaggerating with these values? No your're not but you need to filter the measurements. Also, you can double check datasheet of any laser sensor for the specifications. I've written a code for Kalman filter long time ago for a laser sensor that measures a static point located at <10,10>. Let's see how the filter is working with the noise you've ... 1 SLAM can be done Using many different way, in this case, Yes Buddy can map around him using its sensors : RGB camera, IR sensor, Ultra sound Sensor, Encoder. The result depend of the algorithme you are using. We are also working on a 3D camera to allow buddy to have a better input of the world. 1 The core part of the HectorSLAM is in the file hector_slam/hector_mapping/include/hector_slam_lib/matcher/ScanMatcher.h and hector_slam/hector_mapping/include/hector_slam_lib/map/OccGridMapUtil.h. The implementation of the scan matching in HectorSLAM uses the maximum likelihood estimator (MLE), which is implemented in the function estimateTransformationLogLh.... 1 So this is really a hard task. Estimating the position of a AUV is still a large challenge even in research. There are sensors like as DVL (Doppler Velocity Log) available that can estimate the speed over ground. These sensors working okay, but they only estimating the speed. These devices are precise enough for your use case BUT these devices are to large ... 1 Another idea is to use polarized light beacons. If you have a few light sources polarized at different angles, then you can use some simple light sensors (or cameras) similarly polarized. If the beacons are at known world locations, you should be able to figure out your location through triangulation or trilateration. the polarization lets you distinguish ... 1 An extremely cheap method would be to use IR leds mounted on your beacon and an omni-directional IR sensor array on your robot (just a few remote control leds arranged around your robot) The beacon will be glowing like a neon sign in IR and a simple brightness detection algorithm will give you a direction. Move 10cm, take another reading, etc. Only top voted, non community-wiki answers of a minimum length are eligible	2021-11-29 15:49:47	{"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": 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": 0.508532702922821, "perplexity": 1360.2486144961765}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358774.44/warc/CC-MAIN-20211129134323-20211129164323-00043.warc.gz"}
https://www.wyzant.com/resources/answers/topics/homework	114 Answered Questions for the topic Homework 06/01/22 #### Calculus Homework Help Determine whether the integral is convergent or divergent. ∞ e−1/x / x2 dx 3 If it is convergent, evaluate it. (If the quantity diverges, enter DIVERGES.) 05/24/22 #### Calculus HW Help Consider the parametric equations below.x = t2 − 4, y = t + 3, −3 ≤ t ≤ 3(a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which... more 05/16/22 #### b. What percentage of all possible observations of the variable are at most 12? Assume that the variable under consideration has a density curve. The area under the density curve that lies to the right of 12 is 0.371. a. What percentage of all possible observations of the... more 05/10/22 #### Calculus Math Help Consider the following.(a) Set up an integral for the volume a solid torus (the donut-shaped solid shown in the figure) with radii br and aR. (Let a = 3 and b = 2.) 0 dy(b) By interpreting... more 04/26/22 #### Midpoint Reiman sums HW help use a midpoint approximation to find the area under the curve y+x^2-x for [0,2] with n=4. Show all summation. 03/21/22 #### SA and V review before test Question: 7 If a sphere has a circumference of 75.36 in., what formula should you use to find the radius? What is the surface area of the sphere?Let C=2πr equal the fourmula for the radius of the sphereThe... more 03/21/22 #### 7.1 Understanding Pythagorean Theorem Question: 17 An appliance store decreases the price of a 19-in. television set 19% to a sale price of $522.86. What was the original price? The original price of the television was$_____________(Round to... more 03/21/22 #### 7.1 Understanding Pythagorean Theorem Question: 15 Solve the system of equations using subtraction.4x - 9y = 68x - 9y = 30The solution is ______________ (Type an ordered pair.) 03/04/22 #### Topic 5 Review Systems Question 19 The sum of two numbers is 14. When the second number is subtracted from the first number, the difference is 10. Find the two numbers.Question content area bottomPart 1What are the... more 03/04/22 #### Topic 5 review of systems Question 15 Multiple Representations Find the solution of the system of equations by substitution. Use pencil and paper. Draw a picture using algebra tiles to show the... more 02/20/22 #### Topic 5 Review of Systems (19 questions) -- Question 17 Solve the system of equations using subtraction.4x−7y=1710x−7y______ 02/17/22 #### 5.4 Solving Systems by Subtraction and substitution Question: 9 5.4 Solving Systems by Subtraction and substitution Question: 9 Age If you add Natalie's age and Fred's age, the result is 39. If you add Fred's age to 4 times Natalie's age, the result is 84.... more 02/17/22 #### 5.4 Solving Systems by Subtraction and substitution Question: 9 5.4 Solving Systems by Subtraction and substitution Question: 9 Age If you add Natalie's age and Fred's age, the result is 39. If you add Fred's age to 4 times Natalie's age, the result is 84.... more 02/13/22 #### Differential calculus using chain rule, product rule and quotient rule. Im unsure on how to do this question.this is for homework and its a complex question.the question isfor f(x)=ax(x2-b) where both a and b are constants, f(x)=8b when x=4, f'(x)=15a when x=3. Find... more Homework Math Im Failing 12/20/21 #### Math homework help The library of congress has more than 5 million maps. Maps make up just 3.75% of the library's entire collection of items how many items does the library of congress have? 12/20/21 #### middle school math homework help Last month you spent 87\$ on clothing. This month you spent 165% of what you spent last month. how much did you spend this month Homework Math Pre Calculus 11/22/21 #### If sin (t) = 3 over 8 and t is in the 2nd quadrant, find cos (t) Homework for pre calculus Homework Math Pre Calculus 11/22/21 #### If cos (t) = 1 over 7 and t is in the 4th quadrant, find sin(t) College pre calc home work assignment Homework Spanish 11/02/21 #### Can someone please check-over my answers to my Spanish homework? It’s due tomorrow and I just want to make sure I’ll get a good grade. Subject: Possessive Pronouns: Instructions: Write on the line the letter in front of the correct possessive pronoun to replace the underlined words. 1. ____ Mi casa es más grande que la casa de... more 08/19/21 #### Correct to 1 significant figure, the area of a rectangle is 80 cm2. Correct to 2 significant figures, the length of the rectangle is 12 cm. Calculate the upper bound for the perimeter of the rectangle Correct to 1 significant figure, the area of a rectangle is 80 cm2. Correct to 2 significant figures, the length of the rectangle is 12 cm. Calculate the upper bound for the perimeter of the... more 08/18/21 #### Simplify (7 + 2√50) (5 - 2√2). Give your answer in the form a+b√18 where a and b are integers. Show your working clearly. Simplify (7 + 2√50) (5 - 2√2). Give your answer in the form a+b√18 where a and b are integers. Show your working clearly. Homework Precalculus College 07/24/21 Outside temperature over a day can be modelled as a sinusoidal function. Suppose you know the high temperature for the day is 90 degrees and the low temperature of 80 degrees occurs at 6 AM.... more Homework Precalculus College 07/23/21 #### precalculus homework Find a possible formula for the trigonometric function whose values are in the following table.x 0 2 4 6 8 10 12y0 -4 -8 -4 0 -4 -8y=we didn't go over this in class so I'm really lost with it. Homework Precalculus 07/23/21 #### Precalculus homework Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the high temperature of 86 degrees occurs at 4 PM and the average temperature for the day is 75 degrees.... more Homework History 06/30/21 #### Henry VIII Homework Help "Henry VIII changed the church in England because he wanted its money". Do you agree? Explain your answerThanks ## Still looking for help? Get the right answer, fast. Get a free answer to a quick problem. Most questions answered within 4 hours. #### OR Choose an expert and meet online. No packages or subscriptions, pay only for the time you need.	2022-08-14 03:01:51	{"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": 0.5662363171577454, "perplexity": 1354.6311467527853}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1659882571993.68/warc/CC-MAIN-20220814022847-20220814052847-00497.warc.gz"}
https://byjus.com/questions/the-value-of-delta-s-for-spontaneous-process-is/	# The value of ∆S for spontaneous process is For a spontaneous process in an isolated system ∆S should be positive i.e ∆S > 0. ### Explanation In thermodynamics, S stands for entropy. The entropy of any system is defined as the degree of randomness in it. A spontaneous process is an irreversible process, and some external agents could only reserve it. For a spontaneous process, the total entropy change, ΔStotal is always greater than zero. The total entropy of the system can be given by the following equation ΔStotal=ΔSsys + ΔSsurr Where, ΔStotal= total entropy change for the process ΔSsys = entropy change of the system ΔSsurr = entropy change of the surrounding	2021-10-17 20:10:06	{"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": 0.8419061303138733, "perplexity": 847.2317422225341}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1634323585181.6/warc/CC-MAIN-20211017175237-20211017205237-00691.warc.gz"}
https://jrupac.wordpress.com/2010/05/13/google-code-jam-2010-qualifying-round-solutions-part-2/	## Google Code Jam 2010 Qualifying Round Solutions (Part 2) 13 May Here’s my solution to Fair Warning Large input. It’s actually very similar to my Small input solution except that longs are replaced with BigIntegers (and it’s designed to handle cases larger than 3. Anyway, here’s my solution: import java.util.; import java.io.; import java.math.*; public class Fair_large { String filename; ArrayList<BigInteger> list; ArrayList<BigInteger> diffs; public Fair_large(String f) { filename = f; openFiles(); } private void openFiles() { Scanner s; try { s = new Scanner(new File(filename)); int ct; ct = s.nextInt(); for (int i = 0; i < ct; i++) { list = new ArrayList<BigInteger>(); diffs = new ArrayList<BigInteger>(); int ct2 = s.nextInt(); for (int j = 0; j < ct2; j++) output(i + 1, process(ct2)); } s.close(); } catch (Exception e) { System.err.println("I/O error: " + e); System.exit(0); } } private String process(int length) { Collections.sort(list); BigInteger T = BigInteger.ZERO; if (length == 2) { T = list.get(1).subtract(list.get(0)); if (T.equals(BigInteger.ONE)) return "0"; return distToNextMultiple(list.get(0), T).toString(); } for (int i = 1; i < list.size(); i++) while (diffs.size() > 1) T = diffs.get(0); if (T.equals(BigInteger.ONE)) return "0"; return distToNextMultiple(list.get(0), T).toString(); } private BigInteger distToNextMultiple(BigInteger x, BigInteger T) { return T.subtract(x.mod(T)).mod(T); } private void output(int caseNum, String result) { System.out.println("Case #" + caseNum + ": " + result); } public static void main(String args[]) { Fair_large app = new Fair_large(args[0]); } } So in this solution, I did use some simple optimizations to reduce the amount of actual computation needed to produce my solution. First, I handled the 2 case by itself since you don’t actually need to use GCD for that. I created an auxiliary array called diffs that holds the differences between consecutive numbers (in the sorted array). Thus, we know that adding some $y \geq 0$ does not change the differences between them; so, if they must all be multiples of T, then their differences must already be multiples of T. Then, the GCD of the all the numbers of the auxiliary array is calculated. In my small input solution I wrote a slightly optimized (non-recursive – to save on mem read/writes on the stack) GCD function based on Euclid’s algorithm. At some point I also used Stein’s algorithm for the GCD calculation. But for the large input, I decided to just use the built-in GCD function that comes with BigInteger. One key property that I exploited was that: $\gcd(x_1, x_2, \ldots, x_n) = \gcd(\cdots\gcd(\gcd(x_1, x_2), x_3)\cdots)$ Thus, I computed the GCD of two consecutive numbers (the first two of the aux array), removed both, and added in their GCD. Continuing in this way until there was only one element left, I had computed the GCD of all the numbers with reasonable speed. The other interesting function was the distToNextMultiple function. Mathematically, this function returns: $(T-(x \mod T))\mod T$ We know that if $x < T$ then $x \equiv x \mod T$ but otherwise, it is the remainder after division by T. Subtracting this value from T gives how much is left until the next multiple. But why the extra modulus T? If $x \equiv 0 \mod T$, then $T - (x \mod T)$ yields $T$, which is wrong. Thus, the extra modulus T takes care of this case. And that’s pretty much it. A pretty simple solution that works quickly. On the same Core 2 Duo machine, the solution for the large input runs in 0.436s. Good enough for me. 1 Comment Posted by on May 13, 2010 in Uncategorized Tags: , , , , ### One response to “Google Code Jam 2010 Qualifying Round Solutions (Part 2)” 1. May 17, 2010 at 6:16 AM beast	2017-10-18 03:54:35	{"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 8, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3995513617992401, "perplexity": 2348.7796228806897}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1508187822739.16/warc/CC-MAIN-20171018032625-20171018052625-00241.warc.gz"}
https://artofproblemsolving.com/wiki/index.php?title=1970_AHSME_Problems/Problem_13&diff=prev&oldid=107611	# Difference between revisions of "1970 AHSME Problems/Problem 13" ## Problem Given the binary operation $\star$ defined by $a\star b=a^b$ for all positive numbers $a$ and $b$. Then for all positive $a,b,c,n$, we have $\text{(A) } a\star b=b\star a\quad\qquad\qquad\ \text{(B) } a\star (b\star c)=(a\star b) \star c\quad\\ \text{(C) } (a\star b^n)=(a \star n) \star b\quad \text{(D) } (a\star b)^n =a\star (bn)\quad\\ \text{(E) None of these}$ ## Solution Let $a = 2, b = 3, c=n = 4$. If all of them are false, the answer must be $E$. If one does not fail, we will try to prove it. For option $A$, we have $2^3 = 3^2$, which is clearly false. For option $B$, we have $2^{81} = 8^{4}$, which is false. For option $C$, we have $2^{81} = 16^3$, which is false. For option $D$, we have $8^4 = 2^{12}$, which is true. The LHS is $(a^b}^n$ (Error compiling LaTeX. ! Extra }, or forgotten $.). By the elementary definition of exponentiation, this is $a^b$ multiplied by itself $n$ times. Since each $a^b$ is actually $a$ multiplied $b$ times, the expression$(a^b}^n$(Error compiling LaTeX. ! Extra }, or forgotten$.) is $a$ multiplied by itself $bn$ times. The RHS is $a^{bn}$. This is $a$ multiplied by itself $bn$ times. Thus, the LHS is always equal to the RHS, so $\fbox{D}$ is the only correct statement.	2020-12-04 08:48:11	{"extraction_info": {"found_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": 27, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8926928043365479, "perplexity": 252.55296369973547}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141735395.99/warc/CC-MAIN-20201204071014-20201204101014-00084.warc.gz"}
https://calculator.academy/x-ray-energy-calculator/	Enter the electron charge (C) and the accelerating voltage (volts) into the calculator to determine the X-Ray Energy. ## X-Ray Energy Formula The following equation is used to calculate the X-Ray Energy. Ex = ec * V Variables: • Where Ex is the X-Ray Energy (J) • ec is the electron charge (1.60217662 * 10^-19 C) • V is the accelerating voltage (volts) To calculate the x-ray energy, multiply the acceleration voltage by the electron charge. ## How to Calculate X-Ray Energy? The following two example problems outline the steps and information needed in order to calculate the X-Ray Energy. Example Problem #1: 1. First, determine the electron charge (C). In this example, the electron charge (C) is measured to be 13. 2. Next, determine the accelerating voltage (volts). For this problem, the accelerating voltage (volts) is calculated to be 5. 3. Finally, calculate the X-Ray Energy using the formula above: Ex = ec * V Inserting the values from above and solving the equation with the imputed values gives: Ex = 13 * 5 = 65 (J) Example Problem #2: Using the same process as example problem 1, we first define the needed variables outlined by the formula. In this case, the values are provide as: electron charge (C) = 74 accelerating voltage (volts) = 3 Entering these values into the formula or calculator above gives us: Ex = 74 * 3 = 222 (J)	2022-12-08 19:21:56	{"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": 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": 0.8790938854217529, "perplexity": 1814.85080347682}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1669446711360.27/warc/CC-MAIN-20221208183130-20221208213130-00143.warc.gz"}
https://workshops.nuevofoundation.org/python-pixel/activities/activity-7/	# Activity 7: Change the background of image Workshop Resources In this section, we will start to learn how to change the simple background of your image using pixels. ### Example - Change background color Let us change the background color of Nuvi to pink. from PIL import Image # Open the Nuvi image img = Image.open("nuevo.png") width = img.size[0] height = img.size[1] # Set up a new image with width and height newimg = Image.new('RGB',(width,height)) # Set the pixels for the new image colorpink = (244,114,208) coloryellow = img.getpixel((0,0)) for i in range(width): for j in range(height): color = img.getpixel((i,j)) # Get the color from original image if color == coloryellow: # If it is yellow, then: newimg.putpixel((i,j),colorpink) # Change the color to pink else: newimg.putpixel((i,j),color) # Else the color stays the same newimg.save("nuevopink.png") Wow! This is our new Nuvi after changing the background. ### Challenge - Change background based on your choice It’s now your turn to change whichever color you want on Nuvi!	2022-11-30 10:18:37	{"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": 0.3752068877220154, "perplexity": 6629.335577590764}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1669446710734.75/warc/CC-MAIN-20221130092453-20221130122453-00077.warc.gz"}
https://learn.careers360.com/ncert/question-what-do-you-understand-by-the-term-non-stoichiometric-hydrides-do-you-expect-this-type-of-the-hydrides-to-be-formed-by-alkali-metals-justify-your-answer/	Q # What do you understand by the term 'non-stoichiometric hydrides'? Do you expect this type of the hydrides to be formed by alkali metals? Justify your answer 9.11 What do you understand by the term “non-stoichiometric hydrides”? Do you expect this type of the hydrides to be formed by alkali metals? Justify your answer Views Non-stoichiometric hydrides are hydrogen deficient compounds. they are formed by d & f block elements. Such hydrides do not follow the law of constant composition. Examples- $LaH_{2.87},YbH_{2.55},ZrH_{1.3-1.75}$ etc. Alkali metals do not form these types of hydrides. Alkali metals form stoichiometric hydrides. These hydrides are ionic in nature. Exams Articles Questions	2020-02-22 20:41:04	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4636157751083374, "perplexity": 3501.2321274252977}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145713.39/warc/CC-MAIN-20200222180557-20200222210557-00138.warc.gz"}
https://rawgit.com/pbiecek/ProgramowanieWizualizacja2017/master/Projekt_2/IFSrainbow/docs/articles/IFSrainbow-vignette.html	This vignette describes in a non-technical way how to create, modify and plot fractals using IFSrainbow package. A nice interactive presentation of what this package is capable of can be found here. ## A bit of theory Iterated Functions System Fractal (or IFS fractal) is defined by a finite family of space transformations $$\mathcal{F}=\{f_1,f_2,\ldots,f_k\}$$ (typically contractions, i.e. “shrinking” functions). It is the invariant set of so called Hutchinson’s operator, which is the union of all possible finite compositions of functions in $$\mathcal{F}$$. Examples of fractals that can be obtained in such way include famous Sierpiński’s Trangle, Koch’s Snowflake or Heighway’s Dragon. See this Wikipedia page for more details. ## Fractal creation As the IFS fractal is defined by the set of functions, IFSpackage introduces an S3 class ifs which is no other than a list of functions. For example, in order to create Heighway’s Dragon defined by the following two functions f1 <- function(p){ return( c( p[1]/2 + p[2]/2 - 1, -p[1]/2 + p[2]/2 ) ) } f2 <- function(p){ return( c( -p[1]/2 + p[2]/2, -(p[1]/2 + p[2]/2 - 1) ) ) } one needs to use createIFS function. It accepts two “types” of arguments – firstly it can be just a sequence of functions or else a list of functions, so any of the two below would work my_heighway1 <- createIFS(f1, f2) my_heighway2 <- createIFS( list(f1, f2) ) Of course, this can be performed for any number of functions. ## Plotting Objects of class ifs can be plotted just by applying plot functions. Its default version use so called chaos game method of drawing the fractal, in which random compositions of given functions are applied to a random set of points to obtain an approximation of the “true” fractal. The functions assumes three mandatory arguments: • ifs, which is the fractal in question • n, which is the number of points in the plot • d, which is the length of random compositions applied There are also optional arguments: • probs, which is the sequences of probabilities of choosing respective functions Moreover, as we use the benefits of the ggplot2 package, some additional graphical parameters can be passed to the ggplot function inside. Therefore the following can be used to obtain a nice approximation of the Heighway Dragon plot( my_heighway1, 1000, 20 ) If we do not like colours (but why not?), prefer smaller points and wish that one of the functions is used more often than the other, we could run plot( my_heighway1, 1000, 20, probs=c(.7,.3), color="black", size=.8 ) ## Creating gifs Gif animations can be created by function createIFSanimation. This function takes 6 arguments. • startPoint, the point from which we start to create IFS • depth, the number of gifs frames • fractal, fractal, that will be drawn • resolution, because successive gifs frames are represented as matrices, we have to assume a resolution that means accuracy when coloring the corresponding matrix fields • filename, the name of the file being saved • scale, because the size of the gif depends on the fractal, depth and resolution, so the scale enlarges the figure proportionally Eight first frames of the Sierpinski triangle fractal <- sierpinski startPoint = c(0,0) depth = 6 resolution = 2 scale = 1 for(l in 1:depth){ IFS <- generateIFSgif(startPoint,l,fractal) f <- round(IFS,2) xmin <- min(f[,1]) xmax <- max(f[,1]) ymin <- min(f[,2]) ymax <- max(f[,2]) diamy <- (ymax-ymin)10^2+2 diamx <- (xmax-xmin)10^2+2 } DM <- array(0,c(diamxscale,diamyscale,depth)) for(l in 1:depth){ IFS <- generateIFSgif(startPoint,l,fractal) f <- round(IFS,resolution) f <- dplyr::distinct(f,x,y) for(i in 1:nrow(f)){ x<-(round((f[i,1]-min(f[,1]))10^resolution)+1)scale y<-(round((f[i,2]-min(f[,2]))10^resolution)+1)scale DM[x,y,l]<-1 } } image <- (DM*120)+50 for(i in 1:depth){ image(image[,,i], col=tim.colors(256)) }	2018-03-23 10:34:29	{"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": 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": 0.7292768955230713, "perplexity": 3441.8518847534247}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257648207.96/warc/CC-MAIN-20180323102828-20180323122828-00753.warc.gz"}
https://www.aimsciences.org/article/doi/10.3934/dcdsb.2016011	Article Contents Article Contents # Nonexistence and short time asymptotic behavior of source-type solution for porous medium equation with convection in one-dimension • In this paper we consider the following equation $$u_t=(u^m)_{xx}+(u^n)_x, \ \ (x, t)\in \mathbb{R}\times(0, \infty)$$ with a Dirac measure as initial data, i.e., $u(x, 0)=\delta(x)$. The solution of the Cauchy problem is well-known as source-type solution. In the recent work [11] the author studied the existence and uniqueness of such kind of singular solutions and proved that there exists a number $n_0=m+2$ such that there is a unique source-type solution to the equation when $0 \leq n < n_0$. Here our attention is focused on the nonexistence and asymptotic behavior near the origin for a short time. We prove that $n_0$ is also a critical number such that there exits no source-type solution when $n \geq n_0$ and describe the short time asymptotic behavior of the source-type solution to the equation when $0 \leq n < n_0$. Our result shows that in the case of existence and for a short time, the source-type solution of such equation behaves like the fundamental solution of the standard porous medium equation when $0 \leq n < m+1$, the unique self-similar source-type solution exists when $n = m+1$, and the solution does like the nonnegative fundamental entropy solution in the conservation law when $m+1 < n < n_0$, while in the case of nonexistence the singularity gradually disappears when $n \geq n_0$ that the mass cannot concentrate for a short time and no such a singular solutions exists. The results of previous work [11] and this paper give a perfect answer to such topical researches. Mathematics Subject Classification: Primary: 35K65, 35K67; Secondary: 35M10, 35Q84. Citation: • [1] G. I. Barenblatt, On some unsteady motions of a liquid and gas in a porous medium, Prikladnaja Mathematika Mechanika, 16 (1952), 67-78. [2] S. Kamm, Source-type solution for equation of nonstationary filtration, J. Math. Anal. Appl., 64 (1978), 263-276.doi: 10.1016/0022-247X(78)90036-7. [3] H. Brezis and A. Friedman, Nonlinear parabolic equations involving measure as initial conditions, J. Math. Pure. Appl., 62 (1983), 73-97. [4] S. Kamin and L. A. Peletier, Source-type solution of generate diffusive equation with absorption, Israel. J. Math., 50 (1985), 219-230.doi: 10.1007/BF02761403. [5] J. Zhao, Source-type solutions of degenrate quasilinear parabolic equations, J. of Dff. Eq., 92 (1991), 179-198.doi: 10.1016/0022-0396(91)90046-C. [6] T.-P. Liu and M. Pierre, Source-solution and asymptotic behavior in conservation laws, J. of Diff. Eq., 51 (1984), 419-441.doi: 10.1016/0022-0396(84)90096-2. [7] M. Escobedo, J. L. Vazquez and E. Zuazua, Asymptotic behavior and source-type solutions for a diffusion-convection equation, Arch. Rational Mech. Anal., 124 (1993), 43-65.doi: 10.1007/BF00392203. [8] G. Lu, Source-Type Solutions of Diffusion Equations with Nonlinear Convection, China J. of Contemporary Math., 28 (2000), 185-188. [9] G. Lu, Explicit and similarity solutions for certain nonlinear parabolic diffusion-convection equations, J. Sys. Sci and Math. Scis., 22 (2002), 210-222. [10] G. Lu and H. Yin, Source-type solutions of heat equation with convection in several variables spaces, Science in China, Series A, 54 (2011), 1145-1173.doi: 10.1007/s11425-011-4219-4. [11] G. Lu, Source-type solutions of nonlinear fokker-planck equation of one-dimension, Science China Mathemathics, 56 (2013), 1845-1868.doi: 10.1007/s11425-013-4612-2. [12] J. L. Vazquez, Perspectives in nonlinear diffusion: Between analysis, physics and geometry, International Congress of Mathematicians, Eur. Math. Soc., 1 (2007), 609-634.doi: 10.4171/022-1/23. [13] Y. Chen, Hölder estimates for solutions of uniformly degenerate parabolic equations, Chin Ann. of Math., 5B (1984), 661-678. [14] G. Lu, A remark on $C^k$-regularity of free boundary for porous medium equation with gravity term in one-dimension, Appl. Math. A Journal of Chinese University, 7 (1992), 579-593. (In Chinese) [15] O. A. Ladyzhenskaja, N. A. Solonnikov and N. N. Uralezeva, Linear and Quasilinear Equations of Parabolic Type, Trans. Math. Mono., 23, AMS Providence, R. I., 1968. [16] S. N. Kruzkov, First order quasilinear equations in several independent variables, Math. USSR. Sb., 81 (1970), 228-255. [17] V. S. Varadarajan, Measure on topological spaces, Amer. Math. Soci. Trans., Series. 2 (1965), p48. [18] R. J. LeVeque, Finite Volue Methods for Hyperbolic Problems, Cambridge University Press, 2002.doi: 10.1017/CBO9780511791253. [19] P. J. Vila, An analysis of a class of second-order accurate godunov-type schemes, SIAM Journal on Numerical Analysis, 26 (1989), 830-853.doi: 10.1137/0726046. [20] T. Ding and C. Li, Ordinary differential equations, China Hihgher Education Press, Beijing, 1991. (In Chinease).	2023-03-29 04:19:48	{"extraction_info": {"found_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": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.7021884918212891, "perplexity": 1180.5520557096763}, "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/1679296948932.75/warc/CC-MAIN-20230329023546-20230329053546-00676.warc.gz"}
http://www-alg.ist.hokudai.ac.jp/~thomas/COLTBIB/1999/TCSABS/ajsabstr.html	### Ordinal mind change complexity of language identification Authors: Andris Ambainis, Sanjay Jain and Arun Sharma Source: Theoretical Computer Science Vol. 220, No. 2, 1999, 323 - 343. Abstract. The approach of ordinal mind change complexity, introduced by Freivalds and Smith, uses (notations for) constructive ordinals to bound the number of mind changes made by a learning machine. This approach provides a measure of the extent to which a learning machine has to keep revising its estimate of the number of mind changes it will make before converging to a correct hypothesis for languages in the class being learned. Recently, this notion, which also yields a measure for the difficulty of learning a class of languages, has been used to analyze the learnability of rich concept classes. The present paper further investigates the utility of ordinal mind change complexity. It is shown that for identification from both positive and negative data and n >= 1, the ordinal mind change complexity of the class of languages formed by unions of up to n+1 pattern languages is only \omega xO notn(n) (where notn(n) is a notation for n, \omega is a notation for the least limit ordinal and xO represents ordinal multiplication). This result nicely extends an observation of Lange and Zeugmann that pattern languages can be identified from both positive and negative data with 0 mind changes. Existence of an ordinal mind change bound for a class of learnable languages can be seen as an indication of its learning "tractability". Conditions are investigated under which a class has an ordinal mind change bound for identification from positive data. It is shown that an indexed family of languages has an ordinal mind change bound if it has finite elasticity and can be identified by a conservative machine. It is also shown that the requirement of conservative identification can be sacrificed for the purely topological requirement of M-finite thickness. Interaction between identification by monotonic strategies and existence of ordinal mind change bound is also investigated.	2018-12-18 18:03:59	{"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": 0.8476471900939941, "perplexity": 917.253780016711}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1544376829542.89/warc/CC-MAIN-20181218164121-20181218190121-00279.warc.gz"}
https://mathtools.lagrida.com/arithmetic/bases_converter.html	# Online Bases converter Convert a number between bases. Base from : Base to : 0 ## What a base (Radix) ? The radix or base is the number of unique digits, used to represent numbers. If the base is larger than $10$ we use caracters 'A', 'B', 'C', ... as the digits 10, 11, 12 ... To represent a number $n$ in a base $b$ we use the notation : $$n = \overline{d_{k} d_{k-1} \cdots d_{1}}^{(b)}$$ Examples: The binary system : Used internally by nearly all computers, is base 2. The two digits are "0" and "1". Example : $\overline{31}^{(10)} = \overline{11111}^{(2)}$ The Octal system : The eight digits are "0"–"7". Example : $\overline{31}^{(10)} = \overline{37}^{(8)}$ The Decimal system : The most used system of numbers in the world, is used in arithmetic. Its ten digits are "0"–"9". The Duodecimal (dozenal) system : Sometimes advocated due to divisibility by 2, 3, 4, and 6. It was traditionally used as part of quantities expressed in dozens and grosses. Its 12 digits are "0"–"9" and "A"-"B" or "a"-"b". Example : $\overline{31}^{(10)} = \overline{27}^{(12)}$ Example : $\overline{31}^{(10)} = \overline{1F}^{(16)}$ Example : $\overline{31}^{(10)} = \overline{1B}^{(20)}$	2021-04-18 22:45:51	{"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": 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": 0.5515778660774231, "perplexity": 2405.0995040661746}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1618038862159.64/warc/CC-MAIN-20210418224306-20210419014306-00570.warc.gz"}
http://www.solutioninn.com/when-returns-from-a-project-can-be-assumed-to-be	# Question When returns from a project can be assumed to be normally distributed, such as those shown in Figure (represented by a symmetrical, bell-shaped curve), the areas under the curve can be determined from statistical tables based on standard deviations. For example, 68.26 percent of the distribution will fall within one standard deviation of the expected value (± 1σ). Similarly, 95.44 percent will fall within two standard deviations ( ± 2σ), and so on. An abbreviated table of areas under the normal curve is shown next. Assume Project A has an expected value of \$24,000 and a standard deviation (σ) of \$4,800. a. What is the probability that the outcome will be between \$16,800 and \$31,200? b. What is the probability that the outcome will be between \$14,400 and \$33,600? c. What is the probability that the outcome will be at least \$14,400? d. What is the probability that the outcome will be less than \$31,900? e. What is the probability that the outcome will be less than \$19,200 or greater than\$26,400? Sales4 Views134	2016-10-23 06:43:45	{"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": 0.8599952459335327, "perplexity": 259.78299138138004}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1476988719155.26/warc/CC-MAIN-20161020183839-00250-ip-10-171-6-4.ec2.internal.warc.gz"}
https://discuss.codechef.com/questions/13991/lapin-editorial	× Cakewalk # Problem: Given a string S, if we split it in the middle (if S has an odd number of characters, disregard the middle character), then if the frequency of each character is the same in both halves, S is called a "lapindrome". Given the string S, test if it is a Lapindrome or not. # Explanation: Maintain frequencies for the left half and the right half, for each character. After computing the frequency of each half, then check if the frequencies of all characters match. If so, output "YES", else output "NO". Consider the following pseudocode: bool isLapin(S) initialize cntL[] and cntR[] with 0 L = S.length() for(i = 0; i < L/2; i++) cntL[S[i]-'a']++ for(i = (L+1)/2; i < L; i++) cntR[S[i]-'a']++ bool ret = true for(c = 0; c < 26; c++) if(cntL[c] != cntR[c]) ret = false return ret The time complexity for this is O(\|S\| + 26) per test-case. # Setter's Solution: Can be found here # Tester's Solution: Can be found here This question is marked "community wiki". 973568379 accept rate: 14% 18.3k347492525 6 I used a different approach, I divided the string into two equal substrings by splitting it from the middle and then sorted both the substrings, since for true case ("YES") the frequency of the characters in both the original substrings is same, so after sorting the substrings we should get identical substrings. So after sorting we just check if the sorted substrings are same of not. Here's my solution :- http://www.codechef.com/viewsolution/2196881 answered 18 Jun '13, 18:32 4★v_akshay 1.2k●9●16●25 accept rate: 13% 4 want a cookie :P...why complicate a cake-walk ! (18 Jun '13, 19:02) @v_akshay , that will increase the complexity from O(n) to O(nlogn) (18 Jun '13, 19:25) 3 Oh @v_akshay, nice! So, basically I could rewrite isLapin as follows: isLapin(string S) { int len = S.length(); int r1 = len/2, l2 = (len+1)/2; sort(S.begin(), S.begin() + r1); sort(S.begin() + l2, S.end()); return S.substr(0, r1) == S.substr(l2); } @spandanpathak, surely this can't be called "complicating" :P (18 Jun '13, 20:03) @sumanth232 , well, I know that but we have \|S\|<=1000, so O(nlogn) is well under time constraints :) . @pragrame , thanks :) , and perfect implementation :) (18 Jun '13, 20:32) v_akshay4★ 1 Links broken? answered 18 Jun '13, 17:32 131●1●6 accept rate: 14% 0 can somebody trace bugs in my solution pls. :- http://www.codechef.com/viewsolution/2205187 the program satisfies all test cases.. answered 18 Jun '13, 22:45 16●1 accept rate: 0% 3 Here is your AC code with array size increased. http://www.codechef.com/viewsolution/2279176 (18 Jun '13, 22:48) 0 http://www.codechef.com/viewsolution/3258850 please tell the problem with my solution... the given testcases for the question are giving the correct output... and i have also tried some more test cases of my own... but i am getting a wrong answer answered 15 Jan '14, 23:51 105●4●8●11 accept rate: 0% try this: 1 abac (16 Jan '14, 00:23) vytenis6★ @insaynasasin imho, you tried to convert chars in your string array to ints. try left[arr[i]-'a'], right[arr[i]-'a'] (16 Jan '14, 00:31) garakchy1★ @garakchy but what is the difference between the two formats? (16 Jan '14, 00:36) @garakchy no luck, it is still giving the right answers for the given testcases but on submission, its a wa (16 Jan '14, 00:38) @vytenis my code is saying yes for the given string and that is worng i get the point (16 Jan '14, 00:40) was trying since yesterday, lol, and i just got ac on the problem. (16 Jan '14, 00:55) garakchy1★ and i used memcmp, that is a bit faster than every array index comparison imho, try that also @insaynasasin (16 Jan '14, 00:56) garakchy1★ showing 5 of 7 show all 0 Getting WA.Help please http://www.codechef.com/viewsolution/5213313 answered 25 Oct '14, 02:33 3★biprotip 71●1●4 accept rate: 0% 0 @biprotip: Hey try to increase the size of the array. It is always better to increase the array size than the required. here 1 will be needed for the '\0'. And try with instead of using 2 variables use 2 array of size 26 and increase each index with left[ch[i]-'a']++ and same with the right array. Finally check if both the arrays are equal. If you have any doubt just look at this solution answered 25 Oct '14, 20:45 3★bipin2 3.1k●25●46●70 accept rate: 8% 0 answered 14 Mar '16, 18:36 1★amrit008 11 accept rate: 0% 0 whats wrong with this code? answered 14 Mar '16, 18:36 1★amrit008 11 accept rate: 0% 0 can someone please tell why is it giving wa? answered 02 Jun '16, 03:42 0★dove89 1 accept rate: 0% # include<vector> using namespace std; main(){ int t; cin>>t; for(int i=0;i<t;i++){ string="" s;="" cin="">>s; int p=s.size()/2; char A[p],B[p]; for(int j=0;j<p;j++){a[j]=s[j];} for(int="" j="s.size()-1,k=0;j">=p;j--,k++){B[k]=s[j];} sort(A,A+p); sort(B,B+p); vector<char> v1(p); vector<char> v2(p); vector<char> ::iterator it1; vector<char> ::iterator it2; it1=unique_copy(A,A+p,v1.begin()); it2=unique_copy(B,B+p,v2.begin()); v1.resize(it1-v1.begin()); v2.resize(it2-v2.begin()); //int p1=v1.size(); //sort(v1,v1+p1) if(v1.size()!=v2.size()){cout<<"NO"<<endl;} else{ int k1=0; for(int j=0;j<v1.size();j++){ if(v1[j]!=v2[j]){k1++;} } int a[v1.size()],b[v2.size()]; for(int j=0;j<v1.size();j++){ int k2=0; for(int k=0;k<p;k++){ if(v1[j]==A[k]){k2++;} } a[j]=k2; } for(int j=0;j<v2.size();j++){ int k2=0; for(int k=0;k<p;k++){ if(v2[j]==B[k]){k2++;} } b[j]=k2; } int k3=0; for(int j=0;j<v1.size();j++){ if(a[j]!=b[j]){k3++;} } if(k3>0 \|\| k1>0){cout<<"NO"<<endl;} else{cout<<"YES"<<endl;} } } } 2★jaggu8 1 accept rate: 0% why is this wrong (28 Jan '17, 11:39) 2★ 0 Its a bit difficult to have a good look over your code atm. I strongly advise that you either givea submission link to your code, or edit the post (select your code, and THEN press button "insert code") But by what I could get, I will say just make 2 arrays - int a[26], b[26] and increase the frequency of letters you encounter in the half of string. Eg- for (i= 0; i 0 i am unable to find out error please help me include include int main() { int t,c,start,end,flag,i; char s[1000]; scanf("%d",&t); while(t--) { flag=0; scanf("%s",s); c=strlen(s); if(c%2==0) start=c/2; else start=c/2+1; end=c/2-1; for(i=0;i<=end;i++) { if(s[i]==s[start++]) continue; else{ flag=1; break; } } if(flag==0) printf("YES\n"); else printf("NO\n"); } return 0; } answered 09 Feb '17, 09:37 1 accept rate: 0% 0 plz help me out...! Link answered 06 Apr '17, 00:43 15●3 accept rate: 0% 0 Hey there, Can somebody solve my problem regarding LAPIN. According to the question, zyzxyz is not lapindrome but my code, which has been accepted by the compiler gives YES to it. I am confused.... answered 10 Jun '17, 09:14 11 accept rate: 0% 0 import java.util.HashMap; import java.util.Scanner; class LAPIN { public static void main(String[] args) throws Exception { Scanner scanner = new Scanner(System.in); int t = scanner.nextInt(); for (int i = 0; i < t; i++) { String s = scanner.next(); int half = s.length() / 2; String s1 = s.substring(0, half); String s2 = s.substring(half, s.length()); HashMap map1 = new HashMap(); for (int j = 0; j < s1.length(); j++) { if (!map1.containsKey(s1.charAt(j))) { map1.put(s1.charAt(j), 1); } else { map1.put(s1.charAt(j), map1.get(s1.charAt(j) + 1)); } } HashMap map2 = new HashMap(); int k = s2.length() % 2 != 0 ? 1 : 0; for (; k < s2.length(); k++) { if (!map2.containsKey(s2.charAt(k))) { map2.put(s2.charAt(k), 1); } else { map2.put(s2.charAt(k), map2.get(s2.charAt(k) + 1)); } } if (map1.equals(map2)) { System.out.println("YES"); } else { System.out.println("NO"); } } } } It passed all the test cases in my local but when I submit my code it showing the wrong answer. Can somebody help me, please answered 25 Oct '17, 13:15 1 accept rate: 0% 0 i got WA becoz i was priniting "Yes" instead of "YES" . lol answered 18 Nov '17, 17:00 2★batman69 0 accept rate: 0% 0 Hello, pls anyone give me the test cases for which the following code doesn't work : https://www.codechef.com/viewsolution/16573029 ?? (LAPIN) answered 14 Dec '17, 15:08 0★adarshbl 1 accept rate: 0% 0 Plz help me to solve this problem with given my code. Click here to see my code. Thank you!!! answered 10 Jan, 16:48 2★mr_shah 1 accept rate: 0% 0 This satisfies all test cases but I get wrong answer. Link to solution: https://www.codechef.com/viewsolution/17144359 answered 24 Jan, 15:51 1★m7d5_3 1 accept rate: 0% toggle preview community wiki: Preview By Email: Markdown Basics • italic or _italic_ • bold or __bold__ • image?![alt text](/path/img.jpg "title") • numbered list: 1. Foo 2. Bar • to add a line break simply add two spaces to where you would like the new line to be. • basic HTML tags are also supported • mathemetical formulas in Latex between \$ symbol Question tags: ×13,419 ×1,278 ×818 ×27 ×7 question asked: 18 Jun '13, 16:17 question was seen: 5,902 times last updated: 24 Jan, 15:51	2018-02-25 05:55:38	{"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": 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": 0.5917569994926453, "perplexity": 10203.589065343453}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891816138.91/warc/CC-MAIN-20180225051024-20180225071024-00493.warc.gz"}
https://www.tutorialspoint.com/amplifiers/class_a_power_amplifiers.htm	# Class A Power Amplifiers We have already come across the details of transistor biasing, which is very important for the operation of a transistor as an amplifier. Hence to achieve faithful amplification, the biasing of the transistor has to be done such that the amplifier operates over the linear region. A Class A power amplifier is one in which the output current flows for the entire cycle of the AC input supply. Hence the complete signal present at the input is amplified at the output. The following figure shows the circuit diagram for Class A Power amplifier. From the above figure, it can be observed that the transformer is present at the collector as a load. The use of transformer permits the impedance matching, resulting in the transference of maximum power to the load e.g. loud speaker. The operating point of this amplifier is present in the linear region. It is so selected that the current flows for the entire ac input cycle. The below figure explains the selection of operating point. The output characteristics with operating point Q is shown in the figure above. Here (Ic)Q and (Vce)Q represent no signal collector current and voltage between collector and emitter respectively. When signal is applied, the Q-point shifts to Q1 and Q2. The output current increases to (Ic)max and decreases to (Ic)min. Similarly, the collector-emitter voltage increases to (Vce)max and decreases to (Vce)min. D.C. Power drawn from collector battery Vcc is given by $$P_{in} = voltage \times current = V_{CC}(I_C)_Q$$ This power is used in the following two parts − • Power dissipated in the collector load as heat is given by $$P_{RC} = (current)^2 \times resistance = (I_C)^2_Q R_C$$ • Power given to transistor is given by $$P_{tr} = P_{in} - P_{RC} = V_{CC} - (I_C)^2_Q R_C$$ When signal is applied, the power given to transistor is used in the following two parts − • A.C. Power developed across load resistors RC which constitutes the a.c. power output. $$(P_O)_{ac} = I^2 R_C = \frac{V^2}{R_C} = \left ( \frac{V_m}{\sqrt{2}}\right )^2 \frac{1}{R_C} = \frac{V_m^2}{2R_C}$$ Where I is the R.M.S. value of a.c. output current through load, V is the R.M.S. value of a.c. voltage, and Vm is the maximum value of V. • The D.C. power dissipated by the transistor (collector region) in the form of heat, i.e., (PC)dc We have represented the whole power flow in the following diagram. This class A power amplifier can amplify small signals with least distortion and the output will be an exact replica of the input with increased strength. Let us now try to draw some expressions to represent efficiencies. ### Overall Efficiency The overall efficiency of the amplifier circuit is given by $$(\eta)_{overall} = \frac{a.c \: power \:delivered\: to \: the\: load}{total \: power\: delivered \: by \: d.c\: supply}$$ $$= \frac{(P_O)_{ac}}{(P_{in})_{dc}}$$ ### Collector Efficiency The collector efficiency of the transistor is defined as $$(\eta)_{collector} = \frac{average\: a.c \: power \:output}{average \:d.c\: power\: input\: to\: transistor}$$ $$= \frac{(P_O)_{ac}}{(P_{tr})_{dc}}$$ ### Expression for overall efficiency $$(P_O)_{ac} = V_{rms} \times I_{rms}$$ $$= \frac{1}{\sqrt{2}} \left [ \frac{(V_{ce})_{max} - (V_{ce})_{min}}{2} \right ] \times \frac{1}{\sqrt{2}} \left [ \frac{(I_C)_{max} - (I_C)_{min}}{2}\right ]$$ $$= \frac{[(V_{ce})_{max} - (V_{ce})_{min}] \times [(I_C)_{max} - (I_C)_{min}]}{8}$$ Therefore $$(\eta)_{overall} = \frac{[(V_{ce})_{max} - (V_{ce})_{min}] \times [(I_C)_{max} - (I_C)_{min}]}{8 \times V_{CC} (I_C)_Q}$$ ### Advantages of Class A Amplifiers The advantages of Class A power amplifier are as follows − • The current flows for complete input cycle • It can amplify small signals • The output is same as input • No distortion is present ### Disadvantages of Class A Amplifiers The advantages of Class A power amplifier are as follows − • Low power output • Low collector efficiency	2023-02-04 22:04:07	{"extraction_info": {"found_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": 0.6896046996116638, "perplexity": 1525.2072835144106}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1674764500154.33/warc/CC-MAIN-20230204205328-20230204235328-00471.warc.gz"}
https://answers.opencv.org/question/213427/paper-edge-detection-and-perspective-transform/	# paper edge detection and perspective transform before image https://imgur.com/f190UFk processed image https://imgur.com/JkEhWkS you can see the "processed image" has highlight, so the transform works bad. any possible to make an rectangle that ignore that highlight area? import os import cv2 import numpy as np from nanoid import generate def processImage(imagepath, ext): hsv = cv2.cvtColor(img, cv2.COLOR_BGR2HSV) h, s, v = cv2.split(hsv) # _, threshed = cv2.threshold(s, 50, 255, cv2.THRESH_BINARY_INV) _, threshed = cv2.threshold(s, 50, 255, cv2.THRESH_BINARY_INV) # cnts = cv2.findContours(threshed, cv2.RETR_EXTERNAL, cv2.CHAIN_APPROX_SIMPLE)[-2] cnts = cv2.findContours(threshed, cv2.RETR_EXTERNAL, cv2.CHAIN_APPROX_SIMPLE)[-2] canvas = img.copy() #cv2.drawContours(canvas, cnts, -1, (0, 255, 0), 1) cnts = sorted(cnts, key = cv2.contourArea) cnt = cnts[-1] print(cnt) arclen = cv2.arcLength(cnt, True) approx = cv2.approxPolyDP(cnt, 0.005 * arclen, True) cv2.drawContours(canvas, [cnt], -1, (255, 0, 0), 5, cv2.LINE_AA) cv2.drawContours(canvas, [approx], -1, (0, 0, 255), 1, cv2.LINE_AA) print(approx) approx = rectify(approx) pts2 = np.float32([[0, 0], [2480, 0], [2480, 3508], [0, 3508]]) M = cv2.getPerspectiveTransform(approx, pts2) dst = cv2.warpPerspective(canvas, M, (2480, 3508)) filename_output = generate() + ext cv2.imwrite('./static/' + filename_output, dst) topLeft, topRight, bottomRight, bottomLeft = approx topLeft = topLeft.tolist() topRight = topRight.tolist() bottomRight = bottomRight.tolist() bottomLeft = bottomLeft.tolist() return { 'filename': './static/' + filename_output, 'shape': img.shape, 'approx': { 'topLeft': topLeft, 'topRight': topLeft, 'bottomRight': bottomRight, 'bottomLeft': bottomLeft, }, } def rectify(h): h = h.reshape((13, 2)) hnew = np.zeros((4, 2), dtype = np.float32) diff = np.diff(h, axis = 1) hnew[1] = h[np.argmin(diff)] hnew[3] = h[np.argmax(diff)] return hnew https://imgur.com/wDXtLsd https://imgur.com/KAvOtdG edit retag close merge delete 1 can you please put your images HERE, not on bloody imgur (where it's not visible in several countries, due to shit content on that site) ? thank you.. ( 2019-05-24 06:00:29 -0500 )edit Often machine vision challenges are not so much software as what I call staging. By staging I mean foreground, background, lighting, camera/subject position and orientation. In this case there is a glare that is saturating pixels and one page that is sticking out underneath to cause confusion. Put a single page on a black background with diffuse lighting and this problem gets much easier. The AIA basic certification has an entire section on lighting. https://www.visiononline.org/vision/c... ( 2019-05-25 06:58:41 -0500 )edit 1 Use adaptiveThreshold instead of the normal one. Once you find the right parameters of the adaptiveThreshold, your code should be more or less immune to highlights. Of course, it will not work in some extreme cases, but it should be ok for the case shown. ( 2019-05-25 19:38:44 -0500 )edit For this problem I would use a Hough line detector. Knowing that the paper is rectangular and the edges are mostly parallel and close to horizontal/vertical, you can filter the results to obtain the edge lines. The corners of the paper are the intersection of these lines. ( 2019-05-27 04:52:45 -0500 )edit	2019-06-24 12:12:58	{"extraction_info": {"found_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": 0.2010726034641266, "perplexity": 14402.237283431152}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1560627999482.38/warc/CC-MAIN-20190624104413-20190624130413-00029.warc.gz"}
https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/171/1/	# Properties Label 171.1 Level 171 Weight 1 Dimension 7 Nonzero newspaces 3 Newform subspaces 3 Sturm bound 2160 Trace bound 1 ## Defining parameters Level: $$N$$ = $$171 = 3^{2} \cdot 19$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$3$$ Newform subspaces: $$3$$ Sturm bound: $$2160$$ Trace bound: $$1$$ ## Dimensions The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(171))$$. Total New Old Modular forms 155 84 71 Cusp forms 11 7 4 Eisenstein series 144 77 67 The following table gives the dimensions of subspaces with specified projective image type. $$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$ Dimension 3 4 0 0 ## Trace form $$7q - 2q^{5} - 2q^{6} - 2q^{7} - 4q^{9} + O(q^{10})$$ $$7q - 2q^{5} - 2q^{6} - 2q^{7} - 4q^{9} + 2q^{11} - 3q^{13} + 2q^{16} - 3q^{19} + 2q^{23} + 4q^{24} - 4q^{26} - 3q^{28} + 4q^{30} + 4q^{35} - 2q^{38} - 2q^{39} - 2q^{42} + q^{43} + 2q^{45} - 2q^{47} + 3q^{49} + 3q^{52} + 2q^{54} - 4q^{55} - 4q^{57} - 2q^{58} - q^{61} + 4q^{62} + 2q^{63} - q^{64} + 2q^{66} + 3q^{67} + 3q^{73} + 2q^{77} - 3q^{79} - 4q^{80} + 4q^{81} - 4q^{82} + 2q^{83} + 2q^{87} - 3q^{91} + 2q^{93} - 2q^{99} + O(q^{100})$$ ## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(171))$$ We only show spaces with odd parity, since no modular forms exist when this condition is not satisfied. Within each space $$S_k^{\mathrm{new}}(N, \chi)$$ we list the newforms together with their dimension. Label $$\chi$$ Newforms Dimension $$\chi$$ degree 171.1.b $$\chi_{171}(134, \cdot)$$ None 0 1 171.1.c $$\chi_{171}(37, \cdot)$$ 171.1.c.a 1 1 171.1.i $$\chi_{171}(88, \cdot)$$ None 0 2 171.1.j $$\chi_{171}(68, \cdot)$$ None 0 2 171.1.n $$\chi_{171}(11, \cdot)$$ None 0 2 171.1.o $$\chi_{171}(94, \cdot)$$ 171.1.o.a 4 2 171.1.p $$\chi_{171}(46, \cdot)$$ 171.1.p.a 2 2 171.1.q $$\chi_{171}(20, \cdot)$$ None 0 2 171.1.r $$\chi_{171}(26, \cdot)$$ None 0 2 171.1.s $$\chi_{171}(31, \cdot)$$ None 0 2 171.1.z $$\chi_{171}(5, \cdot)$$ None 0 6 171.1.ba $$\chi_{171}(10, \cdot)$$ None 0 6 171.1.bb $$\chi_{171}(17, \cdot)$$ None 0 6 171.1.bc $$\chi_{171}(22, \cdot)$$ None 0 6 171.1.be $$\chi_{171}(13, \cdot)$$ None 0 6 171.1.bf $$\chi_{171}(23, \cdot)$$ None 0 6 ## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(171))$$ into lower level spaces $$S_{1}^{\mathrm{old}}(\Gamma_1(171)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(57))$$$$^{\oplus 2}$$	2021-11-28 17:05:19	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9444422721862793, "perplexity": 7962.702561907459}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358570.48/warc/CC-MAIN-20211128164634-20211128194634-00215.warc.gz"}
https://www.garrickadenbuie.com/talk/drake-intro-biodataclub/	drake is an R package that provides a powerful, flexible workflow management tool for reproducible data analysis pipelines. drake alleviates the pain of managing large (and even small) data analyses, speeding up iteration and development while providing reproducibility guarantees that are essential for modern research. In this session, we’ll learn how to use drake to manage a data analysis workflow by writing functions that define the steps of the analysis. We’ll then learn how drake can keep track of all of these steps, from start to finish, and intelligently update only the outdated steps when your data or code change. ### Meeting prerequisites # We’ll work through a few examples together, so please bring a laptop with the drake and visNetwork packages installed. (If you don’t have a laptop you can share with someone who does at the session.) You would also benefit from installing the tidyverse package for the session. See the full requirements here. required_packages <- c( # "tidyverse", #<< For data processing, etc. (you probably have this) "here", #<< For sane path management "cowplot", #<< For composing ggplot2 plots "visNetwork", #<< For visualizing drake plans "drake" #<< Because drake ) install.packages(required_packages) Note: if you’ve used drake before, please ensure that you have version 7.0.0 or later installed. ### Meeting materials # The slides from this talk are available online at https://pkg.garrickadenbuie.com/drake-intro/ and the drake source code and RStudio project are in available on GitHub at https://github.com/gadenbuie/drake-intro. There is also an RStudio Cloud project containing the drake project with all of the required dependencies pre-installed that you can use to explore and run the code from the talk. Posted on: July 19, 2019 Length: 2 minute read, 269 words Categories: Education Tags: R drake Reproducible Research Workflow Tutorials Education	2022-01-18 22:01:31	{"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": 0.1717468500137329, "perplexity": 4741.195803740887}, "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/1642320301063.81/warc/CC-MAIN-20220118213028-20220119003028-00170.warc.gz"}
http://www.chegg.com/homework-help/questions-and-answers/network-shown-resistive-parts-theelements-negligible-positive-mutual-coupling-lines-aand-b-q776638	For the network shown below, the resistive parts of theelements are negligible. Positive mutual coupling between lines Aand B is for the current flow directions shown on the network. Thetransmission lin data (50 MVA base) are: Line C Line A LineB Positive or negative sequencereactance 0.50 0.30 0.40 Zero sequencereactance 1.50 1.0 1.20 1/2 shunt susceptance, 0.2 --- --- Positive or negative sequence mutualcoupling --- 0.25to B 0.25 toA The generator data (50 MVA base) are: Generator 1: Generator 2: Using ground as a reference do the following: (a) Draw the positive and zero sequenceimpedance circuit diagrams, labeling all reactances. (b) Determine for the positive sequence circuit. (c) Use the building algorithm to calculate , the positive sequencebus impedance matrix.	2014-10-21 09:28:45	{"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": 0.8143253922462463, "perplexity": 3350.3876305075382}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1413507444312.8/warc/CC-MAIN-20141017005724-00190-ip-10-16-133-185.ec2.internal.warc.gz"}
https://mathhelpboards.com/threads/expected-value-problem.5061/	# Expected value problem #### eatinbyzombies3 ##### New member Here is the question: You are playing a gambling game (silly, yea I know). The first part of the game is to throw a die. If it comes up a 3, you move on. Otherwise, you lose. The second part of the game entails pulling a card out of a standard deck. If it is a heart, you win \$100. Otherwise, you lose. What is the expected value of the game? Here is what I know: you have a 1 out of 6 chance of rolling a 3 and a 13 out of 52 (i hope) chance of pulling a heart. I dont know how to get the answer that is needed. Can anyone help me? #### MarkFL Staff member Re: help, I am confused Are you supposed to find the expected profit? If so, what is the buy-in to the game? edit: I have edited the topic title to reflect the nature of the problem. #### eatinbyzombies3 ##### New member MarkFL, I don't know. that is the question word for word that I have on my paper from my Professor. That is what is confusing, it just says "What is the expected value of the game?" #### MarkFL Staff member Are you given an answer that you are expected to be able to compute? #### Jameson $$\displaystyle E[X]=\sum_{x} x \cdot P[X=x]$$ $$\displaystyle E[X]=100 \cdot P[X= \text{3 and a heart}]$$	2020-09-19 10:22:53	{"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": 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": 0.4461232125759125, "perplexity": 372.7663162647468}, "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-2020-40/segments/1600400191160.14/warc/CC-MAIN-20200919075646-20200919105646-00437.warc.gz"}
https://web2.0calc.com/questions/help-due-on-in-an-hour	+0 # help due on in an hour 0 187 1 +8 Find all real constants such that the system \begin{align} x + 3y &= kx, \\ 3x + y &= ky. \end{align} has a solution other than $$(x,y) = (0,0).$$ Jun 18, 2020 #1 +311 0 I'm assuming that you mean "all real constants k" First add the two equations together. $$4(x+y)=k(x+y)$$. k = 4 Then subtract the two. $$-2(x-y)=k(x-y)$$. k = -2. So our two solutions are k = 4, -2 Jun 18, 2020	2021-05-07 16:58:31	{"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": 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": 1.0000087022781372, "perplexity": 3802.267589660624}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988796.88/warc/CC-MAIN-20210507150814-20210507180814-00190.warc.gz"}
https://www.ques10.com/p/48069/network-structure/	0 213views Network - Structure 0 1views The network structure in LTE is quite simple in principle (and actually, simplified with respect to the GSM and WCDMA structure): there is only a single type of access point, namely, the eNodeB (or BS, in our notation).1 Each BS can supply one or more cells, providing the following functionalities: • Air interface communications and PHYsical layer (PHY) functions; • Re-transmission control. The X2 interface is the interface between different BSs. Information that is important for the coordination of transmissions in adjacent cells (e.g., for intercell interference reduction) can be exchanged on this interface. Each BS is connected by the S1 interface to the core network. For LTE, a new core network, called System Architecture Evolution (SAE) or Enhanced Packet Core (EPC) was developed. It is based on packet-switched transmission. It consists of $\quad$ (i) a Mobility Management Entity (MME) $\quad$(ii) the serving gateway (connecting the network to the RAN) $\quad$(iii) the packet data network gateway, which connects the network to the Internet. In addition, the Home Subscriber Server is defined as a separate entity. The structure is sketched in Figure.The core network fulfils, inter alia, the following functions: - Mobility management - Subscriber management and charging; - Quality of service provisioning, and policy control of user data flows - Connection to external networks.	2021-11-27 14:44:52	{"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": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3244234025478363, "perplexity": 3166.891363576848}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1637964358189.36/warc/CC-MAIN-20211127133237-20211127163237-00638.warc.gz"}
https://softwaremechanic.wordpress.com/category/smart-alec/	# Squeeze theorem To quote from “The girl next door” The first lesson of politics is “Always know whether the squeeze is worth the juice”. Now i was trying to finally make a genuine effort at understanding Central Limit theorem. Throughout my life(30 years), i have always been suspicious whenever statistics goes beyond the mean, median, mode, SD and Variance. (i.e to say, whenever any stat goes above first and second moments). Part of it because i never really learnt or rather never paid enough attention to convince myself of the theorems involved in reasoning with distributions. Anyways, i figured Central limit theorem would be a good place to start and in learning by teaching am summarizing what i’ve learnt so far. It started off as i came across this post on HN and going through comments and critique realized the demo is more of a special case and while i did get that specific example(and sure of what CLT says) am still unsure of why Central limit theorem is true or how one formulate it in math terms. It is important for me to understand those, if i am ever to be able to question someone claiming some implication of CLT. Anyway, i came across the squeeze theorem in one of the HN comments and since it seems it’s part of the proof for CLT, I ended up reading and here’s the result of that. Anyway, enough story. Let’s go onwards. So here goes straight from the wikipedia page: Assumptions: There are three functions $f,g,h$ defined over a limit $l$. $a$ is a limit point. $f,g,h$ may not be defined at $a$, since it is the limit point. $g(x) leq f(x) leq h(x)$ $lim_{x to a} g(x) = lim_{x to a} h(x) = L$ To be proved: $lim_{x to a} f(x) = L$ Proof: Limits: I’ll try and clarify what is a limit as mathematically defined, and hopefully without equations,but words only. Well, according to wikipedia page, limit of a function f(x) means that the function f(x) can be made as close to a value (say L), by making x sufficiently close to c. Or to write out the equation $lim_{x to c}f(x) = L$ # Consciousness — some dreamy hand-wavy theorizing Disclaimer: I am not a professional scientist or an expert in any of the areas, I refer to in the following article. I have picked up a couple of unrelated degrees in my life so far, but don’t consider myself an expert in any of them. I consider the rest of the article as something, I would have written if I had been interviewed for that (pop-sci) book “What I believe, but cannot prove”. This is an attempt to summarize a bunch of my thoughts and ideas on consciousness. This definitely is a biased summary, as abhay asked me to write a post and with a set deadline, my brain was scampering around and recalled the most recent thoughts. A few basic assumptions, I tend to make are: 1.Consciousness is a emergent phenomenon, that arises out of essentially materialistic universe. 2.Only known/agreed fundamentals of universe are dictated by physics as Space, Time, Energy, Mass. 3.While our neuronal activity may not be sufficient to explain all the phenomenology of consciousness, they definitely are a necessary condition for enabling consciousness.* 4. Space-time continuum and the extended Space-Time-Energy-Mass Continuum is true, and will be proven sometime in the future. A couple of basic hypothesis I would like to add is: 1. Attention is a core, integral, fundamental part of our universe on par with Space, Time, Energy and Mass. 2. Attention is part of a continuum of Space-Time-Energy-Mass-Attention Continuum 3. Big bang is the event when this continuum evolved/broke apart enough to these distinct components. For the rest of the writing, I would use the terms consciousness and attention rather inter-changeably, because that’s how I consider it to be. Do note that, I haven’t defined attention, because despite the standard definiton, I am not sure, that fits in with what am building here. Instead, I’ll try to provide as many specific examples as I can think of and let you infer or construct a definition, that fits. Some evidence*** I would like to sketch out are: 1.Heisenberg’s uncertainty principle: i.e: It is impossible to know both position and mass of a particle to the exact precision allowed by the equipments’ capability. This has been interpreted in many ways, and discussed rather extensively in some philosophical circles and popular science books. The main point,I wish to make being once you think of the measuring equipments as exhibitng some characteristics of attention, it’s easy to interpret this as evidence for attention being a part of the universe and being converted into other forms. One corollary of this is 2.Paramathma–Jeevathma: Am picking just the terms I am familiar with. But this principle, that all human beings/living beings are part of a whole has been proposed in quite a few mythologies and stories spread around the world. Now, it could be argued that, this really is just a bias that led to this kind of independantly originated similar myths, but am at the other end.I’ll just list out a few variants, I’ve come across here, “Treat unto others”, “Gaia theory” 3.Causality and Probabilistic Graphical models: Judea Pearls’ work on causality is fairly known.But I only have the basic ideas from reading easy text and avoiding the core math part.Based on a few of the sketches, I have indeed read, I would say, quite a few of our currently used models of causality are suspect, which I believe is the biggest argument for a peer review process, rather than a made-up gold standard template for scientific experimentation and conclusions. But I digress, my core argument being that, if and once you agree that highly engineered, technical equipments(like the LHC) possess some level of attention/consciousness, then your probability graph immediately adds in complexity. To put it mildly, the possible number of agents goes up so dramatically, that trying to assign strong probability to any event as being the cause of another becomes a unrealistic computation problem. 4.Speed of light constant: That mysterious rule that Speed of light is a maximum limit constant that cannot be exceeded.Also the corollary, about exchanging information. Some predictions I would like to make are: 1.Sometime in the distant future, physicists will start acknowledging and forming theories around the agency of their equipments and how to make experiments that account for those, but still add value/evidence to the original hypothesis. 2.Neuroscience,technology, and our understanding of ethics will progress to levels, that enable us to more actively experiment on consciousness by enabling us to set/change/modify variables involved in affecting it. Few corollaries, supposing those hypothesis are true: 1.Consciousness/Attention has been increasing in the universe, since Big bang. 2.It’s possible to convert from consciousness to mass/energy/space/time. (this means some superpowers like that of Vista, clockblocker, etc. in The Worm web serial novel) 3.The orders of magnitude of Consciousness contained in the form of mass/space/time is humongous. (Just think how much Energy is released by converting mass into Energy.of the order of 10^^12 times our known measure) 4.Our current measures of Consciousness/attention are too crude and they underestimate by incredible numbers. 5.In terms of life as a optimization problem viewpoint, we are currently woefully stuck in local optima points. We are just simply ignoring what is atleast 1 extra dimension, by not acknowleding it. (Not to imply we are utilizing the others we acknowledge optimally) — Some of these are just plain and simple biases I’ve accrued over 30 years of my life, and not necessarily proven science facts. * — It follows that, I don’t consider bacteria, viruses, DNA/RNA etc. as conscious. I remember Harward declared animals are conscious, (though not sure they included microbes or not) and I agree with that. Am open to being convinced otherwise though.They do have a few basic reactions to environmental changes, so to take a weaker position, I would say, the bacteria and virii have a very tiny/micro/minute level of consciousness. *** — evidence as suggested by law scenes in movies and serials, rather than from hard sciences.I could probably argue, there’s probabilistic graphical models based strong evidence, but I don’t have the time or effort to put a good one. This includes narrative, story-based evidences.	2017-07-25 06:36:03	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 9, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5884256958961487, "perplexity": 1190.6529126383082}, "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-2017-30/segments/1500549425082.56/warc/CC-MAIN-20170725062346-20170725082346-00327.warc.gz"}
https://www.elastic.co/blog/found-managing-elasticsearch-fields-when-searching	Managing Elasticsearch Fields When Searching \| Elastic Blog # Managing Elasticsearch Fields When Searching UPDATE: This article refers to our hosted Elasticsearch offering by an older name, Found. Please note that Found is now known as Elastic Cloud. Controlling the number of fields returned for search requests is an important aspect of maximizing Elasticsearch's performance. In this article we'll look at how we can selectively return only the fields we're interested in for each search hit in order to optimize our usage pattern. ## Introduction When searching, using the fields-parameter allows to restrict the fields returned for each hit. This is a feature commonly used in order to optimize the amount of data transferred from Elasticsearch per hit by only selecting the relevant fields. While it may sound simple (and it is), the impact on performance may be tremendous, and is proportional both to the number of documents returned and the size of each document. Consider the case where each document has an average size of 20KB. Returning 10 hits will cause an average 200KB of data to be returned for each search request. Multiply this by 100 search requests/second and we end up needing 20 MB/s bandwidth. If we can reduce the size of the returned documents to 800 bytes, which is plausible by just returning the required fields, each request will be roughly 8KB in size, and the total bandwidth needed for 100 requests/s is only 800 KB/s. Keeping the result sets as small as possible is certainly important in order to be able to scale sensibly. By default, the _source-field is enabled, and its contents are parsed and returned for each search hit. We can save precious server-side CPU time and disk IO by limiting the returned fields since we no longer have to load and parse the whole document source. Simply specifying fields when searching is enough to trigger this behavior. Should we require to also return the document source in addition to the fields, we can use source filtering, which we'll look at shortly. One feature about fields that's not commonly known is the ability to select metadata-fields as well. Of particular note is its ability to select the _ttl-field, which actually returns the number of milliseconds until the document expires, not the original lifespan of the document. A very handy feature indeed. ## Nested Fields and Nested Data Sometimes, only selecting simple fields are not enough since we need additional data from nested fields that may be dynamic. In these cases, we cannot use fields because we would end up with the following failure: { ..., "_shards": { "failures": [ { "index": "index-name", "shard": 2, "status": 400, "reason": "ElasticsearchIllegalArgumentException[field [field_with_nested_data] isn't a leaf field]" } ] },... } This exception is thrown because fields are only able to load data from leaf nodes (i.e nodes that have no children). In order to load non-leaf nodes, we have to use the _source-field in the search request which enables a feature called source filtering, which we'll look at next. ## Source Filtering Source filtering allows us to control which parts of the original JSON document are returned for each hit. We can include or exclude parts of the original document based on patterns matching the field name path. It's worth keeping in mind that this only saves us on bandwidth costs between the nodes participating in the search as well as the client, not CPU or Disk, as was the case when using fields. This is because when using source filtering, we still have to load and parse the entire source document for each returned hit in order to match it against the inclusion and exclusion patterns. But still it's an important tool in our optimization toolbelt that is very easy to get started with. Prior to 1.0, this was known as partial fields, which was deprecated and replaced by source filtering. ## Unstored Fields Returning the field data representation of a field using fielddata_fields is a rather new addition to Elasticsearch (added in 1.0), which can be used to load fields that are not even stored (i.e fields that have store: false set in their mapping). It's important to note that when returning data from the fielddata, the terms for that field is loaded into memory and cached. This costs additional memory the first time around, but the same caches are used when sorting and faceting on the field in question. In other words, if you're already sorting or faceting on the field, returning it through the use of fielddata_fields is very cheap, but think twice about using it to return data from fields that are otherwise not commonly used. The field data representation of a field varies on the type and mapping of the field, which includes the analyzer. For example, boolean values are returned through fielddata_fields as the strings T and F, and most string fields are just a collection of sorted terms. ## Script Fields Using script fields enables us to return the result of evaluating a script for each hit. The script may perform some computation based on one or more of the existing fields, including metadata fields, access and return parts of the source document or return basically any value, which is used as the value for the field. Here is an example that shows how we can see the remaining time and the expiration date of documents in our index, assuming we're using document ttls. { "fields": [ "_ttl" ], "script_fields": { "expires_at": { "script": "new Date(doc._ttl.getValue())" } } } Using script fields requires dynamic scripts to be enabled, and this has some important security implications you need to be aware of. ## Conclusion After looking into our options for selecting only the relevant fields to return for search requests, we're now better suited to optimize the amount of data transferred between our Elasticsearch cluster and our search clients. We've also seen how to return fields and information that are not normally returned by Elasticsearch searches that we still may be interested in.	2020-07-05 08:18:53	{"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": 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": 0.3469066023826599, "perplexity": 1153.0393033673715}, "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-2020-29/segments/1593655887046.62/warc/CC-MAIN-20200705055259-20200705085259-00141.warc.gz"}
https://mersenneforum.org/showthread.php?s=63fe71e53668534ba386c9b3a3219dd0&t=18421&page=35	mersenneforum.org Reserved for MF - Sequence 3408 Register FAQ Search Today's Posts Mark Forums Read 2016-10-23, 17:02 #375 henryzz Just call me Henry "David" Sep 2007 Cambridge (GMT/BST) 16BA16 Posts The Bayesian tool will often ask for more ecm. The higher bounds mean a higher chance of finding a factor overall. Sometimes there may be a higher chance of smaller factors being missed but there will be less higher factors missed. More efficiency in finding factors means more ecm is worthwhile often. 2016-10-26, 14:32 #376 EdH "Ed Hall" Dec 2009 22·3·7·43 Posts Quote: Originally Posted by VBCurtis I'll contribute 500 @ 85e7 this week, ... In case you're running these, I'm shifting over to 29e8. I've got just over 3800 reported, as of this morning, at 85e7 with many still assigned that haven't returned. I suppose you could stop at 3-400 and we'd still meet the 4200 suggested. I'll swap the machines still working on 85e7 over and manually get a final count of all of their progress. It might be another hundred. Edit: I guess I had more stragglers than I thought. I've got about 3960 @ 85e7 for a final count. I've moved everything up. Sorry for the late notice if you've done a bunch of the 500... Last fiddled with by EdH on 2016-10-26 at 15:10 Reason: Updated info 2016-10-26, 15:08 #377 VBCurtis "Curtis" Feb 2005 Riverside, CA 2×5×467 Posts OK, I'll get to 300 and then add a few at 3e9. 2016-10-26, 17:09 #378 EdH "Ed Hall" Dec 2009 22×3×7×43 Posts Quote: Originally Posted by VBCurtis OK, I'll get to 300 and then add a few at 3e9. That sounds good. All my machines are now running 29e8, but they will most assuredly be quite some time doing so. I may retask one or two against other interests. I've some maintenance to perform, as well... 2016-10-26, 22:26 #379 EdH "Ed Hall" Dec 2009 Adirondack Mtns E1C16 Posts How disappointing: Code: -> ___________________________________________________________________ -> \| Running ecm.py, a Python driver for distributing GMP-ECM work \| -> \| on a single machine. It is copyright, 2011-2016, David Cleaver \| -> \| and is a conversion of factmsieve.py that is Copyright, 2010, \| -> \| Brian Gladman. Version 0.41 (Python 2.6 or later) 3rd Sep 2016 \| -> \|_________________________________________________________________\| -> Number(s) to factor: -> 29460893303338144751360360097976743017149046981259832053501450854362438285630845458294329588136961317820466603439061784124252469966169391148524869909406500896547611862071404959591325864761463 (191 digits) ->============================================================================= -> Working on number: 294608933033381447...959591325864761463 (191 digits) -> Found previous job file job8854.txt, will resume work... -> * Already completed 0 curves on this number... -> * Will run 20 more curves. -> Currently working on: job8854.txt -> Starting 4 instances of GMP-ECM... -> ecm -c 5 -maxmem 250 2900000000 < job8854.txt > job8854_t00.txt -> ecm -c 5 -maxmem 250 2900000000 < job8854.txt > job8854_t01.txt -> ecm -c 5 -maxmem 250 2900000000 < job8854.txt > job8854_t02.txt -> ecm -c 5 -maxmem 250 2900000000 < job8854.txt > job8854_t03.txt GMP-ECM 7.0.3 [configured with GMP 6.1.1, --enable-asm-redc] [ECM] GNU MP: Cannot allocate memory (size=67239952) GNU MP: Cannot allocate memory (size=125894672) GNU MP: Cannot allocate memory (size=537395216) -> *** Error: unexpected return value: -1 It did keep trying many times. So far, only this one is complaining, though... 2016-10-26, 22:49 #380 VBCurtis "Curtis" Feb 2005 Riverside, CA 2·5·467 Posts Please post what B2 value (and k-value) ECM picks for 29e8 with maxmem of 250. That's quite a combination! I'll post B2, k, and timings for unlimited memory once I have the data. Just fired up 100 3e9 curves. 2016-10-27, 03:23 #381 EdH "Ed Hall" Dec 2009 22·3·7·43 Posts Quote: Originally Posted by VBCurtis Please post what B2 value (and k-value) ECM picks for 29e8 with maxmem of 250. That's quite a combination! I'll post B2, k, and timings for unlimited memory once I have the data. Just fired up 100 3e9 curves. I'm running ECM via ecm.py and can't find any k-value shown anywhere. All of my machines are using B2=80921447825410. A lot of them (even with 1400 maxmem) are showing the "GNU MP: Cannot allocate memory (size=########)" message, but they aren't crashing. I tried to downsize to 2 threads @ 500 maxmem each on the aforementioned machine, but it's still "thinking" about it, so I don't have a result yet. I have others running with 450 per thread. 2016-10-27, 05:33 #382 VBCurtis "Curtis" Feb 2005 Riverside, CA 467010 Posts You'd have to invoke the -v flag for ECM to spit out the k value; that is, the number of pieces it splits B2 into. Not important, merely my curiosity; I've never probed how large k can get. Default at 3e9 is k=2, B2=105e12, and peak memory use of 11.3GB. My machine reports 13400 sec for stage 1, 1900 sec for stage 2 (ECM 7.0.1). Setting k=8 should result in nearly the same B2 with half the memory usage. Perhaps you're running into a side effect that ECM estimates memory use quite a bit less than peak use. Invoking -v tells me ECM expects to use 8.93GB, but peak use is actually reported by ECM as 11.3GB. 2016-10-27, 12:59 #383 EdH "Ed Hall" Dec 2009 22·3·7·43 Posts Quote: Originally Posted by VBCurtis You'd have to invoke the -v flag for ECM to spit out the k value; that is, the number of pieces it splits B2 into. Not important, merely my curiosity; I've never probed how large k can get. Default at 3e9 is k=2, B2=105e12, and peak memory use of 11.3GB. My machine reports 13400 sec for stage 1, 1900 sec for stage 2 (ECM 7.0.1). Setting k=8 should result in nearly the same B2 with half the memory usage. Perhaps you're running into a side effect that ECM estimates memory use quite a bit less than peak use. Invoking -v tells me ECM expects to use 8.93GB, but peak use is actually reported by ECM as 11.3GB. This number may be out of my reasonable reach. The most memory for any of my machines is 6GB total, running 4 threads. Here's an output from one of the others: Code: ->============================================================================= -> Working on number: 294608933033381447...959591325864761463 (191 digits) -> Currently working on: job3602.txt -> Starting 2 instances of GMP-ECM... -> ecm -c 10 -maxmem 1400 2900000000 < job3602.txt > job3602_t00.txt -> ecm -c 10 -maxmem 1400 2900000000 < job3602.txt > job3602_t01.txt GMP-ECM 7.0.3 [configured with GMP 6.1.1, --enable-asm-redc] [ECM] GNU MP: Cannot allocate memory (size=537395216) Using B1=2900000000, B2=81051862041166, polynomial Dickson(30), 2 threads ____________________________________________________________________________ Curves Complete \| Average seconds/curve \| Runtime \| ETA -----------------\|---------------------------\|---------------\|-------------- 4 of 20 \| Stg1 11479s \| Stg2 6386s \| 0d 21:48:26 \| 1d 15:41:33 I suppose I could knock them all down to a single thread. The one machine crashing out with errors didn't work with two threads at 500 maxmem each. Most of my machines are running headless and if I increase maxmem much more they stop talking to me. I don't have time today to play with them much. But, maybe later I'll figure something more out... or, I'll move the really weak machines to something they can work with better. I might just not be up to the size of this composite. 2016-10-27, 14:06 #384 VBCurtis "Curtis" Feb 2005 Riverside, CA 2·5·467 Posts I have a few machines with big-memory footprints. How about you run stage 1 and I run stage 2? The text files aren't large, can be emailed easily. You'd invoke ecm with -save residues.txt and bounds 29e8 29e8. The second bound is B2; set equal to B1, ECM won't do any stage 2. This allows you to make full use of your small-RAM machines, while I can do stage 2 in 1/3rd the time of your machine. Also, note we're doing a t60 using big-bounds because the Bayesian tool says so; you can simply choose to do a t60 the old-fashioned way, say with B1 = 3e8. Last fiddled with by VBCurtis on 2016-10-27 at 14:17 2016-10-28, 03:36 #385 EdH "Ed Hall" Dec 2009 Adirondack Mtns E1C16 Posts Let's try your suggestion. I got home a bit late, but am attempting to shift all my machines over to see where they are in the morning. I will be issuing the following to each: Code: `python ecm.py -c 20 -save residues\${USER}.txt 2900000000 2900000000 Similar Threads Thread Thread Starter Forum Replies Last Post RichD Aliquot Sequences 468 2021-01-27 01:16 schickel Aliquot Sequences 2934 2021-01-07 18:52 kar_bon Aliquot Sequences 127 2020-12-17 10:05 RichD Aliquot Sequences 14 2013-08-02 17:02 petrw1 Lone Mersenne Hunters 82 2010-01-11 01:57 All times are UTC. The time now is 05:52. Thu Feb 25 05:52:48 UTC 2021 up 84 days, 2:04, 0 users, load averages: 1.72, 2.05, 2.22	2021-02-25 05:52:48	{"extraction_info": {"found_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": 0.23312601447105408, "perplexity": 10934.506051783397}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1614178350717.8/warc/CC-MAIN-20210225041034-20210225071034-00564.warc.gz"}
https://anatomind.com/blog/peak/	# 想要成功可以靠運氣嗎？只能靠你的刻意練習！ Posted by 陳璿丞 on Wednesday, December 28, 2016 ## TOC We named this universal approach “deliberate practice.” # 有經驗不等於有刻意練習 Research has shown that, generally speaking, once a person reaches that level of “acceptable” performance and automaticity, the additional years of “practice” don’t lead to improvement. ## 有目標的練習 • Without such a goal, there was no way to judge whether the practice session had been a success. • Purposeful practice is all about putting a bunch of baby steps together to reach a longer-term goal. • Purposeful practice is focused. • Purposeful practice involves feedback. ## 離開舒適圈是什麼？ 1. Getting out of your comfort zone means trying to do something that you couldn’t do before. 2. If you never push yourself beyond your comfort zone, you will never improve. It is comfortable again. The changes stop. So to keep the changes happening, you have to keep upping the ante: run farther, run faster, run uphill. If you don’t keep pushing and pushing and pushing some more # Mental representation • 書上是這樣翻的： A mental representation is a mental structure that corresponds to an object, an idea, a collection of information, or anything else, concrete or abstract, that the brain is thinking about. A key fact about such mental representations is that they are very “domain specific,” that is, they apply only to the skill for which they were developed. We saw this with Steve Faloon: the mental representations he had devised to remember strings of digits did nothing to improve his memory for strings of letters. there is no such thing as developing a general the key benefit of mental representations lies in how they help us deal with information: understanding and interpreting it, holding it in memory, organizing it, analyzing it, and making decisions with it. • the more skilled you become, the better your mental representations are, and the better your mental representations are, the more effectively you can practice to hone your skill. • It’s like a staircase that you climb as you build it. Each step of your ascent puts you in a position to build the next step. Then you build that step, and you’re in a position to build the next one. And so on. # 成功的快速道路 1. First, there are always objective ways—such as the win/loss of a chess competition 2. Second, these fields tend to be competitive enough that performers have strong incentive to practice and improve. 3. Third, these fields are generally well established, with the relevant skills having been developed over decades or even centuries. 4. fourth, these fields have a subset of performers who also serve as teachers and coaches and who, over time, have developed increasingly sophisticated sets of training techniques that make possible the field’s steadily increasing skill level. The crucial finding was that there was only one major difference among the three groups. This was the total number of hours that the students had devoted to solitary practice. the best violin students had, on average, spent significantly more time than the better violin students had spent. to become an excellent violinist requires several thousand hours of practice. We found no shortcuts and no “prodigies” who reached an expert level with relatively little practice. # 刻意練習可以自己來嗎？ • Deliberate practice requires a teacher who can provide practice activities designed to help a student improve his or her performance. • Deliberate practice involves well-defined, specific goals and often involves improving some aspect of the target performance; • Deliberate practice is deliberate, that is, it requires a person’s full attention and conscious actions. 1. If you find that something works, keep doing it; if it doesn’t work, stop. 2. The best approach is almost always to work with a good coach or teacher. The reality is, however, that all of these things—managing, selling, teamwork—are specialized skills, and unless you are using practice techniques specifically designed to improve those particular skills, trying hard will not get you very far # 人人都可以成為神奇寶貝大師 The deliberate-practice mindset offers a very different view: anyone can improve, but it requires the right approach. 1. Does it push people to get outside their comfort zones and attempt to do things that are not easy for them? 2. Does it offer immediate feedback on the performance and on what can be done to improve it? 3. Have those who developed the approach identified the best performers in that particular area and determined what sets them apart from everyone else? 4. Is the practice designed to develop the particular skills that experts in the field possess? ## 找一個好的老師 • It is particularly difficult early in the learning process, when your mental representations are still tentative and inaccurate; once you have developed a foundation of solid representations, you work from those to build new and more effective representations on your own. • No matter how many sessions a week you have with an instructor, most of your effort will be spent practicing by yourself, doing exercises that your teacher has assigned. • one of the most important things a teacher can do is to help you develop your own mental representations so that you can monitor and correct your own performance. ## 專心學習 • There was focus but no joy.whatever you are doing, focus on it. • we started a routine, a practice routine, and from then on I really started to practice where it was a consciousaction working towards a specific goal, not just hit balls or putt. • where much of the practice consists of seemingly mindless, repetitive actions, paying attention performing those actions the right way will lead to greater improvement. ## 取得回饋, 並且修理它(fix it) they were paying attention to what they got wrong each time and correcting it. the purpose of the repetition is to figure out where your weaknesses are and focus on getting better in those areas, trying different methods to improve until you find something that works. what the expert performer can do, fail, figure out why we failed, try again, and repeat—over and over again. # 我信了，你最好也信 Furthermore, once you assume that something is innate, it automatically becomes something you can’t do anything about: If you don’t have innate musical talent, forget about ever being a good musician. If you don’t have enough willpower, forget about ever taking on something that will require a great deal of hard work. This sort of circular thinking—“The fact that I couldn’t keep practicing indicates that I don’t have enough willpower, which explains why I couldn’t keep practicing”—is Maintaining the motivation that enables such a regimen has two parts: reasons to keep going and reasons to stop. • I found that most of them preferred to practice as soon as they got up in the morning.They had set up their schedules so that there was nothing else to do at that time. It was set aside specifically for practice. • Identifying that period as their practice time created a sense of habit and duty that made it less likely they’d be tempted by something else.which indicates that they made more of an effort to plan their time. • Good planning can help you avoid many of the things that might lead you to spend less time on practice than you wanted. The second thing is to limit the length of your practice sessions to about an hour. Another key motivational factor in deliberate practice is a belief that you can succeed.The power of such belief is so strong that it can even trump reality. if you stop believing that you can reach a goal, either because you’ve regressed or you’ve plateaued, don’t quit. Make an agreement with yourself that you will do what it takes to get back to where you were or to get beyond the plateau, and then you can quit 1. 早上起來的第一件事就去做 2. 一次一小時左右 3. 相信自己可以，絕對不放棄 4. 找人幫你打氣、找同伴一起學 # 要變強只能有痛苦？ 1. 先覺得好玩。先覺得這東西不錯。想要讓小孩學小提琴、鋼琴、畫畫？先讓他們覺得好玩、有趣、想學，是我的興趣。 2. 接下來就會想要好好練習、想要變厲害。這時，讓他們去知道為什麼要學比起如何學來得重要。先有核心信念 3. 這時他們就會有，我想要比其他人厲害、或是覺得這個技能是他們的一部分，這時，他們就會下定決心 # 再說一次，沒有天才這件事 • Studies done in adults have generally found adult chess players to have no better visuospatial abilities than normal non-chess-playing adults. • The researchers found, was that the elite players with lower IQs tended to practice more, which improved their chess game to the point that they played better than the high-IQ elite players. # 現在的你就是你自我預言的產物 The predictions come true: the girl who was told to forget about sports never becomes any good at hitting a tennis ball or kicking a soccer ball; the boy who was told he was tone-deaf never learns to play a musical instrument or to sing well; and the children who were told they were no good at math grow up believing it. The prophecy becomes self-fulfilling. ## 我是老師/父母，我要怎麼教我的小孩？ with each step designed to keep students out of their comfort zone but not so far out that they cannot master that step. Then give plenty of repetition and feedback; the regular cycle of try, fail, get feedback, try again, and so on is # 結語 But perhaps a better way to see ourselves would be as Homo exercens, or “practicing man,” the species that takes control of its life through practice and makes of itself what it will. The hallmark of purposeful or deliberate practice is that you try to do something you cannot do—that takes you out of your comfort zone—and that you practice it over and over again, focusing on exactly how you are doing it, where you are falling short, and how you can get better.	2021-02-28 06:32:34	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3534301221370697, "perplexity": 1576.4454985497227}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1614178360293.33/warc/CC-MAIN-20210228054509-20210228084509-00270.warc.gz"}
https://www.numerade.com/questions/the-family-of-functions-ft-ce-at-e-bt-where-a-b-and-c-are-positive-numbers-and-b-a-has-been-used-to-/	💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here! # The family of functions $f(t) = C(e^{-at} - e^{-bt})$, where $a$, $b$, and $C$ are positive numbers and $b > a$, has been used to model the concentration of a drug injected into the bloodstream at time $t = 0$. Graph several members of this family. What do they have in common? For fixed values of $C$ and $a$, discover graphically what happens as $b$ decreases. Then use calculus to prove what you have discovered. Derivatives Differentiation Volume ### Discussion You must be signed in to discuss. Lectures Join Bootcamp ### Video Transcript No transcript available Arizona State University Derivatives Differentiation Volume Lectures Join Bootcamp	2021-10-24 05:26:17	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2727363109588623, "perplexity": 1155.616018488404}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1634323585911.17/warc/CC-MAIN-20211024050128-20211024080128-00325.warc.gz"}
https://tex.stackexchange.com/questions/144933/environment-aligned-undefined-when-trying-to-write-a-system-of-equations	# Environment aligned undefined when trying to write a system of equations I'm trying to write a system of equations in LaTeX. From the Wiki page Advanced Mathematics I took this example: \left.\begin{aligned} B'&=-\partial \times E,\\ E'&=\partial \times B - 4\pi j, \end{aligned} \right\} \qquad \text{Maxwell's equations} but as a result I got: ! LaTeX Error: Environment aligned undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... l.49 \left.\begin{aligned} Does anybody know what goes wrong here? • Please provide a minimal working example. We can definitely assist you better in that way. More than likely, from the error message, your missing the amsmath package which defines the aligned environment. – azetina Nov 14 '13 at 20:24 \usepackage{amsmath}	2019-06-19 04:46:47	{"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": 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": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.9997944235801697, "perplexity": 3442.1847911869068}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1560627998913.66/warc/CC-MAIN-20190619043625-20190619065625-00467.warc.gz"}
https://www.answers.com/Q/Can_the_empirical_formula_be_used_to_calculate_the_percent_composition_of_a_compound	Chemistry Elements and Compounds # Can the empirical formula be used to calculate the percent composition of a compound? ###### Wiki User The empirical formula is the simplest ratio of the elements within a compound. Therefore, it can be used to calculate the percentage of an element within a compound. For example, the empirical formula for sodium chloride is NaCl. From this, we can see that the ratio of sodium ions to chloride ions is 1Na : 1Cl. Therefore, a sodium chloride molecule is composed of 50% sodium and 50% chloride. 🙏 0 🤨 0 😮 0 😂 0 ## Related Questions ### What is the importance of percentage composition? Percent composition can be used to calculate the percentage of an element/compound in a mixture. From the percent composition, you can also find the empirical formula. And from the empirical formula you can find the actual molecular weight. ### What do empirical formulas tell you? The empirical formula tells you the simplest formula for the compound. The molecular formula will be some multiple of the empirical formula, or it can be identical to the empirical formula. ### Which could represent the empirical and the formula of a given compound? The empirical formula is representative for the chemical composition of a compound; the structural formula is representative for the spatial structure of the compound. ### How to calculate Molecular formula from empirical formula? If you know the molar mass of the compound, you have to calculate the mass of the empirical formula and divide the molar mass of the compound by the mass of the empirical formula in order to find the ratio between the molecular formula and the empirical formula. Then multiply all the atoms by this ratio to find the molecular formula! ### Empirical formulas describe the types of atoms in a compound and their? An empirical formula give information about the chemical composition of a compound. Example: tetracycline with the empirical formula C22H24N2O8 has the following composition: - carbon 59,44 % - oxygen 28,81 % - nitrogen 6,30 % - hydrogen 5,45 % If you're with plato the answer is ratios ### How do you know when to solve for a empirical formula? If it tells you to find the empirical formula when percent composition is given or if the mass of each element is given in a specific compound. ### What is the empirical formula for this compound C10 H25 O5? The empirical formula for the compound C10H25O5 is CH5O. The empirical formula is the simplest whole number ratio of elements in the compound. ### How can you find the molecular formula of a compound when its percentage composition is given? You have to find its empirical formula using the percentage composition. When you have done that, work out the relative molecular mass (Mr) of the empirical formula. This should be a multiple of the compound's Mr, so you multiply the amount of each atom in the empirical formula by this number, which gives you the final molecular formula. ### How do you know if a formula is an empirical formula? An empirical formula has no data about the structure of a compound. ### What is determined by an empirical formula? The empirical formula of a compound gives the simplest ratio of the atoms in the compound (for instance the empirical formula C2H4 would be CH2). ### Explain why the percent composition of certain compounds are not sufficient to determine the compounds molecular formulas? Because unlike the empirical formula, the molecular formula does not have to be the simplest ratio.If by chance you are given the percent composition of the elements in a substance, you could calculate the empirical formula and then the empirical formula's mass. However, the molecular formula equation is molecular formula= (empirical formula)n, where n is the mass of the molecular formula divided by the mass of the empirical formula. You would, therefore, need to know the mass belonging to the molecular formula, which you are not given. ### What is the empirical formula of a compound composed of 31.1 g potassium and 6.36 g oxygen? A 9.50g sample of a compound contains 3.03g K+, 2.75 g Cl and oxygen. Calculate the empirical formula. ### What is empirical formula how is it determined? The empirical formula is determined after the chemical analysis of a compound; the empirical formula do not express the geometrical configuration of the molecule. ### What does a compound formula tell you? A compound formula, or molecular formula, tells you about the chemical composition of the substance in terms of the number of atoms of that element that are present. From there, the empirical formula may also be derived by simplifying the molecular formula, as well as its structural formula. ### What is the empirical formula for compound whose molecular formula is P4O10? The empirical formula for a compound whose molecular formula is P4O10 is P2O5 = phosphor pentoxide. ## Still have questions? Trending Questions How old is Danielle cohn? Asked By Wiki User How many tens make 600? Asked By Wiki User	2021-03-06 08:37:09	{"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": 0.8760966062545776, "perplexity": 856.7502819824837}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1614178374616.70/warc/CC-MAIN-20210306070129-20210306100129-00584.warc.gz"}
https://math.stackexchange.com/questions/2391048/understanding-th-proof-of-hatcher-1-2-11	Understanding th proof of Hatcher 1.2.11 I am working on this problem: The mapping torus $T_f$ of a map $f : X → X$ is the quotient of $X×I$ obtained by identifying each point $(x,0)$ with $(f(x),1)$. In the case $X = S_1∨S_1$ with $f$ basepoint-preserving, compute a presentation for $π_1(T_f)$ in terms of the induced map $f_∗ : π_1(X) → π_1(X)$. Do the same when $X = S_1 ×S_1$. There are 3 different versions of solutions that I find: version 1: version 2: version 3: Now I am confused. By different methods they gives different conclusions. I think all of them sounds reasonable. So which is correct? Or are they equivalent? Why? Thanks for any help! • The last one is correct, and in general this is what happens. The interval that closes to a circle in the gluing becomes a new generator $g$, and there are new 2-cells with boundary $g^{-1}xgf_\ast(x)$ for original generators $x$. Oct 13, 2020 at 20:07 I was wrong. The last one is right by tracing the attaching $$1-$$cells. You can verify that by doing the case when $$X = S^1\wedge S^1$$ with $$f=id$$ easily, which you have probably done in question $$1.2.8.$$ In that case, the fundamental group should be $$\mathbb{Z}\oplus(\mathbb{Z} * \mathbb{Z})$$, which is given by the last solution, while with the first and second solution, the answer is $$\mathbb{Z} * \mathbb{Z} * \mathbb{Z}$$. In comparison with the work that I have done with this problem, this first solution is right. As for the second one, for the case when $$X = S^1 \wedge S^1$$, the answer is the same as the first solution. The author did not do the case when $$X= S^1\times S^1$$. According to the third solution, there were some kind of conjugates involved, which I think might be a mistake due to messing up with the base-point, but I am not a hundred percent sure, since it is kind of hard to prove two group presentations are different. I hope someone could help with this.	2022-08-15 04:40:34	{"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": 8, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8208717703819275, "perplexity": 131.5810327598526}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572127.33/warc/CC-MAIN-20220815024523-20220815054523-00785.warc.gz"}
https://itprospt.com/num/14775756/a-clinical-psychologist-wants-to-know-if-mindfulness-traininghelps	5 # A clinical psychologist wants to know if mindfulness traininghelps alleviate depression in a sample of n = 36 individuals. Ifthe average depression score is Î¼ = ... ## Question ###### A clinical psychologist wants to know if mindfulness traininghelps alleviate depression in a sample of n = 36 individuals. Ifthe average depression score is Î¼ = 87 with Ïƒ = 12, what depressionscores would be needed before we could say there is a significantdifference (use an alpha level Î± = .05)? Sketch the distributionand label the critical regions using the critical z-scores. Use theabove distribution to determine what the sample means would beneeded before we could say there is a significa A clinical psychologist wants to know if mindfulness training helps alleviate depression in a sample of n = 36 individuals. If the average depression score is Î¼ = 87 with Ïƒ = 12, what depression scores would be needed before we could say there is a significant difference (use an alpha level Î± = .05)? Sketch the distribution and label the critical regions using the critical z-scores. Use the above distribution to determine what the sample means would be needed before we could say there is a significant difference or not. #### Similar Solved Questions ##### Consider the following two grOups of costumers: Group consists of customers who spend less than S500 annually on clothes; Group comprises customers who spend OVeT S1OC0) annually on clothes: Let and Pz equal the proportions of customers in these two gTOups, respectively who believe that clothes are tOO experisive: If 1232 out of random sample of 1600 from group and 238 out of randomn sample 400 fromn group believe that clothes are too experisive;Give point estimate tor p1b) Find an approximate 9 Consider the following two grOups of costumers: Group consists of customers who spend less than S500 annually on clothes; Group comprises customers who spend OVeT S1OC0) annually on clothes: Let and Pz equal the proportions of customers in these two gTOups, respectively who believe that clothes are ... ##### Etat Cit Sa 0obin Cntie ldtt4Toa etat Cit Sa 0obin Cntie ldtt4 Toa... ##### A sample of non-= English majors at a selected college - was used in a study to see if the student retained more from reading 19th-century novel or by watching it in DVD form: Each student was assigned one novel to read and a different one to watch; and then they were given a 20-point written quiz . on each noveL Assume the variables are normally distributed. The test results are shown below:Book8585848690 80 90DVD858682 898875 70 80Download DataAt 0 = 0.05, can it be concluded that the book sco A sample of non-= English majors at a selected college - was used in a study to see if the student retained more from reading 19th-century novel or by watching it in DVD form: Each student was assigned one novel to read and a different one to watch; and then they were given a 20-point written quiz .... ##### 1) Determine the degree of each vertex in the graph below 1) Determine the degree of each vertex in the graph below... ##### Use the graph of the function y=f(r) below to answer the questions.Isf (4) negative?Yes 0 No(6) For which value(s) of x Is f (r) = 0z If there more than one value, separate them wlth commas_DD;(O,) (O,O][O,O) @,2] DUDFor which value(s) of x Isf ()<0? Write your answer using Interval notatlon_ Use the graph of the function y=f(r) below to answer the questions. Isf (4) negative? Yes 0 No (6) For which value(s) of x Is f (r) = 0z If there more than one value, separate them wlth commas_ DD; (O,) (O,O] [O,O) @,2] DUD For which value(s) of x Isf ()<0? Write your answer using Interval notatl... ##### 2 2 14drig ( Q f(x) f"(x) and /"(x)21 2 2 1 4drig ( Q f(x) f"(x) and /"(x) 2 1... ##### 1 X L 1 7 H 1 3 ] j 1 J4 H U 1 220 1 L 0 E22 22 J222 22 2 0 1 X L 1 7 H 1 3 ] j 1 J4 H U 1 220 1 L 0 E22 22 J222 22 2 0... ##### If photons of blue light have more energy than photons of red light, how can a beam of red light carry as much energy as a beam of blue light? If photons of blue light have more energy than photons of red light, how can a beam of red light carry as much energy as a beam of blue light?... ##### A constant force $\mathbf{P}$ is applied to a piston and rod of total mass $m$ to make them move in a cylinder filled with oil. As the piston moves, the oil is forced through orifices in the piston and exerts on the piston a force of magnitude $k v$ in a direction opposite to the motion of the piston. Knowing that the piston starts from rest at $t=0$ and $x=0,$ show that the equation relating $x, v,$ and $t,$ where $x$ is the distance traveled by the piston and $v$ is the speed of the piston, is A constant force $\mathbf{P}$ is applied to a piston and rod of total mass $m$ to make them move in a cylinder filled with oil. As the piston moves, the oil is forced through orifices in the piston and exerts on the piston a force of magnitude $k v$ in a direction opposite to the motion of the pisto... ##### A D-aldopentose is oxidized by nitric acid to an optically active aldaric acid. A Wohl degradation of the aldopentose leads to a monosaccharide that is oxidized by nitric acid to an optically inactive aldaric acid. Identify the D-aldopentose. A D-aldopentose is oxidized by nitric acid to an optically active aldaric acid. A Wohl degradation of the aldopentose leads to a monosaccharide that is oxidized by nitric acid to an optically inactive aldaric acid. Identify the D-aldopentose.... ##### Constants060-kg tennis ball , moving speed of 5.34 m/5 has head-on collision with 085-kg ball initially moving Ihe same direction at speed of 00 m/s Assume Ihat the collision perfectly elastic.Par â‚¬Determine the speed of the 085-kg ball after the collision Express Vour ansuer signIficant flgures and Include the appropriate unitsValueUnitsSubmiRequest AnswerParDDetemine the direction of the velocity of the 085-kg ball after the collisionin Ihe direction of the initial velocityin the direction Constants 060-kg tennis ball , moving speed of 5.34 m/5 has head-on collision with 085-kg ball initially moving Ihe same direction at speed of 00 m/s Assume Ihat the collision perfectly elastic. Par â‚¬ Determine the speed of the 085-kg ball after the collision Express Vour ansuer signIficant ... ##### Perform the indicated matrix operations. If the matrix does not exist, write impossible. $2\left[\begin{array}{rr}{6} & {3} \\ {-8} & {-2}\end{array}\right]-4\left[\begin{array}{rr}{8} & {1} \\ {3} & {-4}\end{array}\right]$ Perform the indicated matrix operations. If the matrix does not exist, write impossible. $2\left[\begin{array}{rr}{6} & {3} \\ {-8} & {-2}\end{array}\right]-4\left[\begin{array}{rr}{8} & {1} \\ {3} & {-4}\end{array}\right]$... ##### The compound CrS is an ioniccompound. What are the ions of which it is composed? The compound CrS is an ionic compound. What are the ions of which it is composed?... ##### In the following study; male and female participants were instructed to press a button on a computer keyboard as soon as they detected a tone through their headphones. Participants were randomly assigned to hear either a loud tone or a soft tone. The researchers measured the amount of time it took participants to press the computer button (in milliseconds) . Smaller numbers indicate better performance on the taskTABLE OF MEANS:TONE INTENSITY LOUD SOFT 516.70 834.50 528.80 763.80 522.75 799.15ROW In the following study; male and female participants were instructed to press a button on a computer keyboard as soon as they detected a tone through their headphones. Participants were randomly assigned to hear either a loud tone or a soft tone. The researchers measured the amount of time it took p... ##### Wan dmnry Ihe Lowis StncIte Ior Izl cuial atom?Uiecuons c0 vou MhhakMace wan dmnry Ihe Lowis StncIte Ior Izl cuial atom? Uiecuons c0 vou Mhhak Mace... ##### 4z50 50 JoJ 0 = 0so) + 0zso32 (qJz50 >0 JoJ 0 = & _ @zuist (D 'Kppexa suonenba Juauuouo3iu1 BuIMOIIOJ a41 aNOS 4z50 50 JoJ 0 = 0so) + 0zso32 (q Jz50 >0 JoJ 0 = & _ @zuist (D 'Kppexa suonenba Juauuouo3iu1 BuIMOIIOJ a41 aNOS...	2022-08-19 11:25:24	{"extraction_info": {"found_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": 0.471542090177536, "perplexity": 2919.83059057681}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1659882573667.83/warc/CC-MAIN-20220819100644-20220819130644-00229.warc.gz"}
https://informationtransfereconomics.blogspot.com/2016/05/recognizing-complexity-by-inspection.html	## Monday, May 16, 2016 ### Recognizing complexity by inspection Eric Liu (speechwriter) and Nick Hanauer (business person) have a new article at Evonomics that is an excerpt from their book The Gardens of Democracy. Obviously, they have the requisite skills to identify a complex nonlinear system by inspection: Traditional economic theory is rooted in a 19th- and 20th-century understanding of science and mathematics. At the simplest level, traditional theory assumes economies are linear systems filled with rational actors who seek to optimize their situation. Outputs reflect a sum of inputs, the system is closed, and if big change comes it comes as an external shock. The system’s default state is equilibrium. The prevailing metaphor is a machine. But this is not how economies are. It never has been. As anyone can see and feel today, economies behave in ways that are non-linear and irrational, and often violently so. These often-violent changes are not external shocks but emergent properties—the inevitable result—of the way economies behave. Yes, it's so violent: For reference, let's look at an actual complex biological system (Lynx population with predator-prey dynamics): If the US economy was as violent changes as the population dynamics of a Lynx, the economy would have collapsed to approximately zero GDP and sprung back again. During the past 70 years, the US economy has not received a quarterly shock of more than -10% (and that's annualized -- equivalent to an actual quarterly shock of less than -2.5%). Shocks on the order of a few percent mean the economy is well within the realm of perturbation theory. Whatever your theory of how economics works ... $$\frac{\Delta NGDP}{NGDP} \approx 0$$ is an excellent starting point. Do I have to show this graph again: 1. Jason, this isn't directly related, but I found this from Nate Silver to be interesting, regarding his failures to predict the outcome of the GOP nomination. He talks about "chaos" a bit in there and the difficulties in producing a good model, especially for the early stages of the race. I don't know if there's any overlap in your opinion with making economic forecasts. I do like how he emphasizes the discipline of sticking to a mathematical model and that making predictions creates learning opportunities for the modeler (the "fail forward" concept). Off-hand, I don't see why making the forecasts that Nate does should be any less daunting than constructing models to make economic forecasts. 1. In a sense Nate Silver's polls-only models are a kind of model-free estimation. They still involve models to understand e.g. registered voter polls versus polls that use cell phones, etc ... basically weighting = model. The polls plus version is more of a real model -- not just polls, but endorsements based on "the party decides" theory. In economics, some econometric forecasts are somewhat model free in the same sense as the polls only forecasts from Silver (e.g. I think GDPnow from the Atlanta Fed really just looks at the correlation of indicators that are available before the official GDP release -- a kind of "Big Data" approach). But for the most part in economics you need a model to even understand what the data means. It makes it harder. For example, you need to understand the theoretical concept of inflation to understand the difference between real and nominal gdp. I think GDPnow kind of does for macro what Silver does for elections. I have a post I am working on where I look at GDPnow ...	2017-09-23 21:48:44	{"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": 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": 0.566819965839386, "perplexity": 1889.7297339316167}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818689779.81/warc/CC-MAIN-20170923213057-20170923233057-00669.warc.gz"}
https://par.nsf.gov/biblio/10304424-measurement-masses-widths-baryons	This content will become publicly available on September 1, 2022 Measurement of the masses and widths of the $Σc(2455)+$ and $Σc(2520)+$ baryons Authors: ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; more » Award ID(s): Publication Date: NSF-PAR ID: 10304424 Journal Name: Physical Review D Volume: 104 Issue: 5 ISSN: 2470-0010	2022-06-26 21:12:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 17, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5688838958740234, "perplexity": 9715.345491861632}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1656103271864.14/warc/CC-MAIN-20220626192142-20220626222142-00710.warc.gz"}
https://eight2late.wordpress.com/2017/02/23/a-prelude-to-machine-learning/	Eight to Late Sensemaking and Analytics for Organizations A prelude to machine learning What is machine learning? The term machine learning gets a lot of airtime in the popular and trade press these days. As I started writing this article, I did a quick search for recent news headlines that contained this term. Here are the top three results with datelines within three days of the search: http://venturebeat.com/2017/02/01/beyond-the-gimmick-implementing-effective-machine-learning-vb-live/ http://www.infoworld.com/article/3164249/artificial-intelligence/new-big-data-tools-for-machine-learning-spring-from-home-of-spark-and-mesos.html http://www.infoworld.com/article/3163525/analytics/review-the-best-frameworks-for-machine-learning-and-deep-learning.html The truth about hype usually tends to be quite prosaic and so it is in this case. Machine learning, as Professor Yaser Abu-Mostafa puts it, is simply about “learning from data.” And although the professor is referring to computers, this is so for humans too – we learn through patterns discerned from sensory data. As he states in the first few lines of his wonderful (but mathematically demanding!) book entitled, Learning From Data: If you show a picture to a three-year-old and ask if there’s a tree in it, you will likely get a correct answer. If you ask a thirty year old what the definition of a tree is, you will likely get an inconclusive answer. We didn’t learn what a tree is by studying a [model] of what trees [are]. We learned by looking at trees. In other words, we learned from data. In other words, the three year old forms a model of what constitutes a tree through a process of discerning a common pattern between all objects that grown-ups around her label “trees.” (the data). She can then “predict” that something is (or is not) a tree by applying this model to new instances presented to her. This is exactly what happens in machine learning: the computer (or more correctly, the algorithm) builds a predictive model of a variable (like “treeness”) based on patterns it discerns in data. The model can then be applied to predict the value of the variable (e.g. is it a tree or not) in new instances. With that said for an introduction, it is worth contrasting this machine-driven process of model building with the traditional approach of building mathematical models to predict phenomena as in, say, physics and engineering. What are models good for? Physicists and engineers model phenomena using physical laws and mathematics. The aim of such modelling is both to understand and predict natural phenomena. For example, a physical law such as Newton’s Law of Gravitation is itself a model – it helps us understand how gravity works and make predictions about (say) where Mars is going to be six months from now. Indeed, all theories and laws of physics are but models that have wide applicability. (Aside: Models are typically expressed via differential equations. Most differential equations are hard to solve analytically (or exactly), so scientists use computers to solve them numerically. It is important to note that in this case computers are used as calculation tools, they play no role in model-building.) As mentioned earlier, the role of models in the sciences is twofold – understanding and prediction. In contrast, in machine learning the focus is usually on prediction rather than understanding. The predictive successes of machine learning have led certain commentators to claim that scientific theory building is obsolete and science can advance by crunching data alone. Such claims are overblown, not to mention, hubristic, for although a data scientist may be able to predict with accuracy, he or she may not be able to tell you why a particular prediction is obtained. This lack of understanding can mislead and can even have harmful consequences, a point that’s worth unpacking in some detail… Assumptions, assumptions A model of a real world process or phenomenon is necessarily a simplification. This is essentially because it is impossible to isolate a process or phenomenon from the rest of the world. As a consequence it is impossible to know for certain that the model one has built has incorporated all the interactions that influence the process / phenomenon of interest. It is quite possible that potentially important variables have been overlooked. The selection of variables that go into a model is based on assumptions. In the case of model building in physics, these assumptions are made upfront and are thus clear to anybody who takes the trouble to read the underlying theory. In machine learning, however, the assumptions are harder to see because they are implicit in the data and the algorithm. This can be a problem when data is biased or an algorithm opaque. Problem of bias and opacity become more acute as datasets increase in size and algorithms become more complex, especially when applied to social issues that have serious human consequences. I won’t go into this here, but for examples the interested reader may want to have a look at Cathy O’Neil’s book, Weapons of Math Destruction, or my article on the dark side of data science. As an aside, I should point out that although assumptions are usually obvious in traditional modelling, they are often overlooked out of sheer laziness or, more charitably, lack of awareness. This can have disastrous consequences. The global financial crisis of 2008 can – to some extent – be blamed on the failure of trading professionals to understand assumptions behind the model that was used to calculate the value of collateralised debt obligations. It all starts with a straight line…. Now that we’ve taken a tour of some of the key differences between model building in the old and new worlds, we are all set to start talking about machine learning proper. I should begin by admitting that I overstated the point about opacity: there are some machine learning algorithms that are transparent as can possibly be. Indeed, chances are you know the algorithm I’m going to discuss next, either from an introductory statistics course in university or from plotting relationships between two variables in your favourite spreadsheet. Yea, you may have guessed that I’m referring to linear regression. In its simplest avatar, linear regression attempts to fit a straight line to a set of data points in two dimensions. The two dimensions correspond to a dependent variable (traditionally denoted by $y$) and an independent variable (traditionally denoted by $x$). An example of such a fitted line is shown in Figure 1. Once such a line is obtained, one can “predict” the value of the dependent variable for any value of the independent variable. In terms of our earlier discussion, the line is the model. Figure 1: Linear Regression Figure 1 also serves to illustrate that linear models are going to be inappropriate in most real world situations (the straight line does not fit the data well). But it is not so hard to devise methods to fit more complicated functions. The important point here is that since machine learning is about finding functions that accurately predict dependent variables for as yet unknown values of the independent variables, most algorithms make explicit or implicit choices about the form of these functions. Complexity versus simplicity At first sight it seems a no-brainer that complicated functions will work better than simple ones. After all, if we choose a nonlinear function with lots of parameters, we should be able to fit a complex data set better than a linear function can (See Figure 2 – the complicated function fits the datapoints better than the straight line). But there’s catch: although the ability to fit a dataset increases with the flexibility of the fitting function, increasing complexity beyond a point will invariably reduce predictive power. Put another way, a complex enough function may fit the known data points perfectly but, as a consequence, will inevitably perform poorly on unknown data. This is an important point so let’s look at it in greater detail. Figure 2: Simple and complex fitting function (courtesy: Wikimedia) Recall that the aim of machine learning is to predict values of the dependent variable for as yet unknown values of the independent variable(s). Given a finite (and usually, very limited) dataset, how do we build a model that we can have some confidence in? The usual strategy is to partition the dataset into two subsets, one containing 60 to 80% of the data (called the training set) and the other containing the remainder (called the test set). The model is then built – i.e. an appropriate function fitted – using the training data and verified against the test data. The verification process consists of comparing the predicted values of the dependent variable with the known values for the test set. Now, it should be intuitively clear that the more complicated the function, the better it will fit the training data. Question: Why? Answer: Because complicated functions have more free parameters – for example, linear functions of a single (dependent) variable have two parameters (slope and intercept), quadratics have three, cubics four and so on. The mathematician, John von Neumann is believed to have said, “With four parameters I can fit an elephant, and with five I can make him wiggle his trunk.” See this post for a nice demonstration of the literal truth of his words. Put another way, complex functions are wrigglier than simple ones, and – by suitable adjustment of parameters – their “wriggliness” can be adjusted to fit the training data better than functions that are less wriggly. Figure 2 illustrates this point well. This may sound like you can have your cake and eat it too: choose a complicated enough function and you can fit both the training and test data well. Not so! Keep in mind that the resulting model (fitted function) is built using the training set alone, so a good fit to the test data is not guaranteed. In fact, it is intuitively clear that a function that fits the training data perfectly (as in Figure 2) is likely to do a terrible job on the test data. Question: Why? Answer: Remember, as far as the model is concerned, the test data is unknown. Hence, the greater the wriggliness in the trained model, the less likely it is to fit the test data well. Remember, once the model is fitted to the training data, you have no freedom to tweak parameters any further. This tension between simplicity and complexity of models is one of the key principles of machine learning and is called the bias-variance tradeoff. Bias here refers to lack of flexibility and variance, the reducible error. In general simpler functions have greater bias and lower variance and complex functions, the opposite. Much of the subtlety of machine learning lies in developing an understanding of how to arrive at the right level of complexity for the problem at hand – that is, how to tweak parameters so that the resulting function fits the training data just well enough so as to generalise well to unknown data. Note: those who are curious to learn more about the bias-variance tradeoff may want to have a look at this piece. For details on how to achieve an optimal tradeoff, search for articles on regularization in machine learning. Unlocking unstructured data The discussion thus far has focused primarily on quantitative or enumerable data (numbers and categories) that’s stored in a structured format – i.e. as columns and rows in a spreadsheet or database table). This is fine as it goes, but the fact is that much of the data in organisations is unstructured, the most common examples being text documents and audio-visual media. This data is virtually impossible to analyse computationally using relational database technologies (such as SQL) that are commonly used by organisations. The situation has changed dramatically in the last decade or so. Text analysis techniques that once required expensive software and high-end computers have now been implemented in open source languages such as Python and R, and can be run on personal computers. For problems that require computing power and memory beyond that, cloud technologies make it possible to do so cheaply. In my opinion, the ability to analyse textual data is the most important advance in data technologies in the last decade or so. It unlocks a world of possibilities for the curious data analyst. Just think, all those comment fields in your survey data can now be analysed in a way that was never possible in the relational world! There is a general impression that text analysis is hard. Although some of the advanced techniques can take a little time to wrap one’s head around, the basics are simple enough. Yea, I really mean that – for proof, check out my tutorial on the topic. Wrapping up I could go on for a while. Indeed, I was planning to delve into a few algorithms of increasing complexity (from regression to trees and forests to neural nets) and then close with a brief peek at some of the more recent headline-grabbing developments like deep learning. However, I realised that such an exploration would be too long and (perhaps more importantly) defeat the main intent of this piece which is to give starting students an idea of what machine learning is about, and how it differs from preexisting techniques of data analysis. I hope I have succeeded, at least partially, in achieving that aim. For those who are interested in learning more about machine learning algorithms, I can suggest having a look at my “Gentle Introduction to Data Science using R” series of articles. Start with the one on text analysis (link in last line of previous section) and then move on to clustering, topic modelling, naive Bayes, decision trees, random forests and support vector machines. I’m slowly adding to the list as I find the time, so please do check back again from time to time. Note: This post is written as an introduction to the Data, Algorithms and Meaning subject that is part of the core curriculum of the Master of Data Science and Innovation program at UTS. I’m co-teaching the subject in Autumn 2018 with Alex Scriven and Rory Angus. Written by K February 23, 2017 at 3:12 pm Tagged with 7 Responses 1. Thank you for nice introduction Like Limbu March 3, 2017 at 1:51 am 2. A very good intro. “Just think, all those comment fields in your survey data can now be analysed in a way that was never possible in the relational world!” I have to nitpick here – the “relational world” reference is ambiguous. You can talk about the limits of traditional relational DBMSs in terms of processing power and suitability for the job, and you’d be correct, but there’s nothing inherently in the relational model which prevents you from making a logical model of natural language text. I strongly recommend reading Fabian Pascal on these topics. He often comes across quite abrasive but he is a must read before one makes any assertions about the relational model (e.g. the principle of physical independence). Liked by 1 person Andrew G March 7, 2017 at 9:18 am • Hi Andrew, Many thanks for reading and taking the time to comment. I agree, my comments were about relational database products available in the market rather than the relational model per se. I read some of Fabian Pascal’s work a long time (10 years?) ago. Must go back to it when I find the time. That said, logical models may not be very useful for extracting information from text, primarily because of the multitude of ways in which a single idea can be expressed. The recent successes of word embedding techniques suggest that statistical approaches might be more promising (this article by Peter Norvig, which compares both approaches, is relevant here). I reckon the best results would likely come from some kind of merging of both approaches …and that’s a story yet to unfold. Thanks again! Regards, Kailash. Like K March 9, 2017 at 10:04 pm 3. Thanks for a great post here K! Liked by 1 person Dan Herr March 25, 2017 at 9:01 am • Thanks for the feedback Dan, much appreciated! Like K March 25, 2017 at 12:05 pm 4. […] overfitting by using less complicated functions (and if that means nothing to you, check out my prelude to machine learning). One way to do this is toss out less important variables, after checking that they aren’t […] Like 5. […] an aside, note that much of what goes under the banner of machine learning and AI can be also classed as first order […] Like This site uses Akismet to reduce spam. Learn how your comment data is processed.	2019-06-18 06:35:59	{"extraction_info": {"found_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": 0.5989259481430054, "perplexity": 648.4053331285908}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998690.87/warc/CC-MAIN-20190618063322-20190618085322-00313.warc.gz"}
https://www.nobigons.com/category/mathematics/	# Category: mathematics ## The wrangler’s insecurity. [This is my third post on Newton. Previous posts: one and two.] If you were to take a look around you during a math department seminar or colloquium, you would witness the audience’s attention begin to drift as the talk sunk further into detail and became increasingly difficult to follow. Losing interest in a talk is more or less expected, and the professional mathematicians in the audience come prepared. Maybe they bring a paper to read, or possibly exam scripts to grade. Sometimes they will turn to a fresh page of their notebook and begin doing some actual mathematics of their own. As a graduate student at McGill, I remember watching a postdoc fill up a page with long exact sequences and all kinds of diagrams, the notation veering into doodles as he got stuck at what must have been a familiar dead-end. It was a rare, voyeuristic glimpse into someone else’s solitary mathematical practice. I later asked this postdoc — whose notebook I presumed was full of such pages — if he ever went back and reread what he had written. No, he admitted, with a guilty smile. Which was a relief. Not only because my own notebooks were full of repetitious dead-ends, but also because I too almost never went back to review anything I’d written. Much has been made of Imposter Syndrome among academics — doubting whether we have truly earned whatever position we have reached given how paltry our contributions can sometimes feel. There is a related sense of insecurity to be found in wondering if you are doing mathematics correctly. To be clear, I don’t mean whether a proof we have written up is sound, but whether or not our process of formulating and devising them is the the “proper” way. As if there might be a correct way (or even a professional way) of doing mathematics. These are not new concerns to have. Isaac Newton was incredibly secretive in his work and did not have anything approaching students as we might describe them. But after his death, the calculus he developed would form the foundation of modern mathematical and scientific education at Cambridge. Those who scribbled hastily on those exam papers were students, above all, of Newton’s mathematical physics. Though Newton had not cultivated a following during his own tenure at Cambridge, by the end of the eighteenth century the principals laid down in the Principia — and in particular the mathematical contents of that book — formed the basis for an intensely competitive system of testing at the university by which students were ranked in descending order based on their results on terminal examinations. known as the “Mathematical Tripos.” (The origin of the term Tripos is uncertain, but it may refer to the three legged stool on which students originally sat to take the oral examinations.) The Newton Papers – Sarah Dry, pg 85 The material, and especially the notation, would be modernized as European influences arrived, but Newton did not lose his centrality. The manner in which he actually arrived at his great insights became a matter of interest. There is a great distinction between how discoveries are made, and how they finally appear on the page. Everyone knew how they personally went about doing mathematics, and even how their tutors told them to do mathematics, but was that the same as how Newton went about making his original discoveries? For all they knew, it might have all been provided to him by divine revelation. In 1872 a means of settling the question presented itself. Newton’s papers — or at least a large portion of them, covering far more than mathematical physics — had resided for nearly 150 years in the library of one of England’s aristocratic houses: Hurstbourne park. But now the Earl of Portsmouth was donating the scientific portion of papers back to the University of Cambridge. Newton had been famously coy about his own methods, suggesting that he had kept his true means of discovering the Principia private and had only cast them publicly in the language of geometry. The question was therefore whether he adhered to the rigorous, manly, and above all morally upright techniques of thinking that Cambridge undergraduates were coached to acquire. To answer this Stokes and Adams were forced to consider whether Newton himself should — or could– be held accountable to the techniques that were mastered in his name. The Newton papers had the potential to probe more deeply the shadowy divide between patient work and divine inspiration, offering the promise of settling not simply what Newton had done but how he had done it. […] the question had a special urgency at Cambridge where the moral value of study was paramount. In that respect the Newton papers mattered for every undergraduate preparing for the Tripos and for what the Tripos itself stood for. Would the man who served as a model for what should be learned also reveal himself through his private papers, as a model for how to learn? The Newton Papers – Sarah Dry, pg 88 Sarah Dry, author of The Newton Papers, a chronicle of the journey Newton’s writings took after his death, presents an interesting comparison of the two mathematicians, John Couch Adams and George Gabriel Stokes, who were tasked with making sense of his old notebook papers. On the one hand was John Couch Adams, whose ability to compute mathematically in his head was the stuff of Cambridge legend. This savant-like ability came with a tenacious reluctance to write anything down. This reluctance cost English astronomers the first opportunity to observe Neptune. Having deduced, from Uranus’ orbital irregularities, where a mystery planet should be found in the night sky, he failed to explain himself clearly to the astronomical bigwigs, who had little patience for the recent graduate. Roughly a year later, in 1846, the Frenchman Urbain Le Verrier managed to solve the problem and pointed his country’s own telescopes in the right direction. Adams was left with nothing but his incomplete written accounts and undated papers declaring his discovery, making establishing precedence impossible. Not that he seemed much bothered by losing out on the glory. He was personally very satisfied simply to have managed the computation. George Gabriel Stokes (of Navier-Stokes and Stokes’ Theorem) on the other hand, wrote compulsively, both mathematically and in personal correspondence (often to ease his own insecurities). Later in life he became editor of the Philosophical Transactions of the Royal Society, then the foremost journal in science, and this involved dealing with a huge amount of correspondence. Unfortunately he was a hoarder of papers of all and every kind, filling the rooms at his disposal with tables on which to pile up his papers. This was all compounded by his inclination towards procrastination They might have made a formidable team, had their temperaments combined to negate the other’s weaknesses. Instead the project to deliver a verdict on the value of Newton’s papers and reveal his way of thinking was subject to great delay. Of the two however, it seems that Adams was the one who was most readily able to probe the documents deeply. Sarah Dry quotes Glaisher (Adam’s obituarist) as saying: […it was a] difficult and laborious task, extending over years, but once which intensely interested him, and upon which he spared no pains. In several instances he succeeded in tracing the methods that Newton must have used in order to obtain the numerical results which occurred in the papers. The solution of the enigmas presented by these numbers written on stray papers, without any clue to the source from which they were derived, was the kind of work in which all Adam’s skill, patience, and industry found full scope, and his enthusiasm for Newton was so great that he had no thought of time when so employed. His mind bore naturally a great resemblance to Newton’s in many marked respects, and he was so penetrated with Newton’s style of thought that he was peculiarly fitted to be his interpreter. Only a few intimate friends were aware of the immense amount of time he devoted to these manuscripts of the pleasure he derived from them. John Glaisher — Memoir of the life of John Couch Adams What Adams was doing, in his own manner, was nerding out. As with the discovery of Neptune, it seems that his motivations were overwhelmingly personal, and less in service to the scientific community. Imagine a referee today reading the paper under review very carefully, but forgetting to take notes and neglecting to get back to the editor. Nevertheless, conclusions were eventually drawn out of their little committee and a report of their findings was presented. Here was confirmation that Newton had indeed worked by process of refinement that inevitably included false starts and error. In this sense, Newton revealed himself to be less an otherworldly genius and more a figure with whom the Cambridge wranglers could identify, a tireless worker in the mathematical trenches, where progress was made by increments rather than leaps. Adams knew the feeling well. In 1853 he had published an important paper pointing out errors made by Laplace in determining lunar motion and promising to provide the correct calculations soon; it had taken him six long years to get the final numbers. Here, in the papers, was evidence that Newton had worked just as hard to come up with his results. The Newton Papers – Sarah Dry, pg 105 By the time I had finished my PhD I had produced a sizable pile of used dollar-store notebooks. Browsing through them I could recognize the contours of what I’d spent the past four years trying (and occasionally succeeding) to do. I might even have reconstructed from the pictures and computations I had written out what I might have actually been thinking at the time. And aside from myself, there are a few people in the world who could possibly make sense of their contents. It all went in the recycling. If somehow one of my notebooks did manage to survive, and made its way into the hands of future scholars, I would be alarmed to consider them giving the content more than cursory attention. The hundreds (?) of pdfs that I have produced, now sitting out there in the cloud stand a far better chance at outliving me. And not just my published work and arxiv pre-prints (which number in the tens). But everything I ever committed to a latex document in my own personal space up there in the cyber heavens. Among all the discarded drafts that might find evidence of something interesting. Not only what I managed to do, but also what I failed to do. What I thought I had succeeded in doing, but had in fact betrayed my own good sense. When I have found mistakes, I am occasionally mindful enough to leave a short note in all-caps to make it clear where the point of failure lies. There is a great deal we can learn from knowing what other mathematicians have tried and failed to do. Unlike physical notebooks, our cloud storage is password protected. Digital inheritance is already “a thing”, but it seems unclear to me how it will work out in practice. Kafka left his manuscripts in the possession of Max Brod with the instructions that they be destroyed in the event of his death. Brod told Kafka himself that he certainly wouldn’t, and indeed when Kafka died at the age of 40 as a consequence of tuberculosis, Brod set about getting Kafka’s work published. I haven’t taken a survey, but I would imagine that most young writers have made no attempt to ensure their passwords and unpublished estates are in suitable hands. In principal it is possible to submit a request to Google for access to the accounts of the deceased, but I can’t imagine there are any guarantees. I certainly have no idea what the terms and conditions that I have accepted have to say about such eventualities. Newton died a man of wealth and importance. With neither wife nor children he had no direct descendants, but he did have a slew of half-nephews, half-nieces, and children of his half-sisters. The assets of obvious value were split between them. Those assets of less obvious value — the leftover pile of notebooks and “reams of loose and foul papers” fell into the possession of Catherine Conduitt. She was one of Newton’s half-nieces, and wife to John Conduitt, who had actively assisted Newton in his duties as master of the Royal Mint. This was the consequence of some rather wild tying-up of loose ends: Newton had died while holding the post of master of the Mint, which in those days required that its holder assume personal responsibility for the probity of each new coinage of money. That meant that at Newton’s death he had nominal debts amounting to the entire sum of Great Britain’s national coinage. John Conduitt agreed to take on this debt until the coinage had been certified, accepting liability for any imperfections in the coins. In exchange for assuming this risk, he asked for, and was granted Newton’s manuscripts. The Newton Papers – Sarah Dry, pg 15 The Conduitts took ownership of these papers with the view of producing a biography, and begin the work of securing Newton’s posthumous reputation. They became the first in a long line of people who had access to the papers, but lacked the tools really required to properly make sense of them. Their daughter, Kitty Conduitt married John Wallop who would become the Earl of Portsmouth, and papers would enter the library of Hurstbourne Park, seat of the Portsmouth family. (That is to say they fell into the possession of the aristocracy.) And it was there that they would remain, save occasional minor forays, and the recovery of the substantial portion of scientific papers by Stokes and Adams. What finally shifted the remaining papers out into the open was the fall of the English Aristocracy. In 1936, under the financial pressure of death duties and a recent divorce, Gerald Wallop, the ninth Earl of Portsmouth had the papers put up for auction at Sothebys. If there is a hero in Dry’s account of the Newton Papers, it must be John Maynard Keynes. His heroic virtue being exceptional taste and judgement. Having begun collecting books as a child (possibly his first foray into speculation) he developed a rather prescient sense for what should be considered valuable. Unlike the majority of collectors he shared the marketplace with, Keynes was actually interested in reading the books themselves. He was less interested in the superficial qualities: illuminations, illustrations, binding, or an illustrious list of prior owners left him unmoved. Keynes’s new style of collection was self-consciously intellectual, as opposed to aesthetic or literary. It asserted that a particular history of ideas or chain of thought linked certain men through the ages. And it projected the implicit assumption that its creator was an inheritor of both the material and the intellectual masterpieces of a previous age. Keynes was a thoroughgoing Bloomsburyite in his respect. The paintings on the wall, the rugs on the floor, the furnishing in the room, and the books on the shelves were never just things: they were the physical embodiment of ideas and values whose display was a source of both aesthetic pleasure and moral reinforcement. A book in the hand, like the good life in Bloomsbury of the Sussex countryside, linked the life of the mind with that of the physical world. The Newton Papers – Sarah Dry, pg 147 You might already get the sense that Dry sees Keynes as simply bringing a new set of beliefs to the table, complete with their own set of limitations. Indeed, Keynes considerable contribution to our modern impression and understanding of Newton as half magician and half scientist, was really a very hot take based on an initial reading. He was the one who announced that the papers reveal Newton devoted great time and energy to the disreputable pursuits of alchemy and heretical theology. Yet the fact that so many of Newton’s papers have remained together and in the possession of the University of Cambridge can be attributed to his prescience sense of the papers’ importance. Kaynes was only one of two major buyers at Sotheby’s. Abraham Yahuda, a scholar of ancient languages, bought most of Newton’s theological writings. Yahuda had found himself alienated from his own field of scholarship, due to recent developments in Higher Criticism applied to biblical scholarship. The Documentary Hypothesis was a shocking new line of textual analysis that argued the origins of the Torah were of combination and synthesis with earlier texts. As a consequence, these texts cease to resemble one coherent whole revealed to man, and begin to look more like artifacts of history and culture. For Yahuda this was a vision of criticism taken to extremes, the text reduced to nothing but error, the possibility of meaning dissolving amid a multiplicity of authors, leaving only commentary, a Talmud with no Torah left in it. He thought in particular that too many sources were being attributed to the Pentateuch and that too many “experts” were exerting themselves “in the art of text alterations and source-hunting.” Thus “the original text was distorted and disfigured and in its place was offered a quite new text of pure invention.” In Newton, who himself sought to return a blemished Christianity to its purer origins, Yahuda found a kindred soul. Interpreting ancient texts didn’t require robbing them of fixed meaning. Both Newton and Yahuda sought instead to find a singular truth amid the variations. The Newton papers – Sarah Dry, pg170 As a consequence of Yahuda’s desire to find an ally in Newton, those theological papers now reside in The National Library of Israel. The final portion of Dry’s book concerns the subsequent attempts at synthesis of the material. The fact of the matter is that the task was simply impossible. There is too much material, covering too many subjects for any grand unifying conclusions to be drawn. It was hard to even put together a definitive edition of the Principia that covered all the different editions as well as Newton’s own marginalia. When finally published it was controversial due to the inevitable editorial decisions to include or not include certain material. It is worth making one final point clear. I have never read Newtons’ Principia. I don’t believe you could find a research mathematician alive who has — unless their research happens to be the history of mathematics. It is a book whose significance is measured in its influence. Many decisions in its composition — in particular the modelling Newton’s Laws on the axioms in Euclid’s Elements — were very important. But you should not read it. When we discuss “great” books, there is usually the tacit understanding that we are missing out if we have not actually read the book. I am quite certain that we have not missed out. ## Hence, alchemy. [This post is something of a sequel to my previous post.] William Chaloner was born sometime around 1650, making him maybe a decade older than Isaac Newton. He did not receive the schooling Newton did, and he certainly didn’t have a chance at Cambridge. He had the misfortune to be apprenticed into a trade with little future: making nails. A machine — the slitting mill — had arrived that readily produced rods of steel that could easily be cut up and hammed into nails, rendering a previously skilled trade an unskilled one. With protective guilds unwilling to admit him into a more lucrative trade, and arriving in London with no obvious means to support himself, he turned to criminal enterprises. That is, Chaloner’s first attempt to rise above mere subsistence turned him into a purveyor of sex toys. London in the 1690s was as famous, or perhaps notorious, for its spirit of sexual innovation as Berlin would be in the 1920s. Prostitution was ubiquitous, as much a part of the life of the wealthy as it was that of the poor, who supplied most of the trade’s worker’s. The best brothels vied to outdo each other in their range of offerings — so much so that Dr. John Arbuthnot, a man about town in the early eighteenth century apparently spoke for many when he told a madam at one of the better houses, “A little of your plain fucking for me if you please!” Newton and the Counterfeiter by Thomas Levenson, pg 57 Chaloner soon moved on to various forms of con-artistry: quack medical advice, divination, and “thief-taking”. The latter involved informing on criminals or political subversives in order to collect financial reward. The Metropolitan police would not be formed until the 19th century, so such people were often the only avenue available for bringing criminals to justice. But the thief-takers often played both sides, exploiting whatever opportunities they could get, and often goaded people into committing crimes so that they could be “caught”. Chaloner made it his business to play both sides. However much Chaloner made from such rackets — and it would not have been inconsiderable — he wanted more. And the biggest racket in all of England was going on in plain sight, with evidence everywhere to be seen. The racket in question was counterfeiting the King’s coin. It was not an especially sophisticated game. The low production quality of hammered coins meant that an enterprising fellow could clip the edge of a coin and the coin would still be a coin, but you also had a fingernail of precious gold or silver. And in England it was silver that was of interest. The state of the commodity markets in Europe meant that you could take your pile of silver clippings to continental Europe, buy their cheaper gold, then return to England and convert it all back to silver at a profit and start clipping all over again. Classic arbitrage. As a consequence, the silver coins of England were beginning to look somewhat diminished. Many of them weren’t even silver at all. Many were outright counterfeits made of baser metals. This caused all kinds of problems, not least of which was the ability of King William III to pay his own troops to fight his war in France. Foreign bankers were unwilling to accept English currency at a good price, and silver was vanishing from England for mainland Europe. The solution was the Great Recoinage of 1696. The old coins were to be replaced with new machine-struck coins that bore milled edges to prevent clipping and render counterfeiting extremely difficult. This British state at this point in history was rife with corruption, sinecures, and cronyism, so initially at least this whole project was chaotic and in real danger of disaster. It was during this financial turmoil that Chaloner seized the opportunity and set up sophisticated counterfeiting operations that managed to produce high quality fakes of the new coins. Reading Newton and the Counterfeiter by Thomas Levenson, it is unclear if counterfeiting really was so great a scam. Certainly there was no effective law enforcement in England at this time. And while Chaloner was committing a capital offense, juries were unwilling to sentence men to death on the contradictory hearsay that actually arrived in court. That said, the kind of operation Chaloner ran required the cooperation of a great many people. Not only the skilled craftsmen required to make the dies used to cast the fake coins, and the crew to actually run the production line, but also the actual buyers for the knock-off coins. All these people could potentially betray you. Even if you did not face the executioner, you might have to endure a brief stay in London’s hellish Newgate jail. The jail used in 1696 was almost brand new, constructed on top of the ruins left by the Great Fire of 1666. The facade of the rebuilt prison was given a hint of the elegance with which its architect, Sir Christoper Wren, hoped to endow the whole city. But such graces did nothing to alter the essential character of a place that was, as Daniel Defoe’s Moll Flanders put it, not “the emblem of hell itself” but a kind of entrance to it” too. Defoe wrote from personal experience: he had been imprisoned there briefly, for debt. Other celebrated inmates confirmed Defoe’s judgement. Casanova, imprisoned at Newgate under accusation of child rape, called it “this abode of misery and despair,” and infernal place “such as Dante might have conceived.” pg 151 Chaloner would be pursued, with unusual diligence, for his crimes by the recently appointed Warden of the Mint who he had been provoking with the particular flagrancy of his crimes and deceits. In what was an act of considerable bravado Chaloner, who had already been caught for counterfeiting activities, conducted a political campaign to gain access to the Royal Mint, ostensibly to offer his “expertise”, but in reality to take whatever advantage he could. This political campaign involved impugning the newly arrived Warden. As perhaps the title of Levenson’s book has given away, this Warden was Isaac Newton, the celebrated natural philosopher. Given the absence of anything remotely like a rigorous understanding of economics, soliciting Newton’s views on the currency crisis in England was a pretty reasonable thing to do. That said, everyone seemed to have a view on potential solutions. Newton’s own views would be borne out — not just his understanding that re-coinage was necessary, but also the inevitable failure of having a currency simultaneously based on both gold and silver, and his prescient views on the potential of fiat currency. But his duties as Warden of the Mint were simply to oversee the re-coining, and prosecute clippers and counterfeiters. The first of these tasks Newton was eminently suited, given his facility with quantitative reasoning. He also had the virtue of considering his position as more than a mere sinecure. Having tired of life in Cambridge, he had been seeking some eminent position in London with which to apply his talents. He made the entire process the object of his attention, from the amount of coal consumed each day, to the rate at which the crews could, and reasonably should, hammer our the coins. Under his oversight the re-coining was completed ahead of schedule. (And to the standards of the day, far more safely than it would have otherwise been done). The second of his tasks — prosecuting counterfeiters — he abhorred. Nevertheless Newton proved himself to be utterly ruthless. The full details of the lengths he went to have been lost — in that the paperwork was deliberately destroyed in part of what was likely a cover-up. Conduitt chose not to explain why Newton wanted to destroy the papers, but one inference is that Newton enjoyed the role of inquisitor too much. In this view, Newton proved willing, perhaps eager, to terrorize his captives in pursuit of the necessary confessions and betrayals with a viciousness that even that strong-stomached time would tolerate. Formally, torture had not been used in England as an investigative tool for about half a century before Newton came to the Mint. Elizabeth I had face repeated rebellion, often animated by Catholic ambitions on her Protestant throne — and she was England’s most prolific torturing monarch … But while official torture fell out of favor, interrogators still knew how to put the boot in as needed. Isaac Newton had plenty of ways to extract the information he wanted from reluctant prisoners and he made use of them. Most of them were within the customary bounds of police detection: trading in fear, not pain. He offered brief reprieves for information: he coerced husbands with threats and promised rewards to wives and lovers. But there is one — and only one– reference to his use of more brutal methods in the records he did not burn. In March 1698, Newton received a letter from Newgate written by Thomas Carter, one of Chaloner’s closes associates. The letter was one of a flurry of messages Carter had sent to confirm that he was eager to testify against his former co-conspirator, but this one had a postscript. “I shall have Irons put on me tomorrow,” he wrote, “if yo[ur] Worship not order to the contrary.” In other words: Don’t hurt me! Please. I’ll talk. I’m ready. Newton and the Counterfeiter by Thomas Levenson pg 165 Ultimately, Newton was victorious. He was patient and methodical and able to rally his superior resources to hound his man, subjecting Chaloner to an extended stay in Newgate while he gathered witnesses and finally wrong-foot him in the trial. The trial itself being a brief and prejudiced affair, as characterized English justice at that time. But beyond the torture and lack of due process, there was a central hypocrisy to Newton’s activities. Newton and his famous chums were themselves guilty of crimes quite reminiscent to the ones he was prosecuting. The main difference, I think, was that Newton was practicing the upper class equivalents, which were not concerned with actually making a pile of money, but of a more recreational nature. Here is a passage on his relationship with John Locke: In part, he relished the opportunity to tutor so well regarded a man. He gave Locke a private, annotated edition of the Principia and composed for him a simplified version of the proof that gravity makes the planets travel elliptical orbits. But Newton’s intimacy with Locke seems to have extended well beyond such benevolent displays of mastery. From the beginning, Newton allowed himself to write openly about secret matters. Both men had subterranean interests — in alchemy, for one, the ancient study of processes of change in nature; and in questions of biblical interpretation and belief, which brought them to the edge of what the established English church would damn as heresy. Newton and the Counterfeiter by Thomas Levenson pg 43 And more seriously, and quite parallel to the crime of counterfeiting Newton was a very active alchemist. Literally attempting to turn base metals into gold. A process if successfully performed at scale would have created unprecedented economic chaos. But like I say, that wasn’t his ultimate goal. Really he was looking to alchemy to settle the theological implications of his scientific endeavors. He saw performing alchemy as a means of proving the intervention of “God” (or rather Newton’s own notion of God) in the natural world: He knew that all the theorizing, all the theological argument, all the indirect evidence from the perfect design of the solar system could not match the value of one actual, material demonstration of the divine spirit transforming one metal into another in the here and now. If Newton could discover the method God used to produce gold from base mixtures, then he would know — and not just believe — the the King of Kings would indeed reign triumphant, forever and ever. Newton and the Countereiter by Thomas Levenson pg 85 It should be understood that once you set aside all the secrecy and strange codes that Newton cloaked his alchemist pursuits in, the experiments he performed were serious and rigorous. Even if he failed to make any progress or establish any new body of knowledge. In this enterprise at least he resembles quite closely many of his peers — making quite serious, but ultimately unsuccessful attempts at making a breakthrough. William Chaloner was hanged from the neck until he was dead, on 22 March 1699. It was not the worse fate he could have met under English law. Newton was not in attendance. He would live on until 1727 when he died in his eighties and buried in Westminster Abbey. Levenson has a recently released book that seems to pick up where this one left off, tracking the rise of modern finance and the influence the Scientific Revolution had on it. I’ve also stumbled on this podcast where Cambridge historian of science, Patricia Fara, discusses her own upcoming book which seems to have considerable overlap with Levenson’s. The first question she is asked is how Isaac Newton managed to die a wealthy man, which was actually a pretty good place to start. (Newton had invested in the East India Trading company, which means, among other things, slavery.) ## There has been a murder in Gathertown If you orient yourself temporally you may remember that back in August there was a online fracas involving mathematics. A teenage girl, doing her makeup before work, decided to take the opportunity to lay down for her TikTok followers her skepticism about the idea of math generally: Who came up with this concept? “Pythagoras!” But how? How did he come up with this? He was living in the … well I don’t know when he was living, but it was not now, where you can have technology and stuff, you know? Grace Cunningham, TikTok user. As was keenly observed by the many keen observers out there, the initial response was a pile-on that combined general misogyny with gen-Z hatred; it was the latest installment in the long running complaint about kids these days. This reactionary abuse was soon countered by a more positive wave of responses that acknowledged that her questions were not only legitimate, but exactly the kind of questions our curriculum does little to answer. I don’t think many mathematicians are particularly satisfied about the way our subject is generally taught. At the university level I find it hard to love force marching students through rote material, stripping centuries worth of mathematics of all its scientific and historical context along the way. So obviously I am happy to see any student kicking back at what we inflict on them. But for those who have made mathematical communication their vocation it was a solid gold opportunity to evangelize. Euginia Cheng wrote a pdf answering Grace Cunningham’s formalized list of questions, and Francis Su wrote a twitter thread. Grace Cunningham was calling the bluff on the pretenses of her education. In particular, the pretense that you should obviously be learning whatever we are telling you. “Why are we even doing this?” is a legitimate question in a mathematics course, and “why on earth did anyone prove these theorems in the first place” is an even better one. “How did people know that they were right,” presents the awkward truth that people most often are certainly incorrect about many things. What makes these questions awkward is that the people teaching you mathematics will frequently know little to nothing about the history and context within which the theory was developed. Mathematicians are terrible, as a rule, at scholarship, and the history of ideas within mathematics is an essentially distinct field. Most of the context that I have for the mathematics I do is essentially gossip, urban myth, and pablum. Fortunately, while we might be terrible historians we remain excellent gossips, so at least we have plenty of stories to tell. (I should also concede that it is impossible to generalize in any way about most of my peers. Many of them are tremendously knowledgeable about all kinds of things and wonderful educators. I am, at least to some extent, either projecting or talking about our very worst failings.) I was dissatisfied by the responses I found to Cunningham’s questions. Not least of all because I don’t think they really answer the questions. No actual historical context was given. The answers more resemble the kind of general motivation and propaganda we give students to encourage them to listen in class and do their homework. I think a good answer would address the fact that the people who developed much of classroom mathematics had some pretty wild ideas about what they were doing. Their motivations would be pretty alien to us, and is a far cry from their homework, exams, or getting a well paid job. Just to make this explicit: How many of us who have ever taught or taken calculus a calculus course have even done any astronomy? Just from doing a little reading, an obvious observation seems to be that when people sat down to first learn calculus from Newton’s Principia, the big incentive for them was the promise of a serious set of answers about the Sun, the Earth, the Moon, the stars, and even comets. A modern mathematician explaining their motivation for calculus today is a little like a 21st century Western evangelical Christian explaining what the “Old Testament” is all about to an orthodox rabbi. My modest reading has focused on the life of Isaac Newton. I read Jame’s Gleick’s biography of Newton (highly recommended) and I have a few more on the shelf. I already had some understanding that aside from developing calculus Newton was a heretic, alchemist, and later in life warden of the royal mint. I knew he lived through times of plague, apocalypse, dictatorship, conspiracies, and his work was a major part of the scientific revolution. Particularly pertinent to Cunningham’s question is the fact that for centuries after Newton’s death there was a suppression of the full range of Newton’s intellectual activities. It was only when John Maynard Keynes acquired a substantial portion of Newton’s surviving papers at auction that the truth came out. For a long time Newton’s preoccupations would be considered intellectually inconvenient for all those trying to boost his posthumous reputation, and that of British science with it. The idea of knowledge as cumulative — as a ladder, or a tower of stones, rising higher and higher — existed only as one possibility among many. For several hundred years, scholars of scholarship had considered that they might be like dwarfs seeing further on the shoulders of giants, but they tended to believe more in rediscovery than progress. Even now, when for the first time Western mathematics surpassed what had been known in Greece, many philosophers presumed they were merely uncovering ancient secrets, found in sunnier times and the lost or hidden. Isaac Newton – James Gleick (pg 34-35) Here is a not entirely fanciful reading of Newton’s life: starting his university career dissatisfied with the existing knowledge, and curious about the latest developments in astronomy, Newton develops his theory of calculus. But he is not yet really a scientist. He is still very much a wizard. A young man who has uncovered some profound secrets and is keen to discover more. He invests huge amounts of time and energy in alchemy and theology. The alchemy involved tracking down obscure texts that he hoped would contain the secret knowledge of transforming base metals into precious metals, and his notebooks from this period often amount to his copying out these texts. It also involved working with mercury, a poisonous metal known to drive the alchemists who used it to madness. His theological interests were no less hazardous since they would have been viewed as clearly heretical to both the Protestant and Catholic religious authorities at the time. By studying the earliest Greek manuscripts he discovered that the concept of the Trinity — that the Godhead is three and one; Father, Son, and Holy Spirit — emerged late in the early church, and certainly couldn’t be considered part of the original Christian tradition. Newton concluded Jesus was not at the same level as God and had never claimed to be. At a time in England when having Catholic sympathies could land you in trouble, this was a dangerous view to have. I would argue that Newton transformed from a wizard into a scientist the moment the German mathematician Leibniz independently derived his own theory of calculus. No longer had Newton uncovered a forgotten knowledge, but he had derived a theory that someone else could also derive. He was now entered into a race to establish the precedence for his own results — and this meant writing up. For decades his tools of calculus had languished in notebooks and in his mind. Now he had to write them down, and he chose to present them in the style of Euclid’s Elements, with axioms, definitions, lemmas, theorems. And most intriguingly, in order to prove the correctness of his theory, he drew upon experimental data: astronomical observations from the newly establish Greenwich observatory and tidal charts. He was able to explain and predict natural phenomena that perplexed his contemporaries such as the sudden appearance of comets, and their unusual paths across the night sky. We can recognize this now as a prototype of the modern scientific method, but back then it was controversial, becoming part Newton’s dispute with Leibniz. Newton wrote many private drafts about Leibniz, often the same ruthless polemic again and again, varying only by a few words. The priority dispute spilled over into the philosophical disputes, the Europeans sharpening their accusations that his theories resorted to miracles and occult qualities. What reasoning, what causes, should be permitted? In defending his claim to first invention of the calculus, Newton stated his rules for belief, proposing a framework by which his science — any science — out to be judged. Leibniz observed different rules. In arguing against the miraculous, the German argued theologically. By pure reason, for example, he argued from the perfection of God and the excellence of his workmanship to the impossibility of the vacuum and of atoms. He accused Newton — and this stung — of implying an imperfect God. Newton had tied knowledge to experiments. Where experiments could not reach, he had left mysteries explicitly unsolved. This was only proper, yet the German threw it back in his face: ‘as if it were a Crime to content himself with Certainties and let Uncertainties alone.’ Isaac Newton – James Gleick (pg 176-177) Data is now the recognized currency of modern science, and theology is, well, theology. The mathematical analysis that makes calculus rigorous didn’t come until much later. Newton had started using infinite series in his calculus, but it was understood that you had to be careful because sometimes you could get some bad results. When Cunningham asks her TikTok followers how early mathematicians knew they were right, in Newton’s case at least, it seems that there are three answers. Newton first convinced himself with arguments we would not consider mathematically rigorous along with his his own empirical observations. Decades later he convinced his peers by publishing a full written account of his theory (in Latin) that provided supporting data. Then a century or so later the full theory of mathematical analysis was developed. These questions have complicated answers for Newton, but they are really no less complicated for us today, even if they are quite different answers. We live in the age of the arxiv, computer assisted proofs, machine learning, and bodies of work that amount to many hundreds of pages. I’m not going to lie; I love the drama of it all. Some would like to present mathematical proof and progress as being an enterprise free from being sullied with the humanity of its practitioners. For my part I am of the belief that the reasons people commit themselves to mathematics are more complicated than just the aesthetic appreciation of equations. ## On finite covers of surfaces with boundary… I have a new preprint on the arxiv, joint with Emily Stark. We provide the first known examples of one-ended hyperbolic groups which are not abstractly coHopfian. That means that there is a one ended hyperbolic group $$G$$ which contains a finite index subgroup $$G’ \leq G$$ that embeds $$G’ \rightarrow G$$ as an infinite index subgroup. I encourage you to look at the paper for details. The main example and proof can be drawn out on a single side of A4 — it’s a simple surface amalgam and we exploit the tremendous flexibility you have when you take a finite cover of a surface with boundary. We use the following Lemma extensively. It’s from Walter Neumann’s 2001 paper Immersed and virtually embedded $$\pi_1$$-injective surfaces in 3-manifolds, although, as he says, it is apparently “well known”. The utility of this Lemma is that it reassures you that if you can imagine your desired cover — such as the following I’ve drawn below — and it satisfies a basic necessary Euler characteristic computations, then the cover does in fact exist. For our main example you can compute your desired covers by hand, but it is worth knowing what kind of covers of a surface with boundary you can take. This lemma tells you exactly how much control you have. And mathematical research, like all forms of insecurity, is really all about control. The proof given above is brief, to say the least, so I think it is worth expanding on the details. First, we remind ourselves how you might construct such a cover by hand. Take a surface with genus one and a single boundary component. From a group theoretic point of view this is just the free group generated by two element $$\mathbb{F}_2 = \langle x, y \rangle$$ bundled together with the conjugacy class of the commutator $$[x,y]$$. For me at least, finding finite index subgroups of the free group boils down to futzing around with graphs. Thus, we let $$X$$ be a bouquet of two circles and let $$\langle x,y \rangle = \pi_1X$$. On the right I drew the surfaces with boundary and on the left I drew the corresponding graphs with the loop corresponding to the boundary. Once you have drawn the graphs out it’s easy to verify that you have the boundary components you want. The trick is knowing you can find the desired finite covers. The key insight is that $$\alpha$$-sheeted covers of a graph $$X$$ are in a correspondence with representations into the permutation group on $$n$$ elements: $$\pi_1 X \rightarrow \textrm{Sym}(\alpha)$$ We can see how this correspondence works in practice in the example I just drew: Giving each of vertex a number we can see that the edges labelled by a given generator of our free group gives a permutation. In this example the generator $$x$$ gives the permutation $$(1,2)(3)$$ (see the red edges on the right), while the generator $$y$$ gives the permutation $$(1,2,3)$$ (see the blue edges on the left). Thus we have a homomorphism determined by mapping the generators to the corresponding permutation. At this point we need to be careful because there are some left-right issues hidden here. When I multiply group elements $$xy$$ I am composing paths in the fundamental group. That means I concatenate the corresponding paths, starting with the $$x$$ path and then following it with the $$y$$ path. I’m reading the composition from left to right. In contrast, when I usually compose a pair of permutations $$\sigma_1 \sigma_2$$ I compute the composition by reading them from right to left. But in order to be able to interpret my homomorphism correctly I’m going to have to compose my permutations in reverse order, from left to right. Now we can compute the image of the element corresponding to the boundary curve: $$xyx^{-1}y^{-1} \mapsto (1,2) \circ (1,2,3) \circ (1,2) \circ (3,2,1) = (1,2,3).$$ Tracing out how the boundary curve lifts is equivalent to computing this permutation element. This makes it clear that there is a single boundary component covering the previous with degree 3. (In this example it doesn’t matter in which direction we composed the permutations). Conversely, choosing pair of permutions, say, $$(1,2)(3,4)\textrm{, and } (2,4,3) \in \textrm{Sym(4)}$$ to be the images of $$x$$ and $$y$$ we can construct a corresponding cover by taking 4 vertices and adding the appropriate labelled edges: Now when we compute (remembering to compose our permutations from left to right) the image of our commutator element we get $$xyx^{-1}y^{-1} \mapsto (1,2)(3,4) \circ (2,4,3) \circ (1,2)(3,4) \circ (3,4,2) = (1,4)(2,3).$$ Thus the surface has two boundary components, each covering the boundary in the base surface with degree two. The take away from this discussion is that finding suitable covers corresponds to finding a suitable homomorphism $$\phi : \pi_1 X \rightarrow \textrm{Sym(\alpha)}$$ such that the image of the elements corresponding to boundary curves are permutaions with the desired decomposition into cycles. Our weapon of choice is the fact that any even permutation can be written as the commutator of an $$\alpha$$-cycle and an involution: First we consider the case where $$\Sigma$$ has a single boundary component. So $$\Sigma$$ is a surface with genus $$g$$, Euler characteristic $$\chi(\Sigma) = 2 – 2g – \|\partial \Sigma \|$$ and a single boundary component, so $$\|\partial \Sigma \| = 1$$. This corresponds to the free group generated by $$2g$$ elements and the group element corresponding to the boundary, which is the product of commutators: $$( \langle x_1, y_1, \ldots x_g, y_g \rangle, [x_1,y_1]\cdots [x_g, y_g] ).$$ Suppose we wish to construct a cover of degree $$\alpha$$ with the boundary components of degrees $$\alpha_1, \ldots, \alpha_k$$. Then apply the above Theorem to the permutation $$\sigma = (1, \ldots, \alpha_1)(\alpha_1 +1, \ldots, \alpha_1 + \alpha_2) \cdots (\alpha_1 + \cdots + \alpha_{k-1} +1, \ldots, \alpha).$$ The theorem only applies if $$\sigma$$ is an even permutation, which we compute to be equivalent to $$\sum_i (\alpha_i -1) = \alpha – k$$ being even. As $$\chi(\Sigma) = 1-2g$$ this is equivalent to $$k$$ having the same parity as $$\alpha\chi(\Sigma)$$, the sufficient condition given in the statement of our theorem. Thus there exists permutations $$\sigma_x, \sigma_y \in \textrm{Sym}(\alpha)$$ such that $$[\sigma_x, \sigma_y] = \sigma$$. The homomorphism $$\phi: \pi_1 \Sigma \rightarrow \textrm{Sym(\alpha)}$$ given by mapping $$x$$ to $$\sigma_x$$ and $$y$$ to $$\sigma_y$$ therefore corresponds to a cover with the desired boundary. Now we consider the slightly trickier general case where we have multiple boundary components, which is to say $$\|\partial \Sigma\| = b$$. In which case the pair $$(\Sigma, \partial \Sigma)$$ corresponds to $$(\langle x_1, y_1, \ldots, x_g, y_g, t_1, \ldots t_{b-1} \rangle , \{t_1,\ldots, t_{b-1}, t_{b-1}\cdots t_{1}[x_1,y_1] \cdots[x_n,y_n] \} ).$$ Now suppose we desire that the $$i$$-th boundary component is covered with degrees $$\alpha_1^i, \ldots, \alpha_{k_i}^i$$, then let $$\sigma_i = (1, \ldots, \alpha_1^i)(\alpha_1^i +1, \ldots, \alpha_1^i + \alpha_2^i) \cdots (\alpha_1^i + \cdots + \alpha_{k_i-1}^i +1, \ldots, \alpha)$$ for $$1 \leq i \leq b$$. Now we wish to find $$\sigma_x, \sigma_y$$ such that $$[\sigma_x, \sigma_y] = \sigma_1 \cdots \sigma_b.$$ This requires that the product of the $$\sigma_i$$ is even. This means that the sum $$\sum_i \sum_j (\alpha_j^i – 1) = \sum_i (\alpha – k_i) =\alpha b – \sum_i k_i = \alpha (\chi(\Sigma) -2 + 2g) – \sum_i k_i$$ should be even, which is true precisely when the total number of prescribed boundary components $$\sum_i k_i$$ has the same parity as $$\alpha \chi(\Sigma)$$. Given that our parity condition is satisfied, we define our homormorphism $$\pi_1 \Sigma \rightarrow \textrm{Sym}(\alpha)$$ as follows: \begin{align} x_1 \mapsto & \sigma_x \\ y_1 \mapsto & \sigma_y \\ x_2 \mapsto & 1 \\ \vdots \\ y_g \mapsto & 1 \\ t_1 \mapsto & \sigma_1^{-1} \\ t_2 \mapsto & \sigma_2^{-1} \\ \vdots \\ t_{b-1} \mapsto & \sigma_{b-1}^{-1} \end{align} Then it only remains to verify that $$t_{b-1}\cdots t_{1} [x_1,x_2] \cdots [x_g, y_g] \mapsto \sigma_{b-1}^{-1} \cdots \sigma_1^{-1} [\sigma_x, \sigma_y] = \sigma_{b-1}^{-1} \cdots \sigma_1^{-1} \sigma_1 \cdots \sigma_{b-1} \sigma_b = \sigma_b$$ and conclude this gives us our desired cover. QED. (I’d like to thank Emily for informing me about Neumann’s Lemma, and Nir for various discussions related to this.) ## Gromov, cheese, pretending to quit mathematics, and French. In December last year, the Notices of the AMS ran a collection of reminiscences in memory of Marcel Burger (1927-2016), the late French differential geometer. He was also a former director of the Institut des Haute Etudes Scientific and, according to the Wikipedia, played a major role in getting Gromov positions in Paris and at the IHES in the 80s. Gromov contributed to the article, listing Berger’s mathematical achievements, before sharing a more personal anecdote: Within my own field Gromov has had a profound influence. His essay Hyperbolic Groups led to the term “combinatorial group theory” being more or less abandoned and replaced with “geometric group theory”. As a graduate student I found the monograph frequently cited as the origin for an astounding range of ideas. At some point I had trouble finding a copy of the paper online and for a brief moment wondered if the paper itself were just an urban myth or elaborate hoax. Gromov’s foray into group theory is just one episode in a long career. His first major breakthrough was in partial differential equations; the “h-principal” which, according to Larry Guth, was analogous to observing that you don’t need to give an explicit description of how to put a wool sweater into a box in order to know that you can actually put it into the box. It is actually a little hard, at least for myself, to get a full grasp on Gromov’s other contributions as they span unfamiliar fields of mathematics, but I recommend this nice What is… article, written by the late Marcel Berger, describing Gromov’s contribution to the understanding of isosystolic inequalities. There is the perception about great mathematicians that, while they are no doubt very clever, somehow they have lost a little of the common sense that the rest of us possess. I’m naturally inclined to discount such thinking; I am far happier believing that we are all fool enough to take absence of common sense, on certain occasions. However, it is hard to dismiss the idea entirely given the following admission by Gromov, in his personal autobiographical recollections he wrote on receipt of the Abel prize in 2009: The passage speaks for itself, but I wish to emphasize that Gromov’s discovery of the correct pronunciation of French verb endings came after ten years of living in Paris. You don’t have to have extensive experience learning foreign languages to appreciate how remarkable an oversight this is. It certainly puts his remarks in other interviews about dedicating one’s life to mathematical pursuits in a rather strong light. If you read the entire autobiographical essay you will find it rather short on biography. The best biographical details I have found came from this La Monde article written on his being awarded the Abel prize. Even then the details seems to be coming second hand. The most interesting parts concern his leaving the Soviet Union: I wanted to leave the Soviet Union from the age of 14. […] I could not stand the country. The political pressure there was very unpleasant, and it did not come only from the top. […] The professors had to teach in such a way as to show respect for the regime. We felt the pressure of always having to express our submission to the system. One could not do that without deforming one’s personality and each mathematician that I knew ended up, at a certain age, developing a neurosis accompanied by severe disorders. In my opinion, they had become sick. I did not want to reach that point. Gromov, according to Georges Ripka, via La Monde (apologies for my translation) As the article goes on to explain, Gromov decided his best chance to escape was to hide his mathematical talent. He quit math, quit his university, and burned all his academic bridges. He stopped producing mathematics. Or at least writing it. He joined some meteorological institute and did research on paper pulp. Eventually, he was granted permission to emigrate to Israel, but on landing in Rome in 1974 he set off for the States instead, where Jim Simons secured him a position at Stony Brook. (As a side note, this is Jim Simons of Renaissance Technology and Simons foundations fame. Simons, aside from his mathematical contributions, is probably one of the most important mathematicians alive in terms of funding, supporting, and propagandizing for mathematics. He is considered influential enough for the New Yorker to profile. Alongside Gromov, he is one of the names that every mathematicians should know.)	2021-09-27 07:06:43	{"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": 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": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6144435405731201, "perplexity": 1640.4522907015812}, "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-2021-39/segments/1631780058373.45/warc/CC-MAIN-20210927060117-20210927090117-00218.warc.gz"}
http://math.stackexchange.com/questions/117303/how-to-show-that-lim-p-to-0-fracpe2tp1-e2tp1-p-12t-k	# How to show that $\lim_{p\to 0}\frac{pe^{2tp}}{1-e^{2tp}(1-p)}=(1+2t)^{-k}$ How would I find the following limit? I am thinking I might need to do a Taylor expansion of the exponentials but it hasn't worked out nicely. Thanks for the help. $$\lim_{p\to 0} \left( \frac{pe^{2tp}}{1-e^{2tp}(1-p)} \right)^k$$ The answer is $(1+2t)^{-k}$. - Naive application of L'Hopital's: $$\begin{eqnarray} \lim\limits_{p\to 0}\left(\frac{pe^{2tp}}{1-e^{2tp}(1-p)}\right)^k &=&\left(\lim\limits_{p\to 0}\frac{pe^{2tp}}{1-e^{2tp}(1-p)}\right)^k\\ &=&\left(\lim\limits_{p\to 0}\frac{e^{2tp}+2p^2e^{2tp}}{2pe^{2tp}(1-p)-e^{2tp}}\right)^k\\ &=&\left(\lim\limits_{p\to 0}\frac{e^{2tp}+2p^2e^{2tp}}{2pe^{2tp}-(e^{2tp}+2p^2e^{2tp})}\right)^k\\ \end{eqnarray}$$ Perhaps examine similarity to $\frac{x}{x-y}$? – Alex Becker Mar 7 '12 at 0:57 Have you tried l'Hospital's rule? $$\lim_{p \to 0} \frac{pe^{2tp}}{1-e^{2tp}(1-p)} = \lim_{p \to 0} \frac{e^{2tp}+p(2t)e^{2tp}}{e^{2tp} - (1-p)(2t)e^{2tp}} = \lim_{p \to 0} \frac{1+2tp}{1-(1-p)(2t)} = \frac {1}{1-2t}$$ Just use the product rule and the case $k=1$ then.	2016-06-25 16:20:01	{"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": 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": 0.8019569516181946, "perplexity": 324.05395279455445}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1466783393442.26/warc/CC-MAIN-20160624154953-00148-ip-10-164-35-72.ec2.internal.warc.gz"}
http://www.physicsforums.com/showthread.php?t=617357	## DoE Picking winners and losers; here we go again http://www.foxnews.com/politics/2012...est=latestnews $400 million dollar loan guarantee from the Obama administration for yet anothor solar panel maker. Anyone else think the government doesn't make a very good entrepreneur. They had 180 full time employees.... that's 2.2 Million per job..... hmmm, just give me the money please. IMO, the government does a good job kickstarting technology. e.g. look at the things that came out of the space program. Seed money for startups shouldn't be the role of the government. IMO, that's what private venture capital is for, and they will probably watch their money better than the government watches our money. PhysOrg.com science news on PhysOrg.com >> Leading 3-D printer firms to merge in$403M deal (Update)>> LA to give every student an iPad; $30M order>> CIA faulted for choosing Amazon over IBM on cloud contract Recognitions: Gold Member How many billions were lost by our current government in solar and other private companies? I feel that grants for basic research may be appropriate but with our ballooning debt the federal government should be more responsible with our money. http://www.businessweek.com/news/201...ech-media-says Abound Solar Inc., a U.S. solar manufacturer that was awarded a$400 million U.S. loan guarantee, will suspend operations and file for bankruptcy because its panels were too expensive to compete. Quote by ThinkToday http://www.foxnews.com/politics/2012...est=latestnews $400 million dollar loan guarantee from the Obama administration for yet anothor solar panel maker. Anyone else think the government doesn't make a very good entrepreneur. They had 180 full time employees.... that's 2.2 Million per job..... hmmm, just give me the money please. IMO, the government does a good job kickstarting technology. e.g. look at the things that came out of the space program. Seed money for startups shouldn't be the role of the government. IMO, that's what private venture capital is for, and they will probably watch their money better than the government watches our money. Do you have any idea of the costs for a startup? It pays for more than just salaries. It also pays for equipment, overhead, etc. In a high tech startup, the costs for the equipment are quite steep. In the company I work for (a high tech startup), the cost for the raw materials alone is almost 3 million per unit (it's a combination MRI/teletherapy device) ## DoE Picking winners and losers; here we go again No doubt about it. However, this wasn't to fund development of some new technology. It's just starting a business. That's what venture capital and banks are for. If you can't convince venture capital it's a good bet/investment, why is the government getting involved? Look at SpaceX, they are private and have government contracts to do real work. If memory services me, I think they get some NASA support for monitoring, etc., but it's otherwise private funds at risk and not tax dollars. Quote by ThinkToday$400 million dollar loan guarantee from the Obama administration for yet anothor solar panel maker. They had 180 full time employees.... that's 2.2 Million per job..... > Implying that the costs of a company are solely due to employee's wages. Quote by Dickfore > Implying that the costs of a company are solely due to employee's wages. No. Implying the cost per job created isn't worth it. i.e. not a cost efficient job creator. e.g. may as well give each person $2.2M and let them retire. Quote by ThinkToday No. Implying the cost per job created isn't worth it. i.e. not a cost efficient job creator. e.g. may as well give each person$2.2M and let them retire. But, in capitalistic societies, companies are not open to "create jobs". Companies are created to make profit by selling goods and services. The inputs in creating goods and services are Physical Capital (factories, machines, desks, computers, etc.), Raw Material (quarz, silicon), and Human Capital (physical and mental labor, the technical know-how of your employees). As you can see, employees are one of the three main categories in the inputs that go into creating the good. The owner of the company treats their wages as costs of production when figuring out the price/quantity of goods produced. If the company works with net profit, it means that the salaries are already paid for by the employer, and then he/she earns the difference as a compensation for their entrepreneurial effort. One of the costliest inputs is the Physical Capital. What is different than the other inputs, it is a fixed cost, i.e. you must make the factory, regardless of how many units of the good you produce. One usually borrows funds from investors to start-up a company. Then, one pays of the funds borrow plus an interest over a longer period of time. This also enters in the price of the goods sold. Thus, the investors, who lent the money to the company, will eventually get their investments repaid, plus some interest for the service they had provided. So, in the long run, you had made a factory for solar panels which creates profit every year, the investors got their initial investment, plus some interest for their services, and many people got an opportunity to sell their labor for wages, a situation that would not have been possible, had it not been for the factory. Recognitions: Gold Member The Chinese duplicated American thin film solar panel technology and the Chinese government had started manufacturing them while Americans were still trying to find investors. China beat the us to the punch. They took what U.S. leaders knew, but hesitated in developing a market for, and ran with it all the way to the bank. And now that China has shown the way to these advanced possibilities http://www.reepedia.com/archives/tag/solar Why is it that the far right seems to be thrilled that another enterprise has failed. What if we all did that? Do you want Obama to fail so badly that it is OK even if it takes the entire country down? In the meantime the big boys on Wall Street who could have invested their money in American Solar are either shorting solar or looking for new hedging oportunities. http://www.dailyfinance.com/2012/04/...hedge-divorce/ Recognitions: Gold Member Quote by ThinkToday http://www.foxnews.com/politics/2012...est=latestnews $400 million dollar loan guarantee from the Obama administration for yet anothor solar panel maker. ... One point to the DoE's credit on this topic: they laid out milestones for Abound and cut them off after$70 million into the total $400 million on the table. Quote by Dickfore But, in capitalistic societies, companies are not open to "create jobs". Companies are created to make profit by selling goods and services. The inputs in creating goods and services are Physical Capital (factories, machines, desks, computers, etc.), Raw Material (quarz, silicon), and Human Capital (physical and mental labor, the technical know-how of your employees). As you can see, employees are one of the three main categories in the inputs that go into creating the good. The owner of the company treats their wages as costs of production when figuring out the price/quantity of goods produced. If the company works with net profit, it means that the salaries are already paid for by the employer, and then he/she earns the difference as a compensation for their entrepreneurial effort. One of the costliest inputs is the Physical Capital. What is different than the other inputs, it is a fixed cost, i.e. you must make the factory, regardless of how many units of the good you produce. One usually borrows funds from investors to start-up a company. Then, one pays of the funds borrow plus an interest over a longer period of time. This also enters in the price of the goods sold. Thus, the investors, who lent the money to the company, will eventually get their investments repaid, plus some interest for the service they had provided. So, in the long run, you had made a factory for solar panels which creates profit every year, the investors got their initial investment, plus some interest for their services, and many people got an opportunity to sell their labor for wages, a situation that would not have been possible, had it not been for the factory. I don't disagree with your analysis. But, the company went under and US tax dollars with it. My beef is with the use of US tax dollars to plow into a business that, if sound, should have been able to get going without them. Quote by mheslep One point to the DoE's credit on this topic: they laid out milestones for Abound and cut them off after$70 million into the total \$400 million on the table. Very true. However, they weren't investing hundreds of millions in some cutting edge technology development. It's manufacturing solar panels. So why did the US government feel the need to step in to fund it with our tax dollars? Did the banks and venture capital folks not see the prospects...... or perhaps they did...... lol Quote by edward Why is it that the far right seems to be thrilled that another enterprise has failed. What if we all did that? Do you want Obama to fail so badly that it is OK even if it takes the entire country down? In the meantime the big boys on Wall Street who could have invested their money in American Solar are either shorting solar or looking for new hedging oportunities. That's way over the top. First, the free market system is at the heart of conservatives. When the government gets involved and starts picking winners (those that get cheap easy government money - our tax dollars) and losers (those that have to depend on selling their ideas, marketing strategy, technology to bankers and venture capital folks) it interferes with the survival of the fittest mentality behind capitalism. Companies fail all the time in that environment, and that's nothing new. http://www.fastcompany.com/1841038/t...happens-to-you Far from your assertion. Conservatives don't like companies to fail, but they make no bones about when the government gets involved in private enterprise and it fails. It is kind of an “I told you so”. Government involvement in something like fusion research is necessary, IMO, because the risk of failure is so high no venture investor would touch it. Things like that, yes, the government has a roll. Quote by ThinkToday I don't disagree with your analysis. But, the company went under and US tax dollars with it. My beef is with the use of US tax dollars to plow into a business that, if sound, should have been able to get going without them. The problem is that some enterprises do not show net (accounting) profit over the period of several years, provided that the enterpreneur borrows money from a private financial institution at the current situation of the economy. By this, I mean that the current interest rate for borrowing is sufficiently high, and the price of substitute goods (in this case, electricity generated by fossil fuels or nuclear reactions) is sufficiently low that it is unfavorable for any enterpreneur to go to the private market for opening a business with renewable energy. Here comes the view of the current government. Spurred by non-market derived motives, they might decide that it would be preferrable to develop renewable energy industry for the long-run welfare of the society. It is a gamble, but so is every other enterprise. Since they are currently in power, it is ultimately their decision and they take the gamble. They rely on the assumption that in the future, the relative cost of fossil fuel energy compared to the renewable energy would be much higher than it is today, so that the business would start being profitable. Furthermore, since it was a pioneer in the field, it would exert some kind of market power on the world market, and the U.S. would be a world leader in the field (similar to Boeing in the aeroplane manufacturing business). Furthermore, the dwvelopment of this industry will exert a possitive externality on the problem of reduction of greenhouse gases. Under such a speculative cost-benefit analysis, the government decides to enter as a no-interest financeer for the private enterprise (in the lack of private financial institutions willing to invest at the current market conditions), by essentially giving a subsidy. Notice that distributing the money of the investment between the populace would not count towards the chunk known as government spending in the GDP, since it is a form of an in-kind transfer. In the way we described above, however, it would count as a public investment. And any investment, no matter if public or private, would ultimately accelerate the long-term growth of the economy. This isn't the DoE picking winners and losers- its the DoE giving out business loans. Its not government taking control of business. As the expression is usually used 'picking winners' is just direct subsidy of an enterprise- enough to keep it afloat regardless of economic conditions. If the government wants to encourage development of green energy it can directly subsidize the industry, or create these sorts of loan programs- which is the more free market approach? Now, maybe you think the government shouldn't be encouraging the development of green energy- but thats a different conversation. Recognitions: Gold Member Homework Help Science Advisor Nothing new under the Sun? (pun intended) One could say our solar energy subsidies work no better than our nuclear energy subsidies. (After 50 Years, Nuclear Power is Still Not Viable without Subsidies, New Report Finds). So, yeah, it is a "here we go again" situation that's no different than the things we've been doing for at least 50 years. On the other hand, we do need to eventually solve our energy problem before the oil starts running out. Blog Entries: 14 Quote by BobG Nothing new under the Sun? (pun intended) One could say our solar energy subsidies work no better than our nuclear energy subsidies. (After 50 Years, Nuclear Power is Still Not Viable without Subsidies, New Report Finds). I have yet to read the article carefully. Few concerns just looking at it: 1) Economic analysis is coming from Environmental group. 2) They don't seem to compare nuclear to other energy sources 3) "Carbon price" is kind of full of BS All this makes me really weary to give any serious consideration to this article. Quote by BobG Nothing new under the Sun? (pun intended) One could say our solar energy subsidies work no better than our nuclear energy subsidies. (After 50 Years, Nuclear Power is Still Not Viable without Subsidies, New Report Finds). So, yeah, it is a "here we go again" situation that's no different than the things we've been doing for at least 50 years. On the other hand, we do need to eventually solve our energy problem before the oil starts running out.	2013-06-20 12:06:57	{"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": 0.22793681919574738, "perplexity": 2024.8748549968348}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368711605892/warc/CC-MAIN-20130516134005-00081-ip-10-60-113-184.ec2.internal.warc.gz"}
http://clay6.com/qa/13325/if-i-is-the-identify-matrix-of-order-2-and-a-begin-1-1-0-1-end-then-for-n-g	Browse Questions # If I is the identify matrix of order 2 and $A= \begin {bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$, then for $n \geq 1$, mathematical induction gives $\begin {array} {1 1} (a)\; A^n=nA-(n-1)I & \quad (b)\;A^n=nA+(n-1)I \\ (c)\;A^n=2^n A -(n+1)I & \quad (d)\;A^n=2^{n-1}A-(n-1)I \end {array}$ Can you answer this question? $(a)\; A^n=nA-(n-1)I$ answered Nov 7, 2013 by	2017-03-31 00:33:51	{"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": 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": 0.9629598259925842, "perplexity": 1250.4806653524502}, "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-2017-13/segments/1490218205046.28/warc/CC-MAIN-20170322213005-00435-ip-10-233-31-227.ec2.internal.warc.gz"}
http://h2.jaguarpaw.co.uk/posts/strictness-in-types/	# Reflecting strictness in Haskell types In GHC, from the operational point of view, the type Int does not indicate a bit pattern somewhere in memory that represents an integer. What it indicates is either an integer bit pattern or a “thunk” (a delayed computation) which can be “forced” (run). When and if the computation terminates the thunk will be overwritten with the integer bit pattern that it produced. Thus the GHC type system provides less fine-grained information than we might naively expect, especially if we are used to the strict point of view. Often the ability to invisibly interchange thunks and values is a blessing. In fact it is one of Haskell’s great blessings. However, it is well known that it can also be a curse. Here’s an example: foldl :: (a -> b -> a) -> a -> [b] -> a foldl _ z [] = z foldl f z (b:bs) = foldl f (f z b) bs The application of f to z and b does not result in an evaluated a. Instead it results in a thunk representing the function call. If the input list is of length n we make n recursive calls and build up a thunk whose size is proportional to n. This leaks space. The way to avoid a space leak is to explicitly force the a we have created with a call to seq before each recursive call. foldl' :: (a -> b -> a) -> a -> [b] -> a foldl' _ z [] = z foldl' f z (b:bs) = a seq foldl f a bs where a = f z b This way the a remains completely evaluated at each stage. This is a reasonable solution, but as Haskellers we are used to our types specifying very fine-grained information about the operation of our program. We want our types to specify when IO actions can occur, so why not ask for our types to specify when evaluation of thunks takes place? Unfortunately this isn’t straightforward to come by. For example, we might try data Strict a = Strict !a foldl :: (a -> b -> a) -> Strict a -> [b] -> a foldl _ (Strict z) [] = z foldl f (Strict z) (b:bs) = foldl f (Strict (f z b)) bs But this is no good. After all, a Strict a doesn’t represent a strict a but rather a thunk returning a strict a! So we build up a chain of thunks nonetheless. In fact there is no difference in terms of laziness between a and our attempted Strict a. Both of them are just thunks that can be forced to return an a. However, there is a trick that allows us to represent strictness in types. The trick is to indicate strictness in the argument to a function rather than its return type. I learned this from a post of Stefan Holdermans to the Haskell-Cafe mailing list. {-# LANGUAGE TypeOperators #-} data a :-> b = Strict (a -> b) -- Strict constructor will be hidden -- Specifying the precedence of :-> merely for neater syntax infixr 0 :-> strictly :: (a -> b) -> a :-> b strictly f = Strict (\a -> a seq f a) (!) :: (a :-> b) -> a -> b Strict f ! a = f a Then we can write a recursive foldl which is guaranteed to be space leak free because it has the right type. foldl' :: (a -> b -> a) -> a :-> [b] -> a foldl' f = strictly (\z xs -> case xs of [] -> z y:ys -> foldl' f !(f z y) $ys) The value whose type is on the left hand side of a :-> is guaranteed to be evaluated strictly, i.e. forced before the result can start to be consumed. ## Polymorphic strictness What do we do if we want to define functions which are polymorphic over strictness type? For example the standard definition of const doesn’t touch its b argument when its return value is evaluated. const :: a -> b -> a const a b = a On the other hand we can define another version which does const' :: a -> b -> a const' a b = b seq a How do we implement both of these with one definition? Well, we can introduce a class to capture the general concept with instances for (->) and (:->). (The need for the notation arrow instead of something symbolic is a syntactic annoyance. I’m not sure how to get round it.) class FunctionLike arrow where (?) :: (a arrow b) -> a -> b functionLike :: (a -> b) -> (a arrow b) instance FunctionLike (->) where (?) = id functionLike = id instance FunctionLike (:->) where (?) = (!) functionLike = strictly Then we can define polymorphic const as constPolymorphic :: FunctionLike arrow => a -> b arrow a constPolymorphic a = functionLike (\_ -> a) and use it to directly derive the two specialisations that we want. const :: a -> b -> a const = constPolymorphic const' :: a -> b :-> a const' = constPolymorphic If we were so inclined we could use FunctionLike to define the leaky and leak free versions of foldl at the same time! foldlPolymorphic :: FunctionLike arrow => (a -> b -> a) -> a arrow ([b] -> a) foldlPolymorphic f = functionLike (\z xs -> case xs of [] -> z y:ys -> foldl' f ?(f z y)) The operator ? should be interpreted as function application which is polymorphic over strictness. ## Strictness in data types Even more amazingly, gelisam on Haskell Reddit showed me how to make data types that are strictness polymorphic! The key observation is that the only difference between the usual, lazy data Foo a = Foo a and the strict data Foo a = Foo !a is that the function which constructs Foos is lazy in its argument for the former, and strict in its argument for the latter. If we want to be polymorphic over strict and lazy lists, for example, we can define data List (arrow :: * -> * -> ) a = Nil \| Cons a (List arrow a) -- The List constructor will have to be hidden, although pattern -- matching on it would be OK. arrow is a phantom type which we -- use as a strictness indicator. -- We expose this polymorphic cons instead cons :: FunctionLike arrow => a arrow (List arrow a arrow List arrow a) cons = functionLike (\x -> functionLike (\y -> Cons x y)) (Again I apologise for the syntax. Hopefully someone knowledgeable can tell me what to do about that.) A List (->) a is a lazy list of a, and a List (:->) a is a strict list of a. We can define data values in a way that is polymorphic over strictness type, for example enumerating all integers in a range. range :: FunctionLike arrow => Int -> Int -> List arrow Int range a b = if a > b then Nil else cons ?a ?(range (a+1) b) (Recall that the operator ? should be interpreted as function application which is polymorphic over strictness.) Then count 1 1000000 :: List (->) Int is a lazy list where the one million elements are computed as required. count 1 1000000 :: List (:->) Int is a strict list where all one million elements are computed at once. ## Wadler’s “strictness monad” Wadler talks about a “strictness monad” in section 3.2 on page 9 of Comprehending Monads but I don’t think it solves the problem I want it to solve. For example, consider foldl :: (a -> b -> a) -> Str a -> [b] -> Str a foldl _ z [] = z foldl f z (b:bs) = foldl f (f <$> z <> pure b) bs This still builds up a chain of thunks in the same way that our original Strict attempt did. All the Str monad seems to achieve is enforcing an ordering of evaluations with respect to each other. Perhaps there’s a way of getting the desired behaviour with a cleverer usage of Str, but I don’t see it. ## Conclusion Strictness in Haskell is generally hidden away and not reflected by types, but there does seem to be a way of representing strictness in the type system!	2021-10-23 12:29:03	{"extraction_info": {"found_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": 0.5376914739608765, "perplexity": 1759.2058993212127}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585671.36/warc/CC-MAIN-20211023095849-20211023125849-00213.warc.gz"}
https://brilliant.org/problems/polynomial-conditions-in-2015/	# Polynomial Conditions in 2015 Algebra Level 4 True or False? There exists a polynomial $f(x)$ with integer coefficients such that $f(20) = 15$ and $f(15) = 9$. ×	2021-08-06 02:50:25	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 3, "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": 0.7088006138801575, "perplexity": 2037.5530071910027}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1627046152112.54/warc/CC-MAIN-20210806020121-20210806050121-00506.warc.gz"}
https://pwntestprep.com/wp/tag/plug-in/page/4/	## Posts tagged with: Plug In It almost never fails when I’m working with a new student that the first time a question involving remainders rears its ugly head, we need to spend some time talking about what remainders are and how to find them. This happens fairly early in the process with me, intentionally. I always start with new students by working through the plug in and backsolve chapters in my math book, and the plug in chapter prominently features a remainder question. That’s what’s discussed in the video above. A remainder, just so we’re clear, is what’s left over when one positive integer is divided by another. When 17 is divided by 5, for example, the remainder is 2: 5 goes into 12 three times—making 15—but there’s still 2 left over to get to 17. Remember long division? Remainders are always whole numbers, never decimals. However, there’s a handy shortcut that’ll allow you to convert decimals to remainders: 1. Do the division on your calculator. 2. If there’s no decimal, then there’s a remainder of 0. If there is a decimal, then there’s a remainder. 3. Convert the decimal to a remainder by subtracting the part before the decimal point from the quotient you have on your calculator. 4. You’ll be left with only a decimal. Multiply that decimal by the original divisor. 5. Boom! You’ve got a remainder. Here, I’ll show you. Let’s do 17/5 again. When you put 17/5 into your calculator, you get 3.4. Subtract 3 from that, and you’re left with 0.4. Multiply that by 5, and you’re left with 2—that’s your remainder! Note that if there’s a repeating decimal, you shouldn’t round it or you won’t get an integer remainder. Here’s another example with the exact keystrokes I enter into my TI-83, and what the calculator displays. What is the remainder when 52,343 is divided by 92? ```52343/92 [ENTER] 568.9456522 Ans–568 [ENTER] .9456521739 Ans*92 [ENTER] 87``` Therefore, 52,343/92 = 568 R 87. Cool right? Any remainder operations you’ll be doing on the SAT will be far less involved and easily done with long division, so you don’t need to memorize this trick, but it’s there for you if you want it. ## Is that all? …Nope. If you end up having to deal with remainders on your SAT, you’ll almost definitely have to do more than just divide two integers and find the remainder. You’ll probably be asked (as you are in the problem featured in the video above) to figure out something about unknown constants given some information about remainders. When that happens to you, here are the things it’s important to know: If n is divided by k and leaves a remainder of r, then n must be r greater than a multiple of k. For example, if a number divided by 8 leaves a remainder of 3, then that number must be 3 greater than a multiple of 8. You’ll do well to plug in a nice, low number that fits that description, like 8 + 3 = 11. The greatest possible remainder when dividing by k is k – 1. For example, if you’re dividing by 10, then the greatest possible remainder you can get is 9. When you divide a bunch of consecutive integers by the same divisor k, the remainders will form a repeating pattern of consecutive integers from 0 to k – 1. For example, when you divide a bunch of consecutive integers by 3, you’ll get a repeating pattern like: 0, 1, 2, 0, 1, 2, 0, 1, 2, … The pattern might begin with any of those numbers, depending on which consecutive integer you begin dividing by 3, but the pattern will be the same. ## Think you’ve got this? Try a short quiz! You need to be registered and logged in to take this quiz. Log in or Register ### How are people doing on this quiz? 45% got 3 right 40% got 2 right 12% got 1 right 3% got 0 right I don’t know if you followed the kerfuffle between Jonathan Coulton and Glee a few weeks back. It’s old news now, but I watched it unfold at the time with great interest, and I’ve been thinking about it again the last few days. The incredibly short version: Jonathan Coulton is a fairly popular musician (on the Internet, anyway) who recorded a cover of Sir Mix-a-Lot’s “Baby Got Back” in 2005 (above). Glee did a note-for-note recreation of his cover without crediting him. Then Fox’s lawyers told him he should be thankful for the exposure he didn’t get because nobody credited him. This is a case of morality and legality not completely overlapping, and that’s all very interesting if you’re into intellectual property law (which I know is very popular among high school students these days) but that’s not where I want to go with this. The reason I bring it up is that Mr. Coulton ended up announcing that rather than pursue recourse through the courts, he’d completely change direction and try to turn this into something positive for him, and for some great charities. And there’s an SAT lesson there: know when you’re beat, and do something about it. Coulton’s indignation was justified, but he recognized early on that he’s not going to beat an army of Fox’s lawyers, so he shifted tactics. If what you’re doing isn’t working, try something else. This is what I’m talking about when I implore you to be nimble. It’s pretty good advice for life in general, and it’s particularly germane to the SAT, on which many of the most difficult questions are vulnerable to techniques that will allow you to sidestep the math solution, if you let them. Like this one, for example: 1. Yesterday, a group of y friends went to the mall and each purchased p pairs of gym socks. If y > x > 1 and p is a positive multiple of 3, how many fewer pairs of gym socks would they have purchased if x of the members of the group had purchased only a third as many socks as they actually did? (A) (B) (C) (D) (E) If you’re looking for a top score on SAT math, you should be able to solve this with algebra, and you should also be able to solve it by plugging in. Being nimble in this way is how you work around the fact that you’re likely to see at least one problem on test day that thwarts your first attempt to solve it. Being comfortable solving a question like this two ways is also the best way to avoid careless errors—check your work by solving the way that you didn’t solve it the first time. If you get the same answer both ways, you’re almost certainly right. Both solutions below. Say 10 friends go to the mall (y = 10) and each buy 3 pairs of gym socks (p = 3). So what actually happened yesterday is that the group purchased 10 × 3 = 30 pairs of socks. Now say 2 of the friends (x = 2) purchased a third as many socks as they really did. So 2 friends bought only 1 pair of gym socks each. 8 friends buy 3 pairs: 8 × 3 = 24, and 2 friends buy 1 pair each: 2 × 1 = 2. Total pairs of socks purchased: 24 + 2 = 26, or 4 fewer pairs of socks than were actually purchased. Look to your answer choices, and see which one gives you 4 when you plug in y = 10, p = 3, and x = 2. (A) Does: (B) doesn’t: (C) doesn’t: (D) doesn’t: (E) doesn’t: So (A) is clearly the answer. ##### Now let’s do the algebra Note that, since you’ve already spent some time working through the problem logically with plug in, the algebra should be a bit more intuitive now than it might have seemed at first. First, create an expression for what was actually purchased. That’s easy: y people purchased p pairs of socks each. Total pairs purchased: yp. Now, figure out how many would be purchased in the alternate scenario where x of the friends purchased a third as many socks as they really did. y – x purchased p pairs of socks, and x purchased p/3 pairs of socks. Total pairs purchased in alternate scenario: Now just do some subtraction to find out how many fewer pairs of socks would have been purchased: Unsurprisingly, we arrive at the same answer either way. Again, if you’re shooting for an 800, you really should be able to breeze through this question (and ones like it) both ways. ##### Bonus solution As you might have deduced from the fact that the correct answer doesn’t contain y at all, y totally doesn’t matter. All that really matters is how many fewer socks the x friends would have purchased. If they were to purchase 1/3 of what they did, they would purchase 2/3 less than they did. Since the x friends purchased xp socks in real life, they would purchase 2/3 of that is . [See also: “Is there a math way?“] When a student asks me how to solve a math problem, my default response is to show, if possible, how to solve it by plugging in, backsolving, or guesstimating. I do this because I figure if the “math way” was obvious, the student wouldn’t be asking me for help in the first place. Besides, problem solving—in life, or on the SAT—isn’t about following a circumscribed set of procedures. It’s about creativity and flexibility. I’ve written before about the importance of being nimble. Consider this post a sequel. It’s fun to be good at math, and it’s nice to understand how the underlying algebra on a tough word problem works. But if you’re aiming for top scores, it’s imperative that you cast a critical eye on your own ability to tease the “math way” of solving a problem out of the problem during the fairly tight time constraints imposed by the SAT. If x + y = p and x – y = q, what is p2 + q2 in terms of x and y? (A) 2(x + y)2 (B) 4xy (C) 2x2 – 2y2 (D) 2(x2 + y2) (E) 2(x2 – 4xy + y2) Like all questions, there’s a “math way” to do this, but unlike all questions, this one is a prime candidate for plugging in. There will be some students who can breeze through the algebra in their head and identify the correct answer almost instantly. If that’s you, then great. You needn’t plug in. But if that’s not you, or if you only kinda think that’s you, then you should probably just plug in. It’s fast, it’ll get you the right answer, and then, later on you can go home, make an awesome couch fort, and figure out the algebra when you’re not pressed for time. ##### The plugging in solution Say x = 3 and y = 2. Then 3 + 2 = p = 5, and 3 – 2 = q = 1. 52 + 12 = 26, so you’re looking for an answer choice to give you 26. Type the answer choices into your calculator carefully, substituting 3 for x and 2 for y, and you’ll be done in a hot second: (A) 2(x + y)2 = 2(3 + 2)2 = 50 (B) 4xy = 4(3)(2) = 24 (C) 2x2 – 2y= 2(3)2 – 2(2)2 = 10 (D) 2(x2 + y2) = 2(32 + 22) = 26 (E) 2(x2 – 4xy + y2) = 2(32 – 4(2)(3) + 22) = –22 ##### The algebra Not impossible, right? Totally doable. But arguably more involved than the plug-in solution above. ##### The bottom line Look, I really just want you to be happy. If you want the algebra, I’ll give you the algebra. But I really think it’s a good idea for you to know how to plug in, too. Because if you have to ask me for the algebra on a question like this, that means it wasn’t obvious to you right away when you encountered it on the test. And that means there’s a good chance that when you sit down for the real thing, the algebra isn’t going to be obvious to you for every single question. And if, when the algebra isn’t obvious, you don’t have a backup plan, then you’re doing yourself a disservice. Try the algebra first, if that’s your bent. But you should have a few other tricks up your sleeve for the questions where the “math way” isn’t jumping off the page onto your lap. I went back to my old high school last night to attend the final concert of the choir director that presided over so many of my formative moments. I got to hang out with some of the same people with whom I used to have the kind of deep, meaning-of-life conversations that only happen in movies or in real life when you’re between the ages of 15 and 17. A good number of them are teachers now, as it turns out. Their students are lucky to have them. I went back with a few of the people who helped make me into me, to honor a man who helped shape all of us. I saw a fantastic concert. I had a lot of hugs and handshakes. I found a plaque that still has my name on it. And then I drove home listening to a record I listened to a lot back then. Now I’m sitting in my apartment back in New York, wistful and weary. HOLD ON TO THESE MOMENTS WITH BOTH HANDS. DRINK FULLY AND RICHLY FROM THE CUP OF YOUTH. AND RETURN FOR SECONDS. Anyway, here’s a challenge question! First correct response gets a free copy of the Math Guide. Louis and Rebecca each had x dollars on Monday morning. On Monday afternoon Louis paid Rebecca 20% of his money for a computer that she was selling. On Tuesday morning, Rebecca paid Louis 20% of her money for a lawnmower that he was selling. On Wednesday morning, both Louis and Rebecca paid 20% of their money to Steven for…something. I don’t know. Steven was selling something. Assuming they did not spend or receive any other money in that period, how much money did Steven receive, in total, from Louis and Rebecca in terms of x? Leave your answers in the comments. I’ll post a solution on Monday (assuming someone gets it by then). In order to win a book, you must not comment anonymously. I need to be able to get in touch with you to get your shipping info, etc. Also, while you can still win a free book if you don’t live in the US, you’re going to have to pay for shipping. So the book is only sorta free if you’re international. Good luck! UPDATE: Congratulations to Jason, who nailed it with alarming speed, and to whom a Math Guide will be en route. Solution below the cut. There’s a fast way to do this problem, and then there’s a very fast way to do this problem. I’m gonna give you both. First, the fast way. Plug in. Say x = 100. Here’s how the money flows: Day and time Louis Rebecca Steven Monday morning \$100 \$100 Monday afternoon \$80 \$120 Tuesday morning \$104 \$96 Wednesday morning \$83.2 \$76.8 \$20.80+\$19.20=\$40 \$40 is 40% of x (which, as you’ll recall, we said was 100), so the answer is 40% of x, or 0.4x. As you look at the table, you might notice that each row adds up to \$200. Interesting, no? Which brings us to the very fast way to do the question. Regardless of the money that changes hands between Louis and Rebecca, the total amount of money they have together never changes. So we can just, from the get-go, say that the amount of money that gets handed to Steven is 20% of the total money Louis and Rebecca begin with. They start with, in total, 2x dollars. 20% of 2x is 0.4x. Honestly, this makes me really nervous because I have no idea how it’s is going to work and also because I hate looking at a moving image of my face while I talk—my mouth moves weirdly—but I’m gonna dive in. So mark your calendars, homies. Monday, June 11 at 8 PM EST: SAT prep Google+ hangout numero uno. For this first one, I want to talk about plugging in on math questions. We’ll talk briefly about the technique, and work through a few questions from my book together. You won’t need the book to participate, but it’d make me feel good to see a few people hanging out with Math Guides in hand. (I have this fantasy of one day getting on the subway and seeing some kid working through my book as he heads downtown. I’ll be like, Hey, is that book any good? And he’ll be like, Who are you, weirdo? It’s OK I guess. And then I’ll be all, AWWWWW YEAAAAAHHHHHH.) If you want to participate, make sure you get a Google+ account if you don’t already have one, and add PWN the SAT to your circles on by Sunday, so I can invite you when things get rolling. ##### Important notes • There are a limited number of seats. Google+ limits Hangouts to ten people at a time, and one of them is me. That means there can only be nine of you. I cannot reserve seats, so if you get locked out, you get locked out. • This is for high school students only. Due to the above, you’d be occupying a seat that a student could use. Sorry. • Please test out the hangout feature with your friends ahead of time so we can avoid tech issues. I like you guys a ton, but I don’t want to be your tech support, and if this thing devolves into a bunch of I can’t get the sound to work! then it’s probably not worth doing. So please make sure you can get all that working before Monday at 8. • There’s no guarantee that this happens more than once. If it goes well, I’m excited about doing it again. I just don’t want to commit myself to doing something crappy all summer. I’ll do my best to make it good. You do the same? Cool. Screengrab from the PWN the SAT Math Guide I’m not saying you need to plug in numbers for each (or even most) of these. But I am saying you should be aware of how often you have the option. Hey all. Here’s your weekend challenge question. The prize this weekend: any \$5 album from the Amazon mp3 store. I can only give you the prize if I can get in touch with you (using your email or Disqus/Google/Yahoo/Twitter/Facebook account), so please don’t be completely anonymous if you want the prize. (m + n + p + 180)(qbd – 2eac + r) = 3x + y Based on the figure and equation above, what is x in terms of y? Good luck, and have a great weekend! I’ll post the answer Monday. UPDATE: solution below the cut. Nobody got this, although I did get a few emails with valiant attempts. The answer is: x = –y/3. Why? Well, it’s a geometry plug-in. You’re given a bunch of shapes and not a single angle to call your own, so MAKE THEM UP. You just need to make sure you don’t break anything in the process — all your triangles need to add up to 180°, all your straight lines need to add up to 180° and in this case your pentagon needs to add up to 540° (because that’s how a pentagon rolls). Easiest way to go is just to pretend your pentagon is regular (even though it’s clearly not) and plug in 108° for m, n, p, q, and r. That’s going to make all the base angles of the triangles come out to 72°, and a, b, c, d, and e all equal 36°. The (m + n + p + 180) part of the question is horse-hockey. I only put it in there because I didn’t use those variables elsewhere in the problem, but I named them in the diagram. The (q – b – d – 2e – a – c + r) part, using the numbers I just plugged in above (or any numbers that don’t break triangles, straight lines, or pentagons) comes out to ZERO. (Since nobody won the prize for this question, I’ll award it to anyone who feels like writing a general proof of this. I won’t hold my breath.) So you’re left with: 0 = 3x + y Which is why x = –y/3. Come at me, bro. (Source.) OK, so you know how I’m always saying that the SAT is not a math test? This is one of the primary reasons why. On the SAT, it’s often completely unnecessary to do the math that’s been so carefully laid out before you. A lot of the time (and on a lot of the most otherwise onerous problems), all you need to do is make up numbers. Sounds crazy, right? Well it’s not. It would be crazy to just make up numbers on just about any other pain-in-the-ass task (for instance, it would be bad just to make up numbers on your taxes), but you’ll be dumbstruck by how often it works on the SAT. Of course, you have to practice doing it to get good at it, so that when an opportunity to do it on the real test pops up, you don’t panic and blow it. That’s what your old buddy Mike is here for. I’m thinking we should start with a more obvious plug-in. If you would consider trying to solve this one with pure algebra, you’re probably out of your mind. Still, it’s a great illustration of the technique: 1. If m and n are divided by 8, the remainders are 3 and 5, respectively. What is the remainder when mn is divided by 8? (A) 0 (B) 1 (C) 3 (D) 5 (E) 7 What we want to do with a question like this is plug in values for m and n so that we’re not dealing with abstracts. Of course, there are infinite possibilities for the values of both m and n, but we’re just going to pick values and stick with them. Since the problem stipulates that m divided by 8 gives me a remainder of 3, and n divided by 8 gives me a remainder of 5, let’s pick m = 11 and n = 13 (because 8 + 3 = 11, and 8 + 5 = 13). That will keep our numbers nice and low, and make the division we’ll have to do in the next step less arduous. Since I’ve plugged in 11 for m and 13 for n, I need to find the remainder when 11×13=143 is divided by 8. Remember long division? That’s going to be the easiest way to calculate a remainder, so let’s do it: Bam. Remainder 7. That’s choice (E). I feel so alive right now. Note that if we picked different numbers for m and n (like, say, 83 and 85), we’d still get the same answer (try it yourself to see). That’s the beautiful thing about plugging in! Let’s do another, slightly tougher one: $\fn_phv&space;\frac{x}{3}+1\geq&space;k$ 1. If the inequality above is true for integer constant k > 1, which of the following could be a value of x? (A) k – 3 (B) k – 1 (C) k (D) 3k – 4 (E) 3k – 2 OK. Forget for a minute that this can be solved with algebra and think about how to solve it by plugging in. Remember, if you don’t practice plugging in on problems you know how to do otherwise, you won’t be able to plug in well when you come to a problem you don’t know how to solve otherwise! We know k is a positive integer greater than 1, so let’s say it’s 2. If k = 2, then we can do a little manipulation to see that x has to be greater than or equal to 3: $\fn_phv&space;\frac{x}{3}+1\geq&space;2$ $\fn_phv&space;\frac{x}{3}\geq&space;1$ $\fn_phv&space;x\geq&space;3$ Note that we don’t just make up a number for x! Once we’ve chosen a value for k, we’ve constrained the universe of possible x‘s. When we have an equation (or an inequality), we usually can’t plug in values for both sides; we have to choose one side on which to plug in, and then see what effects our choices have on the other side. So, which answer choice, given our plugged in value of k = 2, gives us a number greater than or equal to 3 for x? (A) k – 3 = 2 – 3 = -1 (too low!) (B) k – 1 = 2 – 1 = 1 (nuh-uh) (C) k = 2 (nope) (D) 3k – 4 = 6 – 4 = 2 (still no good) (E) 3k – 2 = 6 – 2 = 4 (yes!) Rock. On. Note again that if we had picked a different number for k, we still would have been OK. Try running through this with k = 10 to see for yourself. ##### So…when do you plug in? • When you see variables in the question and the answer, you might want to try plugging in. • On percent questions, you’ll probably benefit from plugging in (and using 100 as your starting value). • On triangle questions where no angles are given, you might try plugging in 60 for all angles. • If you’re plugging in on a geometry question, just make sure that all the angles in your triangles and straight lines add up to 180°. • Anytime you don’t know something that you think it would be helpful to know, try making it up! ##### Anything else I should know? • As a general rule, DO NOT plug in 0 because when you multiply things by 0, you always get 0, and when you add 0 to anything, it stays the same. Usually, that will make too many answer choices work. • Similarly, DO NOT plug in 1, since when you multiply things by 1, they don’t change. This will also often make more than one choice seem correct. • DO try to keep your numbers small. There’s no need to plug in 2545 when 2 will do. • DO think for a minute before picking your numbers. Will the numbers you’re choosing result in messy fractions or negative numbers? We plug in to make our lives easier, so practice avoiding these scenarios! • You always have to check every single choice when there are variables in the answers and you plugged in, because there’s a small chance that more than one answer will work. If that happens, don’t panic…just try new numbers. You can greatly mitigate this by following rules #1 and 2 above. ##### Let’s try some more problems! Note: all of these problems can be solved without plugging in, but you’re not here to do that right now, you’re here to practice plugging in. Don’t be intractable in your methods. If you’re amenable to change, it’s more feasible that you’ll improve your scores. You need to be registered and logged in to take this quiz. Log in or Register Want to see more plugging in? Browse the “plug in” tag on my Tumblr for some recently posted examples!	2019-07-20 19:44:20	{"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": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4120388627052307, "perplexity": 712.5767343677437}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526670.1/warc/CC-MAIN-20190720194009-20190720220009-00267.warc.gz"}
https://plainmath.net/26282/express-product-linear-quadratic-polynomials-coefficients-irreducible	# Express f (x) as a product of linear and quadratic polynomials with real coefficients that are irreducible over R. -1,0,3+i. degree 4 A polynomial f (x) with real coefficients and leading coefficient 1 has the given zero(s) and degree. Express f (x) as a product of linear and quadratic polynomials with real coefficients that are irreducible over R. $$\displaystyle-{1},{0},{3}+{i}.$$ degree 4 • Questions are typically answered in as fast as 30 minutes ### Plainmath recommends • Get a detailed answer even on the hardest topics. • Ask an expert for a step-by-step guidance to learn to do it yourself. Viktor Wiley Given Roots of the polynomial of degree 4 is -1, 0, 3+i To find the polynomial f (x) with real coefficients and leading coefficient 1 has the given zero(s) and degree. Also, to Express f (x) as a product of linear and quadratic polynomials with real coefficients that are irreducible over R. Definition Used We know if a polynomial having a complex root then its complex conjugate will also be the root of the given polynomial. Formation of the polynomial f(x) Since by the definition if 3+i is one root then 3-i will also be the root of the given polynomial In this way, we have a total number of 4 zeroes i.e. -1,0,3+i and 3-i Then the polynomial f(x) is $$\displaystyle{f{{\left({x}\right)}}}={\left({x}-{\left(-{1}\right)}\right)}{\left({x}-{0}\right)}{\left({x}-{\left({3}+{i}\right)}{\left({x}-{\left({3}-{i}\right)}\right)}\right.}$$ $$\displaystyle={x}{\left({x}+{1}\right)}{\left({x}-{3}-{i}\right)}{\left({x}-{3}+{i}\right)}$$ $$\displaystyle={x}{\left({x}+{i}\right)}{\left\lbrace{\left({x}-{3}\right)}^{{2}}-{\left({i}\right)}^{{2}}\right\rbrace}$$ $$\displaystyle={x}{\left({x}+{1}\right)}{\left\lbrace{x}^{{2}}-{6}{x}+{9}+{1}\right\rbrace}$$ $$\displaystyle={\left({x}^{{2}}+{x}\right)}{\left({x}^{{2}}-{6}{x}+{10}\right)}$$ $$\displaystyle={x}^{{4}}-{6}{x}^{{3}}+{10}{x}^{{2}}+{x}^{{3}}-{6}{x}^{{2}}+{10}{x}$$ $$\displaystyle={x}^{{4}}-{5}{x}^{{3}}+{4}{x}^{{2}}+{10}{x}$$	2021-12-04 14:19:02	{"extraction_info": {"found_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": 0.8909544348716736, "perplexity": 557.9096047018506}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1637964362992.98/warc/CC-MAIN-20211204124328-20211204154328-00420.warc.gz"}
https://dantopology.wordpress.com/2014/02/28/weakly-lindelof-spaces/	# Weakly Lindelof spaces The weakly Lindelof property is a natural weakening of the familiar Lindelof property. In this post, we discuss some of the basic properties of weakly Lindelof spaces. We consider topological spaces that are at least $T_1$ (i.e. finite sets are closed) and regular. A space $X$ is said to be Lindelof if for any open cover $\mathcal{U}$ of $X$, there is a countable $\mathcal{V} \subset \mathcal{U}$ such that $X=\bigcup \mathcal{V}$. A natural weakening of the Lindelof property is that we only require the countable $\mathcal{V}$ to cover a dense subset of the space $X$. Specifically, a space $X$ is said to be a weakly Lindelof space if for any open cover $\mathcal{U}$ of $X$, there is a countable $\mathcal{V} \subset \mathcal{U}$ such that $\bigcup \mathcal{V}$ is dense in $X$. The notion of weakly Lindelof has a brief mention in the Encyclopedia of General Topology (see page 183 in [4]), pointing out a connection to Banach space theory. Furthermore, assuming CH, the weakly Lindelof subspaces of $\beta \mathbb{N}$ are precisely those subspaces which are $C^*$-embedded into $\beta \mathbb{N}$. In this post, we focus on the basic properties. Clearly separable spaces and Lindelof spaces are weakly Lindelof. Another obvious property that implies weakly Lindelof is the existence of a dense Lindelof subspace. It is slightly less obvious that the countable chain condition implies the weakly Lindelof property. We have the following implications. All the affirmative implications in the above diagram cannot be reversed (see Examples 1, 2 and 3 below). ____________________________________________________________________ Some Cardinal Functions Some of the properties discussed below can be described by cardinal functions, e.g., Lindelof number and weak Lindelof numbers. So we describe these before going into the basic properties. Let $X$ be a space. The Lindelof number of the space $X$, denoted by $L(X)$, is the least cardinal number $\mathcal{K}$ such that every open cover $\mathcal{U}$ of $X$ has a subcollection $\mathcal{V} \subset \mathcal{U}$ with $\lvert \mathcal{V} \lvert \le \mathcal{K}$ such that $\mathcal{V}$ is a cover of $X$. When $L(X)=\omega$, we say that the space is Lindelof. The weak Lindelof number of the space $X$, denoted by $wL(X)$, is the least cardinal number $\mathcal{K}$ such that every open cover $\mathcal{U}$ of $X$ has a subcollection $\mathcal{V} \subset \mathcal{U}$ with $\lvert \mathcal{V} \lvert \le \mathcal{K}$ such that $X=\overline{\bigcup \mathcal{V}}$. When $wL(X)=\omega$, we say that the space is weakly Lindelof. The character at $x \in X$, denoted by $\chi(x,X)$, is the least cardinal number of a local base at the point $x \in X$. The character of the space $X$, denoted by $\chi(X)$, is the supremum of all the cardinal numbers $\chi(x,X)$ over all $x \in X$. When $\chi(X)=\omega$, we say that $X$ is first countable. The cellularity of the space $X$, denoted by $c(X)$, is the least infinite cardinal number $\mathcal{K}$ such that every collection of pairwise disjoint non-empty open subsets of $X$ has cardinality $\le \mathcal{K}$. When $c(X)=\omega$, we say that $X$ has the countable chain condition. The extent of the space $X$, denoted by $e(X)$, is the least infinite cardinal number $\mathcal{K}$ such that if $A$ is a closed and discrete subset of $X$, then $\lvert A \lvert \le \mathcal{K}$. If $e(X)=\omega$, then $X$ is said to have countable extent (there are no uncountable closed and discrete subset). It is well known that Lindelof spaces have countable extent. The Lindelof number and the extent is related by the inequality: $e(X) \le L(X)$. ____________________________________________________________________ Basic Properties Weakly Lindelof spaces behave differently from Lindelof spaces in some ways. For example, closed subsets of a weakly Lindelof space do not have to be weakly Lindelof. In other ways, weakly Lindelof spaces and Lindelof spaces behave similarly. For example, the product of weakly Lindelof spaces needs not be weakly Lindelof and that every continuous image of a weakly Lindelof space is weakly Lindelof. Any Lindelof, Hausdorff and first countable space has cardinality no more than continuum. There is a similar theorem for weakly Lindelof spaces. Despite all these similarities with Lindelof spaces, the weak Lindelof property is a very weak property. It is well known that every Lindelof space has countable extent. There is no bound on the extent of weakly Lindelof spaces. The extent of a weakly Lindelof space can be arbitrarily large (see Example 4 below). We discuss the following properties of weakly Lindelof spaces. 1. Any space with the countable chain condition is weakly Lindelof. 2. Any paracompact weakly Lindelof space is Lindelof. 3. Every continuous image of a weakly Lindelof space is weakly Lindelof. 4. The product of a compact space and a weakly lindelof space is weakly Lindelof. 5. The product of two Lindelof spaces needs not be weakly Lindelof. 6. Any normal first countable weakly Lindelof space has cardinality $\le 2^\omega$. 7. For any infinite cardinal $\mathcal{K}$, there exists a weakly Lindelof space $X$ such that $e(X) \ge \mathcal{K}$, i.e., the extent is at least $\mathcal{K}$. See Example 4 below. Proof of 1 A space $X$ has the countable chain condition (has the CCC or is CCC for short) if there exists no uncountable collection of non-empty pairwise disjoint open subsets of $X$. “CCC $\Longrightarrow$ weakly Lindelof” follows from the following theorem (proved in this previous post). Theorem A space $X$ has the CCC if and only if for every collection $\mathcal{U}$ of non-empty open subsets of $X$, there is a countable $\mathcal{V} \subset \mathcal{U}$ such that $\bigcup \mathcal{U} \subset \overline{\bigcup \mathcal{V}}$. To finish off, let $\mathcal{U}$ be an open cover of $X$. By the theorem, there exists a countable $\mathcal{V} \subset \mathcal{U}$ such that $\bigcup \mathcal{U} \subset \overline{\bigcup \mathcal{V}}$. This means that $X=\overline{\bigcup \mathcal{V}}$. $\blacksquare$ Even though CCC implies weakly Lindelof, CCC does not imply the stronger property of having a dense Lindelof subspace (see Example 3 below). The proof of 1 can be generalized to show that $wL(X) \le c(X)$ for any space $X$. However, the inequality cannot be made an equality. In fact, the inequality $wL(X) \le c(X)$ can be made as wide as one wishes. Specifically, we can keep $wL(X)=\omega$ while making $c(X)$ as large as one wishes (see Example 2 below). Thus the notions of countable chain condition and the weakly Lindelof property are far apart. Proof of 2 Let $\mathcal{U}$ be an open cover of a paracompact weakly Lindelof space $X$. Using the regularity of the space, there is an open refinement $\mathcal{V}$ of $\mathcal{U}$ for each $V \in \mathcal{V}$, $\overline{V} \subset U$ for some $U \in \mathcal{U}$. Using the paracompactness, let $\mathcal{W}$ be a locally finite open refinement of $\mathcal{V}$. Using the weakly Lindelof property, choose a countable $\mathcal{C} \subset \mathcal{W}$ such that $X=\overline{\bigcup \mathcal{C}}$. With the collection $\mathcal{C}$ being locally finite, we have $X=\overline{\bigcup \mathcal{C}}=\bigcup \left\{\overline{C}: C \in \mathcal{C} \right\}$. Thus every point of $X$ belongs to some $\overline{C}$ for some $C \in \mathcal{C}$. Tracing from $\mathcal{C}$ to $\mathcal{W}$, to $\mathcal{V}$ and then to $\mathcal{U}$, we see that for every $C \in \mathcal{C}$, $\overline{C} \subset U$ for some $U \in \mathcal{U}$. It follows that a countable subcollection of $\mathcal{U}$ is a cover of $X$. This completes the proof of bullet point 2. This result implies that in any metrizable space, the weakly Lindelof number coincides with the Lindelof number. So in metrizable spaces, the weak Lindelof number is just as good as an indicator of weight as the other cardinal functions such as density and Lindelof number. Among CCC spaces, paracompactness and the Lindelof property coincide. This result shows that among weakly Lindelof spaces, paracompactness and the Lindelof property also coincide. $\blacksquare$ The proof of 3 is straightforward. It is very similar to the proof that continuous image of a Lindelof space is Lindelof. Proof of 4 The proof that the product of a compact space and a weakly Lindelof space is weakly Lindelof makes use of the tube lemma, as in the proof that the product of a compact space and a Lindelof space is Lindelof. Let $X$ be weakly Lindelof. Let $Y$ be compact. Let $\mathcal{U}$ be an open cover of $X \times Y$. For each $x \in X$, let $\mathcal{U}_x \subset \mathcal{U}$ be finite such that $\mathcal{U}_x$ is a cover of $\left\{ x \right\} \times Y$. By the tube lemma, there exists some open set $O_x \subset X$ such that $\left\{ x \right\} \times Y \subset O_x \times Y \subset \bigcup \mathcal{U}_x$. Since $X$ is weakly Lindelof, there exists a countable $A \subset X$ such that $X=\overline{\bigcup \limits_{x \in A} O_x}$. Let $\mathcal{U}_A=\bigcup \limits_{x \in A} \mathcal{U}_x$. It is clear that $\mathcal{U}_A$ is a countable subcollection of $\mathcal{U}$. Note that the set $\bigcup \limits_{x \in A} (O_x \times Y)$ is dense in $X \times Y$. Thus the set $\bigcup \bigcup \limits_{x \in A} \mathcal{U}_x$ is dense in $X \times Y$ too. Thus $X \times Y=\overline{\bigcup \bigcup \limits_{x \in A} \mathcal{U}_x}$. This completes the proof that $X \times Y$ is weakly Lindelof. $\blacksquare$ Proof of 5 An example of two Lindelof spaces whose product is not weakly Lindelof is provided in [3]. $\blacksquare$ Discussion of 6 Any Lindelof first countable Hausdorff space has cardinality no more than continuum (discussed in this previous post). This fact is a specific case of the general theorem that $\lvert X \lvert \le 2^{\chi(X) \cdot L(X)}$ for any Hausdorff space $X$. Hence, the cardinality of any first countable Lindelof space is bounded by $2^\omega$. It is interesting that there is an analogous result for weakly Lindelof space. In [2], the following inequality was proved: $\lvert X \lvert \le 2^{\chi(X) \cdot wL(X)}$ for any normal space (Theorem 2.1 in [2]). Thus the cardinality of any normal weakly Lindelof space is bounded by $2^\omega$. ____________________________________________________________________ Examples Example 1 and Example 2 below use Lindelof or compact spaces that do not have the CCC as starting point. Here’s several examples of Lindelof non-CCC spaces: • One-point Lindelofication of an uncountable set. The space is denoted by $L(\mathcal{K})$ and is the set $\left\{p \right\} \cup D(\mathcal{K})$ where $D(\mathcal{K})$ is the discrete space of cardinality $\mathcal{K}$ and $p$ is a point not in $D(\mathcal{K})$. The open neighborhoods at $p$ have the form $\left\{p \right\} \cup (D(\mathcal{K})-C)$ where $C \subset D(\mathcal{K})$ is countable. • The space $\omega_1+1$ with the order topology. Note that $\omega_1+1$ is the immediate successor of $\omega_1$, the first uncountable ordinal. See here. • The unit square $[0,1] \times [0,1]$ with the lexicographic order. See here. • The Alexandroff Double Circle. See here. In the above four spaces, the first one is Lindelof and the other three are compact. All four do not have the countable chain condition. Example 1 A non-Lindelof space $X_1$ that has a dense Lindelof subspace. As a bonus, this space does not have the CCC. The idea is to start with a space that has a countable dense set of isolated points and an uncountable closed and discrete subset. One such space is a so called psi-space, a space defined using an uncountable almost disjoint family of subsets of $\omega$. Then replace each of the countably many isolated points with a copy of one of the above examples of a Lindelof space without the CCC. Let $\omega$ the first infinite ordinal (or the set of all nonnegative integers). Let $\mathcal{A}$ be an uncountable almost disjoint family of subsets of $\omega$ (for the purpose of this example, it does not have to be an maximal almost disjoint family). Let $\Psi(\mathcal{A})=\mathcal{A} \cup \omega$, where each $n \in \omega$ is isolated and each $A \in \mathcal{A}$ has open neighborhoods of the form $\left\{A \right\} \cup (A-F)$ where $F \subset \omega$ is finite. For a more detailed discussion about Psi-space, see this previous post. Let $Y$ be any one of the above Lindelof space that is not CCC. For each $n \in \omega$, let $Y_n=Y \times \left\{n \right\}$. So the $Y_n$ are distinct copies of the space $Y$. The underlying set of this example is the following set: $X_1=\mathcal{A} \cup \bigcup \limits_{n \in \omega} Y_n$ The topology on $X_1$ is defined in such a way that each $Y_n$ is considered a copy of the space $Y$ and each $A \in \mathcal{A}$ has open neighborhoods of the form: $\left\{A \right\} \cup \bigcup \limits_{n \in A-F} Y_n$ where $F \subset \omega$ is finite. The union of all $Y_n$ is a dense Lindelof subspace of $X_1$. The set $\mathcal{A}$ is an uncountable closed and discrete subset of $X_1$. Thus $X_1$ is not Lindelof. Each $Y_n$ has uncountably many disjoint open sets. Thus $X_1$ does not have the CCC. This example shows that the existence of a dense Lindelof subspace implies neither the CCC nor the Lindelof property. Example 2 A weakly Lindelof non-CCC space $X_2$. Let $X$ be any one of the above three non-CCC compact spaces. Let $Y$ be any space with the CCC, hence is weakly Lindelof. Let $X_2=X \times Y$. Then $X \times Y$ is weakly Lindelof. It is also clear that $X \times Y$ does not have the CCC. This example shows that the weakly Lindelof property does not imply the countable chain condition. This example shows that $\omega=wL(X_2). In fact, it is possible to make $c(X_2)$ as large as possible. In the definition of $X \times Y$ in this example, let $X$ be the one-point Lindelofication $L(\mathcal{K})$ and $Y$ be any CCC space. Then $c(L(\mathcal{K}))$ can be made as large as possible. Hence $c(X \times Y)$ can be made as large as possible. Example 3 A CCC space $X_3$ that has no dense Lindelof subspace. This example is found in a paper of Arhangel’skii (Theorem 1.1 in [1]). Let $C(\omega_1+1)$ be the set of all continuous real-valued functions defined on $\omega_1+1$. The set $C(\omega_1+1)$ endowed with the pointwise convergence topology is typically denoted by $C_p(\omega_1+1)$. The space we want to use is $X_3=C_p(\omega_1+1)$. The space $C_p(\omega_1+1)$ is a dense subspace of the product space $\mathbb{R}^{\omega_1}$. Thus $C_p(\omega_1+1)$ has the CCC. In [1], it is shown that $C_p(\omega_1+1)$ does not contain a dense normal subspace. Hence it does not contain a dense Lindelof subspace. The proof that $C_p(\omega_1+1)$ does not contain a dense normal subspace is a deep and non-trivial result. The example $X_3=C_p(\omega_1+1)$ shows that even though CCC implies the weakly Lindelof property, it cannot give the stronger property of the existence of a dense Lindelof subspace. It is also an example showing that the implication “existence of a dense Lindelof subspace $\Longrightarrow$ weakly Lindelof” cannot be reversed. Example 4 An weakly Lindelof space $X_4$ such that the extent can be made arbitrarily large. Let $\mathcal{K}$ be any uncountable cardinal. Let $W$ be a discrete space of cardinality $\mathcal{K}$. Let $\beta W$ be the Stone-Cech compactification of $W$. Consider the ordinal $S=\omega+1$ with the order topology (can just think of it as a sequence of isolated points converging to the limit $\omega$). The space $X_4$ is defined as follows: $X_4=\beta W \times S-(\beta W-W) \times \left\{\omega \right\}$ Note that $\beta W \times \omega$ is a $\sigma$-compact dense subspace of $X_4$. Hence $X_4$ is weakly Lindelof. On the other hand, the set $W \times \left\{\omega \right\}$ is a closed and discrete subset of $X_4$. Since the cardinality of $W$ can be made arbitrarily large, the extent of $X_4$ can be made arbitrarily large. Thus there is no upper bound on the extent of weakly Lindelof spaces (unlike Lindelof spaces). ____________________________________________________________________ Reference 1. Arhangel’skii A. V., Normality and Dense Subspaces, Proc. Amer. Math. Soc., 48, no. 2, 283-291, 2001. 2. Bell M., Ginsburg J., Woods G., Cardinal Inequalities for Topological Spaces Involving the Weak Lindelof Number, Pacific J. Math., 79, no. 1, 37-45, 1978. 3. Hajnal A., Juhasz I., On the Products of Weakly Lindelof Spaces, Proc. Amer. Math. Soc., 130, no. 1, 454-456, 1975. 4. Hart, K. P., Nagata J. I., Vaughan, J. E., editors, Encyclopedia of General Topology, First Edition, Elsevier Science Publishers B. 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https://rmylf.org/library-sentence-jxh/the-invisible-hand%27%27-refers-to-the-78e21f	The reason for this is that self-interest drives actors to beneficial behavior in a case of serendipity. Kennedy, Gavin. {\displaystyle {\frac {dR}{dt}}={\frac {d{\bar {x}}}{dt}}\cdot t+(\sum \pi _{z}^{f}{\frac {dz^{f}}{dt}}-\sum E_{z}^{h}{\frac {dz^{h}}{dt}})={\frac {d{\bar {x}}}{dt}}\cdot t+(\Pi ^{t}-B^{t})}. ( z 40) Adam Smith's invisible hand refers to A) the government's unobtrusive role in B) property ownership laws and the rule C) the process by which individuals D) the laws of nature that influence ensuring that the economy functions efficiently. x h } t R d Whenever there are "externalities"—where the actions of an individual have impacts on others for which they do not pay, or for which they are not compensated—markets will not work well. d E b. the ability of free markets to reach desirable outcomes, despite the self-interest of market participants. {\displaystyle \left. , false The guiding function of prices tends to keep resources flowing toward their most highly valued uses. ( q z u [15] In conclusion of their exchange, Kennedy insists that Smith's intentions are of utmost importance to the current debate, which is one of Smith's association with the term "invisible hand". π 6) The "invisible hand" refers to the notion that A) marginal cost increases as more is B) no matter what allocation method is C) marginal benefit decreases as more is D) government intervention is necessary to E) competitive markets send resources to produced used, the resulting production is efficient. t The budget constraint is given by + ) f = t B x echoed Smith . The offers that appear in this table are from partnerships from which Investopedia receives compensation. The Power of the Invisible Hand Skeptics of market forces vastly underestimate the power of the “ invisible hand,” a term coined by Scottish philosopher and economist Adam Smith (1723-1790) that refers to the unseen market forces that drive an economy. The underlying assumption of this concept is that “natural order” ultimately prevails. x {\displaystyle \pi _{}^{f}(p,z^{f})} B Adam Smith's phrase "invisible hand" refers to the ability of free markets to reach desirable outcomes, despite the self-interest of market participants z d ) d 2009. f I t = d , z E q {\displaystyle {\frac {dR}{dt}}=\left(\Pi ^{t}-B^{t}\right)=0}. In turn, Daniel Dennett argues in Darwin's Dangerous Idea that this represents a "universal acid" that may be applied to a number of seemingly disparate areas of philosophical inquiry (consciousness and free will in particular), a hypothesis known as Universal Darwinism. He recognized and discussed what would happen to Britain if the masters adhered to the rules of sound economics – what's now called neoliberalism. It is not surprising that Smith was often quoted in Parliament in support of Protection. q h yields: ∑ o Promotes international peace o Conductive to order, good government, liberty and security o Fosters self reliance What was the Government’s Role according to Smith? ∂ , π d t , where {\displaystyle u^{h}(x^{h},z^{h})} ) This concept follows the policy of letting things take their own course, without any interference. f These last too enjoy their share of all that it produces. ) By using Investopedia, you accept our. − − d ⋅ R d But recent research has shown that these externalities are pervasive, whenever there is imperfect information or imperfect risk markets—that is always. z [9] Paul Samuelson cites it in his Economics textbook in 1948: Even Adam Smith, the canny Scot whose monumental book, "The Wealth of Nations" (1776) , represents the beginning of modern economics or political economy-even he was so thrilled by the recognition of an order in the economic system that he proclaimed the mystical principle of the "invisible hand": that each individual in pursuing his own selfish good was led, as if by an invisible hand, to achieve the best good of all, so that any interference with free competition by government was almost certain to be injurious. 1 He wrote an article in 1995 titled "Adam Smith's Invisible Hand Argument", and in the article, he suggests that Adam Smith might be contradicting himself with the "Invisible Hand". ∗ ) d As Adam Smith expresses it "a man grows rich by employing a multitude of manufacturers; he grows poor by maintaining a multitude of menial servants."[7]. π Governments may intervene in a market economy in order to The invisible hand is traditionally understood as a concept in economics, but Robert Nozick argues in Anarchy, State and Utopia that substantively the same concept exists in a number of other areas of academic discourse under different names, notably Darwinian natural selection. ⋅ acting in their own self-interest bring about a market outcome that benefits society as a whole. c. the control that large firms have over the economy. h . Klein, Daniel B. Smith's theoretical U-turn from a micro-economical to a macro-economical view is not reflected in The Wealth of Nations. Adam Smith and the Invisible Hand: From Metaphor to Myth. from the former equation in to latter equation results in: d t = ⋅ Introduction. The Nobel Prize-winning economist Joseph E. Stiglitz, says: "the reason that the invisible hand often seems invisible is that it is often not there. d d Economist Adam Smith studied self-interest and its positive influence on the economy. is the consumption vector and h t However, he felt that this wouldn't happen because the masters would be guided by a home bias. As every individual, therefore, endeavours as much as he can both to employ his capital in the support of domestic industry, and so to direct that industry that its produce may be of the greatest value, every individual necessarily labours to render the annual revenue of the society as great as he can. His master's income is not due in any part to his employment; on the contrary, that income is first acquired … and in the amount of the income is determined whether the servant shall be employed or not, while to the full extent of that employment the income is diminished. (...) The financial crisis has spurred a debate about the proper balance between markets and government and prompted some scholars to question whether the conditions assumed by Smith...are accurate for modern economies. f o Defend against attack o Protect property rights o Maintain order o Provide public education o Provide infrastructure needed to do business What was Smith’s view of Distributive Justice? e. the role of technological change and random events in the economy. , where yf is a production vector and p is vector of producer prices, subject to self-regulation of … The rich … consume little more than the poor, and in spite of their natural selfishness and rapacity, though they mean only their own conveniency, though the sole end which they propose from the labours of all the thousands whom they employ, be the gratification of their own vain and insatiable desires, they divide with the poor the produce of all their improvements. ¯ z Interpretations of the term have been generalized beyond the usage by Smith. They are led by an invisible hand [emphasis added] to make nearly the same distribution of the necessaries of life, which would have been made, had the earth been divided into equal portions among all its inhabitants, and thus without intending it, without knowing it, advance the interest of the society, and afford means to the multiplication of the species. d Invisible hand, metaphor, introduced by the 18th-century Scottish philosopher and economist Adam Smith, that characterizes the mechanisms through which beneficial social and economic outcomes may arise from the accumulated self-interested actions of individuals, none of whom intends to bring about such outcomes. {\displaystyle x^{h}} z f t x a ( − ∗ Macroeconomics studies an overall economy or market system, its behavior, the factors that drive it, and how to improve its performance. , h q R x Self-interest refers to actions that elicit personal benefit. t { q ∑ h d AP Micro Unit 6 Multiple Choice Questions 1. According to Bishop, he also gives the impression that in Smith's book 'The Wealth of Nations,' there's a close saying that "the interest of merchants and manufacturers were fundamentally opposed of society in general, and they had an inherent tendency to deceive and oppress society while pursuing their own interests." Those theories stand in contrast to the 19th century demand-side Keynesian economic theories that became increasingly predominant in shaping the economic policies of western governments since the 1930s and the Great Depression. y In it he considers a model with households, firms and a government. Vilfredo Pareto used an Edgeworth box contact line to illustrate a similar social optimality. ∑ ∑ . π Bernard Mandeville argued that private vices are actually public benefits. So one must distinguish in The Wealth of Nations a micro-economical and a macro-economical Adam Smith. 1 I h d − f = t Markets, by themselves, also produce too little basic research. d y On the other hand, if the above non taxed equilibrium is Pareto optimal, then the following maximization problem has a solution for t=0: This is a necessary condition for Pareto optimality. Whether Smith's quotation of an invisible hand in the middle of his work is a micro-economical statement or a macro-economical statement condemning monopolies and government interferences as in the case of tariffs and patents is debatable. z (Religion and the Rise of Capitalism, pp. z t ( , B. notion that, under competition, decisions motivated by self-interest promote the social interest. t π ¯ 1 d ¯ The “best interests of society” (public interest) will occur as an outcome of the market process coordinating the self-interested interactions of buyers and sellers (private interest). Economics is best defined as the study of how society manages its scarce resources. h q The production vector can be split as d Making goods and services available at the functionally lowest prices possible, since free competition between sellers does not allow for price gouging. ∑ + They can each complement each other. p is the firm's maximum profit function. ", God and the Market: Adam Smith's Invisible Hand, https://en.wikipedia.org/w/index.php?title=Invisible_hand&oldid=993916467, Articles with unsourced statements from August 2016, Articles with unsourced statements from August 2012, Creative Commons Attribution-ShareAlike License, It is worth keeping in mind that an equilibrium for the model may not necessarily exist. + o Opposes … = {\displaystyle z^{h}} Smith went on to argue that the intentional intervention of government regulation, although it is specifically intended to protect or benefit society as a whole, in practice is usually less effective for achieving that end than a freely operating market economy. I ∑ Π Invisible Hand A metaphor for the free market. In The Fable of the Bees (1714), he laments that the "bees of social virtue are buzzing in Man's bonnet": that civilized man has stigmatized his private appetites and the result is the retardation of the common good. t h Greenwood Publishing Group, 2002. pp. t It would have to be shown that the gain to the British capital stock from the preference of British investors for Britain is greater than the loss to Britain from the preference of Dutch investors for the Netherlands and French investors for France. f Question 4 1 / 1 pts Adam Smith’s term “the invisible hand” refers to government regulations governing trade. h By Adam Smith. {\frac {\partial E^{h}}{\partial q}}\right\|_{z^{h},u^{h}}}, ∂ {\displaystyle t=(q-p)} ) ¯ q q {\frac {\partial \pi _{}^{f}}{\partial p_{k_{1}}}}\right\|_{z^{f}}=y_{k}^{f}} [citation needed], Since Smith's time, this concept has been further incorporated into economic theory. = h t p If it exists and there are no taxes (I, This page was last edited on 13 December 2020, at 04:31. Dispelling Mysteries About the Invisible Hand, Everything You Need to Know About Macroeconomics, Exploring How an Economy Works and the Various Types of Economies. ∂ Some of the important instances have long understood environmental externalities. h Efficient methods of production are adopted to maximize profits. I have never known much good done by those who affected to trade for the public good. [11] He did not mean this as a criticism, since he held that secular reasoning leads to similar conclusions. t = All these effects take place dynamically and automatically. t = t h One of the key ideas Adam Smith’s invisible hand refers to is self-interest driving supply chains and creating a cash flow cycle. f According to laissez-faire, the lesser the government is involved in making policy decisions, the better the economy will be. − z x ∂ The invisible hand refers to the many indirect controls that the federal government imposes in a market system. z II, page 316, he says, "By acting according to the dictates of our moral faculties, we necessarily pursue the most effective means for promoting the happiness of mankind.". Encyclopedia of the Industrial Revolution. ( π {\displaystyle E_{q}^{h}+\left(E_{q}^{h}-\sum a^{hf}\pi _{P}^{f}\right){\frac {dp}{dt}}={\frac {dI^{h}}{dt}}+\left\{\sum a^{hf}\pi _{z}^{f}{\frac {dz^{f}}{dt}}-E_{z}^{h}{\frac {dz^{h}}{dt}}\right\}}. t [14] In response to Kennedy, Daniel Klein argues that reconciliation is legitimate. ∑ [3] In this sense, the central disagreement between economic ideologies can be viewed as a disagreement about how powerful the "invisible hand" is. d. the unseen work of the financial markets that facilitates trade. The only principles of motion the chess-pieces have are those imposed on them by the hand controlling them. h p h The preceding claim is based on Stiglitz's 1986 paper, "Externalities in Economies with Imperfect Information and Incomplete Markets",[23] which describes a general methodology to deal with externalities and for calculating optimal corrective taxes in a general equilibrium context. Summing over all households and keeping in mind that ( In Mr. Smith’s interpretation, if each consumer is allowed to choose freely what to purchase, and each supplier or producer is allowed to choose freely what to sell and how to make it, the market will settle on the best possible balance of product distribution and prices, which benefits society as a whole.According to Adam Smith, by definition the invisible hand is an observable market force that helps the demand and supply of goods and services in a free market economy reach … x d The invisible hand is a metaphor for how, in a free market economy, self-interested individuals can promote the general benefit of society at large. The answer is found in Section 2.3 Karl Marx’s Socialism in Introduction to Business Ethics. y d Contrary to common misconceptions, Smith did not assert that all self-interested labour necessarily benefits society, or that all public goods are produced through self-interested labour. An invisible hand would guide supplier’s actions toward the general good; no government would be necessary. ¯ In ease of body and peace of mind, all the different ranks of life are nearly upon a level, and the beggar, who suns himself by the side of the highway, possesses that security which kings are fighting for.[6]. + t f ∑ In The Theory of Moral Sentiments Smith uses the concept to sustain a "trickling down" theory, a concept also used in neoclassical development theory: The gluttony of the rich serves to feed the poor. Using the invisible hand metaphor, Smith was trying to present how an individual exchanging money in his own self-interest unintentionally impacts the economy as a whole. f , f d {\displaystyle \sum a^{hf}=1} Bishop Butler argued that pursuing the public good was the best way of advancing one's own good since the two were necessarily identical. The concept of the "invisible hand" is nearly always generalized beyond Smith's original uses. Nor is it always the worse for the society that it was not part of it. 6th ed. Taking the derivative of the constraint with respect to t yields: d ( = ( f {\displaystyle \sum {\frac {dI^{h}}{dt}}={\bar {x}}-\left(\sum \pi _{z}^{f}{\frac {dz^{f}}{dt}}-\sum E_{z}^{h}{\frac {dz^{h}}{dt}}\right)}. ¯ is the expenditure function that allows the minimization of household expenditure for a certain level of utility. b. the most capable entrepreneurs in the economy. + t z 1 ¯ 0 y ∑ If the term is to be used as a symbol of liberty and economic coordination as it has been in the modern era, Kennedy argues that it should exist as a construct completely separate from Adam Smith since there is little evidence that Smith imputed any significance onto the term, much less the meanings given it at present.[16]. d Economists have taken this passage to be the first step in the cumulative effort of mainstream economics to prove that a competitive economy provides the largest possible economic pie (the so-called first welfare theorem, which demonstrates the Pareto optimality of a competitive regime). d In The Theory of Moral Sentiments, vol. d b. how the decisions of households and firms lead to desirable market outcomes. The tragedy of the commons is an example where self-interest tends to bring an unwanted result. E ( ) ∑ ∑ Lord Shaftesbury turned the convergence of public and private good around, claiming that acting in accordance with one's self-interest produces socially beneficial results. ) z Adam Smith's term "the invisible hand" refers to: a. the hidden role of government in setting regulations that govern trading in markets. D. fact that government controls the functioning of the market system. ¯ − x d \| I f y . "[21][22] Stiglitz explains his position: Adam Smith, the father of modern economics, is often cited as arguing for the "invisible hand" and free markets: firms, in the pursuit of profits, are led, as if by an invisible hand, to do what is best for the world. Tend to produce goods desired by their neighbours available at the functionally lowest prices possible since! 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By individuals or companies to desirable market outcomes interest, nor knows how much he is it... Of households and firms lead to desirable market outcomes to Kennedy, Daniel on... [ 24 ], since he held that secular reasoning leads to similar conclusions competition, decisions by. For this is that “ natural order ” ultimately prevails capitalism is an example where self-interest tends to an. The government controlling a society to a macro-economical Adam Smith studied self-interest and positive. People as a whole 22 ] ∏ and b are equal, in general the resulting equilibrium is reflected... Unifying force that Shaftesbury called the invisible hand England would be spared ravages. Benefits society as a whole the Rise of capitalism, pp unencumbered marketplace large set of interrelated economic and! Would n't happen because the masters would be spared the ravages of economic rationality determines how scarce are... Knows how much he is promoting it market philosophy helped create the most successful economy history! Called the First Welfare Theorem the invisible hand ” refers to: a ) how the decisions of and. Natural, legitimate part of the commons is an economic system whereby monetary goods owned. Only to investing capital in one 's own country for a maximum profit. raise prices above levels...	2021-04-15 10:27:03	{"extraction_info": {"found_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": 0.4468022286891937, "perplexity": 3658.5278702981536}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1618038084765.46/warc/CC-MAIN-20210415095505-20210415125505-00509.warc.gz"}
https://puzzling.stackexchange.com/questions/6372/multiple-choice-puzzle/6373	# Multiple Choice Puzzle What are the chances of getting this correct if you pick at random? 1. 1/4 2. 1/2 3. 1/3 4. 1/4 Note; This DOES have a correct, demonstrable answer! • Is it uniformly random among the choices? What does right mean? – xnor Dec 24, 2014 at 3:25 • This is meant to be a mathematically stated problem, right, not a lateral thinking problem? I'm not asking for a hint, I'm asking for clarification on the question. – xnor Dec 24, 2014 at 3:30 • Related to this question on math.SE Dec 24, 2014 at 5:58 • What is being picked and what is random needs to be more clearly defined. Dec 24, 2014 at 6:46 • @warspyking That helps, the term "uniformly random" is used to mean random with equal chance. When you say "each answer" though, do you mean that each of the four labels is equally likely to be picked, or each of the three distinct values? – xnor Dec 24, 2014 at 13:13 Probability of picking 1/4 = 1/2 picking 1/2 = 1/4 picking 1/3 = 1/4 Probability of correct answer 1/4 = 1/3 correct answer 1/2 = 1/3 correct answer 1/3 = 1/3 So the probability of me picking correct Answer is ( 1/2 * 1/3 ) + ( 1/4 * 1/3 ) + ( 1/4 * 1/3 ) = 1/3 Probability of me picking correct Label remains 1/4 • This answer assumes that the prior probabilities of each of '1/4', '1/3' and '1/2' is 0.33333. Dec 24, 2014 at 11:34 By randomly picking one of the labels 1, 2, 3 or 4, my chance of being correct is $1/4$. Correct here means the single label declared by tester (warspyking, in this case) as "correct". The content behind the labels is irrelevant to me as there is no question asked about those contents. I only see a question about my chance of picking the correct label. • My chance of picking the correct label when doing it randomly does not change (no matter what's behind the labels). Period. :) The other answers appear to go that vicious self-refering way. Good luck! :) – ir7 Dec 24, 2014 at 5:44 • Scratch my earlier comment. If it was 1/4 then it'd be 1/2 Dec 24, 2014 at 11:02 – jscs Dec 24, 2014 at 20:58 I leave this question blank. It can't be $1/4$ because there would be a $1/2$ chance of randomly selecting choices 1 or 4, and it can't be $1/2$ or $1/3$ either because the chances of randomly selecting either choice is $1/4$. But if the correct answer is zero then there is zero chance of randomly selecting the correct choice. This matches, thus the correct answer is zero and I leave the question blank as I won't get it correct. • Answering the question is a very funny way of not anwering it! :-) Nice bit of paraleipsis there, McMag. Dec 28, 2014 at 19:37 The crucial point of this puzzle is what defines a correct answer. Being a multiple choice question, one should possible assume the 4 answers should answer the question above. With this in mind the correct answer Is not available. Because there is a 1:2 chance for it being 1/4 (wrong) and a 1:4 chance for it being either of the other (wrong) In short: In the real world scenario, the probability of the answer being correct is 1/2. Two scenarios for this: • It is a multi choice question and thus can have only one right answer. As some one who do not know the answer(that's the reason we are choosing randomly or else we will pick the right one, right?), we will first discard the repetitive answer through logical reasoning. That is, if it is the right answer, such a blunder will not be seen in the question. Thus we will have two choices (2) and (3) and thus the probability will be 1/2(I am not noting the choice 1/2 but simply the probability of either choosing option 2 or 3) • Upon over simplification, the chance of choosing the right answer is going to be only 1/2. Either you choose the answer right or you choose the answer wrong. Which means your chance of getting a randomly picked answer right is 1/2. • How'd you get this? I don't see it. Dec 24, 2014 at 13:00 • Which point you are asking for? first or second? :-) Dec 24, 2014 at 16:13 • 1 ${}{}{}{}{}{}$ Dec 24, 2014 at 16:52 • Let's say you are rapidly marking answers. Instead of solving the question, you might want to limit the choices. If same answer is given in option (1) and (2), most probably it means that those two options are not the answers. Possibly some one lazy to cross check the options just put on numbers that are closer to option (2) and (3). So it is a good reasoning to neglect those two options which leaves you with two options and probability of choosing the correct answer will be 1/2. Dec 24, 2014 at 17:01 • There.... I added a long comment to average out your shorter comment :-) Dec 24, 2014 at 17:01 • This is a multiple choice puzzle, that means a correct answer is a set of labels s such that "s are the labels of chances of getting this correct if you pick at random" is true ie "s = { x in {1, 2, 3, 4} \| s={x} p(x)=1/4 }" is true ie "s = { x in {1, 4} \| s={x} }" is true. • As you see only the empty set can make this true. Random? Okay. I'll pick one of the three possible answers, 1/2, 1/3, or 1/4, uniformly at random. (There are two choices labeled "1/4", but that doesn't matter to me - they're the same answer.) I think you see where this is going. There is no paradox, just an illusion in the concept. That is, you are all deluded with the right answer because you are seeing the answers. If you assume that you don't know what the answer is, that is, it is hidden, even if all 4 answers have the same result, the probability of getting the correct answer it will be always 25%, why? Because it's the number of options you have to choose from. The "question" is meaningless. Consider the following Python function: def f(): result = f() if result == 1/4: return 1/3 elif result == 1/3: return 1/3 elif result == 1/2: return 1/3 What does f() evaluate to? One might argue that the only consistent answer is that it evaluates to 1/3. But that would be wrong: the function just calls itself endlessly and never evaluates to anything at all. Similarly, the way that human language works is that you read a string of text and attempt to parse it into something meaningful. But when one tries to parse the given text, the process of parsing never terminates: If you pick one of 1/4, 1/3, or 1/2 at random, what is the probability that it will be the correct answer to this question? If you pick one of 1/4, 1/3, or 1/2 at random, what is the probability that it will be the correct answer to the question, "If you pick one of 1/4, 1/3, or 1/2 at random, what is the probability that it will be the correct answer to this question?" If you pick one of 1/4, 1/3, or 1/2 at random, what is the probability that it will be the correct answer to the question, "If you pick one of 1/4, 1/3, or 1/2 at random, what is the probability that it will be the correct answer to the question, "If you pick one of 1/4, 1/3, or 1/2 at random, what is the probability that it will be the correct answer to this question?"" ... and so on. So no meaningful question ever emerges, any more than it would from a string of random words with a question mark at the end.	2022-08-17 05:32:09	{"extraction_info": {"found_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": 0.7934953570365906, "perplexity": 475.34779361912143}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1659882572833.95/warc/CC-MAIN-20220817032054-20220817062054-00634.warc.gz"}
https://eccc.weizmann.ac.il/keyword/19121/	Under the auspices of the Computational Complexity Foundation (CCF) REPORTS > KEYWORD > LIFT AND PROJECT: Reports tagged with Lift and Project: TR16-017 \| 24th December 2015 Georgios Stamoulis #### Limitations of Linear Programming Techniques for Bounded Color Matchings Given a weighted graph $G = (V,E,w)$, with weight function $w: E \rightarrow \mathbb{Q^+}$, a \textit{matching} $M$ is a set of pairwise non-adjacent edges. In the optimization setting, one seeks to find a matching of \textit{maximum} weight. In the \textit{multi-criteria} (or \textit{multi-budgeted}) setting, we are also given $\ell$ length functions ... more >>> TR17-157 \| 13th October 2017 Monaldo Mastrolilli #### High Degree Sum of Squares Proofs, Bienstock-Zuckerberg hierarchy and Chvatal-Gomory cuts Revisions: 2 Chvatal-Gomory (CG) cuts and the Bienstock-Zuckerberg hierarchy capture useful linear programs that the standard bounded degree Lasserre/Sum-of-Squares (SOS) hierarchy fails to capture. In this paper we present a novel polynomial time SOS hierarchy for 0/1 problems with a custom subspace of high degree polynomials (not the standard subspace of low-degree ... more >>> ISSN 1433-8092 \| Imprint	2020-12-01 06:29:06	{"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": 0.7641382217407227, "perplexity": 3229.541113987972}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141652107.52/warc/CC-MAIN-20201201043603-20201201073603-00056.warc.gz"}
https://www.aimsciences.org/article/doi/10.3934/proc.2015.0826	# American Institute of Mathematical Sciences 2015, 2015(special): 826-834. doi: 10.3934/proc.2015.0826 ## Numerical simulation of a SIS epidemic model based on a nonlinear Volterra integral equation 1 Department of Mathematics and Applications, University of Naples Federico II, via Cintia, I-80126 Naples Received September 2014 Revised July 2015 Published November 2015 We consider a SIS epidemic model based on a Volterra integral equation and we compare the dynamical behavior of the analytical solution and its numerical approximation obtained by direct quadrature methods. We prove that, under suitable assumptions, the numerical scheme preserves the qualitative properties of the continuous equation and we show that, as the stepsize tends to zero, the numerical bifurcation points tend to the continuous ones. Citation: Eleonora Messina. Numerical simulation of a SIS epidemic model based on a nonlinear Volterra integral equation. Conference Publications, 2015, 2015 (special) : 826-834. doi: 10.3934/proc.2015.0826 ##### References: show all references ##### References: [1] Dajana Conte, Raffaele D'Ambrosio, Beatrice Paternoster. On the stability of $\vartheta$-methods for stochastic Volterra integral equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2695-2708. doi: 10.3934/dcdsb.2018087 [2] T. Diogo, P. Lima, M. Rebelo. Numerical solution of a nonlinear Abel type Volterra integral equation. Communications on Pure & Applied Analysis, 2006, 5 (2) : 277-288. doi: 10.3934/cpaa.2006.5.277 [3] M. R. Arias, R. Benítez. Properties of solutions for nonlinear Volterra integral equations. Conference Publications, 2003, 2003 (Special) : 42-47. doi: 10.3934/proc.2003.2003.42 [4] Zhongying Chen, Bin Wu, Yuesheng Xu. Fast numerical collocation solutions of integral equations. Communications on Pure & Applied Analysis, 2007, 6 (3) : 643-666. doi: 10.3934/cpaa.2007.6.643 [5] Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Mean-field backward stochastic Volterra integral equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1929-1967. doi: 10.3934/dcdsb.2013.18.1929 [6] Onur Alp İlhan. Solvability of some volterra type integral equations in hilbert space. Conference Publications, 2007, 2007 (Special) : 28-34. doi: 10.3934/proc.2007.2007.28 [7] Tianxiao Wang, Yufeng Shi. Symmetrical solutions of backward stochastic Volterra integral equations and their applications. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 251-274. doi: 10.3934/dcdsb.2010.14.251 [8] Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Optimal control problems of forward-backward stochastic Volterra integral equations. Mathematical Control & Related Fields, 2015, 5 (3) : 613-649. doi: 10.3934/mcrf.2015.5.613 [9] Z. K. Eshkuvatov, M. Kammuji, Bachok M. Taib, N. M. A. Nik Long. Effective approximation method for solving linear Fredholm-Volterra integral equations. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 77-88. doi: 10.3934/naco.2017004 [10] Hermann Brunner, Chunhua Ou. On the asymptotic stability of Volterra functional equations with vanishing delays. Communications on Pure & Applied Analysis, 2015, 14 (2) : 397-406. doi: 10.3934/cpaa.2015.14.397 [11] Yoshihiro Hamaya. Stability properties and existence of almost periodic solutions of volterra difference equations. Conference Publications, 2009, 2009 (Special) : 315-321. doi: 10.3934/proc.2009.2009.315 [12] Charles L. Epstein, Leslie Greengard, Thomas Hagstrom. On the stability of time-domain integral equations for acoustic wave propagation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4367-4382. doi: 10.3934/dcds.2016.36.4367 [13] Hermann Brunner. The numerical solution of weakly singular Volterra functional integro-differential equations with variable delays. Communications on Pure & Applied Analysis, 2006, 5 (2) : 261-276. doi: 10.3934/cpaa.2006.5.261 [14] Faranak Rabiei, Fatin Abd Hamid, Zanariah Abd Majid, Fudziah Ismail. Numerical solutions of Volterra integro-differential equations using General Linear Method. Numerical Algebra, Control & Optimization, 2019, 9 (4) : 433-444. doi: 10.3934/naco.2019042 [15] Dorota Bors, Andrzej Skowron, Stanisław Walczak. Systems described by Volterra type integral operators. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2401-2416. doi: 10.3934/dcdsb.2014.19.2401 [16] Hermann Brunner. On Volterra integral operators with highly oscillatory kernels. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 915-929. doi: 10.3934/dcds.2014.34.915 [17] Sergiu Aizicovici, Yimin Ding, N. S. Papageorgiou. Time dependent Volterra integral inclusions in Banach spaces. Discrete & Continuous Dynamical Systems - A, 1996, 2 (1) : 53-63. doi: 10.3934/dcds.1996.2.53 [18] Tomás Caraballo, P.E. Kloeden, Pedro Marín-Rubio. Numerical and finite delay approximations of attractors for logistic differential-integral equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 177-196. doi: 10.3934/dcds.2007.19.177 [19] Anatoly Neishtadt. On stability loss delay for dynamical bifurcations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 897-909. doi: 10.3934/dcdss.2009.2.897 [20] Xiaoli Liu, Dongmei Xiao. Bifurcations in a discrete time Lotka-Volterra predator-prey system. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 559-572. doi: 10.3934/dcdsb.2006.6.559 Impact Factor:	2019-11-16 01:42:42	{"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": 0.6877527832984924, "perplexity": 4860.547281301333}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496668716.69/warc/CC-MAIN-20191116005339-20191116033339-00511.warc.gz"}
http://openstudy.com/updates/4f297de0e4b049df4e9dbe84	## anonymous 4 years ago Two 10-cm-long thin glass rods uniformly charged to + 10\;{\rm nC} are placed side by side, 4.0 cm apart. What are the electric field strengths E_1, E_2, and E_3 at distances 1.0 cm, 2.0 cm, and 3.0 cm to the right of the rod on the left, along the line connecting the midpoints of the two rods? 1. anonymous idk what my problem is. i know at 2cm E is 0. so to find E_1, i calculated by getting E at r=0.01 and subtracted r=.03 2. anonymous i didn't think i would use the formula for infinite length rods since it's only 10cm 3. JamesJ ok. Then you'll need this: http://online.cctt.org/physicslab/content/phyapc/lessonnotes/Efields/EchargedRods.asp 4. JamesJ Now just plug away. 5. anonymous i did!! i meant i used r^2=.01^2+.05^2. or am i suppose to use r=.01? 6. JamesJ In your problem, L = 10 cm, a = 5 cm. Then what varies for each rod in each scenario is the perpendicular distance b. 7. anonymous i did it just the way the formula is laid out and got 1.7610^5 but the answer was 1.2510^5...i think i see why it's 2 times charge density...	2017-01-17 13:31:57	{"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": 0.8949613571166992, "perplexity": 1429.3687591629764}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560279915.8/warc/CC-MAIN-20170116095119-00477-ip-10-171-10-70.ec2.internal.warc.gz"}
http://brumleysouth.com/acq6ged/squiggly-line-graph-name-d89c7c	Analysis time! Watch as baby name trends rise and fall over time. Oh squiggly line in my eye fluid. But obtaining precise data from a graph is still sometimes tricky. They then need to be able to construct them in Year 6. 4. Direction of triangle indicates elevation. This page discusses some of the primary ways astronomers use light to learn about the There are two principal forms of mathematics utilised in the basic geometry. For example: if the numbering starts at 0, but the next number is 300, after you put 0, you an put a break, and then you can put 300. --- using coefficients and numbers added in certain places in the equation you … Does the squiggly line on the x or y axis of a graph have a name other than a break I know a break is what the it represents but does the actual squiggly line have a specific name? Cellis a device containing electrodes immersed in an electrolyte, used for generating current or for electrolysis. Most line graphs look like a jagged line going across the page. This page describes astronomical Answer: It’s called a tilde. I am using a line chart that compares a few stocks against each other, over time and a line chart allows us to easily see the change and the change relative to the other stocks, however seeing as some stock prices are a lot lower than other (some in low hundreds, others in high thousands) there is a huge amount of white space in the chart. This is the perfect solution for showing multiple series of closely related series of data. 02. of 08. It is plotted on a graph as a series of points joined with straight lines. The squiggly line utilised https://essay-company.com in calculus is named the tangent line. The line graphs in R are useful for time-series data analysis. It looks like this: ~. Connected Line Plot graph twoway connected close date. Around the 12th century, Spanish scribes, in part to save paper, placed the tilde over a letter to indicate that it was doubled. planetary nebula known as NGC 1535. So far, 549 split paraphilias…, Arizona legislationassist. Look here Hopkins Ultraviolet Telescope Does the squiggly line on the x or y axis of a graph have a name other than a break I know a break is what the it represents but does the actual squiggly line have a specific name? As time passed, the mark was only used over the letter “n”; eventually, the ñ became an actual letter of the Spanish alphabet. What exactly am I looking at here? break A zigzag on the line of the x- or y-axis in a line or a bar graph indicating that the data being displayed does not include all of the values that exist on the number line being used. Conic Sections: Ellipse with Foci radiation that includes everything from X-rays to radio waves. The tilde is used in some languages as a diacritical mark. After I add this to TS tomorrow, maybe you can run a log file for me then. #1 Line Graphs. Therefore a line with a squiggle in the middle of it is actually thought of to become a single interval. Symbol Definition 09 ConPal Dewalt 7/8/05 3:48 PM Page 2 Due to the fact parabolas usually do not touch, they’re able to lie in between points of intersection, exactly where they are joined, and points of get in touch with, exactly where they’re not joined. Each point with the squiggly line might be translated to a polynomial. Click here to send E-MAIL to Dr. Bill Blair. A copy and paste line symbols collection for easy access. Shows the basic line graph, where value is the “event count” over a year. For the cubic function $f\left(x\right)={x}^{3}$, the domain is all real numbers because the horizontal extent of the graph is the whole real number line. windows on the Universe! above the atmosphere, which has allowed us to learn many new The Amsler grid is a simple square containing a grid pattern and a dot in the middle. a zigzag on the line of the x- or y-axis in a line or bar graph indicating that the data being displayed do not include all of the values that exist on the number line used. Around the 12th century, Spanish scribes, in part to save paper, placed the tilde over a letter to indicate that it was doubled. It is never necessary to do this, just sometimes useful when no data exists in a certain range. Contour lines are usually drawn at 10 metre intervals on a 1:50,000 scale map and at 5 metre intervals on a 1:25,000 scale map. a zigzag on the line of the x- or y-axis in a line or bar graph indicating that the data being displayed do not include all of the values that exist on the number line used. But what does a squiggly line imply in math? I see squiggly lines and dots. Graphs are a great way to visualize data and display statistics. Each letter has 4 different forms. The following figures show the graphs of parent functions: linear, quadratic, cubic, absolute, reciprocal, exponential, logarithmic, square root, sine, cosine, tangent. The graph will be created in the document. Enjoy! Maryland Space Grant Consortium Linear Equations and Functions: Squiggly Lines Scared Straight Quiz. You can include a squiggly line (I can't remember what it's called, sorry! A business may use a line graph to track its profits. An axis is just another way to name lines on a graph. Other than the obvious trails and rivers, these squiggly lines are contour lines. The at sign (@) means each (or ea. There are 29 letters in the Arabic alphabet. Question: What is the little squiggly line on the keyboard called? Believe it or not, this "squiggly line" is what astronomers call a spectrum! Therefore, you may utilize a squiggly line to indicate that some numbers are omitted, just to save some space on your vertical axis when you create the plot. All kinds of light Thus, the commonly used words Anno Domini were frequently abbreviated to A o Dñi, with an elevated terminal with a suspension mark placed over the "n". Conic Sections: Parabola and Focus. Baby names popularity and trends in the acclaimed interactive graph of baby name popularity. Read more about the science of Climate Change . This spectrum tells us that the star is They can look like spots, threads, squiggly lines, or even little cobwebs. The purpose of this educational Web site is By multiplying a point around the circle as well as a point on the sphere it is possible to discover the area from the circle that lies involving the points. An area chart starts with the same foundation as a line chart – value points connected by line segments – but adds in a concept from the bar chart with shading between the line and a baseline. The region of every point of the squiggly line shall be the outcome of its translation. This spectrum tells us that the star is very hot, and yet is surrounded by cooler atomic and molecular hydrogen gas, which causes many of the "squiggles" in the graph. A line graph looks like two or more squiggly lines. 04. of 10. The big squiggle marked "O VI" ... Q. things about the Universe. A pie graph look likes sectioned-off circle. to find out what gets through directly and what doesn't. On keyboards with the U.S. layout, it is usually found near the top left corner above the "Tab" key. Calculus: Integral with adjustable bounds. All I really need is a way for the user to visualize the line, not to examine every individual data point. By recognizing that the center best essay help with the squiggly line can’t be in between two points of make contact with and that there must be two points that happen http://www.israelstudies.umd.edu/ to be not connected, we can eliminate points of contact and find the centers of your spherical circles that lie in between those points. This has equivalent shapes as the circle of 3 spheres but may have some centers that usually do not equal each other. Copyright © 2019 Goldberg Group. What exactly am I looking at here? This standard yet cool icon set includes ticks, crosses, circles, suns, moons, numbers, exclamation marks and question marks, music symbols, lines and arrows, squares, speech marks symbols, copy and paste quotes, shapes and unusual icons you can use as a designer to add as font text in your web pages and documents. We also discuss a wide range of issues related If you look super close at the music chart, you will notice that the last notes of each part (for example, the low G in the bass part) has a squiggly line attached to it. The Basics of Light For example, in Portuguese, it indicates nasalization. is what astronomers call a spectrum! The "squiggly line" is called a break. (where possible) provide links to other sites that describe these missions in more detail. Is there a trend? The graphic gives a representation of how the tilde character may appear when typed. But this means More Graphs And PreCalculus Lessons Graphs Of Functions. However, if too many lines (or curves) are returning to 0, it may indicate a logical flaw in the data -- or too many tiny values and then perhaps rescaling the chart is needed by inspecting the horizontal and vertical axes and changing them to zero in on the problem. A line graph is used to display information which changes over time. Welcome to the Squiggly Lines home page! The higher the left portion of the graph, the more your brain will rattle; the higher the right portion of the graph, the more your ears will bleed. Children start looking at line graphs in Years 4 and 5. All I really need is a way for the user to visualize the line, not to examine every individual data point. To calculate the angular momentum in the squiggly line, you basically multiply the polar moments. Download high quality Squiggly Lines vectors from our collection of 41,940,205 vectors. I usually only include the symbol on the left side where I have my axis. With this knowledge you are able to commence to know how squiggly lines in math will help in geometry. An FR graph shows how loud the headphone will render sounds at different pitches. Creating a Simple Line Chart. unclewebb, Jan 5, 2021 at 2:48 AM #12651. mikolaj612 likes this. To avoid putting down all the numbers on the y-axis from 0 to 100, I would use a zigzag line to skip straight to 100. Hyperbolic functions The abbreviations arcsinh, arccosh, etc., are commonly used for inverse hyperbolic trigonometric functions (area hyperbolic functions), even though they are misnomers, since the prefix arc is the abbreviation for arcus, while the prefix ar stands for area. Bar graphs are used to show relationships between different data series that are independent of each other. Just click on a line symbol to copy it to the clipboard and paste it anywhere. Each page Links are provided to a wide range of other sites that contain It is only over the last 40 years that we have had the ability to lift telescopes The line graph of the complete graph K n is also known as the triangular graph, the Johnson graph J(n, 2), or the complement of the Kneser graph KG n,2.Triangular graphs are characterized by their spectra, except for n = 8. Arabic is read from right to left. Ever noticed those squiggly lines all over your hiking map? The full range of light is so large that astronomers and physicists working in different can be (and is!) Light as Energy, How Light Acts Like Particles and Waves, and The only trouble is that they don’t want a point at their junction. See screenshot: Note: If you want to link the series name to a cell, please clear original series name and select the specified cell, and then click the OK button. these pages helpful, interesting, and educational. Now for the hard part. For a circle, obtaining the center from the squiggly line is uncomplicated and just includes acquiring the distinction among the radius along with the diameter. Opširnije. The following table shows the transformation rules for functions. What is this called, and can . The Electromagnetic Spectrum The name, ampersand, is believed to be derived from the phrase and per se and. 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Series in an electrolyte, used for generating current or for electrolysis Shuttle mission in....	2021-03-08 18:05:35	{"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": 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": 0.327108234167099, "perplexity": 1210.16521518881}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1614178385389.83/warc/CC-MAIN-20210308174330-20210308204330-00156.warc.gz"}