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https://gmatclub.com/forum/in-the-figure-above-the-measure-of-angle-aob-is-120-degrees-if-the-286700.html	GMAT Question of the Day - Daily to your Mailbox; hard ones only It is currently 22 Oct 2019, 07:29 ### 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 # In the figure above, the measure of angle AOB is 120 degrees. If the Author Message TAGS: ### Hide Tags Math Expert Joined: 02 Sep 2009 Posts: 58432 In the figure above, the measure of angle AOB is 120 degrees. If the [#permalink] ### Show Tags 17 Jan 2019, 04:20 00:00 Difficulty: 15% (low) Question Stats: 87% (02:15) correct 13% (01:57) wrong based on 54 sessions ### HideShow timer Statistics In the figure above, the measure of angle AOB is 120 degrees. If the radius of OB is x, then arc AB is what fraction of the area of sector AOB, in terms of x? A π/x B √3/x C 2/x D 3/x E 4/x Attachment: 2019-01-17_1517.png [ 9.6 KiB \| Viewed 815 times ] _________________ VP Status: Learning stage Joined: 01 Oct 2017 Posts: 1012 WE: Supply Chain Management (Energy and Utilities) Re: In the figure above, the measure of angle AOB is 120 degrees. If the [#permalink] ### Show Tags 17 Jan 2019, 04:31 Bunuel wrote: In the figure above, the measure of angle AOB is 120 degrees. If the radius of OB is x, then arc AB is what fraction of the area of sector AOB, in terms of x? A π/x B √3/x C 2/x D 3/x E 4/x Attachment: 2019-01-17_1517.png Length of arc $$AB=2πx(\frac{120}{360})=2πxk$$ (where $$k=\frac{120}{360}$$) Area of sector $$AOB=πx^2(\frac{120}{360})$$=$$πx^2k$$ Now, measure of arc AB: Measure of area of sector AOB=$$2πxk:πx^2k$$=$$2:x$$ Ans. (C) _________________ Regards, PKN Rise above the storm, you will find the sunshine Manager Joined: 28 Aug 2018 Posts: 226 Re: In the figure above, the measure of angle AOB is 120 degrees. If the [#permalink] ### Show Tags 17 Jan 2019, 04:32 Imo C 2/x 2πX 120/360 / πx2 120/360 2/x Posted from my mobile device _________________ "Press +1 KUDOS" Be very critical of my Post Ask as many questions of my post possible Director Joined: 09 Mar 2018 Posts: 994 Location: India Re: In the figure above, the measure of angle AOB is 120 degrees. If the [#permalink] ### Show Tags 17 Jan 2019, 05:34 Bunuel wrote: In the figure above, the measure of angle AOB is 120 degrees. If the radius of OB is x, then arc AB is what fraction of the area of sector AOB, in terms of x? A π/x B √3/x C 2/x D 3/x E 4/x Attachment: 2019-01-17_1517.png Just a quick question, for this figure why are we not considering the outer sector with theta as $$240^o$$ _________________ If you notice any discrepancy in my reasoning, please let me know. Lets improve together. Quote which i can relate to. Many of life's failures happen with people who do not realize how close they were to success when they gave up. e-GMAT Representative Joined: 04 Jan 2015 Posts: 3078 In the figure above, the measure of angle AOB is 120 degrees. If the [#permalink] ### Show Tags 17 Jan 2019, 07:11 1 Solution Given: • Angle AOB = 120 degrees To find: • Length of arc, AB/area of sector AOB Approach and Working: • Length of arc, AB = $$\frac{120}{360} * 2ᴨ * x = \frac{2ᴨx}{3}$$ • Area of sector AOB = $$\frac{120}{360} * ᴨ * x^2 = \frac{ᴨx^2}{3}$$ Therefore, the answer is $$\frac{2ᴨx}{3}/\frac{ᴨx^2}{3} = \frac{2}{x}$$ Hence, the correct answer is Option C _________________ GMAT Club Legend Joined: 18 Aug 2017 Posts: 5031 Location: India Concentration: Sustainability, Marketing GPA: 4 WE: Marketing (Energy and Utilities) Re: In the figure above, the measure of angle AOB is 120 degrees. If the [#permalink] ### Show Tags 17 Jan 2019, 08:35 Bunuel wrote: In the figure above, the measure of angle AOB is 120 degrees. If the radius of OB is x, then arc AB is what fraction of the area of sector AOB, in terms of x? A π/x B √3/x C 2/x D 3/x E 4/x Attachment: 2019-01-17_1517.png area of sector : 120/360 * pi * x2 and arc AB is 120/360 * 2 * pi * x ratio would give us 2/x IMO C Target Test Prep Representative Status: Founder & CEO Affiliations: Target Test Prep Joined: 14 Oct 2015 Posts: 8125 Location: United States (CA) Re: In the figure above, the measure of angle AOB is 120 degrees. If the [#permalink] ### Show Tags 21 Jan 2019, 19:11 Bunuel wrote: In the figure above, the measure of angle AOB is 120 degrees. If the radius of OB is x, then arc AB is what fraction of the area of sector AOB, in terms of x? A π/x B √3/x C 2/x D 3/x E 4/x Attachment: 2019-01-17_1517.png Since the radius is x, the area is πx^2, and since AB corresponds to a 120 degree angle, the area of sector AOB is 120/360 * πx^2 = 1/3 * πx^2 = (πx^2)/3. Next we determine the circumference of the entire circle and the arclength of arc AB. Circumference = 2xπ, so arc AB is 120/360 * 2xπ = 1/3 * 2xπ = 2xπ/3. Finally arc AB is (2xπ/3)/(πx^2/3) = 6xπ/3πx^2 = 2/x of the area of sector AOB. _________________ # Scott Woodbury-Stewart Founder and CEO Scott@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. Manager Joined: 22 Sep 2018 Posts: 240 Re: In the figure above, the measure of angle AOB is 120 degrees. If the [#permalink] ### Show Tags 24 Jan 2019, 13:34 Bunuel wrote: In the figure above, the measure of angle AOB is 120 degrees. If the radius of OB is x, then arc AB is what fraction of the area of sector AOB, in terms of x? A π/x B √3/x C 2/x D 3/x E 4/x If x is the radius that means $$2pi * x$$ is the perimeter and $$x^2 pi$$ is the area ARC AB is $$\frac{1}{3} of the total perimeter$$ and area of AOB is also $$\frac{1}{3}$$ of the total area. This means that perimeter : area = (2pix/3) : (x^2pi/3) = $$\frac{2}{x}$$ Re: In the figure above, the measure of angle AOB is 120 degrees. If the [#permalink] 24 Jan 2019, 13:34 Display posts from previous: Sort by	2019-10-22 14:29:35	{"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": 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": 0.9520766735076904, "perplexity": 4575.326508482015}, "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-2019-43/segments/1570987822098.86/warc/CC-MAIN-20191022132135-20191022155635-00255.warc.gz"}
https://www.physicsforums.com/threads/helium-balloon-levitation.369427/	# Helium Balloon Levitation 1. Jan 13, 2010 ### skilet I wonder if it is possible to make a helium ballloon to Levitate in the air without touching the ceiling? Something like this. Thanks, Last edited by a moderator: Apr 24, 2017 2. Jan 14, 2010 ### jmatejka Neutral Buoyancy 3. Jan 14, 2010 ### rcgldr In the video, the balloon is being suspended by a stream of air, similar to this ping pong ball demo (the spinning is accomplished by angling the stream): In the case of a helium balloon, as it rises into lower density, lower pressure air, it expands a bit, reducing it's density as well, allowing it to rise further, and expand more, until eventually, the balloon surface expands beyond it's elastic range, and becomes rigid enough to resist significant further expansion, keeping the internal density near constant, and the balloon will then hover at a specific altitude. Another method is to simply leave the balloon open at the bottom, so once it expands to full size, the expanding helium escapes out the bottom. For a relatively low ceiling, you'd need to use a precise ratio of air and helium in the balloon and fill it to a specific amount, and perhaps use some small amount of weight. Density change of the air is tiny over a small change in altitude. Last edited by a moderator: Apr 24, 2017 4. Jan 14, 2010 ### Danger The trick to do it for real is indeed to achieve neutral buoyancy, which can be done with ballast which will have to be experimentally determined. In the case of the video, though, it's an illusion perpetrated by a professional's means. I know how it's done, but I'm not going to tell you because it would ruin the illusions of a few thousand 'magicians'. (I'll give you a hint, though; somewhere in this site is a thread started by me that references the technique involved. ) Good luck searching. edit: Aww, you sneaked in on me, Jeff. Come to think of it, that wasn't in a thread that I started; just one that I responded to. Anyhow, the post was about my Johnny Astro toy that I had as a kid. 5. Jan 14, 2010 ### rcgldr Considering how many levitating ping pong ball videos there are at youtube, plus store displays that levitate beach balls over household fans, I doubt it could be considered a secret. I decided to combine hovering, spinning, and "shooting" ping pong ball sequences in a one take video. Last edited: Jan 14, 2010 6. Jan 14, 2010 ### Danger I wasn't putting you down on that, Jeff. I'm still a net 'newbie', so I didn't realize that it was currently common knowledge. Even with the Johnny Astro on the market when I was a kid, it absolutely astounded people. I could make it take off, fly around, hover... and it even had a little hook on the bottom to pick stuff up and bring it back to the landing pad. I've incorporated parts of that old sucker into a couple of different projects and Hallowe'en costumes over the past 40 years or so. Here's a link to the toy itself: http://johnnyastro.com/" [Broken] Last edited by a moderator: May 4, 2017 7. Jan 14, 2010 ### Yeti08 Yes, I have (accidentally) made helium balloons levitate a few times. It happened, I believe, for two reasons - the baloon had lost some helium having been filled a few days prior, and the balloon was still warming up having just been exposed to liquid nitrogen. 8. Jan 15, 2010 ### nucleus Must be a special balloon. Helium balloons are used in the weather service for two purposes. First as radiosondes where it carries a radio transmitter aloft. The maximum altitude to which the balloon ascends is determined by the diameter and thickness of the balloon. Balloon sizes can range from 150 grams to 3000 grams. As the balloon ascends through the atmosphere, the pressure decreases, causing the balloon to expand. Eventually, the balloon will expand to the extent that its skin will break, terminating the ascent. An 800 gram balloon will burst at about 21 kilometres (69,000 ft). Second as a ceiling balloon, where it measures the height of the cloud base. http://en.wikipedia.org/wiki/Ceiling_balloon Both of these balloons rise until they break. 9. Jan 15, 2010 ### skilet I heard that if you fill the balloon with 1/2 of helium and 1/2 air it will stay in the air instead of going all the way up. Will that work? 10. Jan 15, 2010 ### diazona As the previous posters have said, there is some ratio of helium to air that would make the balloon neutrally buoyant. Generally it wouldn't be 1/2 and 1/2, though. In fact, at least approximately, you could calculate it. If $f_\text{He}$ is the fraction of helium in the balloon by mass, you'd need $$mg + f_\text{He}\rho_\text{He}g V + (1 - f_\text{He})\rho_\text{air} g V = \rho_\text{air} g V$$ Weight on the left, buoyant force on the right. $$mg + f_\text{He}(\rho_\text{He} - \rho_\text{air}) g V = 0$$ or $$f_\text{He} = \frac{m}{V(\rho_\text{air} - \rho_\text{He})}$$ For a regular latex party balloon (mass 2g, volume 14L) I get about 14% helium. Obviously this number would vary for different kinds of balloons. And even if you did get the exact fraction of helium required for neutral buoyancy, in practice it would be a very delicate balance because a slight air current or any other force would be enough to start the balloon moving one way or another. 11. Jan 15, 2010 ### rcgldr I meant as one that could levitate, not a weather balloon. A better example of a near neutral balloons are model and full size blimps. 12. Jan 15, 2010 ### Phrak It's not a helium ballon. Watch closely. Pay attention! It's a common balloon full of air. (mutter, mutter, observational physics, mutter, mutter) 13. Jan 17, 2010 ### Danger Agreed, nor was the Johnny Astro. I should have specified that in my previous posts, but I was trying to work the possibility of ballasting a helium balloon without giving away the (apparently not so secret) method that was used in the video.	2018-02-23 08:36:14	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.501828134059906, "perplexity": 1791.0345478307484}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1518891814538.66/warc/CC-MAIN-20180223075134-20180223095134-00564.warc.gz"}
https://socratic.org/questions/5934e0beb72cff62545a465d	# Question a465d Jun 5, 2017 Approx. $3.5 \cdot g$ #### Explanation: We need (i), a stoichiometric equation...... $K C l {O}_{3} \stackrel{M n {O}_{2} , \Delta}{\rightarrow} K C l \left(s\right) + \frac{3}{2} {O}_{2} \left(g\right)$ Note that the reaction is catalyzed by a bit of $M n \left(I V\right)$ salt. Heating without the catalyst would result in incomplete reduction to $K C l O$. And (ii) we need equivalent quantities of the reagents. Given $0.215 \cdot m o l$ of chlorate we should generate 3/2 equivs of dioxygen gas.............. 3/2xx0.215molxx32.00gmol^-1=??g#. This is a very convenient lab synthesis of dioxygen gas. How many litres of dioxygen would you get under standard conditions of $1 \cdot a t m$, and $298 \cdot K$?	2021-07-25 11:52:43	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 8, "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": 0.8258814215660095, "perplexity": 5578.832180660839}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1627046151672.96/warc/CC-MAIN-20210725111913-20210725141913-00420.warc.gz"}
http://www.theinfolist.com/html/ALL/s/Jung's_theorem.html	TheInfoList In geometry, Jung's theorem is an inequality between the diameter of a set of points in any Euclidean space and the radius of the minimum enclosing ball of that set. It is named after Heinrich Jung, who first studied this inequality in 1901. Algorithms also exist to solve the smallest-circle problem explicitly. Statement Consider a compact set :$K\subset \mathbb^n$ and let :$d = \max_ \\| p - q \\|_2$ be the diameter of ''K'', that is, the largest Euclidean distance between any two of its points. Jung's theorem states that there exists a closed ball with radius :$r \leq d \sqrt$ that contains ''K''. The boundary case of equality is attained by the regular ''n''-simplex. Jung's theorem in the plane Most common is the case of Jung's theorem in the plane, that is ''n'' = 2. In this case the theorem states that there exists a circle enclosing all points whose radius satisfies :$r \leq \frac.$ No tighter bound on ''r'' can be shown: when ''K'' is an equilateral triangle (or its three vertices), then :$r = \frac.$ General metric spaces For any bounded set ''S'' in any metric space, ''d''/2 ≤ ''r'' ≤ ''d''. The first inequality is implied by the triangle inequality for the center of the ball and the two diametral points, and the second inequality follows since a ball of radius ''d'' centered at any point of ''S'' will contain all of ''S''. In a ''uniform metric space'', that is, a space in which all distances are equal, ''r'' = ''d''. At the other end of the spectrum, in an injective metric space such as the Manhattan distance in the plane, ''r'' = ''d''/2: any two closed balls of radius ''d''/2 centered at points of ''S'' have a nonempty intersection, therefore all such balls have a common intersection, and a radius ''d''/2 ball centered at a point of this intersection contains all of ''S''. Versions of Jung's theorem for various non-Euclidean geometries are also known (see e.g. Dekster 1995, 1997). References * * * * * *	2021-10-22 23:21:08	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 5, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9315241575241089, "perplexity": 307.6336557909017}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1634323585522.78/warc/CC-MAIN-20211022212051-20211023002051-00261.warc.gz"}
https://cs.stackexchange.com/questions/65568/why-is-the-time-complexity-of-insertion-sort-not-brought-down-even-if-we-use-bin	# Why is the time complexity of insertion sort not brought down even if we use binary sort for the comparisons? There are two factors that decide the running time of the insertion sort algorithm : the number of comparisons, and the number of movements. In the case of number of comparisons, the sorted part (left side of $j$) of the array is searched linearly for the right place of the $j^{th}$ element. If instead, we use a binary search, then the time complexity of finding a place for the $j^{th}$ element comes down from $\operatorname{O}(n)$ to $\operatorname{O}(\log n)$. So, for all the $n$ elements, the time complexity for comparisons becomes $\operatorname{O}(n \log n)$. Even so, the number of movements is still going to take $\operatorname{O}(n)$ time, and the total time complexity isn't brought down and remains $\operatorname{O}(n^2)$. Why is that? Are any of my statements wrong assumptions? Edit Can a possible explanation be: the total time complexity isn't brought down and remains $\operatorname{O}(n^2)$. This is because to search an element (using binary search) it takes $\operatorname{O}(\log n)$ time, and to move the elements it takes $\operatorname{O}(n)$ time. Total cost is $\operatorname{O}(\log n)+\operatorname{O}(n)=\operatorname{O}(n)$ time. To do this for $n-1$ elements, it takes $n(n-1)=\operatorname{O}(n^2)$ time.? For the $j^{th}$ element, you would do ~ $\log j$ comparisons and (in the worst case) ~$j$ shifts. Summing over $j$, you get $$\sum_{j = 1}^{n} (j + \log j) = \frac{n(n+1)}{2} + \log (n!) = O(n^2 + n \log n) = O(n^2)$$ The idea is that the linear work of shifting trumps the logarithmic work of comparing. You end up doing less comparisons, but still a linear amount of work per iteration. So the complexity does not change. • This explains it perfectly. Thank you.. – Somenath Sinha Nov 5 '16 at 2:33 • Also, can I state it as: The total time complexity isn't brought down and remains $\operatorname{O}(n^2)$. This is because to search an element (using binary search) it takes $\operatorname{O}(\log n)$ time, and to move the elements it takes $\operatorname{O}(n)$ time. Total cost is $\operatorname{O}(\log n)+\operatorname{O}(n)=\operatorname{O}(n)$ time. To do this for $n-1$ elements, it takes $n(n-1)=\operatorname{O}(n^2)$ time. – Somenath Sinha Nov 5 '16 at 2:36 • @SomenathSinha yes, but because in this case we know that linear work sums up to be quadratic (the sum over $j$ ends up being quadratic), one would have to be careful for other kinds of functions that sum up differently. But in this case you can say that $O(j) + O(\log j) = O(j)$ and that sums up to $O(n^2)$. – aelguindy Nov 5 '16 at 2:49 The "possible explanation" after the edit in the question is exactly correct. That's why the time complexity is not improved. On the other hand, unless the array is already mostly sorted, or if the array is very small, using binary search to find where to insert an array element is very likely to make the sorting almost twice as fast. On the other hand, for large n where sorting an array using insertion sort is unacceptably slow, making it twice as fast still leaves it unacceptably slow.	2019-07-21 23:14:57	{"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": 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": 0.703630805015564, "perplexity": 401.7313960585934}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195527396.78/warc/CC-MAIN-20190721225759-20190722011759-00100.warc.gz"}
https://bookstore.ams.org/view?ProductCode=STML/37	An error was encountered while trying to add the item to the cart. Please try again. The following link can be shared to navigate to this page. You can select the link to copy or click the 'Copy To Clipboard' button below. Copy To Clipboard Successfully Copied! $p$-adic Analysis Compared with Real Svetlana Katok Pennsylvania State University, University Park, PA Available Formats: Softcover ISBN: 978-0-8218-4220-1 Product Code: STML/37 List Price: $34.00 Individual Price:$27.20 Electronic ISBN: 978-1-4704-2148-9 Product Code: STML/37.E List Price: $32.00 Individual Price:$25.60 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version. List Price: $51.00 Click above image for expanded view$p$-adic Analysis Compared with Real Svetlana Katok Pennsylvania State University, University Park, PA Available Formats: Softcover ISBN: 978-0-8218-4220-1 Product Code: STML/37 List Price:$34.00 Individual Price: $27.20 Electronic ISBN: 978-1-4704-2148-9 Product Code: STML/37.E List Price:$32.00 Individual Price: $25.60 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version. List Price:$51.00 • Book Details Student Mathematical Library Volume: 372007; 152 pp MSC: Primary 11; 26; 12; The book gives an introduction to $p$-adic numbers from the point of view of number theory, topology, and analysis. Compared to other books on the subject, its novelty is both a particularly balanced approach to these three points of view and an emphasis on topics accessible to undergraduates. In addition, several topics from real analysis and elementary topology which are not usually covered in undergraduate courses (totally disconnected spaces and Cantor sets, points of discontinuity of maps and the Baire Category Theorem, surjectivity of isometries of compact metric spaces) are also included in the book. They will enhance the reader's understanding of real analysis and intertwine the real and $p$-adic contexts of the book. The book is based on an advanced undergraduate course given by the author. The choice of the topic was motivated by the internal beauty of the subject of $p$-adic analysis, an unusual one in the undergraduate curriculum, and abundant opportunities to compare it with its much more familiar real counterpart. The book includes a large number of exercises. Answers, hints, and solutions for most of them appear at the end of the book. Well written, with obvious care for the reader, the book can be successfully used in a topic course or for self-study. This book is published in cooperation with Mathematics Advanced Study Semesters. Undergraduate and graduate students interested in $p$-adic numbers. • Chapters • Chapter 1. Arithmetic of the $p$-adic numbers • Chapter 2. The topology of $\mathbb {Q}_p$ vs. the topology of $\mathbb {R}$ • Chapter 3. Elementary analysis in $\mathbb {Q}_p$ • Chapter 4. $p$-adic functions • Answers, hints, and solutions for selected exercises • Reviews • ...the book gives a good impetus to students to study the "p-adic worlds" more deeply. This role of the book is not only supported by carefully selected material but also by the fact that it is written in a very lively and lucid style. • I think that the reading of this book could animate some students to start to do research $p$-adic work. A good decision from my point of view! Zentralblatt MATH • Request Review Copy • Get Permissions Volume: 372007; 152 pp MSC: Primary 11; 26; 12; The book gives an introduction to $p$-adic numbers from the point of view of number theory, topology, and analysis. Compared to other books on the subject, its novelty is both a particularly balanced approach to these three points of view and an emphasis on topics accessible to undergraduates. In addition, several topics from real analysis and elementary topology which are not usually covered in undergraduate courses (totally disconnected spaces and Cantor sets, points of discontinuity of maps and the Baire Category Theorem, surjectivity of isometries of compact metric spaces) are also included in the book. They will enhance the reader's understanding of real analysis and intertwine the real and $p$-adic contexts of the book. The book is based on an advanced undergraduate course given by the author. The choice of the topic was motivated by the internal beauty of the subject of $p$-adic analysis, an unusual one in the undergraduate curriculum, and abundant opportunities to compare it with its much more familiar real counterpart. The book includes a large number of exercises. Answers, hints, and solutions for most of them appear at the end of the book. Well written, with obvious care for the reader, the book can be successfully used in a topic course or for self-study. This book is published in cooperation with Mathematics Advanced Study Semesters. Undergraduate and graduate students interested in $p$-adic numbers. • Chapters • Chapter 1. Arithmetic of the $p$-adic numbers • Chapter 2. The topology of $\mathbb {Q}_p$ vs. the topology of $\mathbb {R}$ • Chapter 3. Elementary analysis in $\mathbb {Q}_p$ • Chapter 4. $p$-adic functions • Answers, hints, and solutions for selected exercises • ...the book gives a good impetus to students to study the "p-adic worlds" more deeply. This role of the book is not only supported by carefully selected material but also by the fact that it is written in a very lively and lucid style. • I think that the reading of this book could animate some students to start to do research $p$-adic work. A good decision from my point of view!	2023-02-05 10:36:52	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.28060048818588257, "perplexity": 686.8103299258039}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1674764500251.38/warc/CC-MAIN-20230205094841-20230205124841-00656.warc.gz"}
http://ilnumerics.net/ilnumerics-licensing-faq.html	ILNumerics - Technical Computing Tools High Performance Framework for Visualization and Computing in Industry and Science tgt # Licensing FAQ This page lists some of the most common questions regarding the ILNumerics licensing. If you have a specific question not answered here, please get in touch with us: sales@ilnumerics.net # My assembly runs fine on my machine but produces a license exception on other machines. Make sure that the application was licensed and unlocked correctly on your developer machine. During build inspect the Output window, Build tab in Visual Studio. There will be a note saying: This means that your assembly is unlocked for any machine. If there is an error message / warning instead, read the message and follow any instructions provided. See this page if you need further help. # How to find out Which Modules are required for my application? The required modules are easily identified by inspecting the list of references for your application. Each ILNumerics module corresponds to a single assembly. Example: Assuming that the application references ILNumerics.Computing.dll, ILNumerics.Drawing.dll and ILNumerics.Toolboxes.Statistics.dll. In this case the modules “Computing Engine”, “Visualization Engine” and the “Statistics Toolbox” are required. Note that the module ILNumerics.Core.dll is always required and comes free of charge. # How can I deactivate a developer seat? I need to assign the license to another developer. There is no technical procedure for deactivating a developer seat. Once activated it remains activated. However, if your developer left the company / project and you need to reassign a new person get in touch with us! We will support you with the reassignment. # I have to upgrade my licensed computer. Will ILNumerics work afterwards? Yes. Just make sure that you use the same computer name and do not change the OS user account name. ILNumerics will pick up the license after the upgrade and keep working normally. # I am not working with Visual Studio. What are the steps I have to perform manually in order to license my application? Here is what Visual Studio performs on your project transparently. This list is for very rare situations only. It is not recommended to perform those steps manually. With very few exceptions Visual Studio should be used instead: • Add a folder named 'ILNumerics_deploy' with the following files: ILHelper.cs, ILNImports1.targets, ILNImports2.targets, ilnumerics.lic. Use Visual Studio to generate this folder or download one example from here (version 4.11). • Make sure that ILHelper.cs is a regular code file of the project. • Make sure that ilnumerics.lic is included in the project as embedded resource. • In the project file import the ILNImports1.targets and ILNImports2.targets files as msbuild projects. All this can be done by adding the following snippet to the source code of the project file (.csproj, .vbproj): <ItemGroup> <Compile include="ILNumerics_deploy\ILNHelper.cs"/> <EmbeddedResource include="ILNumerics_deploy\ilnumerics.lic"/> <None Include="ILNumerics_deploy\ILNImports1.targets"/> <None Include="ILNumerics_deploy\ILNImports2.targets"/> </ItemGroup> <Import Condition="$(ILNLicCompile) != 'true'" project="ILNumerics_deploy\ILNImports1.targets"/> <Import Condition="$(ILNLicCompile) == 'true'" project="ILNumerics_deploy\ILNImports2.targets"/> Afterwards, the project will create and maintain embedded licenses for execution on non-licensed machines when building with Visual Studio, Team Server or on the developer command line on a licensed seat. Note that we do provide support for regular projects which were set up via Visual Studio and the ILNumerics extension package only. # Suddenly my activation is lost and Visual Studio asks me to reactivate my developer seat? In version 4.8 this could have been caused by certain major updates to Windows. This was fixed in version 4.9. However, an activation is only valid for one single OS account on one single machine. Note that the account name is case sensitive! It has been reported that on some computers the letter case of the username changed due to the way how to login to the machine (VPN). A new activation is required in this case. A simple work around is to make sure to always use the same account name, including the case of the letters.	2018-03-25 05:23:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.2030927538871765, "perplexity": 2727.298636598723}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1521257651820.82/warc/CC-MAIN-20180325044627-20180325064627-00624.warc.gz"}
https://www.physicsforums.com/threads/vector-space-over-the-rationals.199737/	# Vector space over the rationals 1. Nov 21, 2007 ### matheinste Hello all. I came across this problem in Halmos, Finite-Dimensional Vector Spaces, page 16. Is the set R of all real numbers a finite-dimensional vector space over the field Q of all rational numbers. There is a reference to a previous example which says that with the usual rules of addition and multiplication by a rational R becomes a rational vector space. My answer to the question would be that R is not a finite-dimensional vector space over the field Q. The author goes on to say that the question is not trivial and it helps to know something about cardinal numbers. Can anyone please expand on this. Thanks Matheinste. 2. Nov 21, 2007 ### HallsofIvy Staff Emeritus You are correct that the set of all real numbers, as a vector space over the rational numbers, is NOT finite-dimensional. If it were finite dimensional, then there finite basis, say ${r_1, r_2, ..., r_n}$. Then every real number would be of the form $a_1r_1+ a_2r_2+ \cdot\cdot\cdot+ a_nr_n}$ where each $a_i$ is a rational number. Then each set of numbers {$a_ir_i$} would be countable because the set of rational numbers is countable. The set of all real numbers would then be a Cartesian product of countable sets. That would imply that the set of all real numbers is countable- but it isn't. 3. Nov 21, 2007 ### matheinste Thanyyou HallsofIvy. I know just enough to follow your argument but would not have reasoned it out for myself. That completely answers my query. Thanks again. Mateinste. 4. Nov 21, 2007 ### andytoh I don't think that set of all real numbers, as a vector space over the rational numbers, is even countable-dimensional, because (using the exact same argument as HallsovIvy), R would then be a countable cartesian product of countables sets, which is not necessarily countable (only a finite cartesian product of countable sets is countable). Last edited: Nov 21, 2007 5. Nov 21, 2007 ### HallsofIvy Staff Emeritus Yes, that's right- the dimension of the real numbers, as a vector space over the rational numbers, is not countable. However, the original question just asked about the proof that it was not finite dimensional! 6. Nov 21, 2007 ### andytoh Is there a quick proof to why the vector space of reals over the rationals has uncountable dimension? A countable cartesian product of countable sets is not necessarily countable, but it is not necessarily uncountable either. All that is needed is to construct one such (Hamel) basis, show that it is an uncountable basis. Then all other bases would have the same cardinality and hence be uncountable as well. Last edited: Nov 21, 2007 7. Nov 21, 2007 ### morphism If B={r_i} (i in some infinite index set I) is a basis for R/Q, then each real number can be written as a finite linear combination of the r_i's over Q. Let F_n be the set of real numbers expressible as a linear combination of n elements of B. Then R = $\cup_n$ F_n. On the other hand, \|F_n\| <= \|Q^n\| \|B^(<w)\| = $\aleph_0$ \|I\| = \|I\| (where B^(<w) denotes the finite subsets of B). Whence \|R\| <= $\aleph_0$\|I\| = \|I\|. I think this is alright. (Although to be completely rigorous, I think we need to employ the well-ordering theorem on B. Try to see where this is needed. Edit: On second thought, I suppose this can be avoided if we assume for a contradiction that I is countably infinite - this way we can give B the induced well-ordering from N.) Last edited: Nov 21, 2007 8. Nov 21, 2007 ### andytoh I didn't know what your I stood for. Using your notation, don't we simply have \|R\| = \|F_1\|+\|F_2\|+\|F_3\|+... = \|Q\|+\|Q^2\|+\|Q^3\|+...= $\aleph_0$ +$\aleph_0$ +$\aleph_0$ +... = $\aleph_0$$\aleph_0$ = $\aleph_0$ ? If so, that is our contradiction. Last edited: Nov 21, 2007 9. Nov 21, 2007 ### morphism No. For example F_1 contains a copy of Q for each r_i in B, and F_2 contains a copy of Q for each pair {r_i, r_j} in B. So if I is uncountable (which I am assuming can happen, modulo my remark towards the end), then \|F_n\| need not be bounded by \|Q^n\|. Also, strictly speaking, R isn't a disjoint union of the F_n's because I didn't specify that the n elements taken from B be distinct. But this is immaterial... 10. Nov 21, 2007 ### andytoh Let F_n be the set of real numbers expressible as a linear combination of n elements of B, with none of the rational coefficients being zero. By the uniqueness of an element expressed as a linear combinations of basis elements, then we have R = \|{0}\|+\|F_1\|+\|F_2\|+..., since the F_n are now disjoint. I'll look into the bounds of the \|F_n\|.... Last edited: Nov 21, 2007 11. Nov 21, 2007 ### andytoh If \|R\|<=\|I\|=\|B^(<w)\|=\|number of subsets of B\|=2^\|B\|=2^$\aleph_0$ = c = \|R\|, where is the contradiction? What if we use Schauder bases (allowing for infinite sums of the basis elements)? There is no injection P(B)-> B, so P(B) is uncountable since B is equivalent to the natural numbers by assumption. I believe there is no injection from the set of all finite subsets to B either, so B^(<w) would have cardinality >= c. Last edited: Nov 21, 2007 12. Nov 21, 2007 ### morphism Can you stop and read my post from the beginning? It seems like you're completely missing the point. (1) I'm taking any Hamel basis B whose cardinality is \|I\|. All we know about \|I\| is that it's infinite (although, as I indicated in the end of my post, we can assume that I is countable and get a contradiction - by going through the argument unchanged: we get \|R\| <= \|I\|). (2) B^(<w) is the set of finite subsets of B, and not the power set of B. It's an easy exercise to prove that if \|B\| is infinite, then \|B^(<w)\|=\|B\|. (3) In regards to what you posted in post #10, we don't really need to write R as a disjoint union of the F_n's. It's perfectly sufficient that \|R\| <= \|F_1\| + \|F_2\| + ..., since I already gave you an upper bound for each \|F_n\|. 13. Nov 21, 2007 ### andytoh I want to believe you, but I don't see it (yet). B^(<w) = the finite subsets of B Let g: B -> B^(<w). Claim: g cannot be surjective. Let K={b in B\| b does not belong to g(b)}. If g is surjective, let g(x) = K. Then x belongs to K iff x does not belong to g(x)=K, a contradiction. Oops, K can be infinite. Ok, I'll try to prove that \|B^(<w)\|=\|B\|. Last edited: Nov 21, 2007 14. Nov 21, 2007 ### morphism As for Schauder bases, well, I'm only familiar with this concept in the scope of Banach spaces. But I looked it up, and Wikipedia says that a Schauder basis is countable by definition. 15. Nov 21, 2007 ### morphism Why is K finite? Here's a sketch you can use to prove \|C\| <= \|B\|: (1) For each n, define f_n : B^n -> C by (b_1, ..., b_n) $\mapsto$ {b_1, ..., b_n}. (2) Extend this to F : $\cup_n$ B^n -> C. (3) \|$\cup_n$ B^n\| = \|B\|. (4) Try to reason that \|C\| <= \|B\|. Alternative path: (1) Let C_n = { A in C : \|A\| = n }. (2) Well-order B. Define f : C_n -> B^n by {b_1 < ... < b_n} $\mapsto$ (b_1, ..., b_n). Deduce that \|C_n\| <= \|B^n\| = \|B\| (well, except when n=0). (3) \|C\| = \|$\cup_n$ C_n\| <= \|B\|. (I essentially used these ideas in post #7. First I decided not to reuse them here, but then I figured I might as well...) Last edited: Nov 22, 2007 16. Nov 22, 2007 ### andytoh I didn't read your proof to why \|C\|=\|B\| yet, but while I slept I thought of the following proof: Let B = {b_1,b_2,...}. Let B_k be the collection of all subsets of B whose element with the highest index is b_k. Then the elements of B_k is mapped bijectively to any set with 2^(k-1) elements (the number of subsets of {b_1,...,b_(k-1)}. Then C = U(B_k) is mapped bijectively to a countable collection of finite sets and hence is countable, and so \|C\|=\|B\|. In my proof, I assumed that B is countable. If B is not countable, I suppose one can just well-order the index of B and use transfinite induction. Last edited: Nov 22, 2007 17. Nov 22, 2007 ### morphism That's fine; it's more or less the second method I posted in #15. 18. Nov 22, 2007 ### andytoh I finished my proof that \|C\|=\|B\| for the uncountable case (using your method, because your f in your second method is injective). I was wondering if a proof using transfinite induction could work too. Not enough exercises in transfinite induction are given in textbooks. For readers not wishing to read the previous posts: B is any uncountable set and C is the collection of all finite subsets of B. Use transfinite induction to prove that \|C\|=\|B\|. Call I the index set for B and well-order I. Let J be all the elements of I such that the collection C(J) of all finite subsets of B with elements indexed by J has cardinality <= B. If the section S_i is a subset of J, then the only new finite subsets created by introducing {i} are just KU{b_i}, where K belongs to C(J). So then \|C(JU{i})\| <= \|C(J)\| + \|C(J)\| = \|C(J)\| <= \|B\|, so that {i} belongs to J. Thus J is inductive so that J = I by the principle of transfinite induction. Is that right? Last edited: Nov 23, 2007 19. Nov 22, 2007 ### morphism I don't see anything wrong with it! 20. Nov 22, 2007 ### andytoh Thanks morphism. This very interesting topic ends perfectly.	2017-12-17 15:59:28	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9081291556358337, "perplexity": 602.2701022993964}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948596115.72/warc/CC-MAIN-20171217152217-20171217174217-00648.warc.gz"}
http://math.stackexchange.com/questions/879179/is-there-any-similar-math-limerick	# Is there any similar math limerick? I found this one $$\frac{(12+144+20)+\left(3 \cdot \sqrt{4}\right)}{7}+(5 \cdot 11)=9^2+0.$$ Which is : A dozen, a gross, and a score Plus three times the square root of four Divided by seven Plus five times eleven Is nine squared and not a bit more. I think this is very entertaining, thus I wonder if there is any similar limerick/math poem. - That's probably not even right. – Quinn Culver Jul 27 '14 at 1:10 There once was a number named e. Who took way too much LSD. She thought she was great. But that fact we debate. We know she wasn't greater than 3. trottermath.net/humor/limricks.html – Eul Can Jul 27 '14 at 1:12 That site also has a good one with an integral that you may like though it's extremely irksome that they've misspelled limericks in the url. – Eul Can Jul 27 '14 at 1:20 @oliveeuler You should post your comment as an answer. Thanks for the link! – SpamIAm Jul 27 '14 at 1:21 $$\int_1^{\sqrt[3]{3}} t^2\mathrm{d}t\cdot\cos\left(\frac{3\pi}{9}\right)=\ln(\sqrt[3]{e})$$ $$\text{Integral t squared dt,}$$ $$\text{from 1 to the cube root of 3,}$$ $$\text{times the cosine,}$$ $$\text{of three pi over 9,}$$ $$\text{equals log of the cube root of e.}$$ You can find some more here: http://www.trottermath.net/humor/limricks.html - Assuming that Schnaderhuepfel are the (south?) German equivalent of limericks, I offer the following, which I heard from my father (but the misspellings are my own): Mir fehlt nur ein Hilfssatz, Dann bin ich ein Gauss. Doch den Hilfssatz, den Hilfssatz, Den krieg ich nicht raus. - 'I lack only a helpful set, I am a Gaussian, But the alternative set - the auxiliary set, I war not out.' - the limits of machine translation! – mistermarko Jul 27 '14 at 5:15 A non-machine translation (not entirely literal and not rhyming): I need only a lemma; then I'm a Gauss. But that lemma, that lemma; I just can't prove it. – Andreas Blass Jul 27 '14 at 5:20 Hilfssatz (capital H) is 'lemma', and hilfssatz (small h) is 'helpful set' - interesting. But what about the last line - to prove something you need to go to war! Makes perfect sense. – mistermarko Jul 27 '14 at 5:26 @mistermarko "Kriegen" can mean to get or to obtain. So the last line literally says "I don't get it [the lemma] out." Also, I've never heard of "hilfssatz" (small h) meaning "helpful set"; in fact, since "set" is a noun, any German word meaning "helpful set" should be capitalized. – Andreas Blass Jul 27 '14 at 5:33 That's why google was confused. – mistermarko Jul 27 '14 at 5:36	2016-06-30 09:06:04	{"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": 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": 0.670256495475769, "perplexity": 3291.4698381596736}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783398216.41/warc/CC-MAIN-20160624154958-00100-ip-10-164-35-72.ec2.internal.warc.gz"}
https://www.eriksmistad.no/making-charts-and-outputing-them-as-images-to-the-browser-in-django/?replytocom=98885	# Making charts and output them as images to the browser in Django Simple graph made with matplotlib Lets say you are working on a website made in Django. And you want to make some nice looking graphs real time, as images from dynamic data. This can be done by using the python 2D graph library matplotlib. The library can be found in the debian package python-matplotlib. A simple graph showing a sine curve, seen to the right, can be generated in regular python using the following code(taken from this example): from pylab import * t = arange(0.0, 2.0, 0.01) s = sin(2pit) plot(t, s, linewidth=1.0) xlabel('time (s)') ylabel('voltage (mV)') title('About as simple as it gets, folks') grid(True) show() #### Output graph to browser from a Django view If you want to output this graph as a PNG image to the browser from a view in Django, you can store the image in a string buffer and output this buffer using the HttpReponse class in Django and set the mime type to image/png. from django.http import HttpResponse from django.shortcuts import render from matplotlib import pylab from pylab import * import PIL, PIL.Image, StringIO def index(request): return render(request, 'yourapp/index.html') def showimage(request): # Construct the graph t = arange(0.0, 2.0, 0.01) s = sin(2pit) plot(t, s, linewidth=1.0) xlabel('time (s)') ylabel('voltage (mV)') title('About as simple as it gets, folks') grid(True) # Store image in a string buffer buffer = StringIO.StringIO() canvas = pylab.get_current_fig_manager().canvas canvas.draw() pilImage = PIL.Image.frombytes("RGB", canvas.get_width_height(), canvas.tostring_rgb()) pilImage.save(buffer, "PNG") pylab.close() # Send buffer in a http response the the browser with the mime type image/png set return HttpResponse(buffer.getvalue(), content_type="image/png") Let’s say you want to display this image on your index page, you can have something like this on your index.html template: <h1>Hello world of django + matplotlib</h1> <img src="/showimage/" alt=""> Assuming you have an urls.py with something like this: from django.urls import include, path from . import views urlpatterns = [ path('', views.index, name='main-view'), path('/showimage/', views.showimage, name='showimage'), ### 27 Responses 1. Anton says: If you use django 1.11+, you can choose django-matplotlib field (http://django_matplotlib.readthedocs.org/). This is virtual field (it doesn’t generate a column in the db) which can be easily integrated to Django Admin. 2. Wonhee says: could see the full view.py and the main html page please? 3. Anonymous says: What would be helpful is an FULL example from a CBV (meaning views.py + template file). 4. Praveen says: Thanks for the Notes. But how to render the image to html file ? I am unable pass it html file. Please help me • gayathri says: same as praveen… how to write the html file for this • Erik Smistad says: Lets say the URL pointing to the view show_image is /show_image/, then you add the following in your HTML file: <img src="/show_image/" alt=""> • sajwal says: I still do not understand. My index page is on localhost:8000/. in my urls.py it is on path(‘ ‘,views.index, name =’index’). I have a function def index(request). what do I write in index.html to display this image there? • Akshay Nimbalkar says: Hi Sajwal, did you resolve this issue? Even. i dont understand what to write in index.html to display the image generated from matplotlib or seaborn. • Akshay Nimbalkar says: @Erik Smistad : can you be more specific? • Erik Smistad says: I have updated the post with some more details. Hopefully it will help you understand how to set up everything. • Anonymous says: It gives me this error when I followed your logic: Terminating app due to uncaught exception ‘NSInternalInconsistencyException’, reason: ‘NSWindow drag regions should only be invalidated on the Main Thread!’ • Sajwal says: f you are running a webserver and using it to save Matplotlib make sure to set the backend to a non-interactive one (matplotlib.use(‘agg’) or matplotlib.pyplot.switch_backend(‘Agg’)) so that your server does not try to create (and then destroy) GUI windows that will never be seen (or if they are will be more of a nuisance). after import matplotlib matplotlib.use(‘agg’) • Sajwal says: I understand now. Thank you. Just somethings that worked for me are that I had to change buffer = StringIO.StringIO() (we can use just StringsIO() in python 3, imported from io) to buffer = BytesIO() which I had to import from io from io import BytesIO also for some reason I had to add matplotlib.use(‘Agg’) so that I did not get that NSInternalInconsistencyException’, reason: ‘NSWindow drag regions should only be invalidated on the Main Thread!’ error Anyways, thank you so much. Great post! 5. ojas says: Couple of suggestions: As of April 2018: from string is replaced with frombytes user content_type instead of mimetype • Erik Smistad says: Thanks! 6. You don’t need io or cStringIO. You can just do: canvas = FigureCanvasAgg(f) response = HttpResponse(content_type=’image/png’) canvas.print_png(response) plt.close(f) return response 7. Anton says: -pylab.cose() +pylab.close() 8. Derek says: Can you show an example that involves creating and sending multiple chart images to the same page – I cannot get this to work. 9. Satvir says: Thanku it was really helpful. 10. noemi says: Hi! Thanks for your website, it is really useful. About the output graph to browser from a Django view, I wonder if instead of using HttpResponse we can use render_to_response and represent the graph in a specific html page. I’ve done something like this: # serialize to HTTP response response = HttpResponse(buffer.getvalue(), mimetype=”image/png”) return render_to_response(‘eQL/dev/showImage.html’, {‘response’:response}) But I don’t know how to plot it in the html file. Thanks • Erik Smistad says: Images on websites are sent in separate http responses, so I don’t think that’s possible • Anonymous says: Then what would you suggest if we needed to display the image in a specific, html template?? 11. masood says: whats the book you prefer to learn python with that • Erik Smistad says: I don’t have a Python book, I use the manual on the web instead 12. Nathan Moos says: you could use cStringIO…its faster • Erik Smistad says: I tried using cStringIO instead of StringIO and measured execution time using datetime. Both buffer types ran in about 0.25 seconds. I think this is because the code only performs one large write to the buffer. If you have a buffer with several writes and reads, cStringIO will probably be notably faster	2020-02-22 10:16:31	{"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": 0.2780899405479431, "perplexity": 9201.04016191346}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "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-2020-10/segments/1581875145657.46/warc/CC-MAIN-20200222085018-20200222115018-00187.warc.gz"}
http://www.physicsforums.com/showthread.php?t=446921	## deriving lorentz transformations from maxwells equations or em field tensor? electromagnetic eqn and tensor are invariant under lorentz group but is it possible to derive lorentz transformations from them? Tags electromagnetism, lorentz	2013-05-25 18:01:29	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.992671012878418, "perplexity": 1223.2678255214926}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1368706009988/warc/CC-MAIN-20130516120649-00018-ip-10-60-113-184.ec2.internal.warc.gz"}
https://deepai.org/publication/approximate-bayesian-computation-via-the-energy-statistic	# Approximate Bayesian computation via the energy statistic Approximate Bayesian computation (ABC) has become an essential part of the Bayesian toolbox for addressing problems in which the likelihood is prohibitively expensive or entirely unknown, making it intractable. ABC defines a quasi-posterior by comparing observed data with simulated data, traditionally based on some summary statistics, the elicitation of which is regarded as a key difficulty. In recent years, a number of data discrepancy measures bypassing the construction of summary statistics have been proposed, including the Kullback--Leibler divergence, the Wasserstein distance and maximum mean discrepancies. Here we propose a novel importance-sampling (IS) ABC algorithm relying on the so-called two-sample energy statistic. We establish a new asymptotic result for the case where both the observed sample size and the simulated data sample size increase to infinity, which highlights to what extent the data discrepancy measure impacts the asymptotic pseudo-posterior. The result holds in the broad setting of IS-ABC methodologies, thus generalizing previous results that have been established only for rejection ABC algorithms. Furthermore, we propose a consistent V-statistic estimator of the energy statistic, under which we show that the large sample result holds. Our proposed energy statistic based ABC algorithm is demonstrated on a variety of models, including a Gaussian mixture, a moving-average model of order two, a bivariate beta and a multivariate g-and-k distribution. We find that our proposed method compares well with alternative discrepancy measures. ## Authors • 23 publications • 13 publications • 1 publication • 9 publications 02/09/2015 ### K2-ABC: Approximate Bayesian Computation with Kernel Embeddings Complicated generative models often result in a situation where computin... 10/28/2019 ### Approximate Bayesian Computation with the Sliced-Wasserstein Distance Approximate Bayesian Computation (ABC) is a popular method for approxima... 06/13/2020 ### γ-ABC: Outlier-Robust Approximate Bayesian Computation based on Robust Divergence Estimator Making a reliable inference in complex models is an essential issue in s... 10/08/2015 ### Learning Summary Statistic for Approximate Bayesian Computation via Deep Neural Network Approximate Bayesian Computation (ABC) methods are used to approximate p... 09/21/2018 ### Parameter inference and model comparison using theoretical predictions from noisy simulations When inferring unknown parameters or comparing different models, data mu... 01/19/2021 ### Selection of Summary Statistics for Network Model Choice with Approximate Bayesian Computation Approximate Bayesian Computation (ABC) now serves as one of the major st... 11/22/2021 ### Approximate Bayesian Computation via Classification Approximate Bayesian Computation (ABC) enables statistical inference in ... ##### This week in AI Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday. ## 1 Introduction In recent years, Bayesian inference has become a popular paradigm for machine learning and statistical analysis. Good introductions and references to the primary methods and philosophies of Bayesian inference can be found in texts such as Press (2003), Ghosh et al. (2006), Koch (2007), Koop et al. (2007), Robert (2007), Barber (2012), and Murphy (2012). In this article, we are concerned with the problem of parametric, or classical Bayesian inference. For details regarding nonparametric Bayesian inference, the reader is referred to the expositions of Ghosh & Ramamoorthi (2003), Hjort et al. (2010), and Ghosh & van der Vaart (2017). When conducting parametric Bayesian inference, we observe some realizations of the data that are generated from some data generating process (DGP), which can be characterized by a parametric likelihood, given by a probability density function (PDF) , determined entirely via the parameter vector . Using the information that the parameter vector is a realization of a random variable , which arises from a DGP that can be characterized by some known prior PDF , we wish to characterize the posterior distribution π(θ\|x)=f(x\|θ)π(θ)c(x), (1) where In very simple cases, such as cases when the prior PDF is a conjugate of the likelihood (cf. Robert, 2007, Sec. 3.3), the posterior distribution (1) can be expressed explicitly. In the case of more complex but still tractable pairs of likelihood and prior PDFs, one can sample from (1) via a variety of Monte Carlo methods, such as those reported in Press (2003, Ch. 6). In cases where the likelihood function is known but not tractable, or when the likelihood function has entirely unknown form, one cannot exactly sample from (1) in an inexpensive manner, or at all. In such situations, a sample from an approximation of (1) may suffice in order to conduct the user’s desired inference. Such a sample can be drawn via the method of approximate Bayesian computation (ABC). It is generally agreed that the ABC paradigm originated from the works of Rubin (1984), Tavaré et al. (1997), and Pritchard et al. (1999); see Tavaré (2019) for details. Stemming from the initial listed works, there are now numerous variants of ABC methods. Some good reviews of the current ABC literature can be found in the expositions of Marin et al. (2012), Voss (2014, Sec. 5.1), Lintusaari et al. (2017), and Karabatsos & Leisen (2018). The volume of Sisson et al. (2019) provides a comprehensive treatment regarding ABC methodologies. The core philosophy of ABC is to define a quasi-posterior by comparing data with plausibly simulated replicates. The comparison is traditionally based on some summary statistics, the choice of which being regarded as a key challenge of the approach. In recent years, data discrepancy measures bypassing the construction of summary statistics have been proposed by viewing data sets as empirical measures. Examples of such an approach is via the use of the Kullback–Leibler divergence, the Wasserstein distance, or a maximum mean discrepancy (MMD) variant. In this article, we develop upon the discrepancy measurement approach of Jiang et al. (2018), via the importance sampling ABC (IS-ABC) approach which makes use of a weight function (see e.g., Karabatsos & Leisen, 2018). In particular, we report on a class of ABC algorithms that utilize the two-sample energy statistic (ES) of Szekely & Rizzo (2004) (see also Baringhaus & Franz, 2004, Szekely & Rizzo, 2013, and Szekely & Rizzo, 2017). Our approach is related to the maximum MMD ABC algorithms that were implemented in Park et al. (2016), Jiang et al. (2018), and Bernton et al. (2019). The MMD is a discrepancy measurement that is closely related to the ES (cf. Sejdinovic et al., 2013). We establish new asymptotic results that have not been proved in these previous papers. In the IS-ABC setting and in the regime where both the observation sample size and the simulated data sample size increase to infinity, our theoretical result highlights how the data discrepancy measure impacts the asymptotic pseudo-posterior. More specifically, under the assumption that the data discrepancy measure converges to some asymptotic value , we show that the pseudo-posterior distribution converges almost surely to a distribution proportional to : the prior distribution times the IS weight function evaluated at , where stands for the ‘true’ parameter value associated to the DGP that generates observations . Although devised in settings where likelihoods are assumed intractible, ABC can also be cast in the setting of robustness with respect to misspecification, where the ABC posterior distribution can be viewed as a special case of a coarsened posterior distribution (cf. Miller & Dunson, 2018). The remainder of the article proceeds as follows. In Section 2, we introduce the general IS-ABC framework. In Section 3, we introduce the two-sample ES and demonstrate how it can be incorporated into the IS-ABC framework. Theoretical results regarding the IS-ABC framework and the two-sample ES are presented in Section 4. Illustrations of the IS-ABC framework are presented in Section 5. Conclusions are drawn in Section 6. ## 2 Importance sampling ABC Assume that we observe independent and identically distributed (IID) replicates of from some DGP, which we put into . We suppose that the DGP that generates is dependent on some parameter vector , a realization of from space , which is random and has prior PDF . Denote to be the PDF of , given , and write f(xn\|θ)=n∏i=1f(xi\|θ), where is a realization of , and each is a realization of (). If were known, then we could use (1) to write the posterior PDF π(θ\|xn)=f(xn\|θ)π(θ)c(xn), (2) where is a constant that makes . When evaluating is prohibitive and ABC is required, then operating with is similarly difficult. We suppose that given any , we at least have the capability of sampling from the DGP with PDF . That is, we have a simulation method that allows us to feasibly sample the IID vector , for any , for a DGP with PDF f(yn\|θ)=m∏i=1f(yi\|θ). Using the simulation mechanism that generates samples and the prior distribution that generates parameters , we can simulate a set of simulations , where and is the transposition operator. Here, for each , is an observation from the DGP with joint PDF , hence each is composed of a parameter value and a datum conditional on the parameter value. We now consider how and can be combined in order to construct an approximation of (2). Following the approach of Jiang et al. (2018), we define to be some non-negative real-valued function that outputs a small value if and are similar, and outputs a large value if and are different, in some sense. We call the data discrepancy measurement between and , and we say that is the data discrepancy function. Next, we let be a non-negative, decreasing (in ), and bounded (importance sampling) weight function (cf. Section 3 of Karabatsos & Leisen, 2018), which takes as inputs a data discrepancy measurement and a calibration parameter . Using the weight and discrepancy functions, we can propose the following approximation for (2). In the language of Jiang et al. (2018), we call πm,ϵ(θ\|xn)=π(θ)Lm,ϵ(xn\|θ)cm,ϵ(xn) (3) the quasi-posterior PDF, where Lm,ϵ(xn\|θ)=∫Xmw(D(xn,ym),ϵ)f(ym\|θ)dym is the approximate likelihood function, and cm,ϵ(xn)=∫Tπ(θ)Lm,ϵ(xn\|θ)dθ is a normalization constant. We can use (3) to approximate (2) in the following way. For any functional of the parameter vector of interest, say, we may approximate the posterior Bayes estimator of via the expression E[g(Θ)\|xn]≈∫Tg(θ)π(θ)Lm,ϵ(xn\|θ)dθcm,ϵ(xn), (4) where the right-hand side of (4 ) can be unbiasedly estimated using via (5) We call the process of constructing (5), to approximate (4), the IS-ABC procedure. The general form of the IS-ABC procedure is provided in Algorithm 1. ###### Algorithm 1. IS-ABC procedure for approximating . Input: a data discrepancy function , a weight function , and a calibration parameter . For ; sample from the DGP with PDF ; generate from the DGP with PDF ; put into . Output: and construct the estimator . ## 3 The energy statistic (ES) Let define a metric and let and be two random variables that are in a metric space endowed with , where . Furthermore, let and be two random variables that have the same distributions as and , respectively. Here, , , , and are all independent of one another. Upon writing we can define the original ES of Baringhaus & Franz (2004) and Szekely & Rizzo (2004), as a function of and , via the expression , where is the metric corresponding to the (). Thus, the original ES statistic, which we shall also denote as , is defined using the Euclidean norm . The original ES has numerous useful mathematic properties. For instance, under the assumption that , it was shown that E(X,Y)=Γ(d+12)π(d+1)/2∫Rd\|φX(t)−φY(t)\|2∥t∥d+12dt, (6) in Proposition 1 of Szekely & Rizzo (2013), where is the gamma function and (respectively, ) is the characteristic function of (respectively, ). Thus, we have the fact that for any , and if and only if and are identically distributed. The result above is generalized in Proposition 3 of Szekely & Rizzo (2013), where we have the following statement. If is a continuous function and are independent random variables, then it is necessary and sufficient that is strictly negative definite (see Szekely & Rizzo, 2013 for the precise definition) for the following conclusion to hold: for any , and if and only if and are identically distributed. We observe that there is thus an infinite variety of functions from which we can construct energy statistics. We shall concentrate on the use of the original ES, based on , since it is the most well known and popular of the varieties. ### 3.1 The V-statistic estimator Suppose that we observe and , where the former is a sample containing IID replicates of , and the latter is a sample containing IID replicates of , respectively, with and being independent. In Gretton et al. (2012), it was shown that for any , upon assuming that , the so-called V-statistic estimator (cf. Serfling, 1980, Ch. 5 and Koroljuk & Borovskich, 1994) Vδ(Xn,Ym)=2mnn∑i=1m∑j=1δ(Xi,Yj)−1n2n∑i=1n∑j=1δ(Xi,Xj)−1m2m∑i=1m∑j=1δ(Yi,Yj), (7) can be proved to converge in probability to , as and , under the condition that , for some constant (see also Gretton et al., 2007). We note that the assumption of this result is rather restrictive, since it either requires the bounding of the space or the function . In the sequel, we will present a result for the almost sure convergence of the V-statistic that depends on the satisfaction of a more realistic hypothesis. It is noteworthy that if the ES is non-negative, then the V-statistic retains the non-negativity property of its corresponding ES (cf. Gretton et al., 2012). That is, for any continuous and negative definite function , we have . ### 3.2 The ES-based IS-ABC algorithm From Algorithm 1, we observe that an IS-ABC algorithm requires three components. A data discrepancy measurement , a weighting function , and a tuning parameter . We propose the use of the ES in the place of the data discrepancy measurement , in combination with various weight functions that have been used in the literature. That is we set D(Xn,Ym)=Vδ(Xn,Ym), in Algorithm 1. In particular, we consider original ES, where . We name our framework the ES-ABC algorithm. In Section 4, we shall demonstrate that the proposed algorithm possesses desirable large sample qualities that guarantees its performance in practice, as illustrated in Section 5. ### 3.3 Related methods The ES-ABC algorithm that we have presented here is closely related to ABC algorithms based on the maximum mean discrepancy (MMD) that were implemented in Park et al. (2016), Jiang et al. (2018), and Bernton et al. (2019). For each positive definite Mercer kernel function (, the corresponding MMD is defined via the equation MMD2χ(X,Y)=E[χ(X,X′)]+E[χ(Y,Y′)]−2E[χ(X,Y)], where are random variable such that and are identically distributed to and , respectively. The MMD as a statistic for testing goodness-of-fit was studied prominently in articles such as Gretton et al. (2007), Gretton et al. (2009), and Gretton et al. (2012). It is clear that if , the forms of the ES and the squared MMD are identical. More details regarding the relationship between the two classes of statistics can be found in Sejdinovic et al. (2013). We note two shortcomings with respect to the applications of the MMD as a basis for an ABC algorithm in the previous literature. Firstly, no theoretical results regarding the consistency of the MMD-based methods have been proved. And secondly, in the application by Park et al. (2016) and Jiang et al. (2018), the MMD was implemented using the unbiased U-statistic estimator, rather than the biased V-statistic estimator. Although both estimators are consistent, in the sense that they can be proved to be convergent to the desired limiting MMD value, the U-statistic estimator has the unfortunate property of not being bounded from below by zero (cf. Gretton et al., 2012). As such, it does not meet the criteria for a data discrepancy measurement. ## 4 Theoretical results ### 4.1 General asymptotic analysis We now establish a consistency result for the quasi-posterior density (3), when and approach infinity. Our result generalizes the main result of Jiang et al. (2018) (i.e., Theorem 1), which is the specific case when the weight function is restricted to the form (8) where is the Iverson bracket notation, which equals 1 when the internal statement is true, and 0, otherwise (cf. Graham et al., 1994). The weighting function of form (8), when implemented within the IS-ABC framework, produces the common rejection ABC algorithms, that were suggested by Tavaré et al. (1997), and Pritchard et al. (1999). We extended upon the result of Jiang et al. (2018) so that we may provide theoretical guarantees for more exotic ABC procedures, such as the kernel-smoothed ABC procedure of Park et al. (2016), which implements weights of the form w(d,ϵ)=exp(−dq/ϵ), (9) for . See Karabatsos & Leisen (2018) for further discussion and examples. In order to prove our consistency result, we require Hunt’s lemma, which is reported in Dellacherie & Meyer (1980), as Theorem 45 of Section V.5. For convenience to the reader, we present the result, below. ###### Theorem 1. Let be a probability space with increasing and let . Suppose that is a sequence of random variables that is bounded from above in absolute value by some integrable random variable , and further suppose that converges almost surely to the random variable . Then, almost surely, and in mean, as . Define the continuity set of a function as C(w)={d:w is continuous at d}. Using Theorem 1, we can now prove the following result regarding the asymptotic behavior of the quasi-posterior density function (3). ###### Theorem 2. Let and be IID samples from DGPs that can be characterized by PDFs and , respectively, with corresponding parameter vectors and . Suppose that the data discrepancy converges to some , which is a function of and , almost surely as , for some . If is piecewise continuous and decreasing in and for all and any , and if D∞(θ0,θ)∈C(w(⋅,ϵ)), then we have (10) almost surely, as . ###### Proof. Using the notation of Theorem 1, we set . Since , for any , we have the existence of a such that is integrable, since we can take . Since converges almost surely to , and is continuous at , we have with probability one by the extended continuous mapping theorem (cf. DasGupta, 2011, Thm. 7.10). Now, let be the generated by the sequence . Thus, is an increasing , which approaches . We are in a position to directly apply Theorem 1. This yields E[w(D(Xn,Ym),ϵ)\|Xn]→E[w(D∞(θ0,θ),ϵ)\|X∞], almost surely, as , where the right-hand side equals . Notice that the left-hand side has the form E[w(D(Xn,Ym),ϵ)\|Xn]=Lm,ϵ(Xn\|θ) and therefore , almost surely, as . Thus, the numerator of (3) converges to π(θ)w(D∞(θ0,θ),ϵ), (11) almost surely. To complete the proof, it suffices to show that the denominator of (3) converges almost surely to ∫Tπ(θ)w(D∞(θ0,θ),ϵ)dθ. (12) Since and , we obtain our desired convergence via the dominated convergence theorem, because . An application of a Slutsky-type theorem yields the almost sure convergence of the ratio between (11) and (12) to the right-hand side of (10), as . ∎ The following result and proof guarantees the applicability of Theorem 2 to rejection ABC procedures, and to kernel-smoothed ABC procedures, as used in Jiang et al. (2018) and Park et al. (2016), respectively. ###### Proposition 1. The result of Theorem 2 applies to rejection ABC and importance sampling ABC, with weight functions of respective forms (8) and (9). ###### Proof. For weights of form (8), we note that is continuous in at all points, other than when . Furthermore, and is hence non-negative and bounded. Thus, under the condition that , we have the desired conclusion of Theorem 2. For weights of form (9), we note that for fixed , is continuous and positive in . Since is uniformly bounded by 1, differentiating with respect to , we obtain , which is negative for any and . Thus, (9) constitutes a weight function and satisfies the conditions of Theorem 2. ### 4.2 Asymptotic of the energy statistic Let and be arbitrary elements of and , respectively. That is and arise from DGPs that can be characterized by PDFs and , respectively. Under the assumption , Proposition 1 of Szekely & Rizzo (2013) states that we can write the ES as E(X,Y)=Γ(d+12)π(d+1)/2∫Rd\|φ(t;θ0)−φ(t;θ)\|2∥t∥d+12dt, (13) where is the characteristic function corresponding to the PDF . We write . From Szekely & Rizzo (2004) we have the fact that for arbitrary , Vδ(Xn,Ym)=1n2m2n∑i1=1n∑i2=1m∑j1=1m∑j2=1κδ(Xi1,Xi2;Yj1,Yj2), where κδ(xi1,xi2;yj1,yj2)=δ(xi1,yj1)+δ(xi2,yj2)−δ(xi1,xi2)−δ(yj1,yj2) is the kernel of the V-statistic that is based on the function . The following result is a direct consequence of Theorem 1 of Sen (1977), when applied to V-statistics constructed from functionals that satisfy the hypothesis of Szekely & Rizzo (2013, Prop. 3). ###### Lemma 1. Make the same assumptions regarding and as in Theorem 2. Let be a continuous and strictly negative definite function. If E(\|κδ(X1,X2;Y1,Y2)\|log+\|κδ(X1,X2;Y1,Y2)\|)<∞, (14) then converges almost surely to , as , where and are arbitrary elements of and , respectively. Furthermore, if and only if and are identically distributed. We may apply the result of Lemma 1 directly to the case of in order to provide an almost sure convergence result regarding the V-statistic . ###### Corollary 1. Make the same assumptions regarding and as in Theorem 2. If and are arbitrary elements of and , respectively, and (15) and if , then converges almost surely to , of form (13). ###### Proof. By the law of total expectation, we apply Lemma 1 by considering the two cases of (14): when and when , separately, to write (16) where and . The first term on the right-hand side of (16) is equal to zero, since , whenever . Thus, we need only be concerned with bounding the second term. For , , thus E(∣∣κδ2∣∣log+∣∣κδ2∣∣\|∣∣κδ2∣∣>1)≤E(∣∣κδ2∣∣2\|∣∣κδ2∣∣>1) The condition that is thus fulfilled if , which is equivalent to E(∣∣κδ2∣∣2)=p0E(∣∣κδ2∣∣2\|∣∣κδ2∣∣≤1)+p1E(∣∣κδ2∣∣2\|∣∣κδ2∣∣>1)<∞, by virtue of the integrability of implying the existence of E(∣∣κδ2∣∣2\|∣∣κδ2∣∣≤1), since it is defined on a bounded support. Next, by the triangle inequality, , and hence ∣∣κδ2∣∣2 ≤4(∥X1∥22+∥X2∥22+∥Y1∥22+∥Y2∥22) +8(∥X1∥2∥X2∥2+∥X1∥2∥Y1∥2+∥X1∥2∥Y2∥2 Since are all pairwise independent, and and are identically distributed to and , respectively, we have E(∣∣κδ2∣∣2) +32[E∥X1∥2E∥Y1∥2], which concludes the proof since is satisfied by the hypothesis and implies . We note that condition (15) is stronger than a direct application of condition (14), which may be preferable in some situations. However, condition (15 ) is somewhat more intuitive and verifiable since it is concerned with the polynomial moments of norms and does not involve the piecewise function . It is also suggested in Zygmund (1951) that one may replace by if it is more convenient to do so. Combining the result of Theorem 2 with Corollary 1 and the conclusion from Proposition 1 of Szekely & Rizzo (2013) provided in Equation (13) yields the key result below. This result justifies the use of the V-statistic estimator for the energy distance within the IS-ABC framework. ###### Corollary 2. Under the assumptions of Corollary 1. If , then the conclusion of Theorem 2 follows with D(Xn,Ym)→Γ(d+12)π(d+1)/2∫Rd\|φ(t;θ0)−φ(t;θ)\|2∥t∥d+12dt=D∞(θ0,θ), almost surely, as , where and , if and only if . ## 5 Illustrations We illustrate the use of the ES on some standard models. The standard rejection ABC algorithm is employed (that is, we use Algorithm 1 with weight function of form (8)) for constructing estimators (5). The proposed ES is compared to the Kullback–Leibler divergence (KL), the Wasserstein distance (WA), and the maximum mean discrepancy (MMD). Here, the ES is applied using the Euclidean metric , the Wasserstein distance using the exponent (cf. Bernton et al., 2019) and the MMD using a Gaussian kernel . The Gaussian kernel is commonly used in the MMD literature, and was also considered for ABC in Park et al. (2016) and Jiang et al. (2018). Details regarding the use of the Kullback–Leibler divergence as a discrepancy function for ABC algorithms can be found in Jiang et al. (2018, Sec. 2). We use to denote that the random variable has probability law . Furthermore, we denote the normal law by , where states that the DGP of is multivariate normal distribution with mean vector and covariance matrix . We further denote the uniform law, in the interval , for , by . We consider examples explored in Jiang et al. (2018, Sec. 4.1). For each illustration below, we sample synthetic data of the same size as the observed data size, , whose value is specified for each model below. We consider only the rejection weight function, and the number of ABC iterations in Algorithm 1 is set to . The tuning parameter is set so that only the smallest discrepancies are kept to form ABC posterior sample. We postpone the discussion of the results of our simulation experiments to Section 5.5 The experiments were implemented in R, using in particular the winference package (Bernton et al., 2019) and the FNN package (Beygelzimer et al., 2013). The Kullback–Leibler divergence between two PDFs is computed within the -nearest neighbor framework (Boltz et al., 2009). Moreover, the -d trees is adopted for implementing the nearest neighbor search, which is the same as the method of Jiang et al. (2018). For estimating the -Wasserstein distance between two multivariate empirical measures, we propose to employ the swapping algorithm (Puccetti, 2017), which is simple to implement, and is more accurate and less computationally expensive than other algorithms commonly used in the literature (Bernton et al., 2019). Regarding the MMD, the same unbiased U-statistic estimator is adopted as given in Jiang et al. (2018) and Park et al. (2016). For reproduction of the the experimental results, the original source code can be accessed at https://github.com/hiendn/Energy_Statistics_ABC. ### 5.1 Bivariate Gaussian mixture model Let be a sequence of IID random variables, such that each has mixture of Gaussian probability law Xi∼pN(μ0,Σ0)+(1−p)N(μ1,Σ1), (17) with known covariance matrices Σ0=[0.5−0.3−0.30.5] and Σ1=[0.25000.25]. We aim to estimate the generative parameters consisting of the mixing probability and the population means and . To this end, we perform ABC using observations, sampled from model (17) with , and . A kernel density estimate (KDE) of the ABC posterior distribution is presented in Figure 1. ### 5.2 Moving-average model of order 2 The moving-average model of order , MA(), is a stochastic process defined as Yt=Zt+q∑i=1θiZt−i, with being a sequence of unobserved noise error terms. Jiang et al. (2018) used a MA model for their benchmarking; namely . Each observation corresponds to a time series of length . Here, we use the same model as that proposed in Jiang et al. (2018), where follows the Student- distribution with degrees of freedom, and . The priors on the model parameters and are taken to be uniform, that is, and . We performed ABC using samples generated from a model with the true parameter values . A KDE of the ABC posterior distribution is displayed in Figure 2. ### 5.3 Bivariate beta model The bivariate beta model proposed by Crackel & Flegal (2017) is defined with five positive parameters by letting V1=U1+U3U5+U4, and V2=U2+U4U5+U3, (18) where , for , and setting and . The bivariate random variable has marginal laws and . We performed ABC using samples of size , which are generated from a DGP with true parameter values . The prior on each of the model parameters is taken to be independent . A KDE of the ABC posterior distribution is displayed in Figure 3. ### 5.4 Multivariate g-and-k distribution A univariate -and- distribution can be defined via its quantile function (Drovandi & Pettitt, 2011): F−1(x)=A+B[1+0.81−exp(−g×zx)1+exp(−g×zx)](1+z2x)kzx, (19) where parameters respectively relate to location, scale, skewness, and kurtosis. Here, is the th quantile of the standard normal distribution. Given a set of parameters , it is easy to simulate observations of a DGP with quantile function (19), by generating a sequence of IID sample , where , for . A so-called -dimensional -and- DGP can instead be defined by applying the quantile function (19) to each of the elements of a multivariate normal vector , where is a covariance matrix. In our experiment, we use a 5-dimensional -and- model with the same covariance matrix and parameter values for as that considered by Jiang et al. (2018). That is, we generate samples of size from a -and- DGP with the true parameter values and the covariance matrix Σ=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣1ρ000ρ1ρ000ρ1ρ000ρ1ρ000ρ1⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦, where . Marginal KDEs of the ABC posterior distributions is presented in Figure 4. ### 5.5 Discussion of the results and performance For each of the four experiments and each parameter, we computed the posterior mean , posterior median , mean absolute error and mean squared error defined by MAE=1MM∑k=1\|θk−θ0\|, andMSE=1MM∑k=1\|θk−θ0\|2, where denotes the pseudo-posterior sample and denotes the true parameter. Here since and is chosen as to retain of the samples. Each experiment was replicated ten times by keeping the same fixed (true) values for the parameters and by sampling new observed data each of the ten times. The estimated quantities , , and errors MAE and were then averaged over the ten replications, and are reported along with standard deviations in columns associated with each estimator and true values for each parameter in Tables 1, 2, 3 and 4. Upon inspection, Tables 1, 2, 3 and 4 showed some advantage in performance from WA on the bivariate Gaussian mixtures, some advantage from the MMD on the bivariate beta model, and some advantage from the ES on the -and- model, while multiple methods are required to make the best inference in the case of the MA experiment. When we further take into account the standard deviations of the estimators, we observe that all four data discrepancy measures essentially perform comparatively well across the four experimental models. Thus, we may conclude that there is no universally best performing discrepancy measure, and one must choose the right method for each problem of interest. Alternatively, one may also consider some kind of averaging over the results of the different discrepancy measures. We have not committed to an investigation of such methodologies and leave it as a future research direction.	2022-01-22 15:02:36	{"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": 0.9211328625679016, "perplexity": 734.3329416502501}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1642320303864.86/warc/CC-MAIN-20220122134127-20220122164127-00138.warc.gz"}
https://chrishicks.io/blog/?tag=aut64	## Dismantling the AUT64 Automotive Cipher, Part II This is the second (rather lengthy) part of a two-part series on the AUT64 automotive cipher – you can read the first part here. AUT64 is a proprietary 64 bit block cipher used by a number of major automotive manufacturers. We reverse engineered the design from an immobiliser firmware image and exposed a number of weaknesses in the design. Since posting the first part I have presented the work we did on AUT64 at CHES 2018. The conference attracted over 600 attendees and featured a great side-channel analysis hackathon which I will write about soon. My presentation was recorded and has been put on youtube (see below) – speaking of which, I’m really thankful when conference organisers do this. I find it very helpful to be able to watch the presentation and read the paper. In the remainder of this post I’m going to summarise the key cryptographic weaknesses which we found in AUT64. Briefly, • The AUT64 Feistel network compression function is weak. • In eight-round AUT64, the input to the compression function can be precisely controlled for a subset of the message space. • The AUT64 substitution-permutation network is weak. • AUT64 has certain weak keys. #### The AUT64 compression function is weak (and introduces certain weak keys) The AUT64 compression function takes as input a 64 bit bitstring, a 32 bit compression function key $$k_G$$ and a round number $$r$$ between 0 and 23. The output is 8 bits. The function depends on three key-independent lookup tables $$T_U, T_L$$ and $$T_{\text{offset}}$$. The first two lookup tables $$T_U, T_L$$ prescribe a nibble-wise permutation of the compression function key $$k_G$$ which is unique for each round. For each of the 16 nibbles in the input, a round-dependent nibble from $$k_G$$ is concatenated and the resulting byte is input to $$T_{\text{offset}}$$. $$T_{\text{offset}}$$ is a 8×8 lookup table which substitutes each nibble in the input according to the key $$k_G$$ and the round number $$r$$. The way the function is implemented means that the key nibble selects a row and the input nibble selects a column. One of the first things we noticed was that the first row and the first column contains only zeroes. In practice what this means is that the compression function key should not contain any nibbles with the value zero, else bijectivity is lost (which is used in at least one remote keyless entry application) and the cipher is weakened. The resulting key size is reduced from 32 bits to $$15^8\approx {31.25}$$ bits. (Un)Fortunately there’s a much more significant weakness in this design. In each round, for every byte in the input, two values from $$T_{\text{offset}}$$ are output and stored in the registers $$gu$$ and $$gl$$. Where $$i$$ is the input byte index, $$X^\prime_i$$ is the $$i$$’th input byte and $$uk,lk$$ are the upper and lower key nibbles selected by $$T_U$$ and $$T_L$$, respectively $gu = \bigoplus^{7}_{i = 0} T_{\text{offset}}\Big[uk(k_G, r, i)\parallel un(X’_i) \Big]$$gl = \bigoplus^{7}_{i = 0} T_{\text{offset}}\Big[lk(k_G, r, i)\parallel ln(X’_i) \Big]$ Significantly the XOR sum is a commutative operation – the order of the operands does not change the result. When we combine this with the fact that each round key is a permutation of the same underlying compression function key, we learn that if we can fix all the input nibbles to have the same value then the round output will always be a fixed value. The bijective design of $$T_{\text{offset}}$$ further indicates that if we do this for all 256 byte values then we will get a different fixed output byte for each input. #### In eight-round AUT64, the input to the compression function can be precisely controlled for a subset of the message space. Now we seek to exploit the compression function weakness we just described. To do this we first need to have (somewhat) chosen-plaintext access to the compression function. Fortunately, the input to the compression function is just the output of the byte-wise permutation $$R$$ in the Feistel network design. So provided that each input byte $$x_0\ldots x_7$$ has the same value, the permuted input $$x^\prime_0\ldots x^\prime_7$$ will be unchanged. Great – this means we can control the input to the compression function $$G$$ under 256 different inputs. Better yet, these are exactly the form of inputs which exploit the AUT64 compression function weakness. Next, we need to be able to discern the output from the compression function in the ciphertext output by AUT64. #### The AUT64 substitution-permutation network is weak The AUT64 round function $$F$$ is as follows We’ve shown we can control $$x^\prime_0 \ldots x^\prime_7$$ for an interesting subset of messages and that these inputs will cause the output from the compression function $$G$$ to be distinguishable from random. What follows $$G$$ is a simple substitution-permutation network comprising a single S-box $$S$$ applied both at the input and output and a bitwise permutation $$\sigma_{\text{bit}}$$ applied in between. Rather than a single 16×16 S-Box, $$S$$ is formed from the concatenation of a single 8×8 S-Box which is defined by the 64-bit key part $$k_\tau$$. This means that if the upper and lower 4 bits output from $$G$$ have the same value, the output of $$S$$ will retain their symmetry and output a byte with an equivalent upper and lower 4 bits. This gives us the final piece of our puzzle. We can cause each of the 256 chosen inputs $$0x00\ldots, 0x00$$ to $$0xFF,\ldots , 0xFF$$ to be input to G, we know the output for each will be a unique byte and we know that of these, the sixteen bytes $$0x00,0x11,\ldots , 0xFF$$ which are symmetric will also have symmetric outputs from the substitution-permutation network into the final ciphertext. #### Breaking semantic security We can now break semantic security (shown above) as follows. We target distinguishing the first round of encryption as this is when we can fully control the input to the compression function with our chosen plaintexts. We create the set $$\mathbb{P}$$ of all 256 plaintexts with the property that every byte has the same value. $\mathbb{P} = \Big \{n^8 : n\in \{0,\ldots,255\} \Big \}$ Then we encrypt every plaintext in $$\mathbb{P}$$ using 8 round AUT64 and save the resulting set of ciphertexts $$\mathbb{C}$$. Our goal is to distinguish the output from the first round by looking at each of the eight byte positions over all 256 ciphertexts. Next, we create 8 new sets $$\mathbb{C}_0 \ldots\mathbb{C}_7$$, In each new set we store the 256 ciphertext bytes corresponding to each byte position in every ciphertext in the ciphertexts set $$\mathbb{C}$$. The output from the first round will be contained within a single set because the byte permutation $$R$$ which mixes the output from each round is fixed. The output from the first round will also be a uniform distribution over the byte-value space. i.e. the set corresponding to the first round will contain exactly one of each byte value $$0, \ldots, 255$$! Semantic security is now broken with an advantage of 1 as we can always distinguish the output of 8 round AUT86 from a random permutation. #### Breaking AUT64 Now we’ve broken semantic security, we’d also like to break the cipher more pragmatically. The nominal key size of AUT64 is 120 bits and is comprised from three parts: a 32-bit compression function key $$k_G$$, a 64-bit S-Box key $$k_\tau$$ and a 24-bit permutation key $$k_\sigma$$. The compression function key is a simple bit-string, the S-Box key defines a 4×4 S-Box and the permutation key defines an 8-element permutation. The maximum entropy of an AUT64 key is therefore $$232×16!×8!≈91.55$$ bits – too much for us to brute force. In our paper we describe a purely cryptanalytic attack which can recover an AUT64 key in fewer than $$\approx 2^{49.6}$$ encryptions! First, encrypt $\mathbb{P}_1 = \Big \{n^8 : n\in \{0,\ldots,255\} \Big \}$ to get $$\mathbb{C}_0 \ldots\mathbb{C}_7$$ (as described above) and then identify the set $$\mathbb{C}_i$$ which is equal to the set of all byte values $$\mathbb{Z}_{256}$$ and therefore identifies the byte position $$i$$ corresponding to the first round of encryption. Once the position $$i$$ is known, construct the second set of chosen plaintexts $\mathbb{P}_2 = \Bigg \{ \Big(n\ll (64-8\times i)\Big) : n\in \{0,\dots,15\} \Bigg \}$ such that the nibble counter $$n\in\{0,\ldots , 15\}$$ is placed in the lower nibble of byte position $$i$$. Encrypting $$\mathbb{P}_2$$ will cause the output of the compression function $$G$$ to be dependent (in the first round) on only one nibble from the substitution table $$T_{\text{offset}}$$. This is because for 15 of the 16 input nibbles, the value 0 will be selected from $$T_{\text{offset}}$$ and XORed into the output registers. Now all that remains (see paper for specifics of key sizes remaining) is to brute-force the 15 possible key-nibble values, the possible $$\approx 2^{37.3}$$ S-Box values and the $$\approx 2^{8.4}$$ permutation values. The resulting attack takes just $$15\times 2^{37.3} \times 2^{8.4} \approx \mathbf{2^{49.6}}$$ encryptions to recover the full 120 bit AUT64 key! I found reverse engineering and breaking AUT64 very exciting, particularly once it became practical. I’ve since (for predominantly diplomatic reasons) moved on to working on PKI solutions in the V2X domain so this may be the last cryptanalysis post I make for a while, although I did recently make an interesting observation about the DES round function I would like to afford some time to. — Happy new years! ## Dismantling the AUT64 Automotive Cipher, Part 1 I recently got my first academic paper published! This was a big moment for me and it feels great to have my hard work validated and preserved in the literature. My paper was published in the IACR Transactions on Cryptographic Hardware and Embedded Systems 2018, at which I’ll be presenting the work in September later this year 🙂 In this post I’ll give a very brief overview of the AUT64 cipher which we reverse engineered from an immobiliser box firmware. In a follow-up post I’ll go into more detail about the cryptographic and implementational weaknesses which we found (read the paper for spoilers!). My PhD supervisor Flavio Garcia has a long history of exposing weak cryptography in embedded applications, perhaps most prominently the MIFARE classic smartcard which, at one time, comprised 70% of the wireless smartcard market and was widely used for public transport payment schemes such as the Oyster Card in London and the OV-chipkaart in The Netherlands. It is also widely used for access control in office and governmental buildings. The attacks on MIFARE classic allowed the recovery of all secret keys, typically in less than a second. AUT64 is an automotive block cipher which was used by a number of automotive manufacturers including Volkswagen group and Mazda. AUT64 has been used both for traditional vehicle immobiliser systems (which prevent hotwiring) and the more-recent remote keyless entry systems which allow the doors of the vehicle to be unlocked remotely. AUT64 was first discovered in an earlier paper by Garcia et al. which showed how the majority of Volkswagen group vehicles between 1995 and 2016 were undermined, in addition to significant cryptographic issues, by frankly laughably inadequate key management practices in which global master keys were used across entire product families. In the paper, I reverse engineer an AUT64 immobiliser system, find a number of cryptographic and key management weaknesses and then combine them to show how the system is critically flawed. A common wisdom in cryptography is that `security by obscurity’ doesn’t work. The details of AUT64 were proprietary and obscured away beneath patents, datasheets and compiled implementations. My first task was to take the datasheets, the patent and a firmware dump from a vehicle immobiliser box and to work out exactly how the cipher operated. I reverse engineered the firmware using IDA. Once the correct processor type had been identified it was relatively easy to identify the routines of interest, it took a lot (lot lot..) longer to discern all the logic. At a high-level, this is what I found AUT64 is a 64-bit block cipher which uses a 120-bit key. This means it takes an input message of length eight bytes $$x_0\ldots x_7$$ and outputs a ciphertext of eight bytes. This specific type of block cipher is called a Feistel construction, the cipher is iterated in rounds which gradually modify the output until it is (ideally!) indistinguishable from a random string. In each round, a byte permutation $$\sigma_{byte}$$ is applied to the input and then the Feistel function $$F$$ is applied to the permuted bytes $$x’_0\ldots x’_7$$. The Feistel function $$F$$ comprises a compression function $$G$$, an S-Box $$S$$ and a bit-wise permutation $$\sigma_{\text{bit}}$$ applied as follows The compression function takes as input the permuted eight bytes $$x’_0\ldots x’_7$$ and outputs just a single byte. The substitution-permutation network composed from the S-Box, the bit-wise permutation and then the S-Box again is used to provide the final output byte $$x{”}_7$$. Finally, the compression function $$G$$ is composed as follows The compression function operates nibble-wise (on 4-bit values) and uses a three of key-independent look-up tables $$T_U, T_L$$ and $$T_{\text{offset}}$$ which define the key schedule and the diffusion property, respectively. What makes AUT64 particularly interesting is that a great deal of the construction is dependent on the key. A 120-bit AUT64 $$K$$ is comprised from three parts: a 32-bit compression function key $$k_G$$, a 64-bit S-Box key $$k_\tau$$ and a 24-bit permutation key $$k_\sigma$$. The compression function key is a simple bit-string, the S-Box key defines a $$4×4$$ S-Box and the permutation key defines an 8-element permutation. The maximum entropy of an AUT64 key is therefore $$2^{32}\times 16! \times 8!\approx 91.55$$ bits. Unlike some of the earlier work on weak proprietary cryptography, this is far too large to be exhaustively searched. A second interesting property of AUT64 is that it is an unbalanced Feistel construction. Consequently, in each round of AUT64 just one-byte of the ciphertext is changed, the other bytes are only rearranged. This is in contrast to the traditional Feistel construction (i.e. DES) in which half of the ciphertext is changed in each round. This means we need to apply at least eight rounds of AUT64 before the ciphertext could ever hope to be statistically indistinguishable from the plaintext. A comprehensive taxonomy of Feistel networks by Schneier et al. can be found here. To summarise, AUT64 is a 64-bit block cipher with an unbalanced Feistel-network type construction. The key has around 96-bits of entropy, the round size is at least eight and the cryptographic properties of the cipher are key-dependent to an unusually large degree. That’s all for now. In the follow up post on this paper I’ll highlight the key cryptographic and implementational weaknesses which let us break AUT64!	2020-02-23 07:18:22	{"extraction_info": {"found_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": 0.6602458953857422, "perplexity": 2152.77983903345}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1581875145747.6/warc/CC-MAIN-20200223062700-20200223092700-00283.warc.gz"}
https://questions.examside.com/past-years/jee/jee-advanced/mathematics/application-of-derivatives	Mathematics Application of Derivatives Previous Years Questions ## MCQ (More than One Correct Answer) Let $$\alpha=\sum\limits_{k = 1}^\infty {{{\sin }^{2k}}\left( {{\pi \over 6}} \right)}$$ Let $g:[0,1] \rightarrow \mathbb{R}$ be the function d... Let f : R $$\to$$ R be given by$$f(x) = (x - 1)(x - 2)(x - 5)$$. Define$$F(x) = \int\limits_0^x {f(t)dt}$$, x > 0Then which of the following opt... Let, $$f(x) = {{\sin \pi x} \over {{x^2}}}$$, x > 0Let x1 < x2 < x3 < ... < xn < ... be all the points of local maximum of f and y1 ... f : R $$\to$$ R is a differentiable function such that f'(x) > 2f(x) for all x$$\in$$R, and f(0) = 1 then If $$f(x) = \left\| {\matrix{ {\cos 2x} & {\cos 2x} & {\sin 2x} \cr { - \cos x} & {\cos x} & { - \sin x} \cr {\sin x} &... Let f: R$$ \to \left( {0,\infty } \right)$$and g : R$$ \to $$R be twice differentiable functions such that f'' and g'' are continuous functions on... Let$$f, g :\left[ { - 1,2} \right] \to R$$be continuous functions which are twice differentiable on the interval$$(-1, 2)$$. Let the values of... The function$$f(x) = 2\left\| x \right\| + \left\| {x + 2} \right\| - \left\| {\left\| {x + 2} \right\| - 2\left\| x \right\|} \right\|$$has a local minimum o... A rectangular sheet of fixed perimeter with sides having their lengths in the ratio$$8:15$$is converted into an open rectangular box by folding afte... If$$f\left( x \right) = \int_0^x {{e^{{t^2}}}} \left( {t - 2} \right)\left( {t - 3} \right)dt$$for all$$x \in \left( {0,\infty } \right),$$then For the function$$$f\left( x \right) = x\cos \,{1 \over x},x \ge 1,f(x)$$is cubic polynomial with$$f(2)=18$$and$$f(1)=-1$$. Also$$f(x)$$has local maxima at$$x=-1$$and$$f'(x)$$has local minima at$$x=0$$, t... Let$$f\left( x \right) = \left\{ {\matrix{ {{e^x},} & {0 \le x \le 1} \cr {2 - {e^{x - 1}},} & {1 < x \le 2} \cr {x - e,} &am... The function $$f\left( x \right) = \int\limits_{ - 1}^x {t\left( {{e^t} - 1} \right)\left( {t - 1} \right){{\left( {t - 2} \right)}^3}\,\,\,{{\left( {... Let$$h\left( x \right) = f\left( x \right) - {\left( {f\left( x \right)} \right)^2} + {\left( {f\left( x \right)} \right)^3}$$for every real number ... If$$f\left( x \right) = \left\{ {\matrix{ {3{x^2} + 12x - 1,} & { - 1 \le x \le 2} \cr {37 - x} & {2 < x \le 3} \cr } } \right... If the line $$ax+by+c=0$$ is a normal to the curve $$xy=1$$, then ## MCQ (Single Correct Answer) Consider the rectangles lying the region $$\left\{ {(x,y) \in R \times R:0\, \le \,x\, \le \,{\pi \over 2}} \right.$$ and $$\left. {0\, \le \,y\, \le... Which of the following options is the only INCORRECT combination? Which of the following options is the only CORRECT combination? Which of the following options is the only CORRECT combination? The least value of a$$ \in R$$for which$$4a{x^2} + {1 \over x} \ge 1,$$, for all$$x>0$$. is Let$$f:\left[ {0,1} \right] \to R$$(the set of all real numbers) be a function. Suppose the function$$f$$is twice differentiable,$$f(0) = f(1)=0$... Let $$f:\left[ {0,1} \right] \to R$$ (the set of all real numbers) be a function. Suppose the function $$f$$ is twice differentiable, $$f(0) = f(1)=0... Let$$f\left( x \right) = {\left( {1 - x} \right)^2}\,\,{\sin ^2}\,\,x + {x^2}$$for all$$x \in IR$$and let$$g\left( x \right) = \int\limits_1^x {... Let $$f\left( x \right) = {\left( {1 - x} \right)^2}\,\,{\sin ^2}\,\,x + {x^2}$$ for all $$x \in IR$$ and let $$g\left( x \right) = \int\limits_1^x {... Consider the two curves$${C_1}:{y^2} = 4x,\,{C_2}:{x^2} + {y^2} - 6x + 1 = 0$$. Then, The total number of local maxima and local minimum of the function$$f\left( x \right) = \left\{ {\matrix{ {{{\left( {2 + x} \right)}^3},} & {... Let the function $$g:\left( { - \infty ,\infty } \right) \to \left( { - {\pi \over 2},{\pi \over 2}} \right)$$ be given by $$g\left( u \right) = 2{... The tangent to the curve$$y = {e^x}$$drawn at the point$$\left( {c,{e^c}} \right)$$intersects the line joining the points$$\left( {c - 1,{e^{c -... If a continuous function $$f$$ defined on the real line $$R$$, assumes positive and negative values in $$R$$ then the equation $$f(x)=0$$ has a root i... If a continuous function $$f$$ defined on the real line $$R$$, assumes positive and negative values in $$R$$ then the equation $$f(x)=0$$ has a root i... If a continuous function $$f$$ defined on the real line $$R$$, assumes positive and negative values in $$R$$ then the equation $$f(x)=0$$ has a root i... If $$P(x)$$ is a polynomial of degree less than or equal to $$2$$ and $$S$$ is the set of all such polynomials so that $$P(0)=0$$, $$P(1)=1$$ and $$P... If$$f\left( x \right) = {x^3} + b{x^2} + cx + d$$and$$0 < {b^2} < c,$$then in$$\left( { - \infty ,\infty } \right)$$If$$f\left( x \right) = {x^a}\log x$$and$$f\left( 0 \right) = 0,$$then the value of$$\alpha $$for which Rolle's theorem can be applied in$$\lef... In $$\left[ {0,1} \right]$$ Languages Mean Value theorem is NOT applicable to Tangent is drawn to ellipse $${{{x^2}} \over {27}} + {y^2} = 1\,\,\,at\,\left( {3\sqrt 3 \cos \theta ,\sin \theta } \right)\left( {where\,\,\theta \... The length of a longest interval in which the function$$3\,\sin x - 4{\sin ^3}x$$is increasing, is The point(s) in the curve$${y^3} + 3{x^2} = 12y$$where the tangent is vertical, is (are) If$$f\left( x \right) = x{e^{x\left( {1 - x} \right)}},$$then$$f(x)$$is The triangle formed by the tangent to the curve$$f\left( x \right) = {x^2} + bx - b$$at the point$$(1, 1)$$and the coordinate axex, lies in the fi... Let$$f\left( x \right) = \left( {1 + {b^2}} \right){x^2} + 2bx + 1$$and let$$m(b)$$be the minimum value of$$f(x)$$. As$$b$$varies, the range o... Consider the following statements in$$S$$and$$RS:\,\,\,$$Both$$\sin \,\,x$$and$$\cos \,\,x$$are decreasing functions in the interv... If the normal to the curve$$y = f\left( x \right)$$and the point$$(3, 4)$$makes an angle$${{{3\pi } \over 4}}$$with the positive$$x$$-axis, the... Let$$f\left( x \right) = \int {{e^x}\left( {x - 1} \right)\left( {x - 2} \right)dx.} $$Then$$f$$decreases in the interval Let$$f\left( x \right) = \left\{ {\matrix{ {\left\| x \right\|,} & {for} & {0 < \left\| x \right\| \le 2} \cr {1,} & {for} & {... For all $$x \in \left( {0,1} \right)$$ The function $$f(x)=$$ $${\sin ^4}x + {\cos ^4}x$$ increases if The number of values of $$x$$ where the function $$f\left( x \right) = \cos x + \cos \left( {\sqrt 2 x} \right)$$ attains its maximum is If $$f\left( x \right) = {{{x^2} - 1} \over {{x^2} + 1}},$$ for every real number $$x$$, then the minimum value of $$f$$ If $$f\left( x \right) = {x \over {\sin x}}$$ and $$g\left( x \right) = {x \over {\tan x}}$$, where $$0 < x \le 1$$, then in this interval The function $$f\left( x \right) = {{in\,\left( {\pi + x} \right)} \over {in\,\left( {e + x} \right)}}$$ is On the interval $$\left[ {0,1} \right]$$ the function $${x^{25}}{\left( {1 - x} \right)^{75}}$$ takes its maximum value at the point The slope of the tangent to a curve $$y = f\left( x \right)$$ at $$\left[ {x,\,f\left( x \right)} \right]$$ is $$2x+1$$. If the curve passes through t... The function defined by $$f\left( x \right) = \left( {x + 2} \right){e^{ - x}}$$ Which one of the following curves cut the parabola $${y^2} = 4ax$$ at right angles? The smallest positive root of the equation, $$\tan x - x = 0$$ lies in Let $$f$$ and $$g$$ be increasing and decreasing functions, respectively from $$\left[ {0,\infty } \right)$$ to $$\left[ {0,\infty } \right)$$. Let $$... Let$$P\left( x \right) = {a_0} + {a_1}{x^2} + {a_2}{x^4} + ...... + {a_n}{x^{2n}}$$be a polynomial in a real variable$$x$$with$$0 < {a_0} &l... If $$a+b+c=0$$, then the quadratic equation $$3a{x^2} + 2bx + c = 0$$ has $$AB$$ is a diameter of a circle and $$C$$ is any point on the circumference of the circle. Then The normal to the curve $$\,x = a\left( {\cos \theta + \theta \sin \theta } \right)$$, $$y = a\left( {\sin \theta - \theta \cos \theta } \right)$$ a... If $$y = a\,\,In\,x + b{x^2} + x$$ has its extreamum values at $$x=-1$$ and $$x=2$$, then ## Numerical For each positive integer n, let $${y_n} = {1 \over n}(n + 1)(n + 2)...{(n + n)^{{1 \over n}}}$$. For x$$\in$$R, let [x] be the greatest integer les... A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of $$V$$ $$m{m^3}$$, has... The slope of the tangent to the curve $${\left( {y - {x^5}} \right)^2} = x{\left( {1 + {x^2}} \right)^2}$$ at the point $$(1, 3)$$ is Let $$f:IR \to IR$$ be defined as $$f\left( x \right) = \left\| x \right\| + \left\| {{x^2} - 1} \right\|.$$ The total number of points at which $$f$$ att... Let $$p(x)$$ be a real polynomial of least degree which has a local maximum at $$x=1$$ and a local minimum at $$x=3$$. If $$p(1)=6$$ and $$p(3)=2$$, t... Let $$f$$ be a real-valued differentiable function on $$R$$ (the set of all real numbers) such that $$f(1)=1$$. If the $$y$$-intercept of the tangent ... Let $$f$$ be a function defined on $$R$$ (the set of all real numbers) such that $$f'\left( x \right) = 2010\left( {x - 2009} \right){\left( {x - 201... Let$$p(x)$$be a polynomial of degree$$4$$having extremum at$$x = 1,2$$and$$\mathop {\lim }\limits_{x \to 0} \left( {1 + {{p\left( x \right)} \... The maximum value of the function $$f\left( x \right) = 2{x^3} - 15{x^2} + 36x - 48$$ on the set $${\rm A} = \left\{ {x\|{x^2} + 20 \le 9x} \right\}$$... The maximum value of the function $$f\left( x \right) = 2{x^3} - 15{x^2} + 36x - 48$$ on the set $${\rm A} = \left\{ {x\|{x^2} + 20 \le 9x} \right\}$$... ## Subjective For a twice differentiable function $$f(x),g(x)$$ is defined as $$4\sqrt {65} g\left( x \right) = \left( {f'{{\left( x \right)}^2} + f''\left( x \righ... If$$\left\| {f\left( {{x_1}} \right) - f\left( {{x_2}} \right)} \right\| < {\left( {{x_1} - {x_2}} \right)^2},$$for all$${x_1},{x_2} \in R$$. Fin... If$$p(x)$$be a polynomial of degree$$3$$satisfying$$p(-1)=10, p(1)=-6$$and$$p(x)$$has maxima at$$x=-1$$and$$p'(x)$$has minima at$$x=1$$. ... Using Rolle's theorem, prove that there is at least one root in$$\left( {{{45}^{1/100}},46} \right)$$of the polynomial$$P\left( x \right) = 51{x^... Prove that for $$x \in \left[ {0,{\pi \over 2}} \right],$$ $$\sin x + 2x \ge {{3x\left( {x + 1} \right)} \over \pi }$$. Explain the identity if any ... Using the relation $$2\left( {1 - \cos x} \right) < {x^2},\,x \ne 0$$ or otherwise, prove that $$\sin \left( {\tan x} \right) \ge x,\,\forall x \i... Find a point on the curve$${x^2} + 2{y^2} = 6$$whose distance from the line$$x+y=7$$, is minimum. If the function$$f:\left[ {0,4} \right] \to R$$is differentiable then show that (i)$$\,\,\,\,\,$$For$$a, b\,\, \in \left( {0,4} \right),{... If $$P(1)=0$$ and $${{dp\left( x \right)} \over {dx}} > P\left( x \right)$$ for all $$x \ge 1$$ then prove that $$P(x)>0$$ for all $$x>1$$.... Let $$- 1 \le p \le 1$$. Show that the equation $$4{x^3} - 3x - p = 0$$ has a unique root in the interval $$\left[ {1/2,\,1} \right]$$ and identify ... Suppose $$p\left( x \right) = {a_0} + {a_1}x + {a_2}{x^2} + .......... + {a_n}{x^n}.$$ If $$\left\| {p\left( x \right)} \right\| \le \left\| {{e^{x - 1}... A curve$$C$$has the property that if the tangent drawn at any point$$P$$on$$C$$meets the co-ordinate axes at$$A$$and$$B$$, then$$P$$is the ... Suppose$$f(x)$$is a function satisfying the following conditions (a)$$f(0)=2,f(1)=1$$, (b)$$f$$has a minimum value at$$x=5/2$$, and (c) for all... Let$$a+b=4$$, where$$a<2,$$and let$$g(x)$$be a differentiable function. If$${{dg} \over {dx}} > 0$$for all$$x$$, prove that$$\int_0^a {... Let $$f\left( x \right) = \left\{ {\matrix{ {x{e^{ax}},\,\,\,\,\,\,\,x \le 0} \cr {x + a{x^2} - {x^3},\,x > 0} \cr } } \right.$$ Where... A curve $$y=f(x)$$ passes through the point $$P(1, 1)$$. The normal to the curve at $$P$$ is $$a(y-1)+(x-1)=0$$. If the slope of the tangent at any po... Determine the points of maxima and minima of the function $$f\left( x \right) = {1 \over 8}\ell n\,x - bx + {x^2},x > 0,$$ where $$b \ge 0$$ is a ... Let $$(h, k)$$ be a fixed point, where $$h > 0,k > 0.$$. A straight line passing through this point cuts the possitive direction of the coordina... The curve $$y = a{x^3} + b{x^2} + cx + 5$$, touches the $$x$$-axis at $$P(-2, 0)$$ and cuts the $$y$$ axis at a point $$Q$$, where its gradient is $$3... The circle$${x^2} + {y^2} = 1$$cuts the$$x$$-axis at$$P$$and$$Q$$. Another circle with centre at$$Q$$and variable radius intersects the first ... Find the equation of the normal to the curve$$y = {\left( {1 + x} \right)^y} + {\sin ^{ - 1}}\left( {{{\sin }^2}x} \right)$$at$$x=0$$Let$$f\left( x \right) = \left\{ {\matrix{ { - {x^3} + {{\left( {{b^3} - {b^2} + b - 1} \right)} \over {\left( {{b^2} + 3b + 2} \right)}},} & ... A cubic $$f(x)$$ vanishes at $$x=2$$ and has relative minimum / maximum at $$x=-1$$ and $$x = {1 \over 3}$$ if $$\int\limits_{ - 1}^1 {f\,\,dx = {{14}... What normal to the curve$$y = {x^2}$$forms the shortest chord? In this questions there are entries in columns$$I$$and$$II$$. Each entry in column$$I$$is related to exactly one entry in column$$II$$. Write th... A window of perimeter$$P$$(including the base of the arch) is in the form of a rectangle surmounded by a semi circle. The semi-circular portion is f... Show that$$2\sin x + \tan x \ge 3x$$where$$0 \le x < {\pi \over 2}$$. A point$$P$$is given on the circumference of a circle of radius$$r$$. Chord$$QR$$is parallel to the tangent at$$P$$. Determine the maximum possi... Find all maxima and minima of the function$$\$y = x{\left( {x - 1} \right)^2},0 \le x \le 2$$Also determine the area bounded by the curve$$y = x{\... Investigate for maxima and minimum the function $$f\left( x \right) = \int\limits_1^x {\left[ {2\left( {t - 1} \right){{\left( {t - 2} \right)}^3} ... Find the point on the curve$$\,\,\,4{x^2} + {a^2}{y^2} = 4{a^2},\,\,\,4 < {a^2} < 8$$that is farthest from the point$$(0, -2)$$. Find all the tangents to the curve$$y = \cos \left( {x + y} \right),\,\, - 2\pi \le x \le 2\pi ,$$that are parallel to the line$$x+2y=0$$. Let$$f\left( x \right) = {\sin ^3}x + \lambda {\sin ^2}x, - {\pi \over 2} < x < {\pi \over 2}.$$Find the intervals in which$$\lambda $$sho... Show that$$1+xIn\left( {x + \sqrt {{x^2} + 1} } \right) \ge \sqrt {1 + {x^2}} $$for all$$x \ge 0$$Find the coordinates of the point on the curve$$y = {x \over {1 + {x^2}}}$$where the tangent to the curve has the greatest slope. If$$f(x)$$and$$g(x)$$are differentiable function for$$0 \le x \le 1$$such that$$f(0)=2$$,$$g(0)=0$$,$$f(1)=6$$;$$g(1)=2$$, then show that th... If$$a{x^2} + {b \over x} \ge c$$for all positive$$x$$where$$a>0$$and$$b>0$$show that$$27a{b^2} \ge 4{c^3}$$. Use the function$$f\left( x \right) = {x^{1/x}},x > 0$$. to determine the bigger of the two numbers$${e^\pi }$$and$${\pi ^e}$$Let$$x$$and$$y$$be two real variables such that$$x>0$$and$$xy=1$$. Find the minimum value of$$x+y$$. For all$$x$$in$$\left[ {0,1} \right]$$, let the second derivative$$f''(x)$$of a function$$f(x)$$exist and satisfy$$\left\| {f''\left( x \right)... Prove that the minimum value of $${{\left( {a + x} \right)\left( {b + x} \right)} \over {\left( {c + x} \right)}},$$ $$a,b > c,x > - c$$ is $${... ## Fill in the Blanks Let$$C$$be the curve$${y^3} - 3xy + 2 = 0$$. If$$H$$is the set of points on the curve$$C$$where the tangent is horizontal and$$V$$is the set ... Let$$P$$be a variable point on the ellipse$${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$with foci$${F_1}$$and$${F_2}$$. If$$A$$is ... The set of all$$x$$for which$$in\left( {1 + x} \right) \le x$$is equal to .......... The larger of$$\cos \left( {In\,\,\theta } \right)$$and$$In \left( {\cos \,\,\theta } \right)$$If$${e^{ - \pi /2}} < \theta < {\pi ... The function $$y = 2{x^2} - In\,\left\| x \right\|$$ is monotonically increasing for values of $$x\left( {x \ne 0} \right)$$ satisfying the inequalities... ## True or False For $$0 < a < x,$$ the minimum value of the function $$lo{g_a}x + {\log _x}a$$ is $$2$$. If $$x-r$$ is a factor of the polynomial $$f\left( x \right) = {a_n}{x^4} + ..... + {a_0},$$ repeated $$m$$ times $$\left( {1 < m \le n} \right)$$,... EXAM MAP Joint Entrance Examination	2023-03-25 23:34:58	{"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": 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": 0.9990050196647644, "perplexity": 881.8835411530673}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1679296945376.29/warc/CC-MAIN-20230325222822-20230326012822-00498.warc.gz"}
https://testbook.com/question-answer/the-figure-shows-drawing-of-a-part-with-dimensions--608a356ba7a6393fb62b43d8	# The figure shows drawing of a part with dimensions and tolerances, both in mm. The permissible tolerance for slot A (rounded off to one decimal place) in mm is ± _____ This question was previously asked in GATE PI 2020 Official Paper View all GATE PI Papers > ## Detailed Solution Concept: L = LA + LB + LC Calculation: Given: L = L+ LB + LC (100 ± 0.5) + (40 ± 0.2) = (40 ± 0.1) + LA + (40 ± 0.2) LAmax = 100 + 0.5 - (40 - 0.1) = 60.6 mm LAmin = 100 - 0.5 - (40 + 0.1) = 59.4 mm Nominal dimension of slot A = 100 - 40 = 60 mm Dimension of slot A can be given as, L = $$60^{+0.6}_{-0.6}$$ Hence, tolerance for slot A can be given as ± 0.6 mm Important Points The dimension of slot A i.e LA is independent Las it lies outside the overall dimension of 100 ± 0.5.	2021-09-27 13:16:50	{"extraction_info": {"found_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": 0.6101376414299011, "perplexity": 3574.6616837738093}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1631780058450.44/warc/CC-MAIN-20210927120736-20210927150736-00032.warc.gz"}
https://www.gamedev.net/forums/topic/440514-usefulobscure-net-classes/	[.net] Useful/obscure .Net classes This topic is 4073 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic. Recommended Posts I've been tasked with a doing small learning session at work (less than 10 people, maybe 15 minutes). I was thinking that often times people may write code to perform tasks already covered by classes within the .Net framework. So I thought I'd present a few .Net classes that people may be not be aware of that might prove useful in their every day lives. I'd just present the basics of the class, what it does, some of its more useful methods, maybe some of its downfalls, and a small example. So does anyone have any suggestions? Examples of classes (or interfaces or whatever) that make your coding life easier in one way or another? They don't have to necessarily be obscure or necessarily the most absolutely useful thing in the world. Just something that people may not know about that might take some time and effort out of their work? Thanks! Share on other sites Depending on what kind of projects/How many people work o differet projects, you might want to present MSBuild and the accompanying classes. There's also a whole slew of cool stuff in System.Diagnostics. Stuff like DebuggerDisplayAttribute can be useful if you end up debugging a given type a lot and looking for the same info often. Share on other sites Thanks Jfclavette. I'll look into System.Diagnostic and MSBuild. Any others to look into? I was thinking maybe System.Collections and doing a run-down of the advantages of different containers. Share on other sites System.CodeDom and System.CodeDom.Compiler can be used to parse source files and compile them into assemblies - couple this with System.Reflection to load the output and you can create scriptable applications quite easily. Share on other sites I find that a lot of programmers aren't aware/proficient at using .Net Serialization and often recreate their own serialization framework. Share on other sites I would avoid System.Collections and instead work on System.Collections.Generics. The generic collection classes are much neater and more type safe (and if you do need to have a variable type container, you can always explicitly use Object). If they have used C/C++ a lot, then this construct would be familiar: enum AnEnum{ Item1, Item2};const char *const AnEnumNames [] ={ "Item1", "Item2"}; which you'd use a lot for loading/saving data. In .Net (this is the C++/CLI version): enum class AnEnum{ Item1, Item2};for each (String ^item_name in Enum::GetNames (AnEnum::typeid)){ // item name is the string form of the enum value}AnEmum enum_value = AnEnum::Item2;String ^enum_value_name = enum_value.ToString (); And finally with enums, don't forget the FlagsAttribute specifier for enums. Don't use the form designer in DevStudio - it produces horrible code, especially with C++/CLI source. It also doesn't support localisation. And it's in code which requires a recompile to make changes. Consider building a XAML type system, i.e. a data driven framework for form design. Can't think of anything else off the top of my head. Skizz Share on other sites Quote: Original post by benryvesSystem.CodeDom and System.CodeDom.Compiler can be used to parse source files and compile them into assemblies - couple this with System.Reflection to load the output and you can create scriptable applications quite easily. Not forgetting of course to ensure the source of the script is trusted, i.e. it hasn't been sent to your web server from a HTTP POST request. Skizz Share on other sites I (with a bit of direction stumbled across the Dictionary and found it to be quite useful. I also second intrest86's suggestion of serialization. Share on other sites System.Collections.ObjectModel has useful base classes if you want to expose a collection from a class. Some of the classes let you receive notifications of collection changes, and there are read-only collections as well. Share on other sites You might want to go into some of the neat tricks you can do with reflection if they're people coming from C/C++. For me it was the kind of thing I would never, ever use unless some one showed me what it could do since the whole concept doesn't make any sense to a C/C++ programmer. I also have fun writing sloppy code all over the place with anonymous delegates. Making a whole new method is just so tiresome sometimes. • 10 • 16 • 9 • 13 • 41	2018-05-26 17:58:29	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2614845931529999, "perplexity": 2477.822390055314}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794867841.63/warc/CC-MAIN-20180526170654-20180526190654-00227.warc.gz"}
https://www.physicsforums.com/threads/solving-the-shm-differential-equation.1047968/	# Solving the SHM differential equation Callumnc1 Homework Statement: Relevant Equations: x(t) = Ae^(αt) I am trying to solve this homogenous linear differential equation . Since it is linear, I can use the substitution . Which gives, (line 1) (line 2) (line 3) (line 4) (line 5) Which according to Morin's equals, (line 6) However, could someone please show me steps how he got from line 5 to 6? Also was is line 4 is it not: ? In other words, why dose B ≠ A? Many thanks! Last edited: Homework Helper Gold Member All of the expressions below are general solutions of your equation 1. ##x=C_1e^{i\omega t}+C_2e^{-i\omega t}## 2. ##x=A\sin\omega t+B\cos\omega t## 3. ##x=D\sin(\omega t+\phi)## You can verify that this is so by substituting in your ODE. Note that each expression has two arbitrary constants that are determined by the initial conditions, usually the values of ##x## and ##\frac{dx}{dt}## at ##t=0## that are appropriate to a particular situation.. You are asking how to go from 5 to 6 which is essentially going from my item 2 to 3. It is more obvious to see how to go from 3 to 2. Once you see that, you can reverse the algebra, if you wish. Using a well known trig identity for the sine of a sum of angles, $$D\sin(\omega t+\phi)=D\cos\phi \sin\omega t+D\sin\phi \cos\omega t.$$ If you identify $$A\equiv D\cos\phi~~\text{and}~~B\equiv D\sin\phi,$$you have item 2 above. Callumnc1 Callumnc1 All of the expressions below are general solutions of your equation 1. ##x=C_1e^{i\omega t}+C_2e^{-i\omega t}## 2. ##x=A\sin\omega t+B\cos\omega t## 3. ##x=D\sin(\omega t+\phi)## You can verify that this is so by substituting in your ODE. Note that each expression has two arbitrary constants that are determined by the initial conditions, usually the values of ##x## and ##\frac{dx}{dt}## at ##t=0## that are appropriate to a particular situation.. You are asking how to go from 5 to 6 which is essentially going from my item 2 to 3. It is more obvious to see how to go from 3 to 2. Once you see that, you can reverse the algebra, if you wish. Using a well known trig identity for the sine of a sum of angles, $$D\sin(\omega t+\phi)=D\cos\phi \sin\omega t+D\sin\phi \cos\omega t.$$ If you identify $$A\equiv D\cos\phi~~\text{and}~~B\equiv D\sin\phi,$$you have item 2 above. Thanks for your reply @kuruman ! Why don't you have and imaginary unit when going from line 1 to line 2? I though from Euler's identity it should be: . However, are you assuming that the constant B accounts for that? Many thanks!	2023-01-29 02:30:19	{"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": 0.8835283517837524, "perplexity": 815.8264128660313}, "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-2023-06/segments/1674764499697.75/warc/CC-MAIN-20230129012420-20230129042420-00488.warc.gz"}
http://www.gradesaver.com/textbooks/science/physics/physics-principles-with-applications-7th-edition/chapter-26-the-special-theory-of-relativity-questions-page-766/15	## Physics: Principles with Applications (7th Edition) Published by Pearson # Chapter 26 - The Special Theory of Relativity - Questions: 15 #### Answer No, there is no upper limit to the electron’s momentum. #### Work Step by Step The relativistic momentum of the electron is given by equation 26-4. At low speeds (relative to c), this is basically the same as p=mv. As v approaches c, the value of $\gamma = \frac{1}{\sqrt{1-v^{2}/c^{2}}}$ approaches infinity. After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.	2017-09-21 10:45:37	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7815912365913391, "perplexity": 1289.612838406389}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818687740.4/warc/CC-MAIN-20170921101029-20170921121029-00126.warc.gz"}
https://www.nature.com/articles/s41929-020-0426-0?error=cookies_not_supported&code=f70c101b-024a-43ef-aa92-f794240d08ce	SOLID OXIDE FUEL CELLS # Pathway for electrochemical O2 incorporation ### Subjects Identifying the rate-determining step (RDS) for oxygen incorporation into mixed ionic and electronic conducting electrodes is very challenging, particularly since the local composition changes during the reaction. Now, a generally applicable method for identifying the RDS is presented, with the example of a Pr0.1Ce0.9O2–x electrode. Electrochemically driven reactions in which both ions and electrons cross the interface between phases, involve a large variety of technologically important processes, including metal electroplating, lithium intercalation in batteries or gas incorporation in sensors and fuel cells. These mixed ion and electron transfer (MIET) reactions are far more complex than reactions that depend solely on electron transfer. One of the most extensively studied and commonly used MIET reactions is electrochemical O2 incorporation (ionization to O2–). This occurs at the solid–gas interface in oxygen sensors (there is at least one in every automobile), oxygen permeation membranes, solid oxide fuel cells and electrolysers. Over the last two decades, mixed ionic/electronic conductors containing a large concentration of oxygen vacancies, for example, doped ceria and transition metal perovskite oxides, have become prime candidates to serve as electrode materials supporting oxygen incorporation reactions (OIRs): $${\mathrm{O}}_2\left( {{\mathrm{gas}}} \right) + 4e^ - \to 2{\mathrm{O}}^{2 - }\left( {{\mathrm{electrode}}\;{\mathrm{interior}}} \right)$$ The presence of mobile oxygen vacancies in a solid material with electronic conductivity enables OIRs to take place over the entire solid–gas interface, favourably distinguishing these materials from traditional electrodes that require an active triple-phase boundary: gas, ionic and electronic conductors. OIRs involve a series of steps: adsorption of an oxygen molecule, its dissociation, electron transfer, incorporation of the ions into the surface crystal lattice and finally, diffusion of these ions into the bulk of the electrode (Fig. 1a). It is difficult to identify the RDS for such a lengthy chain of reactions, particularly because the electrode undergoes local compositional changes during the OIR process. Among the large arsenal of techniques employed in the study of OIRs1,2,3,4,5, measuring current–voltage curves provides the most direct information concerning mass and charge transfer across the interface6,7. Interpretation of these data is challenging because voltage application alters the activity of oxygen ions ($$a_{{\mathrm{O}}_2}$$) in the electrode interior and, consequently, also at its surface. Simply put, concentrations of intermediate species participating in OIRs have been very difficult to track experimentally. Now, reporting in Nature Catalysis, a team of scientists from Stanford, MIT and Lawrence Berkeley National Laboratory have used Pr-doped ceria (Pr0.1Ce0.9O2–x, 0 ≤ x ≤ 0.05), a promising solid oxide fuel cell cathode material with well-characterized point defect chemistry, to demonstrate a combined experimental and analytical method for direct determination of the RDS, as well as the reaction order for each step of the OIR8 (Fig. 1a,b). The authors measured current density–overpotential curves at 450 °C or 600 °C while controlling oxygen gas partial pressure $$\left( {P_{{\mathrm{O}}_2}} \right)$$ and oxygen activity in the electrode interior ($$a_{{\mathrm{O}}_2}$$). They unambiguously identified the reaction pathway based on two sets of data: one was acquired at constant $$P_{{\mathrm{O}}_2}$$ with varying $$a_{{\mathrm{O}}_2}$$, and the second with constant $$a_{{\mathrm{O}}_2}$$ under varying $$P_{{\mathrm{O}}_2}$$. The key advantage of this approach is the ability to determine oxygen activity in the electrode interior as a function of applied voltage, without the necessity of postulating that the surface is electrically neutral, and without assuming the nature of the adsorbed species and pathways. Of the four ionic species at the surface — Pr3+, Pr4+, oxygen vacancies and oxygen ions — the concentrations of Pr3+ and Pr4+ cations were measured directly at the Pr M4,5-edge with 100 mTorr O2 ambient pressure. Measurements were taken with operando X-ray absorption spectroscopy (Fig. 1b) as a function of applied overpotential. X-ray photoelectron spectroscopy of the O 1s photoelectron binding energy was measured as a function of applied voltage, providing the authors with insight into the surface potential and an independent determination of the reaction orders with respect to $$P_{{\mathrm{O}}_2}$$ and $$a_{{\mathrm{O}}_2}$$ (ref. 5). Previously, researchers would have needed to treat these parameters together9,10. Distinguishing reaction orders with respect to $$P_{{\mathrm{O}}_2}$$ and $$a_{{\mathrm{O}}_2}$$ is crucial for reaction pathway reconstruction. The reaction order with respect to $$P_{{\mathrm{O}}_2}$$ is ~1 if the RDS involves adsorbed oxygen molecules and ~0.5 if the RDS involves adsorbed oxygen atoms. The reaction order with respect to $$a_{{\mathrm{O}}_2}$$ depends on the number of participating ionic and electronic lattice defects. In this case, the overall OIR can be represented in terms of stoichiometric coefficients describing possible reaction pathways. In general, there are four stoichiometric coefficients, leading to 108 possible pathways for OIR, reflecting the number of ions and electrons that participate before, during, and after the RDS. Numerical fitting of these stoichiometric coefficients to the experimentally measured reaction orders revealed that only one RDS can explain the experimental data: dissociation of neutral molecular oxygen into neutral oxygen atoms. The most surprising implication of this finding is that electrons are not involved before or during the RDS. Therefore, neither lowering of the electron transfer barrier nor increasing the oxygen vacancy concentration would be expected to improve surface catalytic activity. Instead, the authors encourage focusing on decreasing the barrier height for O2 dissociation and for subsequent oxygen incorporation into vacancies. Considering that the kinetics of oxygen incorporation is a bottleneck in a variety of technologies, this recommendation could not come too soon. Finally, we note that the approach presented for studying MIET8 is not limited to oxygen incorporation reactions, but is also applicable when identifying the RDS for a variety of reactions occurring on mixed conducting electrodes. These reactions may involve other ions, such as Li+, Na+, OH or H+. However, one should keep in mind that application of the proposed method requires that the concentration of the active species at the electrode surface is measured independently of the activity of external reactants. We may expect that the next application of this powerful method will be in the fields of polymer fuel cells and lithium batteries. ## References 1. 1. Kilner, J. A., DeSouza, R. A. & Fullarton, I. C. Solid State Ion. 86–88, 703–709 (1996). 2. 2. Gopal, C. B. & Haile, S. M. J. Mater. Chem. A 2, 2405–2417 (2014). 3. 3. Baumann, F. S. et al. J. Electrochem. Soc. 154, B931–B941 (2007). 4. 4. Guan, Z. X., Chen, D. & Chueh, W. C. Phys. Chem. Chem. Phys. 19, 23414–23424 (2017). 5. 5. Schmid, A., Rupp, G. M. & Fleig, J. Phys. Chem. Chem. Phys. 20, 12016–12026 (2018). 6. 6. Mizusaki, J., Amano, K., Yamauchi, S. & Fueki, K. Solid State Ion. 22, 313–322 (1987). 7. 7. Kawada, T. et al. in Solid Oxide Fuel Cells (SOFC VI) (eds Singhal, S. C. & Dokiya, M.) 396–403 (The Electrochemical Society, 1999). 8. 8. Chen, D. et al. Nat. Catal. https://doi.org/10.1038/s41929-019-0401-9 (2020). 9. 9. Merkle, R. & Maier, J. Phys. Chem. Chem. Phys. 4, 4140–4148 (2002). 10. 10. De Souza, R. A. Phys. Chem. Chem. Phys. 8, 890–897 (2006). ## Author information Authors ### Corresponding author Correspondence to Igor Lubomirsky. ## Rights and permissions Reprints and Permissions Wachtel, E., Lubomirsky, I. Pathway for electrochemical O2 incorporation. Nat Catal 3, 94–95 (2020). https://doi.org/10.1038/s41929-020-0426-0	2021-01-21 12:33:10	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6072177886962891, "perplexity": 3179.9413735920475}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703524743.61/warc/CC-MAIN-20210121101406-20210121131406-00547.warc.gz"}
https://mymathforum.com/threads/divergence-time.347669/	# Divergence time #### idontknow Which one diverges faster ? $$\displaystyle s_1 =\sum_{i=1}^{\infty } \dfrac{1}{i}$$ and $$\displaystyle s_2 =\sum_{i=1}^{ \infty } \dfrac{1}{\sqrt{i}}$$. Last edited: #### romsek Math Team The magnitude of each sum is identical for every index. They diverge at the same rate. idontknow	2020-02-17 15:18:54	{"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": 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": 0.8570199012756348, "perplexity": 13782.39629037323}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1581875142603.80/warc/CC-MAIN-20200217145609-20200217175609-00303.warc.gz"}
http://mathhelpforum.com/discrete-math/22390-primes-more-more-sparse.html	# Math Help - Primes more and more sparse? 1. ## Primes more and more sparse? Is there some way to prove or show that the prime numbers occure more and more sparse at the number line? 2. what do you mean by "more and more sparse"? 3. Originally Posted by TriKri Is there some way to prove or show that the prime numbers occure more and more sparse at the number line? Google for the "Prime Number" theorem. It shows that the number of primes less than $n$ is approximatly $n/\ln(n)$, hence the density of primes near $n$ is $O(1/\ln(n))$ RonL	2015-07-29 22:38:00	{"extraction_info": {"found_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": 0.9229723215103149, "perplexity": 881.4089243795227}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042986646.29/warc/CC-MAIN-20150728002306-00175-ip-10-236-191-2.ec2.internal.warc.gz"}
https://mathsmadeeasy.co.uk/ks3-revision/ks3-comparing-percentages/	## What you need to know Things to remember: • It doesn’t matter which number you choose as the percentage $x\% \text{ of } y = y\% \text{ of } x$ We have two methods for finding percentage amounts: To do use this method, you need to remember how to find some simple percentages. Percentage Division 100% $\div1$ 50% $\div2$ 25% $\div4$ 20% $\div5$ 10% $\div10$ 5% $\div20$ 4% $\div25$ 2% $\div50$ 1% $\div100$ To do this method, we need to remember that we can add and subtract percentages to make other ones. This method is best when you don’t have a calculator. $$9\%=5\%+4\%$$ $$28\%=20\%+10\%-2\%$$ Find 35% of 250 $$35\%=25\%+10\%$$ Step 2: Find the value of these percentages $$25\%\text{ of } 250 = 250\div4=62.5$$ $$10\%\text{ of } 250 = 250\div10=25$$ $$35\%=25\%+10\%$$ $$35\%\text{ of } 250=62.5+25=87.5$$ Multiplication works too. $$35\%=7\times5\%$$ Step 2: Find the value of these percentages $$5\%\text{ of } 250 = 250\div20=12.5$$ $$35\%=7\times5\%$$ $$35\%\text{ of } 250=7\times12.5=87.5$$ Our second method is the most powerful can we can find any percentage amount in two steps. This is useful for decimal amounts and you have a calculator. Method 2: Multiplying from 1% Find 12.56% of 300. Step 1: Divide the number you’re finding the percentage of by 100 to find 1%. $$300\div100=3$$ $$1\%=3$$ Step 2: Multiply the value of 1% found in Step 1 by your starting percentage. $$12.56\%=12.56\times1\%$$ $$12.56\%\text{ of }300=12.56\times3=37.68$$ We can now use these two methods to compare two percentage amounts. Now that we have the strategies, we can look at some questions. Which is bigger, 55% of 20 or 40% of 30? Find 55% of 20 Method 1 $$55\%=50\%+5\%$$ Step 2: Find the value of these percentages $$50\%\text{ of }20=20\div2=10$$ $$5\%\text{ of }20=20\div20=1$$ $$55\%=50\%+5\%$$ $$55\%\text{ of }20=10 + 1 =11$$ Find 40% of 30 Method 2 Step 1: Divide the number you’re finding the percentage of by 100 to find 1%. $$30\div100=0.3$$ $$1\%=0.3$$ Step 2: Multiply the value of 1% found in Step 1 by your starting percentage. $$40\%=40\times1\%$$ $$40\%\text{ of }30=40\times0.3=12$$ Which is bigger, 70% of 55 or 55% of 70? Find 70% of 55 Method 1 $$70\%=50\%+20\%$$ Step 2: Find the value of these percentages $$50\%\text{ of }55=55\div2=27.5$$ $$20\%\text{ of }55=55\div5= 11$$ $$70\%=50\%+20\%$$ $$70\%\text{ of }55= 27.5+11=38.5$$ Find 55% of 70 Method 2 Step 1: Divide the number you’re finding the percentage of by 100 to find 1%. $$70\div100=0.7$$ $$1\%=0.7$$ Step 2: Multiply the value of 1% found in Step 1 by your starting percentage. $$55\%=55\times1\%$$ $$55\%\text{ of }70=55\times0.7=38.5$$ So, these two percentages are the same. It is worth remembering that whenever you find a percentage of something, it is the same as using the number as the percent and the percent as the number. 55% of 20 = 20% of 55 78.5% of 24 = 24% of 78.5 13.35% of 72.8 = 72.8% of 13.35 ## Example Questions #### Question 1: Which is bigger, 28% of 110 or 27% of 120? 28% of 110 Method 1 $$28\%=25\%+2\%+1\%$$ Step 2: Find the value of these percentages $$25\%\text{ of }110=110\div4=27.5$$ $$2\%\text{ of }110=110\div50=2.2$$ $$1\%\text{ of }110=110\div100=1.1$$ $$28\%=25\%+2\%+1\%$$ $$28\%\text{ of }110=27.5+2.2+1.1=30.8$$ 27% of 120 Method 2 Step 1: Divide the number you’re finding the percentage of by 100 to find 1%. $$120\div100=1.2$$ $$1\%=1.2$$ Step 2: Multiply the value of 1% found in Step 1 by your starting percentage. $$27\%=27\times1\%$$ $$27\%\text{ of }120=27\times1.2=32.4$$ So, 27% of 120 is bigger than 28% of 110.	2019-03-19 10:03:13	{"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": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 53, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.749125599861145, "perplexity": 1135.8253315296463}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1552912201953.19/warc/CC-MAIN-20190319093341-20190319115341-00258.warc.gz"}
https://www.illustrativemathematics.org/content-standards/tasks/1410	# 7.SP – Tetrahedral Dice Alignments to Content Standards: 7.SP.C.8.a 7.SP.C.8.b Many games use dice which are six-sided and fair (meaning each face on the die is equally likely to land face up). Many games also use the sum of two dice rolled at the same time to determine movement of game pieces. However, not all dice are six-sided. Imagine a game in which two fair four-sided (tetrahedral) dice are rolled simultaneously. These dice are in the shape of a pyramid, and when a die is rolled, the outcome is determined by the side that lands face down. Suppose that for these two dice, the possible values (corresponding to the four sides of the die) that can be obtained from each die are as follows: Die #1: 1, 2, 3, or 4 Die #2: 2, 4, 6, or 8 1. A certain game determines the movement of players' game pieces based on the SUM of the numbers on the face down sides when two dice are rolled. There are 10 distinct sum values that can occur, and some of those sums occur more often than others. 1. Using an organized list, table, tree diagram, or method of your choosing, develop a list of all 16 possible outcomes (for example, Die #1 = 1 and Die #2 = 2 for a sum of 3; Die #1 = 1 and Die #2 = 4 for a sum of 5; and so on). 2. From your work in part i, determine the 10 distinct sum values that are possible and calculate the probability of obtaining each sum value. Note: as mentioned above, some values will occur more frequently than others. What is the probability of obtaining a sum of 5? What is the probability of obtaining a sum that is more than 5? What is the probability of obtaining a sum that is at most 5? What is the probability of obtaining a sum that is at least 5? What is the probability of obtaining a sum that is no less than 5? 2. Now consider the case where the DIFFERENCE in the numbers on the face down sides when two dice are rolled is important to the game. Unless the two die values are the same (in which case the difference is 0), the difference for purposes of this game will always be computed as the larger number value rolled minus the smaller number value rolled. In this way, the difference value for any roll of the two dice will always be positive or 0. 1. Using an organized list, table, tree diagram, or method of your choosing, develop a list of all 16 possible outcomes (for example, Die #1 = 1 and Die #2 = 2 for a difference of 1; Die #1 = 1 and Die #2 = 4 for a difference of 3; and so on). 2. From your work in part i, determine the 8 distinct difference values that are possible and calculate the probability of obtaining each difference value. Note: as mentioned above, some values will occur more frequently than others. What is the probability of obtaining a difference of 5? What is the probability of obtaining a difference that is more than 5? What is the probability of obtaining a difference that is less than or equal to 5? ## IM Commentary The purpose of this task is to have students develop an organized list, table, etc. to determine all possible outcomes of a chance experiment and then to use this information to calculate various probabilities. Hopefully, students will note that techniques applicable to six-sided dice and sums are also applicable to four-sided dice and differences. With certain approaches, patterns in the listing of possible outcomes will be more easily recognized. Students will need to make distinctions in the language of "more than" vs. "at least" vs. “at most”, etc. Students should also note that different words can be used to describe the same compound events. If students are familiar with absolute value, you can also use this terminology in describing the variable in question 2 of this task. ## Solution 1. Die #2 2 4 6 Sum 3 5 7 9 4 6 8 10 5 7 9 11 6 8 10 12 2. $X$Probability 3$\frac{1}{16}$ 4$\frac{1}{16}$ 5$\frac{2}{16}$ 6$\frac{2}{16}$ 7$\frac{2}{16}$ 8$\frac{2}{16}$ 9$\frac{2}{16}$ 10$\frac{2}{16}$ 11$\frac{1}{16}$ 12$\frac{1}{16}$ 3. What is the probability of obtaining a sum of 5? $\frac{2}{16} = \frac18$ (see distribution) What is the probability of obtaining a sum that is more than 5? This includes all outcomes $X = 6$ to $X = 12$. This is $\frac{12}{16} = \frac34$ What is the probability of obtaining a sum that is at most 5? This includes all outcomes $X = 3$ to $X = 5$. This is $\frac{4}{16} = \frac14$ Note: this answer is also the complement to the previous question. What is the probability of obtaining a sum that is at least 5? This includes all outcomes $X = 5$ to $X = 12$. This is $\frac{14}{16} = \frac78$ What is the probability of obtaining a sum that is no less than 5? This includes all outcomes $X = 5$ to $X = 12$. This is $\frac{14}{16} = \frac78$ Note: this is another way of saying "at least 5" as in the previous question, hence the similar answer. 1. Die #2 2 4 6 Difference 1 3 5 7 0 2 4 6 1 1 3 5 2 0 2 4 2. $X$Probability 0$\frac{2}{16}$ 1$\frac{3}{16}$ 2$\frac{3}{16}$ 3$\frac{2}{16}$ 4$\frac{2}{16}$ 5$\frac{2}{16}$ 6$\frac{1}{16}$ 7$\frac{1}{16}$ 3. What is the probability of obtaining a sum of 5? $\frac{2}{16} = \frac18$ (see distribution) What is the probability of obtaining a sum that is more than 5? This includes all outcomes $X = 6$ to $X = 7$. This is $\frac{2}{16}$ What is the probability of obtaining a sum that is less than or equal to 5? This includes all outcomes $X = 0$ to $X = 5$. This is $\frac{14}{16}$ Note: this answer is also the complement to the previous question.	2015-03-31 08:24:27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6078785061836243, "perplexity": 220.45077601862747}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427131300444.84/warc/CC-MAIN-20150323172140-00154-ip-10-168-14-71.ec2.internal.warc.gz"}
https://homework.zookal.com/questions-and-answers/when-resisto-systems-inc-was-formed-the-company-was-authorized-942053666	When Resisto Systems, Inc., was formed, the company was authorized to issue 5,000 shares of $100 par value, 8 percent cumulative preferred stock, and 100,000 shares of$2 stated value common stock. Half of the preferred stock was issued at a price of $104 per share, and 56,000 shares of the common stock were sold for$17 per share. At the end of the current year, Resisto has retained earnings of \$382,000. a. Prepare the stockholders’ equity section of the company’s balance sheet at the end of the current year.	2021-06-20 16:38:48	{"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": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.22624137997627258, "perplexity": 3451.2733722430394}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1623488249738.50/warc/CC-MAIN-20210620144819-20210620174819-00178.warc.gz"}
https://math.stackexchange.com/questions/3282128/basis-for-free-module-of-infinite-rank	# Basis for free module of infinite rank In Corollary 19 of Dummit and Foote in section 10.4 it says: Let $$R$$ be a commutative ring and let $$M \cong R^s$$ and $$N \cong R^t$$ be free $$R$$-modules with bases $$m_1,m_2,\ldots,m_s$$ and $$n_1,n_2,\ldots n_t$$. Then, $$M \bigotimes_R N$$ is a free $$R$$-module of rank $$st$$ with basis $$m_i \otimes n_j$$, $$1 \leq i \leq s$$, and $$1 \leq j \leq t$$. That is: $$$$R^s \otimes_R R^t \cong R^{st}$$$$ They give a one line proof based on other corollaries they have proven. I can easily see that the isomorphism is very direct from other material in the section. However, I cannot seem to figure out how it is so clear that $$m_i \otimes n_j$$ is a basis for this tensor product. I would have expected that you needed results like the one I asked here: Spanning lists of the "right length" are a basis for arbitrary $R$-modules. But, they do not prove this anywhere up to this point. How would they expect you to know this if it were not for this result that I mentioned in the post above? Am I missing something extremely obvious? Moreover, I am wondering whether a similar result holds for modules of infinite rank. Namely, if you have modules $$M$$ and $$N$$ of infinite rank with basis $$\{a_i\}_{i \in I}$$ and $$\{b_j\}_{j \in J}$$,respectively, is a basis the set of simple tensors $$a_i \otimes b_j$$? In an exercise they ask you to prove the the rank of two free modules of arbitrary rank over a commutative ring is free, but the exercise gives no mention of a basis for such a free module. Any help to either of these questions would be greatly appreciated. Thank you • Maybe this helps: math.stackexchange.com/questions/1178004/… – Ruben Jul 3 at 19:30 • @Ruben Okay, thank you for the reference. It is for vector spaces, but it looks like the arguments should work for modules. I just find it weird that they claim that the basis from above is in fact the basis for $M \otimes_R N$. It seems very believable, but I have not found a relatively simple proof of this fact. – Mike Jul 3 at 19:43	2019-09-15 07:29:50	{"extraction_info": {"found_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": 19, "wp-katex-eq": 0, "align": 0, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.8241344690322876, "perplexity": 125.20022588839045}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514570830.42/warc/CC-MAIN-20190915072355-20190915094355-00524.warc.gz"}
https://curriculum.illustrativemathematics.org/MS/students/1/1/10/index.html	# Lesson 10 Bases and Heights of Triangles Let’s use different base-height pairs to find the area of a triangle. ### 10.1: An Area of 12 Draw one triangle with an area of 12 square units. Try to draw a non-right triangle. Be prepared to explain how you know the area of your triangle is 12 square units. ### 10.2: Hunting for Heights 1. Here are three copies of the same triangle. The triangle is rotated so that the side chosen as the base is at the bottom and is horizontal. Draw a height that corresponds to each base. Use an index card to help you. Side $$a$$ as the base: Side $$b$$ as the base: Side $$c$$ as the base: Pause for your teacher’s instructions before moving to the next question. 2. Draw a line segment to show the height for the chosen base in each triangle. ### 10.3: Some Bases Are Better Than Others For each triangle, identify and label a base and height. If needed, draw a line segment to show the height. Then, find the area of the triangle. Show your reasoning. (The side length of each square on the grid is 1 unit.) Find the area of this triangle. Show your reasoning. ### Summary A height of a triangle is a perpendicular segment between the side chosen as the base and the opposite vertex. We can use tools with right angles to help us draw height segments. An index card (or any stiff paper with a right angle) is a handy tool for drawing a line that is perpendicular to another line. 1. Choose a side of a triangle as the base. Identify its opposite vertex. 2. Line up one edge of the index card with that base. 3. Slide the card along the base until a perpendicular edge of the card meets the opposite vertex. 4. Use the card edge to draw a line from the vertex to the base. That segment represents the height. Sometimes we may need to extend the line of the base to identify the height, such as when finding the height of an obtuse triangle, or whenever the opposite vertex is not directly over the base. In these cases, the height segment is typically drawn outside of the triangle. Even though any side of a triangle can be a base, some base-height pairs can be more easily determined than others, so it helps to choose strategically. For example, when dealing with a right triangle, it often makes sense to use the two sides that make the right angle as the base and the height because one side is already perpendicular to the other. If a triangle is on a grid and has a horizontal or a vertical side, you can use that side as a base and use the grid to find the height, as in these examples: ### Glossary Entries • edge Each straight side of a polygon is called an edge. For example, the edges of this polygon are segments $$AB$$, $$BC$$, $$CD$$, $$DE$$, and $$EA$$. • opposite vertex For each side of a triangle, there is one vertex that is not on that side. This is the opposite vertex. For example, point $$A$$ is the opposite vertex to side $$BC$$. • vertex A vertex is a point where two or more edges meet. When we have more than one vertex, we call them vertices. The vertices in this polygon are labeled $$A$$, $$B$$, $$C$$, $$D$$, and $$E$$.	2022-05-22 07:09:30	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.788419783115387, "perplexity": 443.890121540842}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662545090.44/warc/CC-MAIN-20220522063657-20220522093657-00775.warc.gz"}
https://math.stackexchange.com/questions/1325780/is-this-proof-of-the-infinitude-of-primes-valid	# Is this proof of the infinitude of primes valid? The current issue (May 2015) of the American Mathematical Monthly has a one-line proof that there are an infinite number of primes, and I don't see why it is correct. Here is the proof: If the set of primes is finite, then $$0 < \prod\limits_{p} \sin\left(\frac{\pi}{p}\right) = \prod\limits_{p} \sin\left(\frac{\pi(1+2\prod_{p'}p')}{p}\right) =0 .$$ (That's the whole proof.) I see why the first equality holds, since, if there are only a finite number of primes, $p \mid \prod_{p'}p'$ for all $p$. But I do not see why the second equality ("$= 0$") holds. None of the terms in the product are zero, and, since there are only a finite number of them, the product is not zero. So, do I not understand the proof, or is the proof incorrect? Thank you. • I really like the comments to Strants' answer. – marty cohen Jun 15 '15 at 18:00 We must have that $1+2\prod_{p'}p'$ is divisible by some prime $q$, so $1+2\prod_{p'}p' = kq$ for some integer $k$. But then, $$\sin\left(\frac{\pi(1+2\prod_{p'}p')}{q}\right) = \sin \pi k = 0$$ which gives the right-hand equality. • @guest : This differs from "Euclid's proof made abstruse" in that Euclid's actual proof did not begin with an assumption that only finitely many primes exist: Euclid's proof was not by contradiction. Euclid showed that (rephrased into modern concepts) for every finite set $S$ of primes (which need not be the smallest $n$ primes for some $n$) the prime divisors of $1+\prod S$ are not in $S$. Thus $S$ can always be extended to a larger finite set of primes. Dirichlet and many later mathematicians erroneously wrote that Euclid's proof was by contradiction. That's a naked emperor. – Michael Hardy Jun 15 '15 at 11:56	2021-04-10 18:47:45	{"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": 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": 0.9659922122955322, "perplexity": 206.88435112039593}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038057476.6/warc/CC-MAIN-20210410181215-20210410211215-00544.warc.gz"}
http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tm&paperid=4003&option_lang=eng	RUS ENG JOURNALS PEOPLE ORGANISATIONS CONFERENCES SEMINARS VIDEO LIBRARY PACKAGE AMSBIB General information Latest issue Forthcoming papers Archive Impact factor Guidelines for authors License agreement Search papers Search references RSS Latest issue Current issues Archive issues What is RSS Trudy MIAN: Year: Volume: Issue: Page: Find Tr. Mat. Inst. Steklova, 2019, Volume 306, Pages 210–226 (Mi tm4003) Feynman Formulas and the Law of Large Numbers for Random One-Parameter Semigroups Yu. N. Orlovab, V. Zh. Sakbaevc, O. G. Smolyanovdb a Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, Moscow, 125047 Russia b Moscow Institute of Physics and Technology (State University), Institutskii per. 9, Dolgoprudnyi, Moscow oblast, 141701 Russia c Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia d Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991 Russia Abstract: We study sequences of compositions of independent identically distributed random one-parameter semigroups of linear transformations of a Hilbert space and the asymptotic properties of the distributions of such compositions when the number of terms in the composition tends to infinity. To study the expectation of such compositions, we apply the Feynman–Chernoff iterations obtained via Chernoff's theorem. By the Feynman–Chernoff iterations we mean prelimit expressions from the Feynman formulas; the latter are representations of one-parameter semigroups or related objects in terms of the limit of integrals over Cartesian powers of an appropriate space, or some generalizations of such representations. In particular, we study the deviation of the values of compositions of independent random semigroups from their expectation and examine the validity for such compositions of analogs of the limit theorems of probability theory such as the law of large numbers. We obtain sufficient conditions under which any neighborhood of the expectation of a composition of $n$ random semigroups contains the (random) value of this composition with probability tending to one as $n\to \infty$ (it is this property that is viewed as the law of large numbers for compositions). We also present examples of sequences of independent random semigroups for which the law of large numbers for compositions fails. Funding Agency Grant Number Russian Science Foundation 19-11-00320 Ministry of Education and Science of the Russian Federation 5-100 The research of V. Zh. Sakbaev (Sections 2 and 3) was supported by the Russian Science Foundation under grant 19-11-00320 and performed at Steklov Mathematical Institute of Russian Academy of Sciences. The research of Yu. N. Orlov and O. G. Smolyanov (Sections 4–6) was performed within the joint project with the Laboratory of Infinite-Dimensional Analysis and Mathematical Physics at the Faculty of Mechanics and Mathematics, Moscow State University, and supported by the Russian Academic Excellence Project “5-100.” DOI: https://doi.org/10.4213/tm4003 Full text: PDF file (276 kB) First page: PDF file References: PDF file HTML file English version: Proceedings of the Steklov Institute of Mathematics, 2019, 306, 196–211 Bibliographic databases: UDC: 517.98:519.2 Revised: May 13, 2019 Accepted: September 9, 2019 Citation: Yu. N. Orlov, V. Zh. Sakbaev, O. G. Smolyanov, “Feynman Formulas and the Law of Large Numbers for Random One-Parameter Semigroups”, Mathematical physics and applications, Collected papers. In commemoration of the 95th anniversary of Academician Vasilii Sergeevich Vladimirov, Tr. Mat. Inst. Steklova, 306, Steklov Math. Inst. RAS, Moscow, 2019, 210–226; Proc. Steklov Inst. Math., 306 (2019), 196–211 Citation in format AMSBIB \Bibitem{OrlSakSmo19} \by Yu.~N.~Orlov, V.~Zh.~Sakbaev, O.~G.~Smolyanov \paper Feynman Formulas and the Law of Large Numbers for Random One-Parameter Semigroups \inbook Mathematical physics and applications \bookinfo Collected papers. In commemoration of the 95th anniversary of Academician Vasilii Sergeevich Vladimirov \serial Tr. Mat. Inst. Steklova \yr 2019 \vol 306 \pages 210--226 \publ Steklov Math. Inst. RAS \publaddr Moscow \mathnet{http://mi.mathnet.ru/tm4003} \crossref{https://doi.org/10.4213/tm4003} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=4040776} \elib{https://elibrary.ru/item.asp?id=43230726} \transl \jour Proc. Steklov Inst. Math. \yr 2019 \vol 306 \pages 196--211 \crossref{https://doi.org/10.1134/S0081543819050171} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000511670100017} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85077383798}	2021-01-25 23:24:38	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5434765815734863, "perplexity": 1618.208473120253}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610704792131.69/warc/CC-MAIN-20210125220722-20210126010722-00260.warc.gz"}
https://soihub.org/resources/journal-papers/improving-on-the-cut-set-bound-for-general-primitive-relay-channels/	• ### Improving on the cut-set bound for general primitive relay channels • Peer Reviewed Conference Papers reported 2016 X. Wu, A. Ozgur. "Improving on the cut-set bound for general primitive relay channels", IEEE International Symposium on Information Theory (ISIT), 1675-1679, 2016 (PDF) Associated Participants In alphabetical order	2019-05-23 16:49:57	{"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": 0.8259523510932922, "perplexity": 9500.514757466777}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232257316.10/warc/CC-MAIN-20190523164007-20190523190007-00532.warc.gz"}
https://eepower.com/news/beacon-power-announces-plans-to-acquire-nxtphase/	News # Beacon Power Announces Plans to Acquire NxtPhase April 25, 2005 by Jeff Shepard Beacon Power Corp. (Wilmington, MA), a designer of advanced products for electric power and grid voltage and frequency regulation, announced that it has entered into an agreement to acquire NxtPhase T&D Corp. (Vancouver, BC, Canada), a supplier of digital and fiber optic products for electric power and grid monitoring and control. Under the agreement, at closing, Beacon will acquire NxtPhase for approximately 15.7 million new common shares of Beacon (subject to adjustment as described in the agreement), which will be distributed to NxtPhase investor shareholders. Also, immediately after closing, Beacon will grant restricted stock units covering approximately 2.7 million new common shares of Beacon to the NxtPhase employees. The agreement results from introductions made by Perseus LLC. An aggregate of $4.4 million of equity financing has also been committed to Beacon and to NxtPhase by a fund affiliated with Perseus. Perseus has committed to invest$2.9 million in Beacon to fund operations, in exchange for approximately 3.5 million newly issued Beacon common shares. Perseus has also committed $1.5 million of equity financing to fund NxtPhase operations during and after the acquisition. Beacon has agreed to issue warrants covering up to 1.22 million shares to Perseus, exercisable at$1.01 per share. In addition, Perseus has paid $100,000 to Beacon Power to extend by two years (until May 23, 2007) preexisting warrants that are already held by Perseus, covering 1,333,333 Beacon shares at an exercise price of$2.25 per share. Perseus, NxtPhase and Beacon Power are affiliates of one another. Beacon Power President and CEO Bill Capp said, "The acquisition of NxtPhase, a leading supplier of advanced grid electronics, is consistent with our commitment to provide the most innovative solutions for today's electrical grid. Both Beacon and NxtPhase share a common vision, with complementary technologies and cultures. We believe the two companies will be stronger as a combined entity in terms of customer base, market access, technology portfolio, product development opportunities, and outside investment potential. This acquisition will bring an immediate increase to Beacon's revenue and, we believe, lead to greater shareholder value."	2022-10-02 17:05:47	{"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": 0.17117935419082642, "perplexity": 10235.874495824217}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337338.11/warc/CC-MAIN-20221002150039-20221002180039-00503.warc.gz"}
http://tex.stackexchange.com/questions/55128/having-difficulty-with-pstricks-box-plot-psaxes-pspicture	# Having difficulty with pstricks box plot, psaxes, pspicture a TeX noob here. I wish to plot performance test results once every two weeks (or sooner) displaying a 2 week window. I saw a similar question: Can I use pgfplots to make a boxplot and the following answer looks very appealing: \documentclass{article} \usepackage{pst-plot} \begin{document} \begin{pspicture}(-1,-1)(12,14) \psset{yunit=0.1,fillstyle=solid} \psaxes[dy=1cm,Dy=10,ticksize=4pt 0,axesstyle=frame](0,0)(12,130) \rput(1,0){\psBoxplot[fillcolor=red!30]{ 100 90 120 115 120 110 100 110 100 90 100 100 120 120 120}}\rput(1,105){2001} \rput(3,0){\psBoxplot[arrowlength=0.5,fillcolor=blue!30]{ 90 120 115 116 115 110 90 130 120 120 120 85 100 130 130}}\rput(3,107){2008} \rput(5,0){\psBoxplot[barwidth=40pt,arrowlength=1.2,fillcolor=red!30]{ 35 70 90 60 100 60 60 80 80 60 50 55 90 70 70}}\rput(5,65){2001} \rput(7,0){\psBoxplot[barwidth=40pt,fillcolor=blue!30]{ 60 65 60 75 75 60 50 90 95 60 65 45 45 60 90}}\rput(7,65){2008} \rput(9,0){\psBoxplot[fillcolor=red!30]{ 20 20 25 20 15 20 20 25 30 20 20 20 30 30 30}}\rput(9,22){2001} \rput(11,0){\psBoxplot[fillcolor=blue!30,linestyle=dashed]{ 20 30 20 35 35 20 20 60 50 20 35 15 30 20 40}}\rput(11,25){2008} \end{pspicture} \end{document} except that it does not quite do what I want it to do. I have obviously tried to modify this example to produce what I want, but I have been so unsuccessful, that I do not have anything else to share, since it won't help a bit. My attempts to re-adjust the coordinates (say to bound a rectangle at left bottom point (12, 40) and right upper point (27,45)) just would not do what I want it to do. They would still start at 0, or would show nothing or garbage. I did not feel like I can change one or two things and get closer to my goal. Too many settings that can potentially conflict, perhaps ... Below is a sample picture that I would like to generate (I drew it manually using Dia). The box plots in my picture all look the same only because it was simpler to copy and paste them. In reality they will be generated with three real numbers - run 1, run 2 and run 3 in minutes, for example: 42.456, 44.123, 43.854. Occasionally a test will not run correctly and I will need to display an error. I would highly appreciate if you could provide me with code that can generate something that looks like the picture below. UPDATE: This worked with MikTeX 2.9 but not MikTeX 2.8 under Windows. UPDATE2: Set Oy=4 if you want to raise the plot. - EDIT I recommend the environment psgraph provided by the package pst-plot. In this way you can simple adjust the graph. I build an example which produces a basis and hope it helps. First the result ;-) To simplify the work I created three new commands. The first one \weekday provided the day of the week whereby 1 represent Monday. In this way the labels of the axis are done. The next two commands draw the rectangles. The command \simpleframe has the following syntax: \simpleframe[options] ( coordinates of lower left corner ) ( coordinates of upper right corner ) {text inside the frame} The second new command is \complexframe which has the following syntax. \complexframe[global options] ( coordinates of lower left corner ) ( coordinates of upper right corner ) [options for the first line] {y value of the first line} [options for the second line] {y value of the second line} {text} The default values are shown in the image. \documentclass{article} \usepackage[landscape]{geometry} \usepackage{pst-plot} \usepackage{xparse} \newcount\daycount \ExplSyntaxOn \makeatletter \DeclareDocumentCommand \weekday { m } { \prg_case_int:nnn { \int_mod:nn { #1 } { 7 } } { { 1 } {Mo} { 2 } {Tue} { 3 } {Wed} { 4 } {Thu} { 5 } {Fri} { 6 } {Sat} { 0 } {Son} } {error} %} } \ExplSyntaxOff \NewDocumentCommand \simpleframe { O{linecolor=black,linewidth=1pt,fillcolor=red!70,fillstyle=solid} r() r() m } { \begingroup \psset{#1} \pst@getcoor{#2}\pst@tempA \pst@getcoor{#3}\pst@tempB \psframe(!% \pst@tempA /YA exch \pst@number\psyunit div def /XA exch \pst@number\psxunit div def XA YA )(! \pst@tempB /YB exch \pst@number\psyunit div def /XB exch \pst@number\psxunit div def XB YB ) \rput(!% \pst@tempA /YA exch \pst@number\psyunit div def /XA exch \pst@number\psxunit div def \pst@tempB /YB exch \pst@number\psyunit div def /XB exch \pst@number\psxunit div def ){#4} \endgroup } \NewDocumentCommand \complexframe { % O{linecolor=black,linewidth=1pt,fillcolor=red!20,fillstyle=solid}% default option r() r() %coordinates of the frame O{linecolor=blue,linewidth=2pt} m %y-value of the first line+option O{linecolor=yellow,linewidth=2pt,linestyle=dashed} m % y-value of the second line+option m %Text } { \begingroup \psset{#1} \pst@getcoor{#2}\pst@tempA \pst@getcoor{#3}\pst@tempB \psframe(!%global frame \pst@tempA /YA exch \pst@number\psyunit div def /XA exch \pst@number\psxunit div def XA YA )(! \pst@tempB /YB exch \pst@number\psyunit div def /XB exch \pst@number\psxunit div def XB YB ) \rput(!%text \pst@tempA /YA exch \pst@number\psyunit div def /XA exch \pst@number\psxunit div def \pst@tempB /YB exch \pst@number\psyunit div def /XB exch \pst@number\psxunit div def ){#8} \begingroup \psset{#4} \pst@getcoor{0,#5}\pst@tempC \psline (!%first line \pst@tempA /YA exch \pst@number\psyunit div def /XA exch \pst@number\psxunit div def \pst@tempC /YC exch \pst@number\psyunit div def /XC exch \pst@number\psxunit div def XA YC)(!% \pst@tempB /YB exch \pst@number\psyunit div def /XB exch \pst@number\psxunit div def \pst@tempC /YC exch \pst@number\psyunit div def /XC exch \pst@number\psxunit div def XB YC ) \endgroup \begingroup \psset{#6} \pst@getcoor{0,#7}\pst@tempC \psline (!%first line \pst@tempA /YA exch \pst@number\psyunit div def /XA exch \pst@number\psxunit div def \pst@tempC /YC exch \pst@number\psyunit div def /XC exch \pst@number\psxunit div def XA YC)(!% \pst@tempB /YB exch \pst@number\psyunit div def /XB exch \pst@number\psxunit div def \pst@tempC /YC exch \pst@number\psyunit div def /XC exch \pst@number\psxunit div def XB YC ) \endgroup \psframe[fillstyle=none](!%global frame again \pst@tempA /YA exch \pst@number\psyunit div def /XA exch \pst@number\psxunit div def XA YA )(! \pst@tempB /YB exch \pst@number\psyunit div def /XB exch \pst@number\psxunit div def XB YB ) \endgroup } \makeatother \begin{document} \psset{% xAxisLabel={Day}, yAxisLabel={Whatever}, xAxisLabelPos={16,-15pt},yAxisLabelPos={-0.4in,c}, } \begin{psgraph}[,Oy=0,labels=y]{->}(0,0)(16,21){0.8\linewidth}{10cm} \psframe(0,0)(15,20.5) \multido{\iA=1+1,\iB=14+1}{14}{% \psxTick(\iA){\weekday{\iA}}% \psxTick[labelsep=20pt](\iA){\iB}% } \simpleframe(3,3)(4,6){foo} \complexframe(10,10)(12,20){14}{18}{foobar} \end{psgraph} \end{document} - Very cool! Final touch - what f I wanted to draw my own box? Since I only have 3 values and their average, it would not make sense for me to have the 25th and the 75th percentile. I would want the similar look and feel though - a colored rectangle with two lines in the middle and text. – The Dude May 11 '12 at 21:04 @TheDude: Do I understand it correct that you simple need a rectangle at a special position with some text? – Marco Daniel May 11 '12 at 21:08 That for the error (in red) plus a rectangle with two lines going across it in the middle of different color. Case 1: red rectangle from (3,3) to (4, 6) with text inside of it. Case 2: pink rectangle from (10, 10) to (12, 20) with some text inside of it, and also a line from (10, 14) to (12, 14) of the same color and thickness as the border plus another line from (10, 18) to (12, 18) of different color, dashed, and of different thickness. I will stop bugging you after that, I swear. – The Dude May 11 '12 at 21:37 @TheDude: I edited my example. – Marco Daniel May 12 '12 at 10:35 Thanks! Can you recommend some documents to go through so that I can understand some of that pstricks macro magic? Whatever is between \begin{document}' and \end{document}` is pretty readable to me. I'd like to find a document or two that would clue me in on syntax and details of the rest of what you did, so that I could modify the behavior of the macros/defs myself. – The Dude May 13 '12 at 15:02	2016-05-31 02:06:27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.500325620174408, "perplexity": 3898.607167559094}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1464051165777.47/warc/CC-MAIN-20160524005245-00151-ip-10-185-217-139.ec2.internal.warc.gz"}
https://en.wikipedia.org/wiki/Feshbach%E2%80%93Fano_partitioning	# Feshbach–Fano partitioning In quantum mechanics, and in particular in scattering theory, the Feshbach–Fano method, named after Herman Feshbach and Ugo Fano, separates (partitions) the resonant and the background components of the wave function and therefore of the associated quantities like cross sections or phase shift. This approach allows us to define rigorously the concept of resonance in quantum mechanics. In general, the partitioning formalism is based on the definition of two complementary projectors P and Q such that P + Q = 1. The subspaces onto which P and Q project are sets of states obeying the continuum and the bound state boundary conditions respectively. P and Q are interpreted as the projectors on the background and the resonant subspaces respectively. The projectors P and Q are not defined within the Feshbach–Fano method. This is its major power as well as its major weakness. On the one hand, this makes the method very general and, on the other hand, it introduces some arbitrariness which is difficult to control. Some authors define first the P space as an approximation to the background scattering but most authors define first the Q space as an approximation to the resonance. This step relies always on some physical intuition which is not easy to quantify. In practice P or Q should be chosen such that the resulting background scattering phase or cross-section is slowly depending on the scattering energy in the neighbourhood of the resonances (this is the so-called flat continuum hypothesis). If one succeeds in translating the flat continuum hypothesis in a mathematical form, it is possible to generate a set of equations defining P and Q on a less arbitrary basis. The aim of the Feshbach–Fano method is to solve the Schrödinger equation governing a scattering process (defined by the Hamiltonian H) in two steps: First by solving the scattering problem ruled by the background Hamiltonian PHP. It is often supposed that the solution of this problem is trivial or at least fulfilling some standard hypotheses which allow to skip its full resolution. Second by solving the resonant scattering problem corresponding to the effective complex (energy dependent) Hamiltonian ${\displaystyle H_{\mathrm {eff} }(E)=QHQ+\lim _{\varepsilon \to 0}QHP{1 \over E+i\varepsilon -PHP}PHQ=QHQ+\Delta (E)-i\Gamma (E)/2,\,}$ whose dimension is equal to the number of interacting resonances and depends parametrically on the scattering energy E. The resonance parameters ${\displaystyle E_{\mathrm {res} }}$ and ${\displaystyle \Gamma _{\mathrm {res} }}$ are obtained by solving the so-called implicit equation ${\displaystyle \det[H_{\mathrm {eff} }(z)-z]=0\,}$ for z in the lower complex plane. The solution ${\displaystyle z_{\mathrm {res} }=E_{\mathrm {res} }-i\Gamma _{\mathrm {res} }\,}$ is the resonance pole. If ${\displaystyle z_{\mathrm {res} }}$ is close to the real axis it gives rise to a Breit–Wigner or a Fano profile in the corresponding cross section. Both resulting T matrices have to be added in order to obtain the T matrix corresponding to the full scattering problem : ${\displaystyle T_{\mathrm {tot} }=T_{\mathrm {background} }+T_{\mathrm {resonances} }.\,}$	2017-03-23 02:22:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 7, "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": 0.8791083693504333, "perplexity": 358.01334118234}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1490218186608.9/warc/CC-MAIN-20170322212946-00357-ip-10-233-31-227.ec2.internal.warc.gz"}
https://www.ademcetinkaya.com/2023/02/bfly-butterfly-network-inc-class-common.html	Outlook: Butterfly Network Inc. Class A Common Stock is assigned short-term Ba1 & long-term Ba1 estimated rating. Dominant Strategy : Sell Time series to forecast n: 21 Feb 2023 for (n+6 month) ## Abstract Butterfly Network Inc. Class A Common Stock prediction model is evaluated with Multi-Task Learning (ML) and Spearman Correlation1,2,3,4 and it is concluded that the BFLY stock is predictable in the short/long term. According to price forecasts for (n+6 month) period, the dominant strategy among neural network is: Sell ## Key Points 1. Reaction Function 2. How can neural networks improve predictions? 3. Can neural networks predict stock market? ## BFLY Target Price Prediction Modeling Methodology We consider Butterfly Network Inc. Class A Common Stock Decision Process with Multi-Task Learning (ML) where A is the set of discrete actions of BFLY stock holders, F is the set of discrete states, P : S × F × S → R is the transition probability distribution, R : S × F → R is the reaction function, and γ ∈ [0, 1] is a move factor for expectation.1,2,3,4 F(Spearman Correlation)5,6,7= $\begin{array}{cccc}{p}_{a1}& {p}_{a2}& \dots & {p}_{1n}\\ & ⋮\\ {p}_{j1}& {p}_{j2}& \dots & {p}_{jn}\\ & ⋮\\ {p}_{k1}& {p}_{k2}& \dots & {p}_{kn}\\ & ⋮\\ {p}_{n1}& {p}_{n2}& \dots & {p}_{nn}\end{array}$ X R(Multi-Task Learning (ML)) X S(n):→ (n+6 month) $∑ i = 1 n a i$ n:Time series to forecast p:Price signals of BFLY stock j:Nash equilibria (Neural Network) k:Dominated move a:Best response for target price For further technical information as per how our model work we invite you to visit the article below: How do AC Investment Research machine learning (predictive) algorithms actually work? ## BFLY Stock Forecast (Buy or Sell) for (n+6 month) Sample Set: Neural Network Stock/Index: BFLY Butterfly Network Inc. Class A Common Stock Time series to forecast n: 21 Feb 2023 for (n+6 month) According to price forecasts for (n+6 month) period, the dominant strategy among neural network is: Sell X axis: Likelihood% (The higher the percentage value, the more likely the event will occur.) Y axis: Potential Impact% (The higher the percentage value, the more likely the price will deviate.) Z axis (Grey to Black): Technical Analysis% ## IFRS Reconciliation Adjustments for Butterfly Network Inc. Class A Common Stock 1. For a discontinued hedging relationship, when the interest rate benchmark on which the hedged future cash flows had been based is changed as required by interest rate benchmark reform, for the purpose of applying paragraph 6.5.12 in order to determine whether the hedged future cash flows are expected to occur, the amount accumulated in the cash flow hedge reserve for that hedging relationship shall be deemed to be based on the alternative benchmark rate on which the hedged future cash flows will be based. 2. If any instrument in the pool does not meet the conditions in either paragraph B4.1.23 or paragraph B4.1.24, the condition in paragraph B4.1.21(b) is not met. In performing this assessment, a detailed instrument-byinstrument analysis of the pool may not be necessary. However, an entity must use judgement and perform sufficient analysis to determine whether the instruments in the pool meet the conditions in paragraphs B4.1.23–B4.1.24. (See also paragraph B4.1.18 for guidance on contractual cash flow characteristics that have only a de minimis effect.) 3. At the date of initial application, an entity is permitted to make the designation in paragraph 2.5 for contracts that already exist on the date but only if it designates all similar contracts. The change in the net assets resulting from such designations shall be recognised in retained earnings at the date of initial application. 4. The decision of an entity to designate a financial asset or financial liability as at fair value through profit or loss is similar to an accounting policy choice (although, unlike an accounting policy choice, it is not required to be applied consistently to all similar transactions). When an entity has such a choice, paragraph 14(b) of IAS 8 requires the chosen policy to result in the financial statements providing reliable and more relevant information about the effects of transactions, other events and conditions on the entity's financial position, financial performance or cash flows. For example, in the case of designation of a financial liability as at fair value through profit or loss, paragraph 4.2.2 sets out the two circumstances when the requirement for more relevant information will be met. Accordingly, to choose such designation in accordance with paragraph 4.2.2, the entity needs to demonstrate that it falls within one (or both) of these two circumstances. International Financial Reporting Standards (IFRS) adjustment process involves reviewing the company's financial statements and identifying any differences between the company's current accounting practices and the requirements of the IFRS. If there are any such differences, neural network makes adjustments to financial statements to bring them into compliance with the IFRS. ## Conclusions Butterfly Network Inc. Class A Common Stock is assigned short-term Ba1 & long-term Ba1 estimated rating. Butterfly Network Inc. Class A Common Stock prediction model is evaluated with Multi-Task Learning (ML) and Spearman Correlation1,2,3,4 and it is concluded that the BFLY stock is predictable in the short/long term. According to price forecasts for (n+6 month) period, the dominant strategy among neural network is: Sell ### BFLY Butterfly Network Inc. Class A Common Stock Financial Analysis* Rating Short-Term Long-Term Senior OutlookBa1Ba1 Income StatementBaa2C Balance SheetCaa2Ba3 Leverage RatiosCBa3 Cash FlowBaa2C Rates of Return and ProfitabilityCB1 Financial analysis is the process of evaluating a company's financial performance and position by neural network. It involves reviewing the company's financial statements, including the balance sheet, income statement, and cash flow statement, as well as other financial reports and documents. How does neural network examine financial reports and understand financial state of the company? ### Prediction Confidence Score Trust metric by Neural Network: 92 out of 100 with 634 signals. ## References 1. Bottou L. 1998. Online learning and stochastic approximations. In On-Line Learning in Neural Networks, ed. D Saad, pp. 9–42. New York: ACM 2. Clements, M. P. D. F. Hendry (1997), "An empirical study of seasonal unit roots in forecasting," International Journal of Forecasting, 13, 341–355. 3. Gentzkow M, Kelly BT, Taddy M. 2017. Text as data. NBER Work. Pap. 23276 4. Athey S. 2017. Beyond prediction: using big data for policy problems. Science 355:483–85 5. uyer, S. Whiteson, B. Bakker, and N. A. Vlassis. Multiagent reinforcement learning for urban traffic control using coordination graphs. In Machine Learning and Knowledge Discovery in Databases, European Conference, ECML/PKDD 2008, Antwerp, Belgium, September 15-19, 2008, Proceedings, Part I, pages 656–671, 2008. 6. Ruiz FJ, Athey S, Blei DM. 2017. SHOPPER: a probabilistic model of consumer choice with substitutes and complements. arXiv:1711.03560 [stat.ML] 7. Bennett J, Lanning S. 2007. The Netflix prize. In Proceedings of KDD Cup and Workshop 2007, p. 35. New York: ACM Frequently Asked QuestionsQ: What is the prediction methodology for BFLY stock? A: BFLY stock prediction methodology: We evaluate the prediction models Multi-Task Learning (ML) and Spearman Correlation Q: Is BFLY stock a buy or sell? A: The dominant strategy among neural network is to Sell BFLY Stock. Q: Is Butterfly Network Inc. Class A Common Stock stock a good investment? A: The consensus rating for Butterfly Network Inc. Class A Common Stock is Sell and is assigned short-term Ba1 & long-term Ba1 estimated rating. Q: What is the consensus rating of BFLY stock? A: The consensus rating for BFLY is Sell. Q: What is the prediction period for BFLY stock? A: The prediction period for BFLY is (n+6 month)	2023-03-26 12:23:14	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 2, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5078452229499817, "perplexity": 6148.642669442064}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1679296945472.93/warc/CC-MAIN-20230326111045-20230326141045-00395.warc.gz"}
https://proxieslive.com/tag/disk/	## Is realization of unit disk graphs hard? It is known that recognizing a unit disk graph is NP-hard [1]. However, the paper does not mention how hard is the realization problem. I have looked up several references [2][3][4]. None of the papers answer whether the following problem is NP-hard: Given a unit disk graph $$G = (V,E)$$, find a configuration of a set $$\mathcal{D}$$ of disks, such that the intersection graph $$G(\mathcal{D})$$ of $$\mathcal{D}$$ is isomorphic to $$G$$. The difference between this problem and the recognition problem is that the input of this problem is guaranteed to be a unit disk. Is there any study that shows the complexity of the above problem? I expect it to be NP-hard, but I am yet to find a full proof. ## Why is disk IO higher on Debian 10 (MariaDB 10.3) with MySQL replication? I have a MySQL/MariaDB master-master replication setup that has been working well for several years, the db and tables are not very large (under 200MB for 18 tables). These were on 2 servers running Debian 9 and MariaDB 10.1.44. Now I’ve spun up 2 new servers running Debian 10 and I’m in the process of moving things over to them, but stopped half-way because I’m seeing much higher disk IO usage on the new servers (about 6x more). So currently, one of the Debian 9 servers and one of the Debian 10 servers are in master-master relationship, with one Debian 9 still being a slave of the master Debian 9 server, and same on the Debian 10 side of things. I didn’t notice the increased disk IO until after all read/write operations were moved to the Debian 10 master. I was trying to browse tables and saw how slow it was outputting the query results, and it felt like I was on a dial-up connection watching the rows scroll across. It turned out there was some disk contention with the virtual host that was partly responsible, and that problem is now mostly gone. Now, as you can imagine, none of this is crashing the server with such a "small" set of tables, but as things continue to grow, I’m concerned that there is some underlying mis-configuration which will rear its ugly head at an inopportune time. On the Debian 9 servers, iotop shows steady write IO at around 300-600Kb/s, but on Debian 10 it spikes as high as 6MB/s, and averages around 3MB/s. Here is the standard config on all 4 servers, everything else is default Debian settings (or MariaDB, as the case may be), full config for Debian 10 at https://pastebin.com/Lk2FR4e3: max_connections = 1000 query_cache_limit = 4M query_cache_size = 0 query_cache_type = 0 server-id = 1 # different for each server log_bin = /var/log/mysql/mysql-bin.log binlog_do_db = optimizer replicate-do-db = optimizer report-host = xyz.example.com #changed obviously log-slave-updates = true innodb_log_file_size = 32M innodb_buffer_pool_size = 256M Here are some other settings I’ve tried that don’t seem to make any difference (checked each one by one): binlog_annotate_row_events = OFF binlog_checksum = NONE binlog_format = STATEMENT innodb_flush_method = O_DIRECT_NO_FSYNC innodb_log_checksums = OFF log_slow_slave_statements = OFF replicate_annotate_row_events = OFF I’ve gone through all the settings here that have changed from MariaDB 10.1 to 10.3, and can’t seem to find any that make a difference: https://mariadb.com/kb/en/replication-and-binary-log-system-variables/ I also did a full listing of the server variables and compared the configs on 10.1 to the 10.3 configuration and didn’t find anything obvious. But either I’m missing something, or the problem lies with Debian 10 itself. Results of SHOW ENGINE INNODB STATUS are here: https://pastebin.com/mJdLQv8k Now, how about that disk IO, what is it actually doing? I include 3 screenshots here to show what I mean by increased disk IO: That is from the Debian 10 master, and you can see where I moved operations back to the Debian 9 server (more on that in a second). Notice the disk IO does go down slightly at that point, but not to the levels that we’ll see on the Debian 9 master. Also note that the public bandwidth chart is pretty much only replication traffic, and that the disk IO far outstrips the replication traffic. The private traffic is all the reads/writes from our application servers. This is the Debian 9 master server, and you can see where I moved all operations back to this server, the private traffic shoots up, but the write IO hovers around 500kB/s. I didn’t have resource graphs being recorded on the old servers, thus the missing bits on the left. And lastly, for reference, here is the Debian 10 slave server (that will eventually be half of the master<–>master replication). There are no direct reads/writes on this server, all disk IO is from replication. Just to see what would happen (as I alluded to above), I reverted all direct read/write operations to the Debian 9 master server. While disk IO did fall somewhat on the Debian 10 server, it did not grow on the Debian 9 server to any noticeable extent. Also, on the Debian 10 slave server, I did STOP SLAVE once to see what happened, and the disk IO went to almost nothing. Doing the same on the Debian 10 master server barely did not have the same drastic effect, though it’s possible there WAS some change that wasn’t obvious; the disk IO numbers on iostat fluctuate much more wildly on the Debian 10 servers than they do on the Debian 9 servers. So, what is going on here? How can I figure out why MariaDB is writing so much data to disk apparently and/or how can I stop it? ## USB Flash Disk with block-chain I am thinking of buying USB flash disk with Security Element(stores Private-Key/Secret-Key/X.509-Certificates) and some encrypted data(8 Megabyte). The disk has to be protected against cloning and possibly maintains logs of insert history with UNIX timestamp & USB Host ID when inserted. I found YubiKey & NitroKey, but they does not have additional storage. Is there a USB Device like that satisfy above requirements. (Or) Is there any cost effective SoC+Security Element Over USB with flash storage available to implement my requirement using Opensource? Thanks ## Call of duty disk space [closed] my friends and I have been wondering about the following thing: Let’s say COD:MW takes 180GB of storage space, then has a 20 GB update. Afterwards, the game takes only 190GB of storage space and not 200GB. Do you you know this happens? I thought maybe it’s because some files are replacing older ones instead of just being “added on top”. Thanks ! ## How fast would a Tenser’s Floating Disk descend if I pulled it over a long drop? So I’m designing a variant human warlock with the wizard ritual caster feat and while considering which rituals to start with I read the description for Tenser’s floating disk and looking through the eldritch invocations I saw the Ascendant step invocation allows levitation on myself at will so if I was to make a floating disk, have a party member or some equipment placed on it and then go down a chasm or hole or off the side of a flying ship/island etc would the disk follow at my levitate speed (20 feet descent or ascent per turn) or my movement speed (30 feet per turn) or would it drop like a rock? I’m picturing using it like a down elevator. Additionally would I be able to hold a wooden tabletop under the disk and levitate up and have it ascend to stay 3 feet above the surface? For ease of reference here is the description of the relevant spells (quoted from D&D Beyond). Tenser’s floating disk: This spell creates a circular, horizontal plane of force, 3 feet in diameter and 1 inch thick, that floats 3 feet above the ground in an unoccupied space of your choice that you can see within range. The disk remains for the duration, and can hold up to 500 pounds. If more weight is placed on it, the spell ends, and everything on the disk falls to the ground. The disk is immobile while you are within 20 feet of it. If you move more than 20 feet away from it, the disk follows you so that it remains within 20 feet of you. It can move across uneven terrain, up or down stairs, slopes and the like, but it can’t cross an elevation change of 10 feet or more. For example, the disk can’t move across a 10-foot-deep pit, nor could it leave such a pit if it was created at the bottom. If you move more than 100 feet from the disk (typically because it can’t move around an obstacle to follow you), the spell ends. Levitate: One creature or loose object of your choice that you can see within range rises vertically, up to 20 feet, and remains suspended there for the duration. The spell can levitate a target that weighs up to 500 pounds. An unwilling creature that succeeds on a Constitution saving throw is unaffected. The target can move only by pushing or pulling against a fixed object or surface within reach (such as a wall or a ceiling), which allows it to move as if it were climbing. You can change the target’s altitude by up to 20 feet in either direction on your turn. If you are the target, you can move up or down as part of your move. Otherwise, you can use your action to move the target, which must remain within the spell’s range. When the spell ends, the target floats gently to the ground if it is still aloft. To be clear I am not asking about whether I can move the disk over a hole, I am aware of that limitation and can easily put a plank over the hole and move the disk over the void, I am only asking about the vertical movement speed of the disk. ## Can second internal hard disk cause infection after reinstall? Lets say I have two internal hard disks, one for the operating system the other for backups. If i make sure to delete the MBR and partition table with dd if=/dev/zero of=/dev/sda bs=512 count=2048 of the disk with the operating system to avoid the possibility of a boot sector virus. If I reinstall in what ways can that second hard disk be used to cause an infection of the primary disk with the operating system? I was reading thata boot sector viruses can even spread to other hard drives you have installed or physical media you have plugged into your system. So what impact can a boot sector virus on a backup drive have on on the primary drive? And any other threats I may have missed? I guess it could effect USB sticks plugged in? ## Disk encryption vs encrypted file container Is there any difference between disk encryption and encrypted file container in terms of security? Which one is better? ## how to set up multiple computers full disk encryption? I want to set up my company’s laptops in a way that all files created on these laptops can only read by these laptops. If it is copy to a usb then that file is only readable when plug that USB in a company laptop. If plug in or copy to another non-authorized laptop then it is not readable. Earn and Young is using this technique to protect their data but I don’t know what is it called and how to set it up. Please help 🙂 thanks guy ## Access SATA disk disabled in BIOS If a remote hacker or a malware gains full root/admin rights on a system, is there any way to access another SATA disk that has been hardware connected but disabled in BIOS ? I am not sure if the disabled disk even has power in that case (I guess it has not) but I found the following post which raised some doubts : https://superuser.com/a/111009 OS considered : Windows or Linux Threat Model : Physical access and BIOS reflash are out of scope as it is game over anyway in such cases. Except this, consider full control of compromised disk system: hacker can issue any command, can modify MBR, kernel, flash the compromised disk firmware, … ## How do recovery tools like Cellebrite work and can they recover data from phones which utilize Full Disk Encryption? How do mobile data recovery tools such as Cellebrite actually work? Are they capable of overcoming full-disk encryption in order to recover data using means other than brute-forcing the unlock pin/password?	2020-07-05 03:41:26	{"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": 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": 5, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2868749797344208, "perplexity": 1909.348591564925}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1593655886865.30/warc/CC-MAIN-20200705023910-20200705053910-00521.warc.gz"}
http://mathematica.stackexchange.com/tags/autocomplete/hot	# Tag Info 38 Preface With Mathematica version 9.0.1 the following answer not valid anymore because the underlying protocol between front end and kernel was changed. Fortunately, we started to implement an open-source Mathematica plugin for IntelliJIDEA which has a full support for camel-hump completion. Please see this post for more information Camel-humps ... 19 Do you mean CtrlShiftK? After typing Plo, press the key combination CtrlShiftK and a window will appear with possible options: As pointed by Yves，CtrlK will also work，but CtrlShiftK will work differently if you finish the function name. For an example, Type Plot3D; Use CtrlShiftK; Mathematica will show: 16 This is obsolete in Mathematica 9, which automatically includes contexts in completions. Undocumented function: use at your own risk, subject to change in future versions, etc.... The function you're interested in is FEFC. It's been around for a while (here's a Mathematica Journal article that references it, near the end) although it has ... 15 There is an undocumented file in the installation directory named specialArgFunctions.tr: NotebookOpen @ FileNameJoin @ { $InstallationDirectory, "SystemFiles", "FrontEnd", "SystemResources" , "FunctionalFrequency", "specialArgFunctions.tr" } This file describes in detail how to attach completion actions to each parameter of listed functions. For ... 15 This took some digging but at least in Version 7 the FrontEnd command is FT, e.g.: FEFT["Plot"] You can read the definition with Definition[FEFT]. If you want only the Box form itself we can modify it accordingly (here for version 7): templateCell[name_String] := Module[{template}, If[! StringQ@ToExpression[name <> "::usage"], ... 13 Changing shortcuts isn't that complicated. All you have to do is change one line in the file KeyEventTranslations.tr in a location in your file system specified by this command: FileNameJoin[{$InstallationDirectory, "SystemFiles", "FrontEnd", "TextResources", $OperatingSystem}] Locate the following line in a text editor and change the key into the one ... 13 Besides the nice real handy option suggested by @yulinlinyu (more here) you can also use text-based interface to find completion for your half-typed function. It is not that fast, but has its own advantages. Try executing this: ?Plot* and you'll get this nice table of possible functions that complete your input. If you click on any you'll get short ... 13 On my system, setting the following to False works (Mac OS X): (in Preferences->Advanced->"Open Option Inspector") 12 To clarify the situation: In Version 9 on Windows and OS X, there is a new Make Template system which supports multiple templates for built-in functions. As part of the new system, unfortunately a bug was introduced which makes it ignore the usage statement for user-defined functions. This bug has been confirmed and we hope to fix it in a future release, ... 9 In the meantime, I was playing around. This is just to add a hotkey (Alt+k in windows) to replace what you have written so far with the partial symbol found. I don't know if it is a useful thing if we don't add it a way to handle multiple findings. Put this in a "init.m" file, inside FileNameJoin[{$UserBaseDirectory, "Applications", "AutocompleteBonus"}] ... 7 You can also use hotstrings as a way of autocompletion. By using such replacements, words are immediately replaced by another word on typing a space after the hotstring: CreateDocument[{}, InputAutoReplacements -> {"sync" -> SynchronousInitialization}] You can set such replacements globally under Option Inspector (CtrlShiftO). Of course no one would ... 5 I also took a crack at this. I think I made it look pretty close to the jquery example you posted. Figuring out how to move the insertion point to the end of the word once a suggestion is selected was a bit of a struggle. As a result, there's a DynamicWrapper in there that may be unstable. Input is the list of possible values from which you'd like to draw ... 5 Sometimes instead of handy shorthands like in the other answers, you'll find more useful Names giving a list of the names of symbols matching the string, (it's case sensitive of course) e.g. Names["GroB"] Names["Grob"] {"GroebnerBasis", "GroupActionBase", "GroupPageBreakWithin"} {"GroebnerBasis", "GroupMultiplicationTable", "GroupOrbits", ... 4 Note: This appears to really slowly in M9, although it works well in M8. It probably is better to use teedr's until it can be figured out what is causing the slow speeds. The following seems to work pretty well. I wrapped the options in a Pane and Framed so the entire row is clickable. ClearAll[AutoInputField]; SetAttributes[AutoInputField, ... 4 Maybe overkill but it was educational to try: DynamicModule[{}, EventHandler[ Overlay[{ Dynamic@Framed[ Row[{Style[x, Transparent, 15, Bold], Style[rest, GrayLevel@.6, 15, Bold]}], ImageSize -> {280, 30}, Alignment -> Top, FrameMargins -> {{5, 0}, {0, 1}}], InputField[Dynamic@x, String, BaseStyle ... 3 You can try (for a single notebook) CreateDocument[{}, NotebookEventActions -> {{"KeyDown", "\t"} :> NotebookWrite[SelectedNotebook[], "\t"]}] or (for global application) SetOptions[$FrontEnd, FrontEndEventActions -> {{"KeyDown", "\t"} :> NotebookWrite[SelectedNotebook[], "\t"]}] Does this meet your needs? For me it had an effect in ... 3 My humble contribution: (* Use this function to style list elements ) listItemStyle[item_] := Mouseover[#, Style[#, Background -> LightBlue]] &@ MouseAppearance[Framed[item], "LinkHand"]; ( This filters the list of data and returns a clickable list *) SetAttributes[autoComplete, HoldFirst]; autoComplete[s_, data_] := If[ StringLength[s] > 0, ... 3 Use Remove aber = {1, 2, 3} Remove@aber 3 This has to do with the Notebook's default context setting in the evaluation menu. If it isn't set to Global, the definitions made in init.m are not seen. As rm-rf says, a good way to put custom definitions in the init.m would be to use Begin and End to create an Init` context and append that context to the context path so that the definitions are ... 3 Partial solution for Linux (Ubuntu 12.04, GNOME 3.4.2) In version 8 I can expand it into a template the same way that normal expansion works. In version 9 this seems to work differently. You have to expand (or type) the full function name first and press then Ctrl+Shift+K xxyyzz::usage = "xxyyzz[x,y]"; Now I type xx press Ctrl+K and then Enter and I ... 3 This problem is fixed in the newest version of Mathematica 9.0.1 (at least for me). Notice: Wolfram published several versions with the name "9.0.1" for Linux and only the current one (md5 sum 7fcbc4d1488757b10ef07740ac30a580) fixed this bug. 2 seems like a bug and should be reported to support@wolfram.com. Noticed when I miss-spelled Plot and wrote PLot instead, the correct auto-completion came up. This tells me the context of another command before on the same line, which is Plot in this case, was confusing the Auto-complete for the next command on the same line. One temporary solution (not ... 2 Seems that you can work around this by modifying for example SyntaxInformation[Plot]={"ArgumentsPattern" -> {_, _, _}}, at the cost of having incorrect syntax highlighting. For some reason, setting SyntaxInformation[Plot]={"ArgumentsPattern" -> {_, {_,_,_}, __}} isn't sufficient to kill the Options[Plot] pattern matching, and I haven't found a form of ... 2 This is a V9 functionality. The closest you can get in V8 are the Ctrl+K (complete selection) and Ctrl+Alt+K (make template) shortcuts to these items in the Edit menu. They also work in V9. 1 Paraphrasing the insights of rm -rf: To make highlighting and autocompletion work in new contexts you have to run: 1) Begin[...]; 2) AppendTo[$ContextPath,...] and 3) End[]. Then you have to open your context again: Begin[...]; Be sure to put each of (1), (2) and (3) in the previous bullet point in separate cells. They have to be evaluated separately or it ... Only top voted, non community-wiki answers of a minimum length are eligible	2014-10-23 06:01:13	{"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": 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": 0.17018166184425354, "perplexity": 3742.825766112883}, "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-2014-42/segments/1413507450767.7/warc/CC-MAIN-20141017005730-00074-ip-10-16-133-185.ec2.internal.warc.gz"}
https://bird.bcamath.org/handle/20.500.11824/13/browse?rpp=20&sort_by=1&type=title&etal=-1&starts_with=N&order=ASC	Now showing items 125-144 of 193 • #### Neuron-glial Interactions (Encyclopedia of Computational Neuroscience, 2020-06-07) Although lagging behind classical computational neuroscience, theoretical and computational approaches are beginning to emerge to characterize different aspects of neuron-glial interactions. This chapter aims to provide ... • #### No Time at the End of the Tunnel (Communications Physics – Nature, 2018-08-21) Modern atto-second experiments seek to provide an insight into a long standing question: “how much time does a tunnelling particle spend in the barrier?” Traditionally, quantum theory relates this duration to the delay ... • #### Non-Markovian models of the growth of a polymer chain (Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2015-12-31) Using simple exactly solvable models, we show that event-dependent time delays may lead to significant non-Poisson effects in the statistics of polymer chain growth. The results are confirmed by stochastic simulation of ... • #### Nonlinear dynamics of shape memory alloy oscillators in tuning structural vibration frequencies (Mechatronics, 2012-12-31) Shape memory alloy (SMA) is one of the novel advanced functional materials that has an increasing range of current and potential applications, including smart materials and structures, bio-medical and nanotechnologies. ... • #### Numerical analysis of complex systems evolution with phase transformations at different spatial scales (Civil-Comp Proceedings, 2010-12-31) This paper shows the existence of a critical dimension for finite length nanowires exhibiting shape memory effects. We give a brief survey of phase transformations, their classifications, and provide the basis of mathematical ... • #### Numerical approximations for fractional elliptic equations via the method of semigroups (ESAIM: Mathematical Modelling and Numerical Analysis, 2020) We provide a novel approach to the numerical solution of the family of nonlocal elliptic equations $(-\Delta)^su=f$ in $\Omega$, subject to some homogeneous boundary conditions $\mathcal{B}(u)=0$ on $\partial \Omega$, where ... • #### Numerical Bifurcation Analysis of Physiologically Structured Populations: Consumer-Resource, Cannibalistic and Trophic Models (Bulletin of Mathematical Biology, 2016-06-22) With the aim of applying numerical methods, we develop a formalism for physiologically structured population models in a new generality that includes con- sumer resource, cannibalism and trophic models. The dynamics at the ... • #### Numerical equilibrium analysis for structured consumer resource models (Bulletin of Mathematical Biology, 2010-12-31) In this paper, we present methods for a numerical equilibrium and stability analysis for models of a size structured population competing for an unstructured resource. We concentrate on cases where two model parameters are ... • #### Numerical Regge pole analysis of resonance structures in elastic, inelastic and reactive state-to-state integral cross sections (Computer Physics Communications, 2014-12-31) We present a detailed description of a FORTRAN code for evaluation of the resonance contribution a Regge trajectory makes to the integral state-to-state cross section (ICS) within a specified range of energies. The ... • #### Numerical simulation of a susceptible-exposed-infectious space-continuous model for the spread of rabies in raccoons across a realistic landscape (Journal of Biological Dynamics, 2013-12-31) We introduce a numerical model for the spread of a lethal infectious disease in wildlife. The reference model is a Susceptible-Exposed-Infectious system where the spatial component of the dynamics is modelled by a diffusion ... • #### Numerical simulation of extreme wave runup during storm events in Tramandaí Beach, Rio Grande do Sul, Brazil (Coastal Engineering, 2015-12-31) We present a high resolution analysis of the interaction of irregular waves with natural and urban structures leading to extreme wave runup. Horizontal runup data, instantaneous flooding maps, and wave propagation beyond ... • #### On a generalization of the global attractivity for a periodically forced Pielou's equation (Journal of Difference Equations and Applications, 2012-12-31) In this paper, we study the global attractivity for a class of periodic difference equation with delay which has a generalized form of Pielou's difference equation. The global dynamics of the equation is characterized by ... • #### On the Analysis of Trajectory-Based Search Algorithms: When is it Beneficial to Reject Improvements? (Algorithmica, 2018) We investigate popular trajectory-based algorithms inspired by biology and physics to answer a question of general significance: when is it beneficial to reject improvements? A distinguishing factor of SSWM (Strong Selection ... • #### On the global stability of a delayed epidemic model with transport-related infection (Nonlinear Analysis: Real World Applications, 2011-12-31) We study the global dynamics of a time delayed epidemic model proposed by Liu et al. (2008) [J. Liu, J. Wu, Y. Zhou, Modeling disease spread via transport-related infection by a delay differential equation, Rocky Mountain ... • #### On the global stability of an SIRS epidemic model with distributed delays (Discrete and Continuous Dynamical Systems- Series A, 2011-12-31) In this paper, we establish the global asymptotic stability of an endemic equilibrium for an SIRS epidemic model with distributed time delays. It is shown that the global stability holds for any rate of immunity loss, if ... • #### On the Perturbation Methods for Vibration Analysis of Linear Time-Varying Systems (International Journal of Applied Mechanics, 2016-01-01) Some perturbation methods in the studying vibrations of the linear time-varying (LTV) system are discussed. Three classical perturbation methods, namely, averaging method, harmonic balance method, and multiple scales method ... • #### Opportunities at the Intersection of Synthetic Biology, Machine Learning, and Automation (ACS Synthetic Biology, 2019-07) Our inability to predict the behavior of biological systems severely hampers progress in bioengineering and biomedical applications. We cannot predict the effect of genotype changes on phenotype, nor extrapolate the ... • #### An optimal scaling to computationally tractable dimensionless models: Study of latex particles morphology formation (Computer Physics Communications, 2020-02) In modelling of chemical, physical or biological systems it may occur that the coefficients, multiplying various terms in the equation of interest, differ greatly in magnitude, if a particular system of units is used. Such ... • #### Optimization of an Externally Mixed Biogas Plant Using a Robust CFD Method (Computers and Electronics in Agriculture, 2020-04) Biogas plants have to be continuously or periodically mixed to ensure the homogenization of fermenting and fresh substrate. Externally installed mixers provide easier access than submerged mixers but concerns of ... • #### Optimized schwarz methods and model adaptivity in electrocardiology simulations (Lecture Notes in Computational Science and Engineering, 2014-12-31) [No abstract available]	2020-09-22 14:34:41	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.36910200119018555, "perplexity": 2740.028567972161}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1600400206133.46/warc/CC-MAIN-20200922125920-20200922155920-00691.warc.gz"}
https://mathematica.stackexchange.com/questions/203393/how-to-use-function-as-a-placeholder-for-codes	# How to use function as a placeholder for codes I want to use function as a placeholder for a bunch of codes while making the function also produce an output Consider A={0,0,0}; f[x_]:=Block[{},A[[x]]=1;A] f[1] this will make A to be {1,0,0} while outputting A. (The Block[] is necessary. I don't know why, but it isn't my main question). My main question is how to make the following code work f[A1_] := Block[{}, A1[[1]] = 1; A1] A={0,0,0}; f[A] I want this changes A to {1,0,0} while outputting it. How can i make this work? Are there alternatives for placeholder of codes? If my question shows an obvious lack on knowledge on some area, such as how defining function works, some reference would be helpful. Give f the Attribute HoldFirst: ClearAll[f]	2020-04-07 04:50:29	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.453220009803772, "perplexity": 1005.4723057199799}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371665328.87/warc/CC-MAIN-20200407022841-20200407053341-00069.warc.gz"}
http://www.maa.org/programs/faculty-and-departments/course-communities/browse?term_node_tid_depth=All&page=62&device=desktop	# Browse Course Communities Displaying 621 - 630 of 791 The lesson begins with an emphasis on isolating the radical expression in a radical equation and then highlights the importance of checking for extraneous solutions that may be generated when t The lesson begins with the product and quotient rules for radicals, highlighting the frequently made mistakes of students by overgeneralizing the rules. The lesson begins by associating the distance between two points with the right triangle that may be formed by joining the points and extending horizontal and vertical lines through the points. Using Kleiber's rule- which linked an animals metabolic rate to its mass- to motivate working with rational exponents, this lesson begins by linking work with $$n$$th roots from the last le Exponential notation for $$n$$th roots and radicals is introduced. To motivate the convention for negative and zero exponents, the lesson begins by observing the halving pattern found in continuously decreasing the exponent of a power of two by one. This lesson focuses on finding appropriate non linear functions to model real world phenomena. The lesson begins with a comparison of data tables and graphs of two functions, one directly proportional (cost of gas) and the other exponential (population), before a definition for direct va A short lesson introducing the cube root and absolute value functions and their graphs. The lesson begins with graphs of the Dow Jones Industrial Average and water levels of Lake Huron where points on the graph are interpreted.	2014-07-30 04:06:29	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.503936767578125, "perplexity": 1029.6426396266636}, "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-2014-23/segments/1406510268533.31/warc/CC-MAIN-20140728011748-00296-ip-10-146-231-18.ec2.internal.warc.gz"}
https://labs.tib.eu/arxiv/?author=X.Luo	• ### Long-Baseline Neutrino Facility (LBNF) and Deep Underground Neutrino Experiment (DUNE) Conceptual Design Report Volume 2: The Physics Program for DUNE at LBNF(1512.06148) Jan. 22, 2016 hep-ex, physics.ins-det The Physics Program for the Deep Underground Neutrino Experiment (DUNE) at the Fermilab Long-Baseline Neutrino Facility (LBNF) is described. • ### Long-Baseline Neutrino Facility (LBNF) and Deep Underground Neutrino Experiment (DUNE) Conceptual Design Report, Volume 4 The DUNE Detectors at LBNF(1601.02984) Jan. 12, 2016 hep-ex, physics.ins-det A description of the proposed detector(s) for DUNE at LBNF • ### An Experimental Exploration of the QCD Phase Diagram: The Search for the Critical Point and the Onset of De-confinement(1007.2613) July 15, 2010 nucl-ex The QCD phase diagram lies at the heart of what the RHIC Physics Program is all about. While RHIC has been operating very successfully at or close to its maximum energy for almost a decade, it has become clear that this collider can also be operated at lower energies down to 5 GeV without extensive upgrades. An exploration of the full region of beam energies available at the RHIC facility is imperative. The STAR detector, due to its large uniform acceptance and excellent particle identification capabilities, is uniquely positioned to carry out this program in depth and detail. The first exploratory beam energy scan (BES) run at RHIC took place in 2010 (Run 10), since several STAR upgrades, most importantly a full barrel Time of Flight detector, are now completed which add new capabilities important for the interesting physics at BES energies. In this document we discuss current proposed measurements, with estimations of the accuracy of the measurements given an assumed event count at each beam energy. • ### QCD with dynamical Wilson fermions - first results from SESAM(hep-lat/9510001) Oct. 2, 1995 hep-lat First results of a recently started simulation of full QCD with two flavours of sea-quarks at a coupling of $\beta = 5.6$ on a $16^3 \times 32$ lattice are presented. Emphasis is laid on the statistical significance that can be achieved by an integrated luminosity'' of 140 TFlop$\times$hrs, for Hybrid Monte Carlo simulations at four intermediate values of $\frac{m_{\pi}}{m_{\rho}}$. The simulation takes place on the Quadrics QH2 at DESY/Zeuthen and DFG/Bielefeld. The performance is optimized by means of BiCGStab and the chronological inversion method of Brower et al. We discuss the systematic errors arising from lack of the molecular dynamic's reversibility on the 32-bit QH2. For plaquette and meson correlators we find integrated autocorrelation times of $< 20$ units of molecular dynamics time and exponential autocorrelation times of about $50$ units. Using these results we perform preliminary measurements of the central potential and $\pi$ and $\rho$ correlators on independent configurations and obtain first estimates of the lattice spacings at three values of the dynamical hopping parameter.	2021-04-16 16:23:59	{"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": 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": 0.6806421875953674, "perplexity": 2415.8310596115657}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2021-17/segments/1618038088245.37/warc/CC-MAIN-20210416161217-20210416191217-00508.warc.gz"}
http://mathoverflow.net/questions/118251/is-the-cup-product-of-holomorphic-n-forms-with-a-fixed-class-injective/118255	# Is the cup product of holomorphic $n$-forms with a fixed class injective? Let $X$ be a compact Kahler manifold of complex dimension $n$. Fix a nonzero class $u \in H^1(X,T_X)$. This gives a linear morphism $$\phi_u : H^0(X,\Omega^n) \to H^{1}(X,\Omega^{n-1}), \quad \sigma \mapsto u \cup \sigma.$$ Is $\phi_u$ injective? It is so for manifolds with $\Omega^n_X = \mathcal O_X$; the proof I've got is not hard but uses Ricci-flat Kahler metrics and the hard Lefschetz theorem so it cannot generalize to other situations. In the examples I know (curves, hypersurfaces in $\mathbb P^n$) we have $h^{n,0} \leq h^{n-1,1}$, so I haven't stumbled upon an obvious counterexample yet. - The answer to this question is negative in dimensions $\ge 3$. For example, take a quintic in $\mathbb CP^4$ and consider its blow up $X$ in $10^{100}$ points (just to be safe). Then the space $H^1(X, T_X)$ will be huge, since it parametrises deformations of the blown up variety and you can move points as you wish. So there will be non-zero $u\in H^1(X, T_X)$ so that you map is trivial. Note that when you blow up the quintic $H^{2,1}$ does not change. Also, this trick with blow ups will not work for Kahler surfaces as is explained for example, in appendix 1 in http://arxiv.org/pdf/1301.0478.pdf - I'd have upvoted just for "$10^{100}$ points (just to be safe)" alone. :D – Gunnar Magnusson Jan 7 '13 at 9:25 I see, that Jason gave a different explanation that works for surfaces :) even though surfaces satisfy $h^{2,0}<h^{1,1}$ – Dmitri Jan 7 '13 at 9:26 The map $\phi_u$ is not always injective. Let $X$ be the blowing up along a large number of points inside a quartic surface in $\mathbb{P}^3$, or any surface $S$ with nonzero $h^{n,0}$. The weight $2$ Hodge structure of $X$ is a direct sum of the weight $2$ Hodge structure of $S$, the image of the pullback morphism, and a finite number of copies of the Hodge structure $\mathbb{Z}(-1)$, the Gysin images of the weight $0$ Hodge structures of the exceptional divisors. So, as you vary the blown up points in $S$, the corresponding variation of Hodge structures is trivial. Thus the Griffiths transversality map $\phi_u$ is zero for every $u$ coming from a variation of the points. -	2014-09-02 14:08:08	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9147156476974487, "perplexity": 152.64194518800466}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1409535922087.15/warc/CC-MAIN-20140909045234-00193-ip-10-180-136-8.ec2.internal.warc.gz"}
https://www.gradesaver.com/textbooks/math/algebra/algebra-2-1st-edition/chapter-2-linear-equations-and-functions-2-3-graph-equations-of-lines-2-3-exercises-skill-practice-page-94/50	# Chapter 2 Linear Equations and Functions - 2.3 Graph Equations of Lines - 2.3 Exercises - Skill Practice - Page 94: 50 The graph is attached. #### Work Step by Step This equation simplifies to x=3. Since any graph in the from $x=constant$ is a vertical line, we know that this is a vertical line passing through 3 on the x axis. After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.	2022-08-16 23:19:27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5149146318435669, "perplexity": 937.292345128316}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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-2022-33/segments/1659882572581.94/warc/CC-MAIN-20220816211628-20220817001628-00192.warc.gz"}
https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Book%3A_Introduction_to_Linear_Time-Invariant_Dynamic_Systems_for_Students_of_Engineering_(Hallauer)/01%3A_Introduction%3B_First_and_Second_Order_Systems%3B_Analysis%3B_MATLAB_Graphing/1.02%3A_LTI_Systems_and_ODEs	# 1.2: LTI Systems and ODEs We consider physical systems that can be modeled with reasonable engineering fidelity as linear, time-invariant (LTI) systems. Such a system is represented mathematically by an ordinary differential equation (ODE), or by a set of coupled ODEs, for which the single independent variable is time, denoted as $$t$$. These ODEs are linear, and they have constant coefficients, so we describe them as linear, time-invariant (LTI), the same as the systems they represent1. For example, suppose we denote a dependent variable as $$x(t)$$, here a general symbol representing some physical dynamic response quantity for which we want to solve. Then an LTI ODE that models an LTI physical system might have the form $\dfrac{dx}{dt} - a\,x =b\,u(t)\label{eqn:1}$ in which $$a$$ and $$b$$ are constant multiplying coefficients, and known function $$u(t)$$ is the excitation and is independent of the response. In the study of systems, an independent excitation $$u(t)$$ is often called an input, and a dependent response $$x(t)$$ is often called an output. Hereafter, we will usually employ the common shorthand dot notation for denoting derivatives with respect to time: $$\dfrac{dx}{dt} = \dot{x}$$, $$\dfrac{d^2 x}{dt^2} = \ddot{x}$$ etc., so that Equation $$\ref{eqn:1}$$ can be written more simply as $\dot{x} - a\,x = b\,u(t) .$ The linearity of Equation $$\ref{eqn:1}$$ is manifested by the linear appearance of $$x(t)$$ and all of its derivatives in the ODE. The following are some similar ODEs that are not linear (they are nonlinear) for obvious reasons: $$\dot{x} - a\,x^2 = b\,u(t)$$; $$sin(\dot{x}) - a\,x = b\,u(t)$$; $$\sqrt(\dot{x}) - a\,tan(x) = b\,u(t)$$. Linear ODEs are almost always easier to solve (at least in closed form, i.e., as equations involving standard functions) than nonlinear ODEs. Moreover, the important principle of superposition applies to linear ODEs, but not to nonlinear ODEs. An example of the application of this principle is: let the response to input $$u_1(t)$$ be $$\ x_1(t)$$, and let the response to another input $$u_2(t)$$ be $$x_1 (t)$$; if a third input is the sum of multiplied terms $$u_3(t) = c_1\,u_1(t) + c_2\,u_2 (t)$$, in which $$c_1$$ and $$c_2$$ are constants, then the response to $$u_3(t)$$ is $$x_3 (t) = c_1\,x_1 (t) + c_2\,x_2 (t)$$. This result is easy to derive just by multiplying two ODEs such as Equation $$\ref{eqn:1}$$ by the constants, then adding the multiplied ODEs. The principle of superposition allows us to solve accurately for the responses of linear systems to any physically realistic inputs. (See Section 8.10 for a derivation of system response to an arbitrary physically realistic input by direct application of superposition.) The time invariance of Equation $$\ref{eqn:1}$$ is manifested by the constant coefficients of $$x(t)$$ and all of its derivatives in the ODE. ODEs with time-invariant coefficients model the behavior of systems assumed to have physical properties that either remain constant in time or vary so slowly and/or slightly that the variation is negligible for engineering purposes. But many practically important systems have time-varying physical properties. For example, a vehicle such as a space shuttle between liftoff and achievement of orbital position has rapidly varying (decreasing) mass as propellant is burned and external fuel tanks and boosters are released. The following is a linear equation somewhat similar to Equation $$\ref{eqn:1}$$, but with an obviously time-varying coefficient: $$\dot{x} -3\,x\,(1-e^(-2\,t))\,x = b\,u(t)$$. The study of systems with time-varying physical properties is generally more complicated, not fundamental, so only time-invariant systems and ODEs are considered in this book. The form of Equation $$\ref{eqn:1}$$, $$\dot{x} - a\,x = b\,u(t)$$, is widely regarded as the standard form for a first order LTI ODE, and we will use it as such in this book. Beginning in the next section, we will study idealized physical systems whose dynamic behaviors are described by equations that are directly analogous to Equation $$\ref{eqn:1}$$. We will express the mathematical constants $$a$$ and $$b$$ in terms of specific physical constants. Also, the roles of input $$u(t)$$ and output $$x(t)$$ in Equation $$\ref{eqn:1}$$ will be assumed by some specific physical quantities, such as force, velocity, voltage, etc., and we will denote them with relevant symbols [often different than $$u(t)$$ and $$x(t)$$] when appropriate. Although only first order ODEs are discussed in this section, we certainly will encounter and study systems and ODEs of second and higher orders. 1LTI ODEs are also sometimes described as linear, constant-coefficient, or LCC.	2020-08-05 20:08:56	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9311240911483765, "perplexity": 309.05856985471297}, "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-2020-34/segments/1596439735964.82/warc/CC-MAIN-20200805183003-20200805213003-00179.warc.gz"}
https://math.stackexchange.com/questions/3686144/set-of-all-positive-rational-numbers-is-countable	# Set of all positive rational numbers is countable In proving set of positive rational numbers is countable, normally we use the way "Connecting the numbers diagonally". Connecting rational numbers "Diagonally" In this way,we can list out all the rational numbers. But may I know what the bijective function that connects Q and N is?, since we need a bijection between two sets to show that they are having same cardinality. Thank you. • Are you asking for an explicit formula for such a bijection? If so, this question has some good answers. – Wyvellum May 22 '20 at 5:00 • Because I am just starting studying Math, and the courses I am studying are elementary. I am not sure if I can understand some deep theories. And this question arises when I learn Discrete Mathematics , and my professor did not show the bijective function. What he said is "In order to prove a set has a cardinality as natural set, we need to order the elements of that set in a sequence/list". But I would appreciate if you can give some explicit formulas, and I will try to understand them. Thank you. – Ho F May 22 '20 at 5:06 • As an alternative, consider the Calkin-Wilf sequence – J. W. Tanner May 22 '20 at 5:18 • OK. Thank you so much. – Ho F May 22 '20 at 5:23 The way you get a bijection is to simply skip the duplicates. The easiest way to do it is to only count the fractions in maximally cancelled form. Note that for the purpose of this post, I assume that $$0\notin\mathbb N$$. So you start with $$1/1$$. That is the totally cancelled form, so it gets assigned the number $$1$$. Next come $$2/1$$ and $$1/2$$. They are also totally cancelled, so they get assigned $$2$$ and $$3$$. The next diagonal is where it gets interesting: $$3/1$$ is still totally cancelled, so we assign is the next number, $$4$$. But $$2/2$$ is not totally cancelled (numerator and denominator have a common factor $$2$$), therefore we skip it and do not assign it a number. If we did, we would have assigned two different numbers to the value $$1=1/1=2/2$$. Since we skip it, the next number considered is $$1/3$$, which again is totally cancelled and therefore gets assigned the next natural number, $$5$$. Note that (apart from the fully-cancelled shortcut) this generally applies: Whenever for any set $$X$$ you have a surjection $$f:\mathbb N\to X$$, you can get a bijection $$g:\mathbb N\to X$$ by simply skipping duplicates. Moreover note that to show that a bijection exists, you don't need to provide an explicit bijection. The Schröder-Bernstein theorem guarantees you that whenever you have an injection both ways, there does exist a bijection. Moreover, if we have a surjection $$X\to Y$$, then there exists also an injection $$Y\to X$$ (assuming the axiom of choice, but in the case of $$X=\mathbb N$$, not even that is needed). The way to get that is basically the skipping process outlined above. Therefore if you can show both an injection and a surjection from 4X\to Y\$ you have proved that a bijection exists. Since the diagonal counting without skipping gives a surjection $$\mathbb N\to\mathbb Q^+$$ (every number is reached), and in injection $$\mathbb N\to\mathbb Q^+$$ is trivial, this already proves the existence of a bijection. • So the function needs not to be an explicit formula, as long as we can assign every elements in Natural set exactly one element in set X and make sure that the function from N to X is also surjective? – Ho F May 22 '20 at 5:22 • @HoF: Yes, exactly, see my edit. Strictly speaking you also need to show that there exists an injection; a surjection alone only proves that there are at most as many positive rationals as naturals. But an injection is trivial (and there is no doubt that the set of rational numbers is indeed infinite). – celtschk May 22 '20 at 5:28 • Ok, I see. Thank you – Ho F May 22 '20 at 5:36	2021-03-01 00:58:19	{"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": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 22, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9198822975158691, "perplexity": 276.7542180431246}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1614178361808.18/warc/CC-MAIN-20210228235852-20210301025852-00131.warc.gz"}
https://quantiki.org/browse?page=10	# Browse Title Post date Two-dimensional non commutative Swanson model and its bicoherent states. (arXiv:1711.10181v2 [math-ph] UPDATED) Thursday, October 11, 2018 - 02:40 Bravyi-Kitaev Superfast simulation of fermions on a quantum computer. (arXiv:1712.00446v3 [quant-ph] UPDATED) Thursday, October 11, 2018 - 02:40 On the Quantum-Mechanics of a Single Photon. (arXiv:1801.00268v3 [math-ph] UPDATED) Thursday, October 11, 2018 - 02:40 Strain-Tunable GaAs Quantum dot: A Nearly Dephasing-Free Source of Entangled Photon Pairs on Demand. (arXiv:1801.06655v3 [quant-ph] UPDATED) Thursday, October 11, 2018 - 02:40 Semiconductor quantum dots as an ideal source of polarization entangled photon pairs on-demand: a review. (arXiv:1804.10472v2 [quant-ph] UPDATED) Thursday, October 11, 2018 - 02:40 Quantum computational supremacy in the sampling of bosonic random walkers on a one-dimensional lattice. (arXiv:1805.01858v3 [quant-ph] UPDATED) Thursday, October 11, 2018 - 02:40 Scrambling and entanglement spreading in long-range spin chains. (arXiv:1806.00022v2 [quant-ph] UPDATED) Thursday, October 11, 2018 - 02:40 Exact calculation of stimulated emission driven by pulsed light. (arXiv:1806.04862v2 [quant-ph] UPDATED) Thursday, October 11, 2018 - 02:40 Enabling a Scalable High-Rate Measurement-Device-Independent Quantum Key Distribution Network. (arXiv:1807.03466v2 [quant-ph] UPDATED) Thursday, October 11, 2018 - 02:40 A tensor network annealing algorithm for two-dimensional thermal states. (arXiv:1809.08258v2 [quant-ph] UPDATED) Thursday, October 11, 2018 - 02:40 Non-Gaussianity and entropy-bounded uncertainty relations: Application to detection of non-Gaussian entangled states Wednesday, October 10, 2018 - 12:00 Internal Quantum Dynamics of a Nanoparticle in a Thermal Electromagnetic Field: a Minimal Model. (arXiv:1807.03811v2 [quant-ph] UPDATED) Wednesday, October 10, 2018 - 03:11 Fake Superoscillations. (arXiv:1810.03607v1 [quant-ph]) Wednesday, October 10, 2018 - 02:41 Stationary Phase Method in Discrete Wigner Functions and Classical Simulation of Quantum Circuits. (arXiv:1810.03622v1 [quant-ph]) Wednesday, October 10, 2018 - 02:41 History operators in quantum mechanics. (arXiv:1810.03624v1 [quant-ph]) Wednesday, October 10, 2018 - 02:41 High-fidelity dissipative engineering using parametric interactions. (arXiv:1810.03631v1 [quant-ph]) Wednesday, October 10, 2018 - 02:41 High-temperature coherent transport in the XXZ chain in the presence of an impurity. (arXiv:1810.03640v1 [cond-mat.str-el]) Wednesday, October 10, 2018 - 02:41 Revisited version of Weyl's limit point-limit circle criterion for essential self-adjointness. (arXiv:1810.03641v1 [math-ph]) Wednesday, October 10, 2018 - 02:41 Convexity and Operational Interpretation of the Quantum Information Bottleneck Function. (arXiv:1810.03644v1 [quant-ph]) Wednesday, October 10, 2018 - 02:41 Many-body effects in quantum metrology. (arXiv:1810.03651v1 [quant-ph]) Wednesday, October 10, 2018 - 02:41 Comments on "Quasi-coherent states for the Hermite oscillator" [J. Math. Phys. {\bf 59}, 062104 (2018)]. (arXiv:1810.03662v1 [quant-ph]) Wednesday, October 10, 2018 - 02:41 Detection of genuine n-qubit entanglement via the proportionality of two vectors. (arXiv:1810.03674v1 [quant-ph]) Wednesday, October 10, 2018 - 02:41 Numerical study of hypergraph product codes. (arXiv:1810.03681v1 [quant-ph]) Wednesday, October 10, 2018 - 02:41 Three-dimensional superconducting resonators at $T < 20$ mK with the photon lifetime up to $\tau=2$ seconds. (arXiv:1810.03703v1 [quant-ph]) Wednesday, October 10, 2018 - 02:41 On the optical nonreciprocity and slow light propagation in coupled spinning optomechanical resonators. (arXiv:1810.03709v1 [quant-ph]) Wednesday, October 10, 2018 - 02:41	2018-10-19 01:32:39	{"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": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.25259581208229065, "perplexity": 12173.61606651808}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583512161.28/warc/CC-MAIN-20181018235424-20181019020924-00285.warc.gz"}
https://www.khanacademy.org/math/pre-algebra/pre-algebra-arith-prop/pre-algebra-arithmetic-properties/a/properties-of-addition	Explore the commutative, associative, and identity properties of addition. In this article, we'll learn the three main properties of addition. Here's a quick summary of these properties: Commutative property of addition: Changing the order of addends does not change the sum. For example, $4 + 2 = 2 + 4$. Associative property of addition: Changing the grouping of addends does not change the sum. For example, $(2 + 3) + 4 = 2 + (3 + 4)$. Identity property of addition: The sum of $0$ and any number is that number. For example, $0 + 4 = 4$. The commutative property of addition says that changing the order of addends does not change the sum. Here's an example: $4 + 2 = 2 + 4$ Notice how both sums are $6$ even though the the ordering is reversed. If Jill starts with $4$ bananas and then gets another $2$ bananas she ends up with the same amount as if she started with $2$ bananas and got $4$ more bananas. Here's another example with more addends: $1 + 2 + 3 + 4 = 4 + 3 + 2 + 1$ Which of these is an example of the commutative property of addition? The associative property of addition says that changing the grouping of the addends does not change the sum. Here's an example: $\blueD{(2 + 3) + 4} = \goldD{2 + (3 + 4)}$ Remember that parentheses tell us to do something first. So here's how we evaluate the left-hand side: $\phantom{=}\blueD{(2 + 3) + 4}$ $= 5 + 4$ $=9$ And here's how we evaluate the right-hand side: $\phantom{=}\goldD{2 + (3 + 4)}$ $= 2 + 7$ $=9$ Notice that both sides sum to $9$ even though we added the $2$ and the $3$ first on the left-hand side, and we added the $3$ and the $4$ first on the right-hand side. Which of these is an example of the associative property of addition? The identity property of addition says that the sum of $0$ and any number is that number. Here's an example: $0 + 4 = 4$ This is true because the definition of $0$ is "no quantity", so when we add $0$ to $4$, the quantity of $4$ doesn't change! The commutative property of addition tells us that it doesn't matter if the $0$ comes before or after the number. Here's an example of the identity property of addition with the $0$ after the number: $6 + 0 =6$	2019-02-18 04:33:35	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 38, "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": 0.787229597568512, "perplexity": 360.33640100750773}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247484648.28/warc/CC-MAIN-20190218033722-20190218055722-00272.warc.gz"}
https://mathoverflow.net/questions/132262/convex-polyhedron-in-the-unit-cube	# convex polyhedron in the unit cube Let $P$ be a given finite set of points within the $n$-dimensional unit cube. A finite set $Q$ of points within the $n$-dimensional unit cube covers $P$ if $\operatorname{conv}(Q) \supseteq P$ where $\operatorname{conv}(Q)$ denotes the convex hull of $Q$. How can one compute a minimal set $Q$ that covers $P$? Trivially, a minimal set $Q$ satisfies $\|Q\| \le \|P\|$ and $\|Q\| \le 2^n$. The minimal size could be computed by appealing to decision procedures for the first-order theory of the reals, but is there a smarter way? This is not a direct answer, but a closely related problem is known to be NP-hard: Das, Goodrich. "On the Complexity of Approximating and Illuminating Three-Dimensional Convex Polyhedra." 1995. (ACM link; PDF download) This paper establishes several results, including this: Theorem 4.2. The problem of fitting a polyhedron with a minimum number of faces between two given nested convex polyhedra is NP-hard. This question was first posed by Victor Klee, and I coauthored a paper that provided an efficient algorithm in $\mathbb{R}^2$. But the above result shows it is already intractable in $\mathbb{R}^3$. I do not remember the Das-Goodrich proof well enough to know if it can achieve the same result with the outer polyhedron a cube. There are many approximation algorithms available, as this is an important practical problem. For example: Mitchell, Suri. "Separation and approximation of polyhedral surfaces." In Proc. 3rd ACM-SIAM Sympos. Discrete Algorithms, pages 296-306, 1992. (CiteSeer link) • Thanks a lot, Joseph. These pointers, especially the NP-hardness result, are extremely helpful. – Stefan Kiefer May 30 '13 at 16:48	2019-05-23 22:08:40	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.889860987663269, "perplexity": 142.80087744061768}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232257396.96/warc/CC-MAIN-20190523204120-20190523230120-00084.warc.gz"}
http://ldtopology.wordpress.com/category/knot-theory/	# Low Dimensional Topology ## May 17, 2013 ### An old corker on the unknotting of knots Filed under: Knot theory — Ryan Budney @ 11:13 am I imagine many readers of this blog are familiar with the fact that you can knot a circle in 3-space, but not in 4-space. If you enjoy thinking about why that is true, please read on! Think of euclidean 3-space, $\mathbb R^3$ as a linear subspace of euclidean 4-space, $\mathbb R^3 = \mathbb R^3 \times \{0\} \subset \mathbb R^4$. So if you have a knotted circle in 3-space, you can consider it as an embedded circle in 4-space. And you can unknot it! I think one of the simplest explanations of of this would be the idea to push the knot up into the 4-th dimension every time a strand is close to being an overcrossing (in a planar diagram). At this stage you could in effect change the crossing to be anything you want, after you’re done modifying the crossings, you could push the knot back into 3-space to get a different knot. ## April 6, 2013 ### New connection between geometric and quantum realms Filed under: Hyperbolic geometry,Knot theory,Quantum topology — dmoskovich @ 9:41 am A paper by Thomas Fiedler has just appeared on arXiv, describing a new link between geometric and quantum topology of knots. http://arxiv.org/abs/1304.0970 This is big news!! (more…) ## April 3, 2013 ### Update on subadditivity of tunnel number Filed under: Heegaard splittings,Knot theory — Jesse Johnson @ 12:54 pm A few months ago, I wrote a blog post about the interesting phenomenon that the tunnel number of a connect sum of two knots may be anywhere from one more than the sum of the tunnel numbers to a relatively small fraction of the sum of the tunnel numbers. Since then, a couple of related papers have been posted to the arXiv, so I thought that justifies another post on the subject. The first preprint I’ll discuss, by João Miguel Nogueira [1], gives new examples of knots in which the tunnel number degenerates by a large amount. The second paper, by Trent Schirmer [2] (who is currently a postdoc here at OSU), gives a new bound on the amount tunnel number and Heegaard genus can degenerate by under connect sum/torus gluing, respectively, in certain situations. ## February 16, 2013 ### The Bridge Spectrum Filed under: 3-manifolds,Heegaard splittings,Knot theory — Jesse Johnson @ 9:37 pm A knot $K$ in a three-manifold $M$ is said to be in bridge position with respect to a Heegaard surface $\Sigma$ if the intersection of $K$ with each of the two handlebody components of the complement of $\Sigma$ is a collection of boundary parallel arcs, or if $K$ is contained in $\Sigma$. The bridge number of a knot $K$ in bridge position is the number of arcs in each intersection (or zero if if $K$ is contained in $\Sigma$) and the genus $g$ bridge number of $K$ is the minimum bridge number of $K$ over all bridge positions relative to genus $g$ Heegaard surfaces for $M$. The classical notion of bridge number is the genus-zero bridge number, i.e. bridge number with respect to a sphere in $S^3$, but a number of very interesting results in the last few years have examined the higher genus bridge numbers. Yo’av Rieck defined the bridge spectrum of a knot $K$ as the sequence $(b_0,b_1,b_2,\ldots)$ where $b_i$ is the genus $i$ bridge number of $K$ and asked the question: What sequences can appear as the bridge spectrum of a knot? (At least, I first heard this term from Yo’av at the AMS section meeting in Iowa City in 2011 – as far as I know, he was the first to formulate the question like this.) ## December 18, 2012 ### Morse-Novikov number and tunnel number Filed under: 3-manifolds,Heegaard splittings,Knot theory,Thin position — Jesse Johnson @ 9:33 am Someone recently pointed out to me a paper by A. J. Pajitnov [1] proving a very interesting connection between circular Morse functions and (linear) Morse functions on knot complements. (A similar result is probably true in general three-manifolds as well.) Recall that a (linear) Morse function is a smooth function from a manifold to the line in which there are a finite number of critical points (where the gradient of the function is zero), and each critical point has one of a number of possible forms. For a two-dimensional manifold the possible forms are the familiar local minimum, saddle or local maximum. This post is about three-dimensional Morse functions, in which case the possible forms are slight generalizations of local minima, maxima and saddles. A circular Morse function is a function with the same conditions on critical points, but whose range is the circle rather than the line. For a three-dimensional manifold, the minimal number of critical points in a linear Morse function is twice the Heegaard genus plus two, and for knot complements it’s twice the tunnel number plus two. (In particular, one can construct a Heegaard splitting or unknotting tunnel system directly from a Morse function, but that’s for another post.) The minimal number of critical points in a circular Morse function is called the Morse-Novikov number, and is equal to the minimal number of handles in a circular thin position for the manifold (usually a knot complement). Pajitnov has a very clever argument to show that the (circular) Morse-Novikov number of a knot complement is bounded above by twice its (linear) tunnel number. Below, I want to outline a slightly different formulation of this proof in terms of double sweep-outs, though I should stress that the underlying idea is the same. ## October 18, 2012 ### Untangling a knot Filed under: Knot theory — dmoskovich @ 8:23 am Chad Musick made a video in which he untangles a complicated trivial knot due to Ochiai. The procedure is described in his paper Recognising Trivial Links in Polynomial Time. My reaction was “Sweet!”. ## September 12, 2012 ### Subadditivity of complexities under gluing Filed under: 3-manifolds,Knot theory — Jesse Johnson @ 1:17 pm A recent talk in our topology seminar by Trent Schirmer (who just joined OSU as a postdoc this year) got me thinking about three closely related (almost equivalent) problems in three-dimensional topology. Trent spoke about the following problem in knot theory: Given two knots in $S^3$ with tunnel numbers $t_1$ and $t_2$, what would you expect the tunnel number of their connect sum to be? Recall that the tunnel number of a knot is the Heegaard genus of the knot complement minus one. With a little work, one can show that the tunnel number of the connect sum is at most $t_1 + t_2 + 1$. However, there are also examples where it is much lower and Trent has constructed links where the connect sum has tunnel number around $\frac{4}{7}(t_1 + t_2)$ [1]. This is fairly interesting on its own, but it turns out there are (at least) two other situations with similar phenomenon that appear to have the same underlying reasons. ## September 8, 2012 ### ICERM Fall 2013: Topology, geometry, and dynamics I’ve mentioned before that the fall semester program at ICERM for 2013 will focus on computation in low-dimensional topology, geometry, and dynamics. You can now apply to be a long-term visitor for this as a graduate student, postdoc, or other. The deadline for the postdoctoral positions is January 14, 2013; the early deadline for everyone else is December 1, 2012 and the second deadline March 15, 2013. There will also be three week-long workshops associated with this, so mark your calendars for these exciting events: 1. Exotic Geometric Structures. September 15-20, 2013. 2. Topology, Geometry, and Group Theory: Informed by Experiment. October 21-25, 2013. 3. Geometric Structures in Low-Dimensional Dynamics. November 18-22, 2013. ## August 22, 2012 ### Bill Thurston is dead at age 65. Bill Thurston passed away yesterday at 8pm, succumbing to the cancer that he had been battling for the past two years. I don’t think it’s possible to overstate the revolutionary impact that he had on the study of geometry and topology. Almost everything we blog about here has the imprint of his amazing mathematics. Bill was always very generous with his ideas, and his presence in the community will be horribly missed. Perhaps I will have something more coherent to say later, but for now here are some links to remember him by: ## August 1, 2012 ### SnapPy 1.6: Now with more links and precision! Filed under: 3-manifolds,Computation and experiment,Hyperbolic geometry,Knot theory — Nathan Dunfield @ 8:37 am Marc Culler and I have released version 1.6 of SnapPy. There are two sets of new features: 1. Creating links formulaically, e.g. via combining tangles algebraically. See our page of examples. 2. Arbitrary precision calculation of certain things (e.g. tetrahedra shapes) and finding associated number fields, a la Snap. Very basic at this point compared to what Snap can to, but here are examples of what we have so far. To use this, you need to install SnapPy in Sage, which should be easy. Next Page » Theme: Rubric. Blog at WordPress.com.	2013-05-22 20:56:16	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 28, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4725610017776489, "perplexity": 911.5381756891387}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1368702447607/warc/CC-MAIN-20130516110727-00005-ip-10-60-113-184.ec2.internal.warc.gz"}
https://www.zbmath.org/?q=ci%3A2173308	## On the area of a polygon inscribed in a circle.(English)Zbl 1142.51016 If $$A$$ is the area of the cyclic $$n$$-gon with side-lengths $$a_1, \dots,a_n$$, and if $$t=16A^2$$, then $$t$$ is a zero of a polynomial $$F_n (T)$$ whose coefficients are symmetric polynomials in the $$a_i$$. A. F. Möbius investigated $$F_n$$ and found its degree in 1828. However, the first to explicitly write down $$F_n$$ for $$n=5$$ and 6 was the late D. P. Robbins in [Discrete Comput. Geom. 12, No. 2, 223–236 (1994; Zbl 0806.52008)]. More work was done on the polynomials $$F_n$$ by F. M. Maley, D. P. Robbins, and J. Roskies in [Adv. Appl. Math. 34, No. 4, 669–689 (2005; Zbl 1088.52005)] and by V. V. Varfolomeev in [Sb. Mat. 194, No. 3, 311–331 (2003; Zbl 1067.51013) and in Sb. Mat. 195, No. 2, 149–162 (2004; Zbl 1064.12001)]. A survey article is written by I. Pak in [Adv. Appl. Math. 34, No. 4, 690–696 (2005; Zbl 1088.52006)]. Unaware of these references, the authors of the paper under review prove that if $$n \geq 5$$, then there is no formula that expresses the area of a cyclic $$n$$-gon in terms of its side-lengths using only arithmetic operations and extracting $$k$$-th roots. They do this by considering the cyclic pentagon with side-lengths 1, 1, 2, 3, 4, writing down the polynomial that defines its area, and showing that its Galois group is the unsolvable group $$S_5$$. In other words, they prove that for the side-lengths 1, 1, 2, 3, 4, $$F_5$$ is not solvable. However, the paper is self-contained and does not make use of the expression of $$F_5$$ found by Robbins. Appendix A of the paper deals with conditions on the positive numbers $$a_1, \dots, a_n$$ that guarantee the existence of a (cyclic) $$n$$-gon whose side-lengths are these numbers. Here, the authors feel that their result is probably not new, but seem to be unaware of any references. This issue is indeed treated on p. 8 of [Z. A. Melzak’s, Invitation to Geometry. New York etc.: John Wiley & Sons, Inc. (1983; Zbl 0584.51001)], and a more rigorous treatment is given by I. Pinelis in [J. Geom. 82, No. 1–2, 156–171 (2005; Zbl 1080.52003)]. ### MSC: 51M25 Length, area and volume in real or complex geometry 52A35 Helly-type theorems and geometric transversal theory 51M04 Elementary problems in Euclidean geometries 51M05 Euclidean geometries (general) and generalizations 12F10 Separable extensions, Galois theory	2022-06-25 04:10:05	{"extraction_info": {"found_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": 0.6140246987342834, "perplexity": 311.5600718919959}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1656103034170.1/warc/CC-MAIN-20220625034751-20220625064751-00092.warc.gz"}
https://samjshah.com/tag/bloggingtwittering/	# New Year, New Blog With the start of a new year, there is no better time to start a new blog! For those of you who have blogs, it is also the perfect time to get inspired to write again! Please join us to participate in this years blogging initiative! To join, all you need to do is write just one post a week for the next four weeks. To make it easier for you, we will post a new prompt every Sunday! Once you have blogged, please fill out the form below. Each week, your blogs will be posted on this site for all to enjoy! This Week’s Theme: My Favorites This week, the blogging theme will be “My Favorites”, where you can post about one (or many) of your favorite things! Called a “My Favorite,” it can be something that makes teaching a specific math topic work really well. It does not have to be a lesson, but can be anything in teaching that you love! It can also be something that you have blogged or tweeted about before. Some ideas of favorites that have been shared are: • A lesson (or part of one) that went great • A game your students love to play • A fun and/or effective way to practice facts • A website or app you love to use in class • An organizational trick or tip that has been life changing • A product that you use in your classroom that you can’t live without! Blog Newbies! If you are brand new to blogging, you can read Starting A Blog from the 2015 initiative. This post will give you specific instructions on how to start a blog. The hardest part about blogging is often coming up with a title. Do not let this detail derail you! A great suggestion is to make your blog address your name. Then, you can title your blog later – or change the title anytime you want! To see what this looks like, check out Sam Shah’s blog. His web address is samjshah.com, but the site name is “Continuous Everywhere But Differentiable Nowhere“. No one cares about your blog name, they just want to read interesting, inspiring, and helpful posts! Hashtag it! #MTBoS #MtbosBlogsplosion Don’t forget to tweet out your blog link and add hashtags so other teachers in the MTBoS community can easily find your post! If you are not tweeting yet, you should be! There is an amazing community of math educators there just waiting to inspire and support you! Check out How To Start a Twitter Account to get started! Also, if you are brand new to Twitter or just want to get more out of it, there are more Twitter tips on Julie Reulbach’s blogpost, Tweet, Connect, Repeat. This year, we are joining up with the #mtbosblogsplosion. Special thanks to Carl Oliver@carloliwitter, for jump starting blogging for many people in our community! Also, if you have a wordpress blog, please re-blog this post to get the word out! Deadline: Press submit by the end of the day Saturday, January 7, 2017. Yes, this is a quick turn around this week – but we don’t want you to put it off or delay! Once you are finished with your blog post, fill out this form and your blog post will be featured on this site [meaning the MTBoS site this is reblogged from] next week! We are starting to gear up for TMC17, which will be at Holy Innocents’ Episcopal School in Atlanta, GA (map is here) from July 27-30, 2017. We are looking forward to a great event! Part of what makes TMC special is the wonderful presentations we have from math teachers who are facing the same challenges that we all are. To get an idea of what the community is interested in hearing about and/or learning about we set up a Google Doc (http://bit.ly/TMC17-1). It’s a GDoc for people to list their interests and someone who might be good to present that topic. The form is still open for editing, so if you have an idea of what you’d like to see someone else present as you’re writing your own proposal, feel free to add it! This conference is by teachers, for teachers. That means we need you to present. Yes, you! In the past everyone who submitted on time was accepted, however, this year we cannot guarantee that everyone who submits a proposal will be accepted. We do know that we need 10-12 morning sessions (these sessions are held 3 consecutive mornings for 2 hours each morning) and 12 sessions at each afternoon slot (12 half hour sessions that will be on Thursday, July 27 and 48 one hour sessions that will be either Thursday, July 27, Friday, July 28, or Saturday, July 29). That means we are looking for somewhere around 70 sessions for TMC17. What can you share that you do in your classroom that others can learn from? Presentations can be anything from a strategy you use to how you organize your entire curriculum. Anything someone has ever asked you about is something worth sharing. And that thing that no one has asked about but you wish they would? That’s worth sharing too. Once you’ve decided on a topic, come up with a title and description and submit the form. The description you submit now is the one that will go into the program, so make sure it is clear and enticing. Please make sure that people can tell the difference between your session and one that may be similar. For example, is your session an Intro to Desmos session or one for power users? This helps us build a better schedule and helps you pick the sessions that will be most helpful to you! If you have an idea for something short (between 5 and 15 minutes) to share, plan on doing a My Favorite. Those will be submitted at a later date. The deadline for submitting your TMC Speaker Proposal is January 16, 2017 at 11:59 pm Eastern time. This is a firm deadline since we will reserve spots for all presenters before we begin to open registration on February 1st. Team TMC17 – Lisa Henry, Lead Organizer, Mary Bourassa, Tina Cardone, James Cleveland, Daniel Forrester, Megan Hayes-Golding, Cortni Muir, Jami Packer, Sam Shah, and Glenn Waddell # #ExpandMTBoS At #TMC16, Tina C. and I led a session called “Breaking Out of Ourselves.” It was a small brainstorming session which started out with us presenting the ways that the online math teacher community (#MTBoS) has started expanding itself — followed by a call to action. Our presentation is here: The crux of the presentation is that we (not just Tina and me, but many in the #MTBoS) have done a lot to make the #MTBoS community more welcoming and accessible to newcomers (the ExploreMTBoS initiative and mentoring program, the mathtwitterblogosphere website).There are conferences (#TMC) and tweetups (all over the place). There is a #MTBoS booth that travels to various (often NCTM) conferences and is manned by #MTBoS participants, to spread the word! Other #MTBoS created things are available that are useful for teachers who don’t participate in the #MTBoS. There are books that have been written by #MTBoS-ers (e.g. Nix The Tricks, The Classroom Chef). There are website that are created by #MTBoS-ers and used by teachers everywhere (e.g. Visual Patterns, Which One Doesn’t Belong, Fraction Talks, Estimation 180, Would You RatherOpen Middle). There are podcasts (e.g. Tales from the Chalkline, Infinite Tangents). There are webinars and the Global Math Department Newsletter which rounds up and distills stuff from the community. There are a number of smaller #MTBoS intiative that have happened pretty organically: A Day in the Life initiative, and the Letters to a First Year Teacher initiative, and Virtual Conferences. And there were fun community building things, like Harlem Shake (Tweep Version) and Twittereen (and the now defunct, for those who remember, “Favorite Tweets”). All of this is to say: for those who are interested, there are many ways to help the community. You just have to find something you love about the #MTBoS, and then come up with a way to create/share/expand it with others. (That often involves breaking the idea into smaller chunks, getting other people on board to help, and actually holding each other accountable.) The #MTBoS doesn’t have a set of leaders. It only works because of the members. You don’t need to ask for permission. You don’t need to have been tweeting or blogging for months/years. You don’t need a “huge” project. You simply need to decide you want to do something, and do it. That is what our session was about. We shared some ideas that we had for places the community could grow, and ways people could actually do it, and then had people share their own thoughts and ideas. Personally, the projects I’d love to see someone take on: (1) Department presentations: I’m all about “packaging” something to make it easier for others to use. So I’d love for a group of people to create 3-4 “Introduction to the #MTBoS” presentations/workshops that math teachers can give to their departments. They can be different styles/lengths, and can have different activities involved. (For example, I made my whole department sign up for the GMD newsletter. At another presentation, I made a #MTBoS scavenger hunt, where different finds/activities were worth different points.) Then, anyone who wants can choose one and adapt it to make it work best when they want to evangelize the #MTBoS to their in-real-life colleagues! [Note: A number of #MTBoS presentations have been archived in the comments here.] (2) A #MTBoS video: I saw PCMI (a math teacher conference I’ve been to) created a video to “sell” the program. I would love it if there were a #MTBoS video which captured the essence of what the community is. Maybe 30-60 seconds. Something professional that evokes feelings and excitement, the emotional essence of #MTBoS, rather than outlining what it all has to offer… Capturing lighting in a bottle, that is what I suppose I’m asking for. But if this can be done well, well… I think it could serve our community well. (3) So you want to have a tweet up…: A number of people have held tweet-ups by now. I think it would be good if there could be “instructions on how to organize a tweet up” — from how to find people and contact them about attending to how to find a space to hold it to what to do at a tweet up. Again, perhaps two or three different “packages” for what tweetups could look like! This might make it easier for someone who might want to organize their own tweet up! (4) NCTM article: I’d love for someone to write an article about the #MTBoS community for Mathematics Teacher (or another NCTM journal) – to share what the community is about, how it has affected someone’s teaching practice, and to show ways for others who might be curious how to get involved. There is also a call for articles for the 2018 Focus Issue which is on Tool Kits for Early Career Teachers which I think a really wonderful article about #MTBoS could be beneficial. I wonder if two newbie #MTBoS-ers and two experienced #MTBoS-ers could collaborate on writing it! I am personally interested in having this happen because I think it is a way to spread the word through more traditional channels, and might just pique the interest of a lot of teachers! (5) Getting Goofy: In addition to things to expand the reach of the #MTBoS, I think there is room for so much more goofy things that can happen (today I saw a tweet that said #keepmtbosweird, copyright @rdkpickle). I don’t know what this might be, but some sort of goofy community building event like twittereen or the great hedgehog sweater run or needaredstamp. A massive picture-based scavenger hunt? A virtual trivia night? A stupid funny poster contest? (6) Appending #MTBoS to Existing Conferences: A number of people who are going to conferences (e.g. CMC south, Asilomar, NCTM) are planning 2-hour meet-ups with #MTBoS-ers. I think it could even be #MTBoS-ers arrive a day early or stay a day late and have a mini-get together (or even a super mini conference in the hotel!). I’d love a “package” that outlines how to organize one of these meet ups. (7) Get more contributors to the One Good Thing blog: I love the One Good Thing blog. I would love for there to be more regular and semi-regular contributors. The more voices we have when talking about the joys of teaching math, the better. It has helped me out so much during my saddest and most down days, when I open the blog and see old things I’ve written. And I love reading the joyous elations that other teachers have. I had one more idea that I have decided I am going to take on… For those who remember them… I am going to bring back Virtual Conferences. I loved the idea of them, and the person who hosted them is no longer doing them… so I’m going to bring them back from the dead! The ideas above are things I’ve been mulling over. The ideas that came up in our meeting, or on twitter afterwards (using hashtag #ExpandMTBoS) are below (in the pictures or in the storify): #ExpandMTBoS Storify These ideas include involving Reddit, making a landing page website/app, creating a MTBoS logo, having teachers tell more of their stories, etc. Choose something small, like presenting the community to your department or manning the #MTBoS booth at NCTM. Choose something huge, like creating your own conference, or website on (topic x), or writing a book. Or choose anything in between. But if you have the time and inclination, think of a way you can help #ExpandMTBoS! If you have an idea of something you want to do, tweet it out with the #ExpandMTBoS hashtag. Get people to help you! And make your idea a reality! # My Takeaways from #TMC16 I have all the feels, coming back from #TMC16, but they also have paralyzed me. There’s a disconnect between all the feels, my making sense of all the feels, and my ability to express all the feels in words. I felt paralyzed because I wanted to express things right. Since that was impossible, I did nothing. But to get past that, and because I need to collate the gems and thoughts from the conference to learn from them, this post is going to be a random collection of thoughts. It’s more for me — to consolidate my thinking and write down all the little things — so apologies if it feels like a confusing brain dump. ### What are you passionate about? Sara VanDerWerf (her blog) gave a keynote that was reminiscent of a keynote last year. She said “What are you an evangelist for?” (For her, one of those things is Desmos, because of the equity and access it allows her kids.) Once you know that thing — the thing you are willing to go to bat for, the thing you want to spread — you should think consciously about how to best evangelize it. That might include having an elevator speech ready for you to give, and being conscious of the different audiences you may be talking with about it (students? parents? teachers? admin?). Being an evangelist isn’t just being passionate… it includes enacting that passion by finding ways to share “the best… with others who can benefit.” Sara’s fabulous calculator museum (mausoleum!)… all your calculators are dead… all hail Desmos! I know I am an evangelist for the #MTBoS. However in terms of math content or math teaching, I don’t quite know what I’m an evangelist for… yet. All this reminds me the end of this blogpost I wrote last year after TMC, where I was trying to figure out what my “brand” was (and came up emptyhanded). But I have faith that with enough time, I’ll figure it out. Speaking of evangilism… Jonathan Claydon (his blog) shared a “my favorite” about Varsity Math, a community he’s built up at his school. I’ve had a teacher crush on this guy for years. There’s something about his energy and style and humor, and the fact that he is good at something I am not (yet) good at (being a “relational” teacher)… he’s a must follow. In any case, Jonathan is an evangelist for changing the way kids look at math at his school. Although ostensibly his goal is to increase the numbers of kids taking AP math classes and increase the AP scores of these students, he’s doing it by building a supportive math community — one that feels like a club. He is doing this by creating “shared experiences.” He knows he has succeeded if he can get kids to say “I love (varsity math). (Varsity math) feels like family. You couldn’t understand because you’re not in (varsity math).” The only way the last statement could make sense is if an entire culture is built around (varsity math). Of course what goes in the parentheses is open. Read about his project here. See a photo of @rawrdimus here: This “my favorite” spoke to me. I’ve been consciously working at my school about raising the math department. Not in terms of teaching and learning (I don’t have much say in that), but in terms of getting kids engaging with math outside of the math classroom. I brought the New York Math League contest to school, I’ve worked (with another teacher) to concertedly increase the number of students taking the American Math Competition each year (from around a dozen to seventy+). I found a non-stressful virtual math team competition that students can compete in so that they can fit it in their busy schedules. I have co-advised math club for years. I started Intersections with a science teacher, a math-science journal for students to submit their works to (it’s now four years old!). Lots of things… I want spaces and times for students to engage in math outside of the classroom. But with all of this, I don’t see a culture of kids who geek out about math. There isn’t a community or culture around doing mathematics at my school. And Jonathan’s talk helped me realize that I have to think intentionally about building a community. It is more than “if you create it, they will come.” It isn’t the event or space that I design, but the “shared experience.” What does this mean? What does this look like? I don’t know yet. But perhaps having a student-created chant before each virtual math team competition, bonding field trips (math movies? museum of math? math scavenger hunt?), swag as proud identifiers, a wall of fame… [Update: I was having trouble figuring out what precisely I want to accomplish in my school. And today is day 3 of a crazy math frenzy day where I’m having fun exploring and writing lesson plans and playing around and coding and getting stuck and getting unstuck and having frustration and elation — so much elation. And then I read this post by Annie Perkins, which talks about a sort-of-crisis I’m having (posts here and here). And in my current haze, I see the glimpses of what I want to achieve. Why do I want kids to engage with math outside of the classroom? Because it’s beautiful and fun to play with and just play mind-blowing cool. But they don’t get that in the classroom — at least not regularly enough. Jonathan created a community of kids who were vested in AP math. I think I want to figure out how to create a community of kids that love to (a) be exposed to interesting/strange things about math, and (b) play with math and explore it. Less “math team tricks” and “competition problem solving strategies” and more pure unadulterated fun. Things like this fold and cut problem that I did in geometry. Or generating and analyzing their own fractals. Taxicab geometry. And I think lecture might be okay for some of this — a lecture on infinity or Godel’s incompleteness theorem. Or following some internet instructions on how to build a planimeter out of a sodacan to calculate the area of a blob just by tracing around it. Or going as a group to a math lecture at the Museum of Math. Or learning about higher dimensions. Whatever! I want to get kids to geek out about how cool and fun math can be. I want a math is cool community, where there is a culture of nerd-sniping and geeking out and regular mind-blowing-ness. The truth is I probably don’t have time this year to come up with a plan to execute this to make it happen this year. I also think that the lack of free time that kids have in their schedules might make any plan of mine totally impossible. But I think it’s worth brainstorming… maybe not for this year… maybe for next year.] ### Desmos Features: At the Desmos preconference, I learned about three things (1) “Listening to graphs.” This feature was included for vision impaired students, but I think many of us teachers started dreaming up other uses for it. To get a sense of it, check out this piece (done by Rachel Kernodle and James Cleveland) playing “Mary Had A Little Lamb” (click on image): To play (at least on a mac), press COMMAND F5 (which enables voiceover), go to the fifth line and press OPTION T (to tell the computer to “read” the graph with sound), and then press H (to play the graph). When it’s done, you can turn off voiceover by pressing COMMAND F5 again. Some thoughts… Have the audio for some periodic and non periodic functions, and have kids do an audio function sort? Play audio of graphs (without telling kids that) and have kids do a notice/wonder (before sharing what they are listening to). Have kids identify if a graph has a horizontal asymptote for end behavior from an audio file? Have kids identify which graphs might have a vertical asymptote from an audiofile? Play sine and cosine (or secant and cosecant) and have kids not be able to tell which is which (because they are just horizontal shifts of each other). Have kids devise their own piecewise functions and play them, while other kids have to graph them. Create a piecewise function and have a student who enjoys singing to sing it? I am not convinced that anything I’ve thought up could help a deeper understanding of any topic, but I also don’t think it could hurt. Some kids might really get into it and enjoy playing with math… (2) Card Sort: You can create card sorts in desmos now! Check a bunch of them out (that were created at the Desmos pre-conference)! Or if you just want to go to one of them, click on the image of Mattie Baker’s card sort on visual sequences: To gain this functionality on your desmos account, go to teacher.desmos.com and click on your name in the upper right hand corner, click on LABS, and then turn on Card Sort. (3) Marbleslides: You can create your own marbleslides in desmos also! Turn it on in labs (see above). Then you have the capability of building your own! If you don’t know about marbleslides, check out this marbleslides activity made by the desmos folk on periodics. At least to me, the use of marbleslides is to help students understand function transformations… so I can see it useful for helping kids gain fluency in transformations. (Anyone see another use for marbleslides, that I’m missing?) ### Showing Student Work Hedge talked about how she uses SnagIt to display student work. She takes a photo of student work on her phone, and using an app called FUSE, transfers it to SnagIt (on the laptop) — as long as both are on the same wifi network. Here’s her blogpost showing it in action! It costs money (\$29.95) but I trust Hedge! I attended PCMI years ago, and I recall Bowen and Darryl using this technique (kids working on problems, taking pictures of different approaches) to facilitate discussion to bring different ideas together. Nearing the end of a session, they would project pictures of student work, people would explain their thinking. Bowen and Darryl would sequence the pictures in a thoughtful way. They wouldn’t focus on those who “got the answer” but on various approaches (visual/algebraic) — whether they worked to get the answer or not. I liked that so much, and I suspect SnagIt could allow that to work for me in that way. ### Getting Triggy With It! Hands On Trigonometry Fouss gave a wonderful hour long session on making trigonometry hands-on for students. Instead of telling us what she did, we got to do some of the activities, and that was powerful. There were activities I’ve read about that I thought “eh, okay, but it would be more efficient to do X, Y, and Z” and then I did them and I saw how the act of doing them could be helpful. Here are three that we got to do: understanding radians with smarties, creating a unit circle with patty paper, and creating a trig wheel to help kids practice converting between radians and degrees and visualize what the size of the angles look like. All her materials are linked to from her presentation, and are easily found on this folder on her google drive. I have to scour them to find my favorites. I did love the radian activity. If you make the radius of the unit circle 7 smarties long, then you can have a good discussion on whether 3 radians is 180 degrees or not… (21 smarties won’t quite make it to 180 degrees… but 22 smarties will fit snugly… nicely giving the 22/7 approximation for $\pi$. Nice!) Some of the ideas linked to from her presentations that I want to steal: (a) Trig Stations (b) Two Truths and a Lie (useful for more than just trig!) (c) #TrigIs (useful for more than just trig!) (d) If I choose to do ferris wheel problems, this ferris wheel comparison [but modified to be more challenging] (e) Desmos’s Polygraph for Sinusoids and Marbleslides for Periodic Functions (f) If I teach trig identities, use this matching game (and have kids check their answers once they are done by graphing on desmos!) (g) Headbandz, trig edition! (for graphing trig functions) ### Variable Analysis Game Joel Bezaire presented a great game that can be used in warmups to help students see relationships and patterns. His video on it is here, showing the game and how it is played: ### Nominations: Making Work Public Kathryn Belmont (@iisanumber) gave a great way to have kids really put forth effort on open-ended assignments without using grades as a stick. She will ask kids to do this assignments, and then put their work on their desks. Each student gets posts its, and as they wander around the room, they put post-its on the works they see… They write two accolades for good things, and two ways to push back or improve the assignment. The way I envision this in my classroom, not everyone will see everyone else’s work, but everyone will see 5-6 other students’s work. After the walk about, the teacher says: “Do you have any nominations”? Jake might reply “I would like to nominate Kiara.” If Kiara feels okay about being nominated and “accepts the nomination,” the teacher takes Kiara’s work and puts it under the document camera. Then Jake might say, “Kiara did … and what I thought was so awesome about it was …” (Her slides for her mini-talk are here. A video of her talk is here.) The teacher is no longer the sole audience member for the work, and kids are defining what good work looks like. In Kathryn’s classroom, she saw a huge increase in kids putting in effort in these open-ended assignments. (I can see this being useful in my own class, especially when I do my explore math mini-explorations.) ### Intentional Talk I went to a session by Jessica Breur (@BreurBreur) which was fantastic. Although it was only one hour, I wish it were a morning session. She wants to have teachers establish a culture where students: • use the group to move the group forward • talk, trust, and depend on classmates and the teacher • persist — even in the face of a challenge • view math as “figure-out-able” and accessible to all She highly recommended Cohen’s Designing Groupwork (a book which I have but haven’t read). To start, over the first week or two, students will be doing lots of groupwork activities. And at the end of them, they will (in their smaller groups) focus on what the group “looks like” “sounds like” and “feels like.” They don’t necessarily need to focus on all three at once — students could focus on “sounds like” during one activity and “feels like” on another. After the week is done, the class comes up with a set of norms in these three categories — where they talk about what successful/good/fun groups look/sound/feel like. We did a lot of hands-on work trying out some of these groupwork activities — and she has included all of those activities in her slides. Here is one of my favorites: This is the red solo cup challenge. A group of 3 or 4 is given 6 red solo cups, stacked inside each other, placed face up on the table (so like a regular drinking up face up). The students are given a rubber band with four strings tied to it (even if 3 students are doing this, keep the four strings). Student must put the solo cups in a pyramid formation. If they finish that, there are other configurations that Jessica includes in her presentations (or students can design their own challenge for others!). Afterwards, the group reflects. Similar tasks can be done, like 100 NumbersSaving Sam, Four 4s [but making an emphasis that we want as many ways to generate the numbers 1-20, not just one for each], Master Designer, or Draw My Picture. For more “math-y” things, you can do a Chalk Talk/Graffiti Board– where students answer questions before a unit to activate some old ideas. For example, “What do you know about the number zero?” [In fact, any sort of talking point/debate-y statement can be used here.] Kids write anything and everything they know on a poster in their group of four. Then hand the posters up and students walk around and read other students’ responses (if time, writing their own comments down). Finally, for closure, you can ask students aloud or using exit slips “What are two things you didn’t think about that you saw on the graffiti boards?” Another more math-y thing is a donut percent task. An example is here but I’m confident it could be modified for trigonometry (values of trig functions, identities, etc.) or rational functions (equations and graphs) or any number of things! The idea behinds this is that each person in the group is given four slips of paper, and as a group, four complete donuts have to be created. Sounds simple? But here’s the rub… group members must follow the rules below to each get their own donut completed. You should keep a poster of the 8 Standards of Mathematical Practice, and every so often during activities or groupwork, ask students which ones they are using. Once norms are established at the start of the year, you consciously need to be doing activities that practice the norms. Be intentional about it. (If you find that kids aren’t listening to each other, find an activity that promotes listening.) I loved this session. However what I need now are a set of activity structures that I can fit actual mathematical work into. So things which develop understanding, or practice solving something, etc. And it would be nice not only to have the activity structures, but the activities themselves all in one place (so, for example, activities for Precalculus!). ### Talk in the Math Classroom My morning session was called “Talk Less, Smile More” and was led by Mattie Baker and Chris Luzniak. In the session, they provided various structures to promote math talk in the classroom. I am going to outline some of the ideas that I can see myself using in my classroom. DEFENSE MECHANISMS & CLASSROOM CULTURE: Most importantly, to get talk in the math classroom involves getting over student defense mechanisms. Students fear being seen as stupid, and they fear being wrong. In order to do this, you have to lower the stakes so kids can temporarily bracket their defense mechanisms to create emotional safety. These could be by doing things like chalk talks (silently writing responses to questions, and responding to other student responses) or doing notice/wonder activities where all responses are honored. Many of the ideas that Chris and Mattie shared in the session do this, by providing a structure for talking, and a bit of a safety net (often where no response is right, or students are required to give a particular answer and justify it). When implementing it, you have to be consistent and do these structures fairly often. Start simple, and then get more complicated with the statements/questions. Give a lot of energy and excitement — especially if a student gives a wrong answer or a right answer (“Oh wow, what an interesting thought… let’s explore that…”). If students turn to the teacher and say “Mr. Shah, what about…” sit down and redirect it to the class. (Remember the teacher is not the center… this is about getting kids to be the center!) As teachers, we have to watch our own facial expressions (a.k.a. don’t make a face when you hear a totally wrong answer). You can avoid this (if it’s a problem for you) by looking down at a clipboard when someone is responding. At the end of a class or a portion of a class with a lot of mathematical talk, do “shout outs” (shout out something they learned, or something someone else said that helped them). And ask kids (to fill out on a card) what they took away from class today (and what questions they still might have). Or “I used to think ____, but now I think _____.” To give students some crutches when talking, have posters with these simple statement starters to help them (on all four walls): TALKING POINTS: In this session I first got to experience Talking Points. I’ve read about them on Elizabeth Statmore’s blog (see links on the right… a bunch of talking points are hosted in one of her google drive folders). But the truth is: I wasn’t sure how much I could get out of them. Now that I’ve participated in one, I feel differently. This is how they work: (1) students in a group of 4 get n statements. The first round involves one person reading the first statement, and then say “agree/disagree/unsure” and then explain why they chose that response. They must give the reason. The next person does the same, then the next, then the last. The important part about this is that no one can comment on another person’s reasons. They can just state their own reasons. They can match someone else’s reasons, but they have to be stated as their own. (2) The second round involves the first person saying “agree/disagree/unsure” (after hearing everyone else’s thoughts) and then they can give reasons involving other people’s thoughts. Others do the same. (3) The third round is quick and short. Each person says “agree/disagree/unsure” and gives no reasons. Then someone records the tally of the responses. Here’s an example of what talking points can look like (when they aren’t about math content): Talking points can also be math content related. Instead of “agree/disagree/unsure,” you can use “always/sometimes/never” or some variation that works for your questions. In our mini-precalculus group, we brainstormed some talking points around trigonometry: After participating in talking points, we as a group came to the following realizations: • Talking points were not as repetitive as we thought they would be. • The more controversial a statement, the more discussion happens. • You were really forced to listen to each other • When the talking point includes “I” statements, you learn about other group members • They are good for pre-assessments (and can be used before a unit starts, as a prelude) • Give n statements, and then leave 3 blank statements. If a group finishes early, they can write their own talking point statements! • Afterwards, you should have a “shout out” round. Kids should shout out something interesting/great they learned, and/or the teacher should shout out something good they heard/witnessed! To debrief: • Don’t go over all of the questions. That debrief will feel boring and repetative. Go over some key things you want to talk about immediately, and then revisit the others during the unit. (You want to make sure that kids don’t leave the unit with misconceptions.) • Use the tally of A/D/U or A/S/N to see where the controversy lies! (You can collect their slips and talk about them later after seeing their responses…) CLAIM AND WARRANT DEBATE: In a math class, you want students to justify themselves. To build that justification as central to the class, you can introduce the notions of an argument which is essentially a statement (a claim) made with sound reasoning (a warrant). (This language comes out of the speech and debate world.) When responding to a question, a student must stand up (even the teacher should sit down) and say “My claim is _______, and my warrant is ________.” If the student messes up, that’s okay, just have them do it again. You have to build this structure as essential to answering questions. (To reduce the fear, you can give students some think time to write something down, or talk in a pair, before doing the claim/warrant step.) When doing this, I am not going to have kids volunteer… I am going to cold call using the Popsicle Sticks of Destiny (names of kids on popsicle sticks… I draw one randomly…). When introducing claim/warrant, make sure you not only teach the structure, but also have kids who aren’t speaking face the speaker and put their eyes on them. Be explicit about the expectation. You can also have kids summarize another student’s point to make sure they’re paying attention. (If you catch a kid not following the audience instructions, you can walk over near them… if not, you can tap them on the shoulder… or kindly talk with them after class about how “it’s really polite to…”) To build this up and create this as a routine and class structure, you should do claim and warrant debates every day or every other day at the start of the school year. Use the language “claim” and “warrant” on assessments too! Types of questions you can ask to get kids started with this: The best movie is ______. The most important math topic is ______. ________ is the best method for solving the system y=2x and y=x+1. [show a Which One Does Belong and say] ______ doesn’t belong. Notice that each of these don’t have a “right” answer. It lowers the barrier of entry for kids. One powerful type of question one can create are “mistake” questions. For example: To extend claim/warrant, you can also create “circle debates” which truly forces listening. One person states a claim/warrant, and then another person summarizes that claim/warrant and then makes their own claim/warrant. This continues. It will sound like: “What I heard is that this statement is sometimes true because …. My claim is ____ and my warrant is ____.” I think only very open ended questions would be good for this structure. Another powerful way to extend claim/warrant is to engage is “point-counterpoint.” Let’s say the statement is: “Would you rather have crayons for teeth or spaghetti for hair?” The first person makes a claim/warrant, and the second person (no matter their true feelings) must disagree and make the opposing claim and give a warrant. Then the third person opposes the second person. Etc. It forces students to think of other points of view. In a question like “_____ is the best way to solve this system of equations” it forces students who might only approach a system in one way to consider other methods and justifications for those other methods. CREATING DEBATE-Y QUESTIONS/STATEMENTS: Use the following words: In the session, we took all types of questions (e.g. Graph $y=8\sin(2x-4)+1$) and came up with debate-y questions based on it (in this example, we said “what number would you change to change the graph the most?” or “what’s the best way to graph a sine function?”). I’m not yet good at this, but I found that even with a little practice and people to bounce ideas around, I’m getting better. We had fun in my group trying to come up with debate-y questions based on this random “do now” that Chris and Mattie found online: I thought it would be impossible, but the group came up with tons of different ways to convert this to a debate-y statement: (a) without solving, which is easiest to solve? (b) which would you give to your worst enemy? (c) which are similar? (d) rank from easiest to hardest? (e) a 5th problem that would fit this set of equations would be ____ (f) a 5th problem that would not fit this set of equations would be ______ (g) which one doesn’t belong? (h) give -4(x+3)=-6 and ask what the most efficient way tot solve it? and then follow up with “how could you change the problem so that method is not the most efficient?” After a month or two, the use of claim/warrant may die down. If kids get the idea and are justifying their statements, that’s okay! It’s not about the structure as much as the idea behind the structure! QUICKWRITE: I love this idea because I make writing integral to my classroom. You give kids a prompt and you tell kids to write nonstop for 2 minutes without editing. They have to continually write. Examples: It can help with vocabulary, but most importantly, I see this as a way to get kids to stop overthinking and looking for “the right” answer, and just write down anything and everything without self-editing of their thinking. It’s like a condensed noticing/wondering done individually. I can be used before a debate — to give kids time to think. Or perhaps depending on the question, kids can “shout out” one part of their quickwrite? But doing it at the start of the year — to help kids get comfortable writing in math class in an non-threatening, non-evaluative manner — is such a great idea! RUMORS: This idea was stolen from Rona Bondi at all-ed.org. On a notecard/paper, everyone write a response to a question or a couple questions (the one we used is “what is our idea setup of our classroom?” but I think it could be used at the end of class with questions like: “One thing I find easy to understand in this unit is… One question I still have about this stuff we’ve been working on…?” or “The most important mathematical idea from today is …?” or “The best way to approach graphing trig functions is…”). After everyone is done writing, everyone finds a parter and reads their card, the other person reads their card, and then they discuss. There is a time limit (maybe 60 seconds). Then they swap papers. Everyone finds a second partner, and they read the card in their hand to the other person, and they discuss what is written on those cards (not their own cards) and then swap. This goes on three or four times. This forces listening, it allows ideas to slowly spread, and the papers can be kept anonymous. ONE INTENTIONAL MISTAKE: [update: a la Kelly O’Shea] Each group of students gets a giant whiteboard and a problem (it could be the same problem as other groups or a different problem). They are asked to solve the problem making one “good” mistake (so nothing like spelling names wrong, transposing a number, or labeling the axes wrong). They then present their solution to another group — playing dumb about their mistake. The other group should ask good questions to help students get at the error. Questions like “don’t you need to add 3 to both sides” is too direct… You need to ask questions which lead the group to see and understand the mistake. So perhaps “what is the mathematical step you used to get from line 2 to line 3, and why is it justified?” might be better. # Interested in Presenting at TMC16? We are starting to gear up for TMC16, which will be at Augsburg College in Minneapolis, MN (map is here) from July 16-19, 2016. We are looking forward to a great event! Part of what makes TMC special is the wonderful presentations we have from math teachers who are facing the same challenges that we all are. To get an idea of what the community is interested in hearing about and/or learning about we set up a Google Doc (http://bit.ly/TMC16-1). It’s a GDoc for people to list their interests and someone who might be good to present that topic. The form is still open for editing, so if you have an idea of what you’d like to see someone else present as you’re writing your own proposal, feel free to add it! This conference is by teachers, for teachers. That means we need you to present. Yes, you! In the past everyone who submitted on time was accepted, however, this year we cannot guarantee that everyone who submits a proposal will be accepted. We do know that we need 10-12 morning sessions (these sessions are held 3 consecutive mornings for 2 hours each morning) and 12 sessions at each afternoon slot (12 half hour sessions that will be on Saturday, July 16 and 48 one hour sessions that will be either Saturday, July 16, Sunday, July 17, or Monday, July 18). That means we are looking for somewhere around 70 sessions for TMC16. What can you share that you do in your classroom that others can learn from? Presentations can be anything from a strategy you use to how you organize your entire curriculum. Anything someone has ever asked you about is something worth sharing. And that thing that no one has asked about but you wish they would? That’s worth sharing too. Once you’ve decided on a topic, come up with a title and description and submit the form. The description you submit now is the one that will go into the program, so make sure it is clear and enticing. Please make sure that people can tell the difference between your session and one that may be similar. For example, is your session an Intro to Desmos session or one for power users? This helps us build a better schedule and helps you pick the sessions that will be most helpful to you! If you have an idea for something short (between 5 and 15 minutes) to share, plan on doing a My Favorite. Those will be submitted at a later date. The deadline for submitting your TMC Speaker Proposal is January 18, 2016 at 11:59 pm Eastern time. This is a firm deadline since we will reserve spots for all presenters before we begin to open registration on February 1st. Team TMC – Lisa Henry, Lead Organizer, Mary Bourassa, Tina Cardone, James Cleveland, Cortni Muir, Jami Packer, Megan Schmidt, Sam Shah, Christopher Smith, and Glenn Waddell # Jump in the online math teaching community! A number of years ago, I had the idea of starting a little program to help those interested in starting a blog do so. And we’ve had some fun variations on a theme in the past few years. Right now, we’re launching it again … but with an awesome twist! There are going to be two things going on simultaneously. Those who are comfortable blogging and tweeting, we’d love for you to sign up to be a mentor for someone just dipping their feet into the online math teacher world! You will be a person that newbies can ask questions to, connect ’em with people and blogs they might find interesting, and be a cheerleader as they get involved. Those who are new are going to have someone help you out. You will be able to have a trusted person to ask questions to, help you find things that will be interesting to you, and encourage you. And through this, you’ll get to see if the online math teacher community has anything to offer that you want. You’ll get to dip your feet in, with no pressure, and a lot of support! In December, we’ll pair up mentors and mentees. And during that time, we’ll all work on introducing those new to the online math teacher world to what we have to offer. In January, we’ll have a 4 week “blogging challenge,” with prompts for both new and experienced bloggers. If you’re interested in finding out more, or you’re ready to sign up to be a mentor or to get your feet wet checking out the online math teacher world (known as #MTBoS which is the unwieldy acronym for mathtwitterblogosphere), check out the exploremtbos website. Huzzah! # The betterQs blog: A new #MTBoS adventure For the past few years, I’ve been (sometimes daily, sometimes sporadically) posting on the one good thing blog. Last year I did it every single day. Often times it was a short post, especially in tough days where it was hard to find some little nugget. But what I loved was that it made me reflect consciously on joy and goodness, and pay attention to it. [1] This year I want to spend some time thinking about how to question well. More specifically, thinking intentionally about what questioning looks like (and how it can be improved) in my classroom — both on my end and on my students’ end. I thought I would blog about it throughout the year, and figured it would be fun to blog with others. @rdkpickle had the same idea! So we figured it was a good idea, and set up a collaborative blog. All this is to say: But more importantly…You are warmly and heartily welcome to join us, and become an author. The blog just started and we’d love to get as many voices and experiences going on the ground floor. Read a few posts. Browse a bit. It’s only a few days old, so there isn’t too much to gander at! And consider joining us. (If you want, there’s a tab at the top of the blog that tells you how to join, or just click here. We’ll add you as an official author!) “But Sam,” you say, “I don’t have time to write every day…” Silly goose, I respond! You can write however frequently works for you. Once a week? Once a month? Three times a year? The point is to take some time — however much of it — to think about questioning in your classroom. “But Sam,” you say, “I don’t have a lot to write about…” Silly turkey, I shoot back! I think it would be cool if you even wrote down a single question that you really loved asking because it provoked discussion. No need to deeply analyze it if you don’t want! Maybe a teacher reading the blog will read that question and think: “YAS! THIS IS EXACTLY THE QUESTION I NEEDED!” And if there were a lot of people just throwing down their good thought-provoking questions, we would soon have an amazing repository. “But Sam,” you say, “I have a blog of my own! Why don’t I just post it there?” Silly quail, I reply! You can post anything to do with questioning both on your own blog, and on this blog. No rule against that! In fact, I did that for my first post on the betterQs blog. And that way, someone reading the betterQs blog might get to know you and your own blog! “But Sam,” you say, “I’m still scared… I don’t want to sign up and then not do it.” Silly emu, I say. Why not take a baby step and just commit yourself to writing one or two things? Just keep a lookout in your school about how you question, or try to script a good question and see how it goes in your classroom, or rewrite a test question and explain how you rewrote it and why… Baby emu steps. And just see how it goes! You just might think: hey, questioning is something I want to pay just a bit more attention to! Or, silly emu, don’t worry about signing up! As I wrote a couple years ago: “You should never feel guilty engaging with the community in ways that make sense to you. We’re all coming at teaching from such different places in our careers, such different backgrounds, and such different environments. We all need and want different things.” In other words, you do you. [1] I also love the fact that because I’ve been using the blog semi-regularly, I can see an archive of so many good things of my own (in addition to seeing everyone else’s good things). On down days, it really helps me remember I’m not as bad as my brain tries to convince me I am.	2017-09-26 00:10:28	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.22930385172367096, "perplexity": 1811.4944585252083}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1505818693866.86/warc/CC-MAIN-20170925235144-20170926015144-00249.warc.gz"}
https://semodi.github.io/nyc-neighborhoods/	# So you want to move to …? Finding (Pareto) optimal neighborhoods in NYC ## Introduction Finding a place to live is never easy, this is especially true in a mega-city like New York. The options seem endless, and different trade-offs need to be considered when hunting for apartments: Do I want to live closer to my workplace but pay a higher rent, or do I possibly want to move to a quieter residential neighborhood sacrificing valuable time during my commute? When it comes to choosing the right place to live, every individual will have different priorities, so there is no “one size fits all” solution. In terms of formal decision-making theory, this problem can be cast as a multi-objective or Pareto optimization. As the name suggests, rather than optimizing a variable (in this case the neighborhood) to minimize a single cost-function (i.e. rent) one wants to simultaneously optimize several cost-functions. While not entirely accurate in the mathematical sense I will refer to these cost-functions loosely as metrics in the remainder of this report and will specify the metrics used in the following section. The goal of this report will be to find Pareto efficient (PE) (aka Pareto optimal) NYC Neighborhoods. The easiest way to understand Pareto efficiency is in terms of a negative example: Let us assume we want to find a neighborhood that is both cheap and safe. The metrics are therefore median rent and crime rate. If a neighborhood is not Pareto efficient, we can always find a different neighborhood that improves at least one of the metrics while not impairing any other ones. Conversely, if this is not possible, the neighborhood is called Pareto efficient. Figure 1. Pareto efficiency for synthetic dataset. The point highlighted in green is not PE as both Cost 1 and Cost 2 can be decreased by moving along the path indicated by the arrow. The benefits of this approach are clear, as simply finding the “best” neighborhood would require one to specify the relative importance of rent and crime rate. This relative importance, however, is highly subjective and will differ from person to person. Pareto efficiency serves as an objective tool to help the apartment-seeker find their best fit. Based on their personal preferences, they can go through the list of PE neighborhoods and choose the one closest to their liking. It should be noted that, from an optimization standpoint, it does not make sense to pick a non-PE neighborhood. ## Data The metrics and associated data used in this report are the following 1. Safety To calculate the average safety of a given neighborhood I will combine data on the number of arrests made and the number of shootings. Both datasets are availabe at NYC Open Data. Ideally the crime rate would be normalized by the population of a neighborhoood. Unfortunately, census data was not readily availabe on a NYC neighborhood level, and data that was available was both outdated and rather coarse-grained. I have therefore chosen to normalize crime rate by the area of a given neighborhood. While not ideal, the area can serve as a proxy for population counts. 2. Rent To gauge rent prices in every neighborhood, I decided to analyze the median rent for a one bedroom apartment. The data was obtained from StreetEasy 3. Venue Density Using the Foursquare API, in particular the “explore” endpoint, one can estimate the venue density in a given neighborhood. I will define venue density as the number of venues returned by Foursquare in a 500m radius around the neighborhood center. 4. Distance from Subway Combining location data on subway entrances with NYC geodata one can determine the average distance to the closest subway entrance for each neighborhood. 5. Distance from Midtown Given the shorter commute time, it might be desirable for some people to live as close as possible to their workplace. Both, the Financial District, Midtown Manhattan are the centers of economic activity in New York. I have chosen the latter to calculate this distance metric. Datasets only cover the year of 2019. Due to the lack of data regarding Staten Island, I disregarded the borough in the final analysis. ## Methods ### NYC Open Data queries Where possible, I obtained data from NYC Open Data through their SODA API. This API allows for filtering, querying and aggregating data using a syntax reminiscient of SQL. For example to obtain data on all shootings that occured in 2019 one would use the following request: https://data.cityofnewyork.us/resource/833y-fsy8.json?$where=occur_date between '2019-01-01T00:00:00.000' and '2019-12-31T00:00:00.000' limit 1000000 ### Safety To calculate the safety metric I combined data on arrests and shootings. Type Total number in 2019 Arrests 214617 Shootings 1716 Along with the location where the incident occured, the arrests data contained information about the type of felony/misdemeanor that was observed. Using a geometric algorithm (ray-casting) I determined which neighborhoods the incidents, both in the case of arrests and shootings, occured in. To get an accurate treatment of distances, the coordinates provided as longitude and latitude first had to be multiplied by appropriate prefactors as so: $d(A,B) = \sqrt{0.52^2(A_{long} - B_{long})^2 + 0.69^2 (A_{lat} - B_{lat})^2 }$ This linear approximation to distances on a sphere works well for the small angles we are interested in. I expressed incident counts for each neighborhood as multiples of the average amount of incidents in all of NYC and normalized them by the neighborhood area. I then combined together data on arrests and shootings. As the normalization occured before the datasets were combined, more weight was automatically given to shootings. The final metric for safety is expressed as a multiple of the crime rate across all of NYC. The following map shows this metric on a log-2 scale: ### Rent Obtaining rent data was challenging as no complete, openly avalaible datasets were obtainable online. The following data from StreetEasy proved to be the most comprehensive, but many neighborhoods are still missing as indicated by the black areas: I decided to fill in the gaps with a predictive model. At this point I also made the decision to disregard Staten Island in the final analysis as data was simply not sufficient to draw reliable conclusions regarding this borough. Several ideas come to mind when trying to build a predictive model for rent prices. An important factor that determines rent prices is certainly location. One can therefore expect that a k-nearest-neighbor regression model will be able to predict rent prices by using information about adjacent neighborhoods. The model clearly becomes less reliable in regions of high rent and performs reasonably well in the 1500 to 2500 dollar regime. Another approach is to predict rent prices from past sales prices. This is possible as we can assume that both are correlated. Moreover sales prices are available for many neighborhoods for which rent prices are not. The data was fitted using Ridge Regression with Polynomial Features up to order 2. The$R^2\$ metric indicates that this model performes better than the one using nearest neighbors. However, sales data is not available for every neighborhood, so the final model will need to be able to take into account other kinds of input as well. For some neighborhoods where rent prices for 2019 are missing, historic prices are still available. For these neighborhoods one can simply model the average increase in rent over the years to get an estimate of current rent prices. Note that rent prices were shifted to zero mean. Among the three models introduced above, only the KNN algorithm can be used for every neighborhood, as both historic data as well as sales data is incomplete. However, it seems that valuable information is contained in those datasets, so we should make use of them wherever possible. The solution to this is to build an ensemble model that takes the predicted prices from the above models as input and outputs one combined price. As model, I used a Random Forest Regressor for its ability to handle missing data. To simulate missing historic and sales data I applied a random mask to both training and test input. The model performs better than the nearest neighbor regressor by itself, and due to the way it was fitted, can be applied to neighborhoods where historic and sales data is either present, partially complete or completely missing. Finally using the ensemble estimator, I can impute missing rent prices into the original dataset: Using my domain knowledge as a NYC resident, I can affirm that none of the imputed prices seem non-sensical. ### Venue Density Venue density in a given neighborhood was estimated as the amount of venues returned by the Foursquare API in a 500m radius around the neighborhood center. ### Distance from Subway To calculate the average distance to the closest subway station for every neighborhood I generated a regular grid across NYC. For every grid point, I determined the distance to the closest subway station. Using a ray casting algorithm, I determined whether a given grid point was inside the polygon spanned by the neighborhood borders. Averaging over grid points in every neighborhood I create the “Distance from Subway” metric. ### Distance from Midtown As a proxy for “Distance from Midtown”, I calculated the distance between every (geometric) neighborhood center and Times Square. ## Results For visualization purposes, I will start by only considering the Pareto frontier for a subset of the metrics introduced above. The following graphic shows neighborhoods that are Pareto optimal with respect to rent, crime rate and average rent. Using all five metrics metioned in the “Data” section, one can measure the relative amount of Pareto optimal neighborhoods by borough: The clear winners are Brooklyn and the Bronx, followed by Manhattan and Queens in last place. What is every boroughs strongest asset? Looking only at Pareto optimal neighborhoods, one can calculate average value for every metric. For visualization purposes, I have normalized every metric to lie between zero and one, with zero being “worst” and one meaning “best”. Not surprisingly, Manhattan is a clear winner when it comes to venue density and distance to Midtown (which, of course, is in Manhattan). With its many residential neighborhoods in the East, Queens can be considered the safest borough (not everywhere!) followed by Brooklyn. Anyone who lives in the Bronx can attest to its unbeatable rent prices. In terms of safety, the Bronx ties with Manhattan. ## Discussion Using a combination of five different metrics and datasets from various sources, I have identified Pareto optimal neighborhoods in New York City. Of course, many factors contribute to an individual’s decision where to live and my set of metrics is certainly far from exhaustive. Possible other factors that could be included in a more detailed or personalized analysis are: Proximity to one’s workplace, proximity to parks/beaches, noise level, air pollution, quality of public schools and many more. Nevertheless, my analysis will hopefully help apartment seekers find their ideal neighborhood. Maybe I can even convince people already living in NYC to upgrade their living situation by moving from a non-PE to a PE neighborhood. I have created a small web-app that, given a NYC neighborhood it will determine whether it is PE and if not recommend PE alternatives. Give it a try: (the app might take a moment to spin up as it is currently hosted on a free Heroku instance)	2021-04-11 13:39:07	{"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": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5134606957435608, "perplexity": 1314.7916019445067}, "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-2021-17/segments/1618038062492.5/warc/CC-MAIN-20210411115126-20210411145126-00354.warc.gz"}
https://techutils.in/blog/tag/confidence-interval/	## #StackBounty: #bayesian #confidence-interval #frequentist #credible-interval When does a confidence interval "make sense" but… ### Bounty: 50 It is often the case that a confidence interval with 95% coverage is very similar to a credible interval that contains 95% of the posterior density. This happens when the prior is uniform or near uniform in the latter case. Thus a confidence interval can often be used to approximate a credible interval and vice versa. Importantly, we can conclude from this that the much maligned misinterpretation of a confidence interval as a credible interval has little to no practical importance for many simple use cases. There are a number of examples out there of cases where this does not happen, however they all seem to be cherrypicked by proponents of Bayesian stats in an attempt to prove there is something wrong with the frequentist approach. In these examples, we see the confidence interval contains impossible values, etc which is supposed to show that they are nonsense. I don’t want to go back over those examples, or a philosophical discussion of Bayesian vs Frequentist. I am just looking for examples of the opposite. Are there any cases where the confidence and credible intervals are substantially different, and the interval provided by the confidence procedure is clearly superior? Get this bounty!!! ## Context This is somewhat similar to this question, but I do not think it is an exact duplicate. When you look for how instructions on how to perform a bootstrap hypothesis test, it is usually stated that it is fine to use the empirical distribution for confidence intervals but that you need to correctly bootstrap from the distribution under the null hypothesis to get a p-value. As an example, see the accepted answer to this question. A general search on the internet mostly seems to turn up similar answers. The reason for not using a p-value based on the empirical distribution is that most of the time we do not have translation invariance. ## Example Let me give a short example. We have a coin and we want to do an one-sided test to see if the frequency of heads is larger than 0.5 We perform $$n = 20$$ trials and get $$k = 14$$ heads. The true p-value for this test would be $$p = 0.058$$. On the other hand if we bootstrap our 14 out of 20 heads, we effectively sample from the binomial distribution with $$n = 20$$ and $$p = frac{14}{20}=0.7$$. Shifting this distribution by subtracting 0.2 we will get a barely significant result when testing our observed value of 0.7 against the obtained empirical distribution. In this case the discrepancy is very small, but it gets larger when the success rate we test against gets close to 1. ## Question Now let me come to the real point of my question: the very same defect also holds for confidence intervals. In fact, if a confidence interval has the stated confidence level $$alpha$$ then the confidence interval not containing the parameter under the null hypothesis is equivalent to rejecting the null hypothesis at a significance level of $$1- alpha$$. Why is it that the confidence intervals based upon the empirical distribution are widely accepted and the p-value not? Is there a deeper reason or are people just not as conservative with confidence intervals? In this answer Peter Dalgaard gives an answer that seems to agree with my argument. He says: There’s nothing particularly wrong about this line of reasoning, or at least not (much) worse than the calculation of CI. Where is the (much) coming from? It implies that generating p-values that way is slightly worse, but does not elaborate on the point. ## Final thoughts Also in An Introduction to the Bootstrap by Efron and Tibshirani they dedicate a lot of space to the confidence intervals but not to p-values unless they are generated under a proper null hypothesis distribution, with the exception of one throwaway line about the general equivalence of confidence intervals and p-values in the chapter about permutation testing. Let us also come back to the first question I linked. I agree with the answer by Michael Chernick, but again he also argues that both confidence intervals and p-values based on the empirical bootstrap distribution are equally unreliable in some scenarios. It does not explain why you find many people telling you that the intervals are ok, but the p-values are not. Get this bounty!!! ### Bounty: 50 Suppose $$Xsim N_3(0,Sigma)$$, where $$Sigma=begin{pmatrix}1&rho&rho^2\rho&1&rho\rho^2&rho&1end{pmatrix}$$. On the basis of one observation $$x=(x_1,x_2,x_3)’$$, I have to obtain a confidence interval for $$rho$$ with confidence coefficient $$1-alpha$$. We know that $$X’Sigma^{-1}Xsim chi^2_3$$. So expanding the quadratic form, I get $$x’Sigma^{-1}x=frac{1}{1-rho^2}left[x_1^2+(1+rho^2)x_2^2+x_3^2-2rho(x_1x_2+x_2x_3)right]$$ To use this as a pivot for a two-sided C.I with confidence level $$1-alpha$$, I setup $$chi^2_{1-alpha/2,3}le x’Sigma^{-1}xle chi^2_{alpha/2,3}$$ I get two inequalities of the form $$g_1(rho)le 0$$ and $$g_2(rho)ge 0$$, where $$g_1(rho)=(x_2^2+chi^2_{alpha/2,3})rho^2-2(x_1x_2+x_2x_3)rho+x_1^2+x_2^2+x_3^2-chi^2_{alpha/2,3}$$ and $$g_2(rho)=(x_2^2+chi^2_{1-alpha/2,3})rho^2-2(x_1x_2+x_2x_3)rho+x_1^2+x_2^2+x_3^2-chi^2_{1-alpha/2,3}$$ Am I right in considering a both-sided C.I.? After solving the quadratics in $$rho$$, I am guessing that the resulting C.I would be quite complicated. Another suitable pivot is $$frac{mathbf1′ x}{sqrt{mathbf1’Sigma mathbf 1}}sim N(0,1)quad,,,mathbf1=(1,1,1)’$$ With $$bar x=frac{1}{3}sum x_i$$, this is same as saying $$frac{3bar x}{sqrt{3+4rho+2rho^2}}sim N(0,1)$$ Using this, I start with the inequality $$left\|frac{3bar x}{sqrt{3+4rho+2rho^2}}right\|le z_{alpha/2}$$ Therefore, $$frac{9bar x^2}{3+4rho+2rho^2}le z^2_{alpha/2}implies 2(rho+1)^2+1ge frac{9bar x^2}{z^2_{alpha/2}}$$ That is, $$rhoge sqrt{frac{9bar x^2}{2z^2_{alpha/2}}-frac{1}{2}}-1$$ Since the question asks for any confidence interval, there are a number of options available here. I could have also squared the standard normal pivot to get a similar answer in terms of $$chi^2_1$$ fractiles. I am quite sure that both methods I used are valid but I am not certain whether the resulting C.I. is a valid one. I am also interested in other ways to find a confidence interval here. Get this bounty!!! ### Bounty: 50 Suppose $$Xsim N_3(0,Sigma)$$, where $$Sigma=begin{pmatrix}1&rho&rho^2\rho&1&rho\rho^2&rho&1end{pmatrix}$$. On the basis of one observation $$x=(x_1,x_2,x_3)’$$, I have to obtain a confidence interval for $$rho$$ with confidence coefficient $$1-alpha$$. We know that $$X’Sigma^{-1}Xsim chi^2_3$$. So expanding the quadratic form, I get $$x’Sigma^{-1}x=frac{1}{1-rho^2}left[x_1^2+(1+rho^2)x_2^2+x_3^2-2rho(x_1x_2+x_2x_3)right]$$ To use this as a pivot for a two-sided C.I with confidence level $$1-alpha$$, I setup $$chi^2_{1-alpha/2,3}le x’Sigma^{-1}xle chi^2_{alpha/2,3}$$ I get two inequalities of the form $$g_1(rho)le 0$$ and $$g_2(rho)ge 0$$, where $$g_1(rho)=(x_2^2+chi^2_{alpha/2,3})rho^2-2(x_1x_2+x_2x_3)rho+x_1^2+x_2^2+x_3^2-chi^2_{alpha/2,3}$$ and $$g_2(rho)=(x_2^2+chi^2_{1-alpha/2,3})rho^2-2(x_1x_2+x_2x_3)rho+x_1^2+x_2^2+x_3^2-chi^2_{1-alpha/2,3}$$ Am I right in considering a both-sided C.I.? After solving the quadratics in $$rho$$, I am guessing that the resulting C.I would be quite complicated. Another suitable pivot is $$frac{mathbf1′ x}{sqrt{mathbf1’Sigma mathbf 1}}sim N(0,1)quad,,,mathbf1=(1,1,1)’$$ With $$bar x=frac{1}{3}sum x_i$$, this is same as saying $$frac{3bar x}{sqrt{3+4rho+2rho^2}}sim N(0,1)$$ Using this, I start with the inequality $$left\|frac{3bar x}{sqrt{3+4rho+2rho^2}}right\|le z_{alpha/2}$$ Therefore, $$frac{9bar x^2}{3+4rho+2rho^2}le z^2_{alpha/2}implies 2(rho+1)^2+1ge frac{9bar x^2}{z^2_{alpha/2}}$$ That is, $$rhoge sqrt{frac{9bar x^2}{2z^2_{alpha/2}}-frac{1}{2}}-1$$ Since the question asks for any confidence interval, there are a number of options available here. I could have also squared the standard normal pivot to get a similar answer in terms of $$chi^2_1$$ fractiles. I am quite sure that both methods I used are valid but I am not certain whether the resulting C.I. is a valid one. I am also interested in other ways to find a confidence interval here. Get this bounty!!! ### Bounty: 50 Suppose $$Xsim N_3(0,Sigma)$$, where $$Sigma=begin{pmatrix}1&rho&rho^2\rho&1&rho\rho^2&rho&1end{pmatrix}$$. On the basis of one observation $$x=(x_1,x_2,x_3)’$$, I have to obtain a confidence interval for $$rho$$ with confidence coefficient $$1-alpha$$. We know that $$X’Sigma^{-1}Xsim chi^2_3$$. So expanding the quadratic form, I get $$x’Sigma^{-1}x=frac{1}{1-rho^2}left[x_1^2+(1+rho^2)x_2^2+x_3^2-2rho(x_1x_2+x_2x_3)right]$$ To use this as a pivot for a two-sided C.I with confidence level $$1-alpha$$, I setup $$chi^2_{1-alpha/2,3}le x’Sigma^{-1}xle chi^2_{alpha/2,3}$$ I get two inequalities of the form $$g_1(rho)le 0$$ and $$g_2(rho)ge 0$$, where $$g_1(rho)=(x_2^2+chi^2_{alpha/2,3})rho^2-2(x_1x_2+x_2x_3)rho+x_1^2+x_2^2+x_3^2-chi^2_{alpha/2,3}$$ and $$g_2(rho)=(x_2^2+chi^2_{1-alpha/2,3})rho^2-2(x_1x_2+x_2x_3)rho+x_1^2+x_2^2+x_3^2-chi^2_{1-alpha/2,3}$$ Am I right in considering a both-sided C.I.? After solving the quadratics in $$rho$$, I am guessing that the resulting C.I would be quite complicated. Another suitable pivot is $$frac{mathbf1′ x}{sqrt{mathbf1’Sigma mathbf 1}}sim N(0,1)quad,,,mathbf1=(1,1,1)’$$ With $$bar x=frac{1}{3}sum x_i$$, this is same as saying $$frac{3bar x}{sqrt{3+4rho+2rho^2}}sim N(0,1)$$ Using this, I start with the inequality $$left\|frac{3bar x}{sqrt{3+4rho+2rho^2}}right\|le z_{alpha/2}$$ Therefore, $$frac{9bar x^2}{3+4rho+2rho^2}le z^2_{alpha/2}implies 2(rho+1)^2+1ge frac{9bar x^2}{z^2_{alpha/2}}$$ That is, $$rhoge sqrt{frac{9bar x^2}{2z^2_{alpha/2}}-frac{1}{2}}-1$$ Since the question asks for any confidence interval, there are a number of options available here. I could have also squared the standard normal pivot to get a similar answer in terms of $$chi^2_1$$ fractiles. I am quite sure that both methods I used are valid but I am not certain whether the resulting C.I. is a valid one. I am also interested in other ways to find a confidence interval here. Get this bounty!!! ### Bounty: 50 Suppose $$Xsim N_3(0,Sigma)$$, where $$Sigma=begin{pmatrix}1&rho&rho^2\rho&1&rho\rho^2&rho&1end{pmatrix}$$. On the basis of one observation $$x=(x_1,x_2,x_3)’$$, I have to obtain a confidence interval for $$rho$$ with confidence coefficient $$1-alpha$$. We know that $$X’Sigma^{-1}Xsim chi^2_3$$. So expanding the quadratic form, I get $$x’Sigma^{-1}x=frac{1}{1-rho^2}left[x_1^2+(1+rho^2)x_2^2+x_3^2-2rho(x_1x_2+x_2x_3)right]$$ To use this as a pivot for a two-sided C.I with confidence level $$1-alpha$$, I setup $$chi^2_{1-alpha/2,3}le x’Sigma^{-1}xle chi^2_{alpha/2,3}$$ I get two inequalities of the form $$g_1(rho)le 0$$ and $$g_2(rho)ge 0$$, where $$g_1(rho)=(x_2^2+chi^2_{alpha/2,3})rho^2-2(x_1x_2+x_2x_3)rho+x_1^2+x_2^2+x_3^2-chi^2_{alpha/2,3}$$ and $$g_2(rho)=(x_2^2+chi^2_{1-alpha/2,3})rho^2-2(x_1x_2+x_2x_3)rho+x_1^2+x_2^2+x_3^2-chi^2_{1-alpha/2,3}$$ Am I right in considering a both-sided C.I.? After solving the quadratics in $$rho$$, I am guessing that the resulting C.I would be quite complicated. Another suitable pivot is $$frac{mathbf1′ x}{sqrt{mathbf1’Sigma mathbf 1}}sim N(0,1)quad,,,mathbf1=(1,1,1)’$$ With $$bar x=frac{1}{3}sum x_i$$, this is same as saying $$frac{3bar x}{sqrt{3+4rho+2rho^2}}sim N(0,1)$$ Using this, I start with the inequality $$left\|frac{3bar x}{sqrt{3+4rho+2rho^2}}right\|le z_{alpha/2}$$ Therefore, $$frac{9bar x^2}{3+4rho+2rho^2}le z^2_{alpha/2}implies 2(rho+1)^2+1ge frac{9bar x^2}{z^2_{alpha/2}}$$ That is, $$rhoge sqrt{frac{9bar x^2}{2z^2_{alpha/2}}-frac{1}{2}}-1$$ Since the question asks for any confidence interval, there are a number of options available here. I could have also squared the standard normal pivot to get a similar answer in terms of $$chi^2_1$$ fractiles. I am quite sure that both methods I used are valid but I am not certain whether the resulting C.I. is a valid one. I am also interested in other ways to find a confidence interval here. Get this bounty!!!	2019-08-18 02:51:38	{"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": 175, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8098755478858948, "perplexity": 250.13356029829743}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027313589.19/warc/CC-MAIN-20190818022816-20190818044816-00018.warc.gz"}
http://tibasicdev.wikidot.com/sinh	The sinh( Command Command Summary Calculates the hyperbolic sine of a value. Command Syntax sinh(value) The sinh( command is only found in the Catalog. Press: 1. 2nd CATALOG to access the command catalog. 3. Scroll down and select sinh(. TI-83/84/+/SE 1 byte Calculates the hyperbolic sine of a value. The hyperbolic trig functions sinh(, cosh(, and tanh( are an analog of normal trig functions, but for a hyperbola, rather than a circle. They can be expressed in terms of real powers of e, and don't depend on the Degree or Radian mode setting. sinh(0) 0 sinh(1) 1.175201194 Like normal trig commands, sinh( works on lists as well, but not on complex numbers, even though the function is often extended to the complex numbers in mathematics. # Formulas The definition of hyperbolic sine is: (1) \begin{align} \sinh{x}=\frac{e^x-e^{-x}}{2} \end{align} .	2014-08-28 09:21:57	{"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": 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": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9678364992141724, "perplexity": 2211.4515247373074}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500830721.40/warc/CC-MAIN-20140820021350-00438-ip-10-180-136-8.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/conservation-law-using-killing-vector.244024/	# Conservation law using Killing vector 1. Jul 8, 2008 ### stephenmitten In Hartle's GR book (p. 177), there is a derivation of $$\xi \cdot u = constant$$, where $$\xi$$ is a Killing vector, $$u$$ is four-velocity along a geodesic in an arbitrary metric, and $$L = (-g_{\alpha\beta}\frac{dx^\alpha}{d\sigma}\frac{dx^\beta}{d\sigma})^\frac{1}{2}$$ The derivation goes: $$\frac{\partial}{\partial \sigma}\frac{\partial L}{\partial \frac{dx^1}{d\sigma}}} = 0 \\ \Rightarrow \frac{\partial L}{\partial \frac{dx^1}{d\sigma}} = -g_{1\beta}\frac{1}{L}\frac{dx^\beta}{d\sigma} = ... = -\xi \cdot u$$ is conserved along the geodesic. (Here the symmetry associated with $$\xi$$ is in $$x^1$$.) It seems to be saying that $$\frac{\partial L}{\partial \frac{dx^1}{d\sigma}} = \frac{1}{2L}({-g_{\alpha 1}\frac{1}{L}\frac{dx^\alpha}{d\sigma}-g_{1\beta}\frac{1}{L}\frac{dx^\beta}{d\sigma}) = {-g_{1\beta}\frac{1}{L}\frac{dx^\beta}{d\sigma}$$ but it appears to me that $$\frac{\partial L}{\partial \frac{dx^1}{d\sigma}}$$ has only seven terms, not eight, since $$-g_{11}\frac{dx^1}{d\sigma}}\frac{dx^1}{d\sigma}}$$ appears only once. I'd appreciate it if someone could point out where I went wrong. Thanks. Last edited: Jul 8, 2008	2018-04-25 09:02:33	{"extraction_info": {"found_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": 0.932105541229248, "perplexity": 437.97682478210527}, "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-2018-17/segments/1524125947759.42/warc/CC-MAIN-20180425080837-20180425100837-00001.warc.gz"}
https://cstwiki.wtb.tue.nl/index.php?title=Embedded_Motion_Control_2013_Group_3	# Embedded Motion Control 2013 Group 3 ## Contact info Vissers, Yorrick (YO) 0619897 y.vissers@student.tue.nl Wanders, Matthijs (MA) 0620608 m.wanders@student.tue.nl Gruntjens, Koen (KO) 0760934 k.g.j.gruntjens@student.tue.nl Bouazzaoui, Hicham (HI) 0831797 h.e.bouazzaoui@student.tue.nl Zhu ,Yifan (YI) 0828010 y.zhu@student.tue.nl ## Meeting hours Mondays 11:00 --> 17:00 Wednesdays 8:45 --> 10:30 Thursday 9:00 --> 10.00 Testing on Pico Meet with tutor: Mondays at 14:00 ## Planning Ma 09 sept: • Finish installation of everything • Go through ROS (beginner) and C++ manual Wo 11 sept: • Finish ROS, C++ manuals • Start thinking about function architecture Ma 16 sept: • Design architecture • Functionality division Do 19 sept: • Finish "state stop" (Koen) • Finish "drive_parallel" (Matthijs, Yorrick) • Creating a new "maze/corridor" in Gazebo (Yifan) • Simulate and build the total code using Gazebo (Hicham) • Testing with robot at 13:00-14:00 Vr 20 sept: • Finish "crash_avoidance" • Coding "gap_detection" (Yifan) • Coding "maze_finished" (Koen, Hicham) Ma 23 sept: • Finish "drive_parallel" • Putting things together • Testing with robot at 12:00-13:00 (Failed due to network down) Unfortunately we encountered some major problems with the Pico robot due to a failing network. We discussed the approach for the corridor competition. At this point the robot is able to drive parallel through the corridor and can look for gaps either left or right. We a gap is reached the robot will make a smooth circle through the gap. This is all tested and simulated. For the corridor competition we will not check for death ends. There isn't enough time to implement this function before Wednesday 25 September. The corresponding actions such as "turn around" won't be finished either. Without these functions we should be alble to pass the corridor competition successfully. Di 24 sept: • Testing with robot at 13:00-14:00 • Finding proper parameters for each condition and state Wo 25 sept: • Finish clean_rotation • Finish gap_handling • Putting things together • Corridor Challenge Week 5: • Design and simulate code • Create a structure "laser_data" which processes the data of the laser. This contains the calculation of (Ma + Yo): • Shortest distance to the wall • Distance to the right wall with respect to Pico • Distance to the left wall with respect to Pico • Edit/improve state "drive_parallel" with a feedback controller according to the angle with respect to the right wall (Ma + Yo). • Create the condition "dead_end". When a dead end is detected switch to the state "turn_around" (Yi + Ko). • Edit/improve the condition "gap_detect_left/right". Make this condition more robust (Yi + Ko). • Start researching the properties of the camera of Pico (Hi). • Testing on Pico (Listed by priorities) 1) Test robustness of the feedback controller and gap_detection 2) Tune the translational and rotational speed (faster) 4) Test strange intersections Week 6: • Design and simulate code • Edit/improve state "drive_parallel" with a feedback controller according to the angle with respect to the right wall (Ma + Yo). • Create the condition "finished". This is the condition when Pico exists the maze. Switch to state "finalize" (Yi + Ko). • Improve the priorities list. Pico needs to make smart decisions when multiply conditions occur. Basically the priorities are listed as (Ma + Yo): 1) Turn right 2) Go straight 3) Turn left • Continue researching the properties of the camera of Pico (Hi) • Testing on Pico (Listed by priorities) 1) Test the state "turn_around" 2) Test the condition "finished" 3) Building a small maze containing a t-intersection and a dead end and test the priorities (Can also be done properly during simulation) Week 7: • Design and simulate code • Edit/improve the state "reset". This states handles the condition when a object is detected close to Pico. • Imaging processing. Detect arrows on the wall (Hi + Yi). • Testing on Pico (Listed by priorities) 1) Build a maze and solve it without using camera. 2) Test camera 3) Test the improved states/conditions. See progress->week6->Testing Week 8: • Design and simulate code • Finalize the code • Implementing the camera node • Testing on Pico (Listed by priorities) 1) Build a maze and solve it using camera. 2) Test camera • Fill in peer assessment (Sending to v.d. Molengraft by Koen) • Upload a video of the Gazebo demo (Yifan) • Updating the wiki: Condition/Node State Name crash_avoidance state_reset Yorrick & Matthijs maze_finished state_finalize Koen maze_exit state_exit Koen gap_detection_right state_gap_handling Yifan gap_detection_left state_gap_handling Yifan go_straight_priority state_drive_parallel Yorrick & Matthijs Camera / Hicham ## Progress Week 1 (All) • Installed and setup all the software • Went through the ros beginner tutorials and c++ tutorials Week 2 (All) • Finishing up the tutorials that weren't yet completed • Initial architecture design: condition and state functions. • The main architectural idea is to keep a clear distinction between conditions and states. • Conditions: In what kind of situation is pico? • State: How should pico act to his current situation? Week 3 (All) • Continued architecture design, and begin of implementation. • Simulation with drive safe example. (MA & YO) • Created state_drive_parallel function (YI) • Created custom mazes in Gazebo (KO) • Created state_stop function Testing on Pico: (All) • First time testing on Pico. Just figuring everything out. Week 4 (MA & YO) • Worked on gap handling • Tuning state_drive_parallel (KO & YI & HI) • Worked on gap detection (All) • Simulating corridor left and right Testing on Pico: (All) • After some tuning drive_parallel worked well. • After some tuning gap_detection worked well (we learned to remove the first and last 20 points that reflect on Pico). • We tuned the radius of gap_handling down. In practise we want pico to take slightly smaller corners. • We tuned down the cornering speed a bit to be very stable in every type of corridor and gap width (from about 60cm to 1m+) • We are ready for the corridor competition. Week 5 (MA & YO) • Removed scanning functionality from the states and conditions and put it in a seperate function that returns a struct "Laser_data" containing info about wall locations and scanner range/resolution etc. Did some tuning on the gap_detection algorithm to make it more robust and made some minor changes on the decision tree. (now only checks nect condition if previous one returns false). • Improved the decision tree to satisfy the required priorities for a wall follower strategy. The priorities for navigation are: 1) Take right gaps 2) Drive par 3) Take left turns. In order to only take left turns when there is nothing interesting ahead or to the right a new condition is introduced: go_straight_priority. When this condition returns false it is ok to take a left turn should there be a gap. • Further improvements (w.r.t robustness) on gap_detection and dead_end_detection. Due to the new decision tree "number_of_spins" are now counted inside of the states (gap_handling), rather than inside the conditions. Testing on Pico: (MA & YO & YI) : We learned that when there's a gap further ahead, Pico's laser doesn't see a clear jump in distance, but instead see several points (reflection?) within the gap. This is taken into account when designing for robustness. • At the time of testing the priorities were not working as intended for the abovementioned reason, this has been improved later that day. Will be tested again next week. • Speeds are tuned for now to x_vel = 0.2 in state_drive_par and z_rot = 0.15 in state_gap_handling and state_turn_around. • Dead_end_detection worked as intended when there was a dead end, but sometimes also detected a dead end when there was none. This has been improved that same day. Also the distance when this condition checks in front has been improved. Will be tested again next week. • state_turn_around was already finished ahead of scedule. Therefore it was already tested on Pico this week and is working as intended. • feedback control in state_drive_parallel was not implemented yet, and therefore is also not tested and tuned. This will be done next testing session. Week 6 (KO) • Modified the condition "maze_finished". Pico will first check in front is there is anything closer then 2 meter (variable). When this is not the case it will continue checking a range of approximately 150 degrees ahead for obstacles. When there is still nothing closer then 2 meters, Pico is done! • When the condition "maze_finished" is satisfied Pico will be in the state "finalize". It will move with a constant translation velocity of 0.3 [m/s] until anyone hit the emergency button. (HI & KO & YO & YI) • Worked on improving gap handling. We implemented 'flexible' corners: when Pico starts to go around a corner, his current orientation is taken into account when determining the counter k that represents the total number of spins pico should rotate. In addition a constant tuning factor is included in k, which helps in simulation. This factor might be retuned when testing this friday. (HI & YO) • Worked on implementing proportional feedback in state_drive_parallel for the purpose of faster settling times (reach setpoint faster) and more stable behaviour while driving straight. We implemented a PD controller that is tuned to work well in the simulation and friday this will be re-tuned to work well on Pico. (HI & YO) • Created a new node for the camera functionality: camera.cpp. This node subscribes to the camera and calls "camera_controller.h" when data is published to the camera topic. The camera controller will process the camera data and detect arrows, which can be published by the camera node to our main node theseus. Hicham will work on processing the camera data and arrow detection. • Note that CMakeLists will now also make an executable of camera.cpp. In addition manifest.xml now also depends on the opencv packages. In order to run both nodes use: rosrun theseus2 theseus camera Testing on pico: (KO & YI & YO) • Things which went good: • Driving parallel to the right wall. We improved/tuned the control parameters on Pico. • Turn around after detecting a dead end. • Gap handling went good after tuning some parameters. • Things to do: • Improve robustness of detecting gaps (either right and left). • Improve exit the maze. It makes a corner to the right when he leaves the maze. • Check this again next week when the other states/conditions are improved. (YO) • In a t-split where pico had an orientation that was slightly rotated to the right (negative theta) he detects the gap on the left first and started cornering to the left. Therefore we split the counter variable number_of_spins into number_of_spins_left and number_of_spins_right. This helps to prevent errors when go_straight priority is not working correctly, because the right cornering takes over. • When testing on Pico we found that when taking a wide corner, after finishing the corner, he would detect another corner if pico was not yet in the next corridor. This is now prevented by a new function reset_corner that requires pico to see a small corridor part before allowing a new corner in the same direction. Week 7 (KO & YO & MA) • Worked on a reset state, that is triggered after crash_avoidance. The reset state should help Pico to continue after a near crash, rather than the previous state stop, that would just end the fun. (YO & MA) • Worked again on robustness of dead_end_detection and gap_handling. Just trying to prevent as many possible problems. (YI & HI) • Worked on the camera functionality. (KO) • Started preparing the presentation. (All) • Merging the camera and theseus nodes Testing on pico (All) • Things that went well • Reset corner works as intended and solves the double corner problem. • Dead end detection seems to work well after tuning. • After tuning, our crash_avoidance + state_reset combination works well in several tests: wall in front, wall to the side. But possibly if pico ends up in a corner, there may be a problem (test again next week). • Gap handling and drive parallel worked fine. • Things to improve • Camera should work from a greater distance. We need to detect arrows sooner in order to make the correct decision. • Perhaps if we detect an arrowshape ahead, we should lower the speed of drive_parallel? Yi & Hi should judge if this may help. • Exit maze still didn't work as we designed it. Very low priority, since pico exits the maze fully which is acceptable in the challenge. With people standing around outside of the maze, the exit maze will nog work anyway. Week 8 Testing on Pico (All) • The camera worked from a bigger distance. Our camera node detected the arrows in time. • There was an error in the communication between our nodes. The bool arrow_left was not published to the topic. After a while we found out we used ros::Spin() instead of ros::SpinOnce(), which prevented communication. Unfortunately this cost us a lot of time during testing. • When this was solved Pico reacted correctly on an arrow. But the reset condition for the arrow detection was very bad (onrobust). The idea was to publish a bool from theseus to the camera node when a corner is finished. (All) • Implemented a more robust reset condition for arrow detection. When a corner is finished the arrow detection is reset. Unfortunately there was no time to test this anymore. • We decided it wasn't necessary to lower the speed if a corner is ahead, because we detect the arrows in time now. (KO) • Created and prepared the presentation. He practiced once for the group, and group gave feedback for improvement. In the end we wanted to emphasize two things in the presentations: • Clear split between conditions (information processing) and conditions (how should pico actuate). • Iterative approach of testing and tuning (and therefore seeing how each change to the code works). ## PICO usefull info • minimal angle = 2,35739 rad • maximal angle = -2,35739 rad • angle increment = 0,00436554 rad • scan.range.size() = 1081 We have defined the following conventions: • We use a right turning orthogonal base w.r.t Pico: • the pos x-axis is pointing forward w.r.t Pico • the pos y axis is pointing to the left w.r.t Pico • the pos z axis is then pointing up w.r.t.Pico • the pos z rotation is then CCW ## Design approach Our approach for this project is to start from a very simple working example. From this starting point an iterative approach of adding, extending or improving functionality and then testing and tuning the new functionality right away. The benefit of this approach is getting instant feedback of the functionality: do the new functions work as intended, are they tuned well (both for simulation and pico), and do they work together with the other parts? Our ambition is to create effective functionality that is challenging out our own level, yet understandable for everybody within the group, rather than trying to sound intensely intelligent with convoluted third party algorithms. Discussing the ideas before implementation and testing after implementation is key in our approach. Some of these concepts are explained in the video on this page: [1] ## Maze solving strategy For navigating through the maze we use a right wall follower strategy. This means that the robot will always stick to the right wall and always find it's way to the exit. As a result we get the following priorities: • 1) Take right turns (if they are there). • 2) Go straight. • 3) Go left (we only go left if we can't go straight or to the right, this is achieved with the go_straight_priority function). But there's a few exceptions to these priorities: • If a gap is a dead end really quickly, we detect that, and skip it. • If there is an arrow to the left detected by the camera node, we prioritize left and act on it as soon as the gap to the left is detected. ## Software architecture • We use one package named Theseus2 • We use a node theseus (20 Hz) that subscribes to pico/laser and pico/cam_data and publishes velocities. • The main architectural idea has been to keep a clear distinction between conditions and states. • Conditions: In what kind of situation is pico? • State: How should pico act to his current situation? ### Theseus node When pico/laser publishes data, theseus_controller is called. theseus_controller contains our main functionality. 1) Process the laserdata for future use. 2) Gather conditions (each conditions is explained elsewhere on the wiki) 3) Based on the conditions a state will be chosen and executed. In some situations (f.e gap_handling) the previous state is also considered to determine the current state (Markov chain principle) 4) The chosen state will set the velocities that can be published by the node. The figure below depicts the simplified architecture of the program at this point. #### General files • datastructures.h: all global variables and datastructures are gathered here. • send_velocities.h: this functions is used for theseus to publish the velocities. • laser_processor.h: this function pre-processes the laserdata for future use. ### Camera node We use a second node camera that spins on 5 Hz, subscribes to pico/camera and puslishes bool arrowleft to the topic pico/cam_data. Because of our right wall follower strategy, we only publish data of left arrows. When pico/camera publishes something camera_controller is called which processes the images and when an arrow to the left is detected sets arrow_left to true. camera_controller contains our main functionality. The method of detecting arrows is explained elsewhere. ## Conditions/States ### Crash and reset functions condition_crash_avoidance: Pico is not allowed to hit the walls, so there when Pico gets too close to the wall, it should stop driving. This is carried out in the next loop: • If the shortest distance from the wall to Pico is less than 32 cm (to make sure rotating doesn't run Pico's rearside into the wall), Pico will go into state_reset. state_reset: This state is being called when Pico is about to crash. After waiting approximately a second to see if the problem might have resolved itself, it will go into the loop depicted in the following picture: In words: Pico scans the laser data to determine the index of the shortest distance. This index is then used to determine what the best course of action is to make sure pica can continue. If the shortest distance is to the left or right, Pico will rotate away from the wall until going forward will increase the shortest distance, such that pico will go away from the wall. If the shortest distance is in front, Pico will drive backwards until a certain threshold is met, and then continue with drive_parallel. ### Maze exit funtions condition_maze_finished: When Pico exists the maze the condition "Finished" is reached. The following steps are carried out to check if Pico behaves in this condition: • Read out the laser data from -135 to +135 degrees w.r.t the front of Pico • Check if all distances are larger than 1.5 meters. When this is the case. Maze finished! state_final: State Final is called when Pico satisfies the condition Finalize. This state will ensure that Pico will stop moving by sending 0 to the wheels for translation and rotation. condition_maze_exit: Pico is always looking for the exit of the maze. This will be done in two steps: 1) Check if the shortest distance, within a range of +- 5 degrees w.r.t. the front of Pico, is larger then 1.5 meters 2) Check if the shortest distance, within a range of 145 degrees w.r.t. the walls, is larger then 1.5 meters. When both steps are true, Pico is at the end of the maze a must switch to the state "exit". We use two steps just to reduce The steps are depicted in the figure below. state_exit: This state ensures that Pico is driving straight out of the maze with a constant velocity. ### Gap detection and Handling functions condition_gap_detection_right: During the going straight state, if there exists a gap to the rightt side, the condition "Gap detection right" is reached. The following steps are carried out to check if Pico behaves in this condition: • Pico will continue checking the laser date of right side between upper and lower index (variables) during going straight. • If the distance within the upper and lower index is larger than the product of sensitivity (variable) and set point ($\displaystyle{ laser.scan.ranges[i] \gt sensitivity * setpoint }$), a internal counter will start to count. • As soon as the counter reaches 35 AND Boolean right_corner_finished is FALSE (this condition will be discussed later), it's for sure that there is a gap to the right side. Pico will jump to state "gap handling" and start to turn. condition_gap_detection_left: This conditions is the same as gap detection right, but obviously for the left side. state_gap_handling: When gap handling is entered, a 90 degrees rotation to the left or right (based on the gap_detection conditions) is made. The overall shortest distance of the laser is used as a radius for the corner. The relation: x_vel = radius * z_rot is used to make a nice corner. When Pico might come close to a wall the radius will be smaller, so also the corner Pico makes will be smaller, which helps in the situation of a small gap in a wide corridor. At the same time a minimum radius is set to ensure that pico will not make the corner too small. When gap handling is entered, we consider Pico's current orientation (theta) and use this to enforce Pico to keep rotating for 90 degrees + theta. This helps to make corners a bit more flexible and ensures that pico will come out of gap_handling with a good orientation. This gives a good performance should several corners come after eachother. condition_reset_corner: This condition is used to ensure that Pico will not turn twice at the same corner. The following steps are carried out to check if Pico behaves in this condition: • After gap_handling finished, Boolean right_corner_finished or left_corner_finished (depends on which turn Pico just takes) will be set to TRUE. • Pico will start to look at the laser range at right or left side, as soon as the distance at it's right or left side is smaller than sensitivity (which is set to 0.8 meter in practice), a counter will start to count. • When the counter reaches 30, the Boolean right_corner_finished or left_corner_finished will be set to FALSE. This means that the corner is fully finished, and Pico is now entering a new corridor. If everything goes well, this will take exactly 1.5 seconds since the spin frequency of the laser is 20 Hz. ### Priorities and driving functions condition go_straight_priority: This condition is necessary because of the right wall follower strategy: when there is a gap to the left, you might want to go straight instead if there is something interesting to the front or right side. In that case go straight priority will return true. Otherwise when there's nothing ahead or to the right (only walls), go straight priority returns false and it's ok to take a left turn. state_drive_parallel: Drive parallel is the state that allows Pico to drive between two walls. In this state a setpoint is calculated as the middle of the closest point to the left and right of pico. On top of that we define the deviation from this setpoint as the error (positive error = right of setpoint), and the current rotation as theta (positive theta = CCW, parallel to wall theta = 0), both relative to closest point left and right. By feeding back the error and theta as the rotation velocity we keep Pico in the middle of corridors which has the benefit that per default it tries not to hit any obstacles. We do this by setting a constant translation velocity, and a PD controlled rotation velocity that feeds back both the error and theta (P-action) and their derivatives (D-action); $\displaystyle{ velocity.Zrot = f(error,derror, theta, dtheta) = kpe * error + kde * derror - kpt * theta - kdt * dtheta }$ When Pico facing a dead end, the condition "Dead end" is reached. The following steps are carried out to check if Pico behaves in this condition: • Check if the condition "Dead end" is active. If active, keeps Pico turning around until Pico facing backward to the dead end. • If the condition "Dead end" is not active, read out the laser data from -5 to +5 degree w.r.t the Pico heading direction. • Check if the distance from -5 to +5 degree are shorter than sensitivity times setpoint to see if there is a dead end in the front. When this is the case, set Boolean "dead_end_infront" to TRUE. • When "dead_end_infront" is TRUE, calculate the differences of two nearby laser distance from -90 to +90 degree with respect to the robot. • When all the difference is less than 0.2 meters, Pico reaches dead end, jump to state "Turn around". state_turn_around: Pico is switched to this state when a dead end is detected. When Pico enters this state he will make a clean rotation on the spot of 180 degrees and is kept in this state untill finished rotating. Turning Pico is done by a simple open loop function. An angular velocity is send to the wheels and for a pre-calculated number of iterations corresponding to 180 degrees. We don't use any feedback to compensate slip and friction etc. In terms of accuracy this will not be the best solution but the small error due to slip and friction will be handled by the state drive_parallel which does contain a feedback controller. ### Camera node functions Arrow detection: At a junction there is a possibility that an arrow is located at the wall. Pico can detect this arrow by using it's camera. When Pico detects the arrow it has to determine whether the arrows points right or left and either turn right or left. Camera_controller: This function contains the code for detecting the arrow and determine whether the arrow is left (or not right). To assure that we don't detect lines or red pixels outside of the corridors of the maze we have bounded Pico's camera view. This is shown in the below picture. In order to detect arrows we first want to detect horizontal lines. After detecting the lines we divided the arrow in a left and a right part. To determine whether an arrow is pointed to the left or right we loop over the red pixels in the left and right part. Because of the shape of the arrow, the red pixels are not equally divided. By counting the red pixels we determine the arrow direction, and we start at the left point of the detected line. However because the camera spins 5 times per second, we get each time different amount of red pixel on the left and right side. To make the code more robuust we let the loop over the red pixels several times and if it detects an arrow to the left for example more than three times, we set the boolean arrow left to true! Function description: - Loop over left and right part of arrow and start at left point of the detected line. - Count the red pixels. - Compare the amount of red pixels on left and right side. - If amount of red pixels on the left is greater than on the right, set the counter counter_arrow to 1. - Loop again over the two parts and keep looping till the counter count_arrow is 3 or greater than 3. - Set the boolean arrow_left true or false. After setting the boolean to true or false, the data is send to the theseus controller function. In this function, when detecting a gap, the camera_controller is called. If: - there is a gap on the right and no arrow, Pico turn right. - there is a gap on the left and no arrow, Pico turns right. - there is both a gap on the left and right and there is an arrow, the function checks if the arrow points to the left, if not turn right. If arrow_left is true, go left. - When detecting an arrow and setting the arrow_boolean to true or false, Pico makes a turn. After finishing that turn, the function resets the counter to 0 and the boolean to false. This is handled in the camera_reset function. Node: The camera function is build in a seperate node. We made a camera topic called cam_data and we subscribed the camera data to this topic. In the main function of the theseus node we published it so the camera_data can be processed in the theseus_controller function. ## Gazebo Simulation Pico is able to solve the maze without the camera node in Gazebo simulation. A video is recorded and shared via YouTube here: Gazebo Simulation Things to note: • When going around a corner (which we call state_gap_handling), the exact number of iterations varies. We consider pico's current orientation when starting gap handling, and use it to adjust the number of iterations such that pico will exit the gap handling state parallel to the new corridor (or next corner). This helps in every corner, but in particular the top right part, with many corners in a small space. • In the "tricky part" on the left crash avoidance is triggered twice to prevent crashes, and help Pico continue his journey. • In the end, the exit_maze condition is shown, which makes pico exit the maze in a straight line, and when he is out of the maze stop. This should also work in practise, but there's always people around, so often Pico will keep going. ## Corridor Competition We succesfully passed the corridor competition with the 2nd fastest time. A video impression is shared here: Corridor competition Theseus ## Maze Competition During the maze competition we had two interesting attempts. In the end we didn't solve the maze, but got very close and were awarded 3rd place. Here is a short report. ### Attempt 1 (with camera node) We started the first run with camera. At the first t-junction with the arrow to the left, we detected it correctly, and Pico took a left turn. However although the counter for resetting arrow detection was reset, the bool wasn't reset. Therefore the priorities in our code werent reset, and pico continued with taking left turns first. Although we didn't hit a wall, we decided to stop, and use our time for a second attempt without camera. ### Attempt 2 (without camera) In attempt 2, Pico followed the right wall, and therefore went right first at the first t-junction. After rotating at the dead end Pico got close to a wall, but didn't hit it. Then we got near the exit, but Pico didn't detect the exit (right gap). Although some of us were standing near, and some of us were watching the terminal output (combined with viewing the video), we didn't understand what went wrong there. Pico detected the gap very late, and therefore got too close to the end of the gap, which caused it to go back to the entrance by following the new right wall. The only thing we can think of is that someone was standing near the exit (without trying to blame anyone, or be sore losers), so that Pico didn't detect it in time. In our gap detection approach it is critical that the exit is free. A video impression of attempt two is shared here. Maze competition Theseus On the positive side: • We got very close to solving the maze. • We were able to have two good attempts demonstating all parts of our functionality. • Our reset state gave a good robust performance. We never hit a wall. • In attempt 1 we demonstrated that we can detect arrows and react to them. On the negative side: • Due to the small corridors compared to what we tested, we went into crash_avoidance quite often. • Camera reset functionality didn't work as intended. We forgot one line of code. ## Presentation Koen has presented the design and approach of our efforts. Because it is difficult to give a lot of detail in 5 minutes, we hope that our wiki can elaborate further on our choices. The slides are available here (in zipped format because ppt isn't allowed): File:EMC03.zip	2023-03-20 22:32:01	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.44564110040664673, "perplexity": 2393.099164689703}, "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/1679296943562.70/warc/CC-MAIN-20230320211022-20230321001022-00454.warc.gz"}
https://forum.math.toronto.edu/index.php?PHPSESSID=dftck8b0bjhducb5knhnfofef3&topic=67.msg346	### Author Topic: Lecture Notes 7 - Code problem (Read 2932 times) #### James McVittie • Full Member • Posts: 20 • Karma: 1 ##### Lecture Notes 7 - Code problem « on: October 13, 2012, 05:56:12 PM » In Lecture Notes 7 - 1D Wave Equation: IBVP, there was code below equation (10) and (15) that didn't get turned into text. It was $Oct$ and (\ref{eq-4}), what do these stand for? Thank you!	2022-08-18 11:40:42	{"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": 0.34311527013778687, "perplexity": 6089.353379607609}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1659882573193.35/warc/CC-MAIN-20220818094131-20220818124131-00374.warc.gz"}
https://dlnp.jinr.ru/en/meetings/video-recordings-of-seminars/2107-flux-integrated-semi-exclusive-cross-sections-for-charged-current-quasielastic-and-neutral-current-elastic-neutrino-scattering-off-argon-and-effects-of-short-baseline-neutrino-oscillations-by-a-v-butkevich	The recording of the DLNP seminar “Flux-Integrated Semi-Exclusive Cross Sections for Charged-Current Quasielastic and Neutral-Current Elastic Neutrino Scattering off Argon and Effects of Short-Baseline Neutrino Oscillations” given by A. V. Butkevich on November 23, 2022. Flux-integrated semi-exclusive differential cross sections for charged-current quasielastic and neutral-current elastic neutrino scattering off argon were analyzed. The cross sections are calculated using the relativistic distorted-wave impulse approximation with values of the nucleon axial mass М_А=1 GeV and 1.2 GeV. The elastic scattering cross sections were also computed for different strange quark contributions to the neutral-current axial form factor. The flux-integrated differential cross sections as functions of reconstructed neutrino energy are evaluated for the near and far detectors of the SBN experiment. The effects of the short-baseline neutrino oscillations are taken into account within the 3+1 model. We found that ratios of the cross sections calculated for the far and near detectors depend on oscillation parameters and can be used to search for muon and active neutrino disappearance in the SNB experiment.	2023-04-01 13:51:45	{"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": 0.9369592666625977, "perplexity": 2772.978283913544}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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-20230401125552-20230401155552-00521.warc.gz"}
http://lists.gnu.org/archive/html/emacs-orgmode/2010-09/msg01304.html	emacs-orgmode [Top][All Lists] Re: [Orgmode] problem with label in latex export From: Indraneel Majumdar Subject: Re: [Orgmode] problem with label in latex export Date: Wed, 22 Sep 2010 15:05:38 +0530 User-agent: Mozilla/5.0 (Windows; U; Windows NT 6.0; en-US; rv:1.9.2.9) Gecko/20100915 Lightning/1.0b2 Thunderbird/3.1.4 Thanks Bastien, this works inline. Please put it in the manual, it's just one line. I've spent looking for it every inch of the manual for more than a day. ``` ``` So will you be putting in \phantomsection for inline targets to work? I guess if someone is using a <<target>> then it's more likely /not/ to be a section heading. Putting it at a section heading doesn't hurt, but not having it inline makes for a useless <<target>>. Off course for Latex only, so probably should be in the exporter. ``` Indraneel On 2010-09-22 14:40, Bastien wrote: ``` ```Hi Indraneel, ``` ```11. #+<<target>> Some text 12. More text [[target][go to]] ``` ```You're right that there is a problem. The usual way of turning radio links invisible is to comment them, but Org comments need to be at the beginning of the line, which breaks list indentation. (Btw, no need for the '+' in '#+' -- '#+' is the syntax prefix for optional elements like blocks, etc.) The workaround here is to add (INVISIBLE) after your<<target>> This works: ``` ```11.<<target>>(INVISIBLE) Some text 12. More text [[target][go to]] ``` ```It exports okay in HTML and LaTeX. I've added a FAQ entry for this -- not sure whether it should go in the manual. Thanks for bringing this up! ``` ``` ```	2015-03-06 11:13:59	{"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": 0.905971109867096, "perplexity": 7773.321706344087}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936468536.58/warc/CC-MAIN-20150226074108-00130-ip-10-28-5-156.ec2.internal.warc.gz"}
https://www.bartleby.com/questions-and-answers/describe-what-is-meant-by-a-reduction-formula.-give-an-example./aeb81a6b-c788-4da4-93e8-ce1ec969ae52	# Describe what is meant by a reduction formula. Give an example. Question Describe what is meant by a reduction formula. Give an example.	2021-07-30 23:02:32	{"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": 0.9667149186134338, "perplexity": 308.51086992174703}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1627046154032.75/warc/CC-MAIN-20210730220317-20210731010317-00400.warc.gz"}
https://www.instasolv.com/question/1-let-a-b-c-be-three-vectors-such-that-a-b-c-b-c-a-c-a-b-0-and-a-1-16-4-c-5m7vxi	1 Let a, b, c be three vectors such... Question # 1 Let a, b, c be three vectors such that → → → a.(b+c)= b.(c+a)= c.(a+b) = 0 and a = 1, 16 = 4, CI=8 then \| a+b+c) = (2) 13 (6)81 (c) 9 (d) 5 th nornendicular to the JEE/Engineering Exams Maths Solution 84 4.0 (1 ratings)	2021-01-20 19:35:21	{"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": 0.8159165978431702, "perplexity": 3030.9106996381515}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703521987.71/warc/CC-MAIN-20210120182259-20210120212259-00309.warc.gz"}
https://math.stackexchange.com/questions/992331/supposed-a-b-in-mathbbz-if-ab-is-odd-then-a2-b2-is-even/992339	# Supposed $a,b \in \mathbb{Z}$. If $ab$ is odd, then $a^{2} + b^{2}$ is even. Supposed $a,b \in \mathbb{Z}$. If $ab$ is odd, then $a^{2} + b^{2}$ is even. I'm stuck on the best way to get this started. My thinking is that I could use cases. i.e. • Case 1: a is even and b is odd • Case 2: a is odd and b is even • Case 3: a is odd and b is odd Would this be my best approach? Or is there an easier way to look at it? Thanks. If $ab$ is odd then $a$ and $b$ must be both odd and then so are $a^2$ and $b^2$. But $a^2+b^2$ is the sum of two odd numbers, so it must be even. $ab$ is odd iff a and b are both odd. That means $a^2$ and $b^2$ are both odd. The sum of two odd numbers is even. Therefore, $a^2+b^2$ must be even if $ab$ is odd. More generally note: $\ ab(a^2+b^2)\,$ is even since ${\rm mod}\ 2\!:\ x^2\equiv x\,$ $\,\Rightarrow\,ab(a^2+b^2)\equiv ab(a+b)\equiv ab+ab\equiv 2ab\equiv 0$	2021-05-17 03:57:21	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.870326578617096, "perplexity": 68.68184135425092}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1620243991557.62/warc/CC-MAIN-20210517023244-20210517053244-00304.warc.gz"}
https://www.proofwiki.org/wiki/Positive_Image_of_Point_of_Continuous_Real_Function_implies_Positive_Closed_Interval_of_Domain	# Positive Image of Point of Continuous Real Function implies Positive Closed Interval of Domain ## Theorem Let $f: \R \to \R$ be a continuous real function. Let $a \in \R$ such that $\map f a > 0$. Then: $\exists k \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \closedint {a - \delta} {a + \delta}: \map f x \ge k$ ## Proof Let $\map f a = l$ where $l > 0$. As $f$ is continuous: $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \size {y - x} < \delta \implies \size {\map f y - \map f x} < \epsilon$ Let $\epsilon = \dfrac l 2 = k$. Then: $\exists \delta' \in \R_{>0}: \forall y \in \R: \size {y - x} < \delta' \implies \size {\map f y - \map f a} < \dfrac l 2$ Thus: $\forall x \in \openint {a - \delta'} {a + \delta'}: \map f x > \dfrac l 2$ Let $\delta = \dfrac {\delta'} 2$ Then $a - \delta \in \openint {a - \delta'} {a + \delta'}$ and $a + \delta \in \openint {a - \delta'} {a + \delta'}$. Thus: $\closedint {a - \delta} {a + \delta} \subseteq \openint {a - \delta'} {a + \delta'}$ and hence the result. $\blacksquare$	2022-08-17 13:27:13	{"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": 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": 0.9822782278060913, "perplexity": 1587.8768936779416}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1659882572908.71/warc/CC-MAIN-20220817122626-20220817152626-00533.warc.gz"}
https://memorize.be/maths/graph/terms/inc.md	# Incidence matrix ¶ Go back This may also be called Matrice d’incidence. It's mainly used in directed graphs as an amelioration of the adjacency matrix because we lost some information. This is a matrix vertex by vertex too, and the values are -1, 0, or 1. If we are at row=A, col=B • -1: an arc is leaving A ($A \to B$) • 1: an arc is entering A ($B \to A$) • 0: no arc ($A \to B$ or $B \to A$) If you can pick either -1 or 1, pick the one you want. ## Example ¶ The incidence matrix for is $\displaylines{ \hspace{0.7cm}\begin{array}{} a&b&c&d&h&i \end{array} \ \ \ \\ \begin{array}{} a\\b\\c\\d\\h\\i \end{array} \begin{pmatrix} 0 & 1 & -1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & -1 & 0 & 1 \\ -1 & -1 & 1 & 0 & 1 & 0 \\ 0 & -1 & 0 & -1 & 0 & 1 \\ 0 & 0 & -1 & 0 & 1 & 0 \\ \end{pmatrix} }$ Note: if you remove all the minus (-) before the ones then you got the adjacency matrix for the undirected graph.	2021-10-15 21:39:19	{"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": 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": 0.7375791072845459, "perplexity": 1295.8153610075758}, "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-2021-43/segments/1634323583083.92/warc/CC-MAIN-20211015192439-20211015222439-00126.warc.gz"}
http://www.ck12.org/book/Basic-Geometry/r1/section/1.5/	<meta http-equiv="refresh" content="1; url=/nojavascript/"> Angle Pairs \| CK-12 Foundation You are reading an older version of this FlexBook® textbook: CK-12 Geometry - Basic Go to the latest version. # 1.5: Angle Pairs Created by: CK-12 ## Learning Objectives • Recognize complementary angles supplementary angles, linear pairs, and vertical angles. • Apply the Linear Pair Postulate and the Vertical Angles Theorem. ## Review Queue 1. Find $x$. 2. Find $y$. 3. Find $z$. Know What? A compass (as seen to the right) is used to determine the direction a person is traveling. The angles between each direction are very important because they enable someone to be more specific with their direction. A direction of $45^\circ \ NW$, would be straight out along that northwest line. What headings have the same angle measure? What is the angle measure between each compass line? ## Complementary Angles Complementary: Two angles that add up to $90^\circ$. Complementary angles do not have to be: • congruent • next to each other Example 1: The two angles below are complementary. $m\angle GHI = x$. What is $x$? Solution: Because the two angles are complementary, they add up to $90^\circ$. Make an equation. $x + 34^\circ & = 90^\circ\\x & = 56^\circ$ Example 2: The two angles below are complementary. Find the measure of each angle. Solution: The two angles add up to $90^\circ$. Make an equation. $8r + 9^\circ + 7r + 6^\circ & = 90^\circ\\15r + 15^\circ & = 90^\circ\\15r & = 75^\circ\\r & = 5^\circ$ However, you need to find each angle. Plug $r$ back into each expression. $m \angle GHI & = 8(5^\circ) + 9^\circ = 49^\circ\\m\angle JKL & = 7(5^\circ) + 6^\circ = 41^\circ$ ## Supplementary Angles Supplementary: Two angles that add up to $180^\circ$. Supplementary angles do not have to be: • congruent • next to each other Example 3: The two angles below are supplementary. If $m\angle MNO = 78^\circ$ what is $m\angle PQR$? Solution: Set up an equation. However, instead of equaling $90^\circ$, now it is $180^\circ$. $78^\circ + m\angle PQR & = 180^\circ\\m\angle PQR & = 102^\circ$ Example 4: What is the measure of two congruent, supplementary angles? Solution: Supplementary angles add up to $180^\circ$. Congruent angles have the same measure. So, $180^\circ \div 2 = 90^\circ$, which means two congruent, supplementary angles are right angles, or $90^\circ$. ## Linear Pairs Adjacent Angles: Two angles that have the same vertex, share a side, and do not overlap. $\angle PSQ$ and $\angle QSR$ are adjacent. $\angle PQR$ and $\angle PQS$ are NOT adjacent because they overlap. Linear Pair: Two angles that are adjacent and the non-common sides form a straight line. $\angle PSQ$ and $\angle QSR$ are a linear pair. Linear Pair Postulate: If two angles are a linear pair, then they are supplementary. Example 5: Algebra Connection What is the measure of each angle? Solution: These two angles are a linear pair, so they add up to $180^\circ$. $(7q-46)^\circ + (3q+6)^\circ &= 180^\circ\\10q - 40^\circ &= 180^\circ\\10q & = 220^\circ\\q & = 22^\circ$ Plug in $q$ to get the measure of each angle. $m\angle ABD = 7(22^\circ) - 46^\circ = 108^\circ \ m\angle DBC = 180^\circ - 108^\circ = 72^\circ$ Example 6: Are $\angle CDA$ and $\angle DAB$ a linear pair? Are they supplementary? Solution: The two angles are not a linear pair because they do not have the same vertex. They are supplementary, $120^\circ + 60^\circ = 180^\circ$. ## Vertical Angles Vertical Angles: Two non-adjacent angles formed by intersecting lines. $\angle 1$ and $\angle 3$ are vertical angles $\angle 2$ and $\angle 4$ are vertical angles These angles are labeled with numbers. You can tell that these are labels because they do not have a degree symbol. Investigation 1-6: Vertical Angle Relationships 1. Draw two intersecting lines on your paper. Label the four angles created $\angle 1, \ \angle 2, \ \angle 3$, and $\angle 4$, just like the picture above. 2. Use your protractor to find $m\angle 1$. 3. What is the angle relationship between $\angle 1$ and $\angle 2$ called? Find $m\angle 2$. 4. What is the angle relationship between $\angle 1$ and $\angle 4$ called? Find $m\angle 4$. 5. What is the angle relationship between $\angle 2$ and $\angle 3$ called? Find $m\angle 3$. 6. Are any angles congruent? If so, write them down. From this investigation, you should find that $\angle 1 \cong \angle 3$ and $\angle 2 \cong \angle 4$. Vertical Angles Theorem: If two angles are vertical angles, then they are congruent. We can prove the Vertical Angles Theorem using the same process we used in the investigation. We will not use any specific values for the angles. From the picture above: $\angle 1 \ \text{and} \ \angle 2 \ \text{are a linear pair} \ \rightarrow m\angle 1 + m\angle 2 & = 180^\circ \qquad \text{Equation} \ 1\\\angle 2 \ \text{and} \ \angle 3 \ \text{are a linear pair} \ \rightarrow m\angle 2 + m\angle 3 & = 180^\circ \qquad \text{Equation} \ 2\\\angle 3 \ \text{and} \ \angle 4 \ \text{are a linear pair} \ \rightarrow m\angle 3 + m\angle 4 & = 180^\circ \qquad \text{Equation} \ 3$ All of the equations $= 180^\circ$, so Equation 1 = Equation 2 and Equation 2 = Equation 3. $m\angle 1 + m\angle 2 = m\angle 2 + m\angle 3 \qquad \text{and} \qquad m\angle 2 + m\angle 3 = m\angle 3 + m\angle 4$ Cancel out the like terms $m\angle 1 = m\angle 3 \qquad \text{and} \qquad m\angle 2 = m\angle 4$ Recall that anytime the measures of two angles are equal, the angles are also congruent. So, $\angle 1 \cong \angle 3$ and $\angle 2 \cong \angle 4$ too. Example 7: Find $m\angle 1$ and $m\angle 2$. Solution: $\angle 1$ is vertical angles with $18^\circ$, so $m\angle 1 = 18^\circ$. $\angle 2$ is a linear pair with $\angle 1$ or $18^\circ$, so $18^\circ + m\angle 2 = 180^\circ$. $m\angle 2 = 180^\circ - 18^\circ = 162^\circ$. Know What? Revisited The compass has several vertical angles and all of the smaller angles are $22.5^\circ, 180^\circ \div 8$. Directions that are opposite each other have the same angle measure, but of course, a different direction. All of the green directions have the same angle measure, $22.5^\circ$, and the purple have the same angle measure, $45^\circ$. $N, \ S, \ E$ and $W$ all have different measures, even though they are all $90^\circ$ apart. ## Review Questions • Questions 1 and 2 are similar to Examples 1, 2, and 3. • Questions 3-8 are similar to Examples 3, 4, 6 and 7. • Questions 9-16 use the definitions, postulates and theorems from this section. • Questions 17-25 are similar to Example 5. 1. Find the measure of an angle that is complementary to $\angle ABC$ if $m\angle ABC$ is 1. $45^\circ$ 2. $82^\circ$ 3. $19^\circ$ 4. $z^\circ$ 2. Find the measure of an angle that is supplementary to $\angle ABC$ if $m\angle ABC$ is 1. $45^\circ$ 2. $118^\circ$ 3. $32^\circ$ 4. $x^\circ$ Use the diagram below for exercises 3-7. Note that $\overline{NK} \perp \overleftrightarrow{IL}$. 1. Name one pair of vertical angles. 2. Name one linear pair of angles. 3. Name two complementary angles. 4. Name two supplementary angles. 1. What is: 1. $m\angle INL$ 2. $m\angle LNK$ 2. If $m\angle INJ = 63^\circ$, find: 1. $m\angle JNL$ 2. $m\angle KNJ$ 3. $m\angle MNL$ 4. $m\angle MNI$ For 9-16, determine if the statement is true or false. 1. Vertical angles are congruent. 2. Linear pairs are congruent. 3. Complementary angles add up to $180^\circ$. 4. Supplementary angles add up to $180^\circ$ 5. Adjacent angles share a vertex. 7. Complementary angles are always $45^\circ$. 8. Vertical angles have the same vertex. For 17-25, find the value of $x$ or $y$. 1. Find $x$. 2. Find $y$. 1. $x+26 = 3x-8\!\\{\;} \quad \ 34 = 2x\!\\{\;} \quad \ 17 = x$ 2. $(7y+6)^\circ = 90^\circ\!\\{\;} \qquad \ 7y = 84^\circ\!\\{\;} \qquad \ \ y = 12^\circ$ 3. $z+ 15 = 5z + 9\!\\{\;} \quad \ \ 6 = 4z\!\\{\;} \quad 1.5 = z$ 8 , 9 , 10 Feb 22, 2012 Aug 21, 2014	2014-12-22 20:48:54	{"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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 105, "texerror": 0, "math_score": 0.781065821647644, "perplexity": 1002.8185187034208}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802776563.13/warc/CC-MAIN-20141217075256-00081-ip-10-231-17-201.ec2.internal.warc.gz"}
http://gaming.stackexchange.com/questions/13774/how-do-i-install-patches-to-battlefield-1942-if-its-on-my-external-hard-drive/18586	# How do I install patches to Battlefield 1942 if it's on my external hard drive? I have a 2TB external hard drive and I want to put Battlefield 1942 on it. I got the game to install and run smoothly on the hard drive, but then I need a patch update. When I install the patch, the patch installer can't find the game, and thus does me no good. How do I get the patch to find the game on the external hard drive so I can run the game? Any help would be great. Thanks. - The BF42 patch installers look for the path to the game in the registry at this location: \HKEY_LOCAL_MACHINE\SOFTWARE\EA Games\Battlefield 1942 in a value named GAMEDIR. On my system that value is d:\Battlefield 1942. You can paste the following lines into a new file in Notepad, change the folder to the game's location on your external drive, save the file as bf42folder.reg, then double-click the file to update the registry. Note that you need to use double backslashes instead of single backslashes in the path to the game. REGEDIT4 [HKEY_LOCAL_MACHINE\SOFTWARE\EA Games\Battlefield 1942] "GAMEDIR"="d:\\Battlefield 1942" If you plan to play on the Internet, you will need a key as well, and that key goes in another place in the registry. Without it you can play single player games or on LAN servers, but not Internet games. - If you cannot find a way to get it to install to a different directory, try this: Symbolic links are a good solution to hosting games on a hard drive that is not your primary drive. They are similar to creating shortcuts, but the difference is that Windows will see them as actual paths, not shortcuts. For example: C:\ is my primary drive. Windows is installed on this drive. I create a symbolic link (very similar to a shortcut) at 'C:\Battlefield 1942' which points to 'E:\Battlefield 1942'. Now, I can install Battlefield 1942 to 'C:\Battlefield 1942', and as far as Windows is concerned, that is where it is installed. But, the files are actually kept on 'E:\Battlefield 1942' since that is where the symbolic link points to! The command for this would be: mklink /D "C:\Battlefield 1942" "E:\Battlefield 1942" For this command to work, the target ("E:\Battlefield 1942") must exist, and the link ("C:\Battlefield 1942") must not exist before the command is executed. If you already have your application installed at the C:\ directory, you could copy it over to the E:\ directory (while it isn't running), and then delete the empty C:\ folder before executing the command. The folder will be created again. You can execute this from windows command line. In Windows 7, just type 'cmd' into the search bar within the Start menu and press Enter to open the command line. In Windows XP, go through Start menu, click 'Run' and then enter 'cmd' for the parameter and press Enter! I only used the name 'E:\Battlefield 1942' as an example, you could name this whatever you want. 'E:\games\bf1942' would work as well, or whatever other path/name you wish to give it. This method is extremely effective when your primary hard drive is a solid state drive without much space, but you want your installed applications to be seen on the primary drive. -	2014-04-16 23:04:33	{"extraction_info": {"found_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": 0.3954637348651886, "perplexity": 2291.4184765797654}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00493-ip-10-147-4-33.ec2.internal.warc.gz"}
https://docs.ai2go.xnor.ai/c/quickstart.html	# Object Detector in 10 Minutes¶ Xnor.ai makes it easy to embed machine learning-powered computer vision into applications on any device. This tutorial shows how to build a simple object detector in C using an Xnor Bundle. ## Downloading the SDK¶ The latest version of the Xnor developer SDK can be found on AI2GO. This SDK includes samples and documentation that support developing applications using Xnor Bundles. Once you’ve downloaded the SDK, extract it to a convenient location. You can use the unzip tool on the command line for this: unzip ~/Downloads/<downloaded SDK name>.zip cd xnor-sdk-<hardware-target> ## Using the SDK¶ The unzipped SDK (and any Xnor Bundle from AI2GO) contains these files, among others: • include/xnornet.h: This is the public header for XnorNet. Include this in your code to use the XnorNet library. • lib/<model>/libxnornet.so: This is an Xnor Bundle compiled for use as a C library. It’s a standard shared object that exports the XnorNet C API. Link against it to use an Xnor Bundle in your application. The code that accompanies this tutorial can be found in samples/c/object_detector.c. ### Models¶ The first step is to load a model. A model is a single “brain” with specific capabilities. For example, some models are designed to do object detection for people, pets, and cars, whereas other models might be able to distinguish different types of fish from each other. Models are loaded using the xnor_model_load_built_in() function: xnor_model* model; panic_on_error(xnor_model_load_built_in(NULL, NULL, &model)); panic_on_error is defined in the example code, and will exit the application with an error message in the unlikely case that the XnorNet library fails to load the model. You can keep this behavior, or instead handle errors using existing mechanisms in your application. ### Inputs¶ Now that you’ve got a model, you’re going to need an image to test it on. The SDK’s data directory contains several sample images. For this example, we’ll use dog.jpg. First read the data into memory: const char* filename = "./samples/test-images/dog.jpg"; /* or elsewhere if desired / uint8_t dog_jpeg; size_t dog_jpeg_length; read_entire_file(filename, &dog_jpeg, &dog_jpeg_length); read_entire_file is a function defined in the example code that opens a file and reads all its contents into memory. Now dog_jpeg points to all the JPEG-encoded bytes of the image, and dog_jpeg_length contains the number of bytes of the JPEG. Next, wrap this data in an input object for Xnor: xnor_input* input; panic_on_error(xnor_input_create_jpeg_image(dog_jpeg, dog_jpeg_length, &input)); Note that xnor_input_create_jpeg_image() holds a reference to the data you give it, and does not make a copy; it is your responsibility to make sure the image data remains valid until after the xnor_input is freed (see Cleaning up below). You can also use other types of input formats. For example, you may have a camera that outputs a raw array of RGB pixels. There are functions to create xnor_input objects for these other formats as well. See the reference for more information. ### Evaluating¶ Once you’ve got a model and an image, you can tell the model to look at the image and tell you what it sees. xnor_evaluation_result* result; panic_on_error(xnor_model_evaluate(model, input, NULL, &result)); Now all that’s left to do is extract the desired data: #define MAX_BOXES 10 xnor_bounding_box boxes[MAX_BOXES]; int num_boxes = xnor_evaluation_result_get_bounding_boxes(result, boxes, MAX_BOXES); if (num_boxes > MAX_BOXES) { /* if there are more than MAX_BOXES boxes, xnor_evaluation_result_get_bounding_boxes will still return the total number of boxes, so we clamp it down to our maximum / num_boxes = MAX_BOXES; } if (num_boxes < 0) { / An error occurred! Maybe this wasn't an object detection model? / fputs("Error: Not an object detection model\n", stderr); return EXIT_FAILURE; } for (int i = 0; i < num_boxes; ++i) { printf("I see a %s\n", boxes[i].class_label.label); } Now boxes contains a list of objects that the model found in the image. All results are standard C structures that you can manipulate as you please. (Note that the label string will be freed and become invalid after you free the evaluation results, so make a copy if you plan to use it for longer than that.) ### Cleaning up¶ Cleaning up objects that are no longer in use will prevent memory leaks that can slow down applications. While this isn’t a serious problem in a tutorial application that only processes a single image, it will begin to cause problems as you scale up.: xnor_input_free(input); free(dog_jpeg); / Note: dog_jpeg must be freed AFTER input is freed! / xnor_evaluation_result_free(result); xnor_model_free(model); When freeing XnorNet objects, always use the provided xnor__free function rather than using free() directly. (dog_jpeg was application-allocated, not an XnorNet object, so the standard free() is used.) ## Compiling and Running¶ The provided Makefile handles compiling and linking the sample application for you. All you need to do is run make from samples/c: $cd samples/c$ make Once compilation is complete, run the executable: $build/object_detector I see a pet I see a vehicle For those curious what the Makefile is doing, here’s what the compiler needs to do to build against an Xnor model: • Locate the xnornet.h header file: Pass -Iinclude to GCC or Clang, or the equivalent in a different compiler. • Locate the libxnornet.so file: Pass -Llib to GCC or Clang, or the equivalent in a different compiler. • Link against libxnornet.so: Pass -lxnornet to GCC or Clang, or the equivalent in a different compiler. • Make sure your operating system knows where to find the libxnornet.so file. This can be done a few different ways: • Pass -Wl,-rpath,\$ORIGIN (commas, backslash, and dollar sign included) to GCC or Clang. This adds a directive to the executable that allows it to look in its own directory for shared objects (such as libxnornet.so) instead of being limited to system library directories. • Set LD_LIBRARY_PATH to the location of libxnornet.so when you run your application • Install libxnornet.so into your system library directory (for example, /usr/lib/x86_64-linux-gnu). The Makefile constructs something like the following gcc command to produce the final executable: $gcc -o object_detector -I../../include -L../../lib -lxnornet -Wl,-rpath,\$ORIGIN/../lib object_detector.c ## What’s Next?¶ • Try using a classification model, which tells you what’s in the image but not where in the image the objects are located. • Try some of the samples, to see how to use camera input. • Read the reference, to see all the possible functions you can call. • Go out and build something, and post it in the showcase!	2019-10-19 14:59:08	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.29419034719467163, "perplexity": 3149.482088387226}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986696339.42/warc/CC-MAIN-20191019141654-20191019165154-00311.warc.gz"}
https://docs.messageix.org/en/master/reporting.html	# Postprocessing and reporting¶ The ix modeling platform provides powerful features to perform calculations and other postprocessing after a message_ix.Scenario has been solved by the associated model. The MESSAGEix framework uses these features to provide zero-configuration reporting of models built on the framework. These features are accessible through Reporter, which can produce multiple reports from one or more Scenarios. A report is identified by a key (usually a string), and may… • perform arbitrarily complex calculations while intelligently handling units; • read and make use of data that is ‘exogenous’ to (not included in) a Scenario; • produce output as Python or R objects (in code), or to files or databases; • calculate only a requested subset of quantities; and • much, much more! Contents: ## Terminology¶ ixmp.reporting handles numerical quantities, which are scalar (0-dimensional) or array (1 or more dimensions) data with optional associated units. ixmp parameters, scalars, equations, and time-series data all become quantities for the purpose of reporting. Every quantity and report is identified by a key, which is a str or other hashable object. Special keys are used for multidimensional quantities. For instance: the MESSAGEix parameter resource_cost, defined with the dimensions (node n, commodity c, grade g, year y) is identified by the key 'resource_cost:n-c-g-y'. When summed across the grade/g dimension, it has dimensions n, c, y and is identified by the key 'resource_cost:n-c-y'. Non-model 1 quantities and reports are produced by computations, which are atomic tasks that build on other computations. The most basic computations—for instance, resource_cost:n-c-g-y—simply retrieve raw/unprocessed data from a message_ix.Scenario and return it as a Quantity. Advanced computations can depend on many quantities, and/or combine quantities together into a structure like a document or spreadsheet. Computations are defined in ixmp.reporting.computations and message_ix.reporting.computations, but most common computations can be added using the methods of Reporter. 1 i.e. quantities that do not exist within the mathematical formulation of the model itself, and do not affect its solution. ## Basic usage¶ A reporting workflow has the following steps: 1. Obtain a message_ix.Scenario object from an ixmp.Platform. 2. Use from_scenario() to create a Reporter object. 3. (optionally) Use Reporter built-in methods or advanced features to add computations to the reporter. 4. Use get() to retrieve the results (or trigger the effects) of one or more computations. >>> from ixmp import Platform >>> from message_ix import Scenario, Reporter >>> >>> mp = Platform() >>> scen = Scenario(scen) >>> rep = Reporter.from_scenario(scen) >>> rep.get('all') Note Reporter stores defined computations, but these are not executed until get() is called—or the results of one computation are required by another. This allows the Reporter to skip unneeded (and potentially slow) computations. A Reporter may contain computations for thousands of model quantities and derived quantities, but a call to get() may only execute a few of these. ## Customization¶ A Reporter prepared with from_scenario() always contains a key scenario, referring to the Scenario to be reported. The method Reporter.add() can be used to add arbitrary Python code that operates directly on the Scenario object: >>> def my_custom_report(scenario): >>> """Function with custom code that manipulates the scenario.""" >>> print('foo') >>> >>> rep.get('custom') foo In this example, the function my_custom_report() could run to thousands of lines; read to and write from multiple files; invoke other programs or Python scripts; etc. In order to take advantage of the performance-optimizing features of the Reporter, however, such calculations can be instead composed from atomic (i.e. small, indivisible) computations. ## Reporter, Key, and Quantity classes¶ ixmp.reporting.Reporter(kwargs) Class for generating reports on ixmp.Scenario objects. ixmp.reporting.Key(name[, dims, tag]) A hashable key for a quantity that includes its dimensionality. ixmp.reporting.Quantity Convert arguments to the internal Quantity data format. The ixmp.Reporter automatically adds keys based on the contents of the ixmp.Scenario argument. The message_ix.reporting.Reporter adds additional keys for derived quantities specific to the MESSAGEix model framework. These include: • out: the product of output (output efficiency) and ACT (activity). • out_hist = output × ref_activity (historical reference activity), • in = input × ACT, • in_hist = input × ref_activity, • emi = emission_factor × ACT, • emi_hist = emission_factor × ref_activity, • inv = inv_cost × CAP_NEW, • inv_hist = inv_cost × ref_new_capacity, • fom = fix_cost × CAP, • fom_hist = fix_cost × ref_capacity, • vom = var_cost × ACT, and • vom_hist = var_cost × ref_activity. • tom = fom + vom. • land_out = land_output × LAND, • land_use_qty = land_use × LAND, • land_emi = land_emission × LAND, • addon conversion, the model parameter addon_conversion (note space versus underscore), except broadcast across individual add-on technologies (ta) rather than add-on types (type_addon), • addon up, which is addon_up similarly broadcast., • addon ACT = addon conversion × ACT, • addon in = input × addon ACT, • addon out = output × addon ACT, and • addon potential = addon up × addon ACT, the maximum potential activity by add-on technology. • price emission, the model variable PRICE_EMISSION broadcast across emission species (e) and technologies (t) rather than types (type_emission, type_tec). Tip Use full_key() to retrieve the full-dimensionality Key for any of these quantities. • <name>:pyam for the above quantities, plus: • CAP:pyam (from CAP) • CAP_NEW:pyam (from CAP_NEW) These keys return the values in the IAMC data format, as pyam objects. • map_<name> as ‘indicator’ quantities for the mapping sets cat_<name>. • Standard reports message:system, message_costs, and message:emissions. • The report message:default, collecting all of the above reports. These automatic features of Reporter are controlled by: class ixmp.reporting.Reporter(kwargs) Class for generating reports on ixmp.Scenario objects. A Reporter is used to postprocess data from from one or more ixmp.Scenario objects. The get() method can be used to: • Retrieve individual quantities. A quantity has zero or more dimensions and optional units. Quantities include the ‘parameters’, ‘variables’, ‘equations’, and ‘scalars’ available in an ixmp.Scenario. • Generate an entire report composed of multiple quantities. A report may: • Read in non-model or exogenous data, • Trigger output to files(s) or a database, or • Execute user-defined methods. Every report and quantity (including the results of intermediate steps) is identified by a utils.Key; all the keys in a Reporter can be listed with keys(). Reporter uses a graph data structure to keep track of computations, the atomic steps in postprocessing: for example, a single calculation that multiplies two quantities to create a third. The graph allows get() to perform only the requested computations. Advanced users may manipulate the graph directly; but common reporting tasks can be handled by using Reporter methods: add(data, args, kwargs) General-purpose method to add computations. add_file(path[, key]) Add exogenous quantities from path. add_product(key, quantities[, sums]) Add a computation that takes the product of quantities. aggregate(qty, tag, dims_or_groups[, …]) Add a computation that aggregates qty. apply(generator, keys, kwargs) Add computations by applying generator to keys. check_keys(keys) Check that keys are in the Reporter. configure([path]) Configure the Reporter. describe([key, quiet]) Return a string describing the computations that produce key. disaggregate(qty, new_dim[, method, args]) Add a computation that disaggregates qty using method. finalize(scenario) Prepare the Reporter to act on scenario. full_key(name_or_key) Return the full-dimensionality key for name_or_key. get([key]) Execute and return the result of the computation key. Return the keys of graph. set_filters(filters) Apply filters ex ante (before computations occur). visualize(filename, kwargs) Generate an image describing the reporting structure. write(key, path) Write the report key to the file path. graph = {'config': {}} add(data, args, kwargs) add() can be called in several ways; its behaviour depends on data; see below. It chains to methods such as add_single(), add_queue(), and apply(), which can also be called directly. Parameters • data (various) – • args (various) – Other Parameters sums (bool, optional) – If True, all partial sums of the key data are also added to the Reporter. Returns Some or all of the keys added to the Reporter. Return type list of Key-like Raises KeyError – If a target key is already in the Reporter; any key referred to by a computation does not exist; or sums=True and the key for one of the partial sums of key is already in the Reporter. add() may be used to: • Provide an alias from one key to another: >>> r.add('aliased name', 'original name') • Define an arbitrarily complex computation in a Python function that operates directly on the ixmp.Scenario: >>> def my_report(scenario): >>> # many lines of code >>> return 'foo' >>> r.finalize(scenario) >>> r.get('my report') foo Note Use care when adding literal str values (2); these may conflict with keys that identify the results of other computations. add_file(path, key=None, kwargs) Reporting the key or using it in other computations causes path to be loaded and converted to Quantity. Parameters • path (os.PathLike) – Path to the file, e.g. ‘/path/to/foo.ext’. • key (str or Key, optional) – Key for the quantity read from the file. Other Parameters • dims (dict or list or set) – Either a collection of names for dimensions of the quantity, or a mapping from names appearing in the input to dimensions. • units (str or pint.Unit) – Units to apply to the loaded Quantity. Returns Either key (if given) or e.g. file:foo.ext based on the path name, without directory components. Return type Key add_load_file(path, key=None, kwargs) Reporting the key or using it in other computations causes path to be loaded and converted to Quantity. Parameters • path (os.PathLike) – Path to the file, e.g. ‘/path/to/foo.ext’. • key (str or Key, optional) – Key for the quantity read from the file. Other Parameters • dims (dict or list or set) – Either a collection of names for dimensions of the quantity, or a mapping from names appearing in the input to dimensions. • units (str or pint.Unit) – Units to apply to the loaded Quantity. Returns Either key (if given) or e.g. file:foo.ext based on the path name, without directory components. Return type Key add_product(key, quantities, sums=True) Add a computation that takes the product of quantities. Parameters • key (str or Key) – Key of the new quantity. If a Key, any dimensions are ignored; the dimensions of the product are the union of the dimensions of quantities. • sums (bool, optional) – If True, all partial sums of the new quantity are also added. Returns The full key of the new quantity. Return type Key add_queue(queue, max_tries=1, fail='raise') Parameters • queue (list of 2-tuple) – The members of each tuple are the arguments (i.e. a list or tuple) and keyword arguments (i.e. a dict) to add(). • max_tries (int, optional) – Retry adding elements up to this many times. • fail ('raise' or log level, optional) – Action to take when a computation from queue cannot be added after max_tries. add_single(key, computation, strict=False, index=False) Add a single computation at key. Parameters • key (str or Key or hashable) – A string, Key, or other value identifying the output of task. • computation (object) – Any dask computation, i.e. one of: 1. any existing key in the Reporter. 2. any other literal value or constant. 3. a task, i.e. a tuple with a callable followed by one or more computations. 4. A list containing one or more of #1, #2, and/or #3. • strict (bool, optional) – If True, key must not already exist in the Reporter, and any keys referred to by computation must exist. • index (bool, optional) – If True, key is added to the index as a full-resolution key, so it can be later retrieved with full_key(). aggregate(qty, tag, dims_or_groups, weights=None, keep=True, sums=False) Add a computation that aggregates qty. Parameters • qty (Key or str) – Key of the quantity to be aggregated. • tag (str) – Additional string to add to the end the key for the aggregated quantity. • dims_or_groups (str or iterable of str or dict) – Name(s) of the dimension(s) to sum over, or nested dict. • weights (xarray.DataArray, optional) – Weights for weighted aggregation. • keep (bool, optional) – Passed to computations.aggregate. • sums (bool, optional) – Passed to add(). Returns The key of the newly-added node. Return type Key apply(generator, keys, *kwargs) Add computations by applying generator to keys. Parameters • generator (callable) – Function to apply to keys. • keys (hashable) – The starting key(s). • kwargs – Keyword arguments to generator. check_keys(keys) Check that keys are in the Reporter. If any of keys is not in the Reporter, KeyError is raised. Otherwise, a list is returned with either the key from keys, or the corresponding full_key(). configure(path=None, config) Configure the Reporter. Accepts a path to a configuration file and/or keyword arguments. Configuration keys loaded from file are replaced by keyword arguments. Valid configuration keys include: Warns UserWarning – If config contains unrecognized keys. default_key = None The default reporting key. describe(key=None, quiet=True) Return a string describing the computations that produce key. If key is not provided, all keys in the Reporter are described. The string can be printed to the console, if not quiet. disaggregate(qty, new_dim, method='shares', args=[]) Add a computation that disaggregates qty using method. Parameters • qty (hashable) – Key of the quantity to be disaggregated. • new_dim (str) – Name of the new dimension of the disaggregated variable. • method (callable or str) – Disaggregation method. If a callable, then it is applied to var with any extra args. If then a method named ‘disaggregate_{method}’ is used. • args (list, optional) – Additional arguments to the method. The first element should be the key for a quantity giving shares for disaggregation. Returns The key of the newly-added node. Return type Key finalize(scenario) Prepare the Reporter to act on scenario. The Scenario object scenario is associated with the key 'scenario'. All subsequent processing will act on data from this scenario. classmethod from_scenario(scenario, kwargs) Create a Reporter by introspecting scenario. Parameters • scenario (ixmp.Scenario) – Scenario to introspect in creating the Reporter. • kwargs (optional) – Passed to Scenario.configure(). Returns A Reporter instance containing: • A ‘scenario’ key referring to the scenario object. • Each parameter, equation, and variable in the scenario. • All possible aggregations across different sets of dimensions. • Each set in the scenario. Return type Reporter full_key(name_or_key) Return the full-dimensionality key for name_or_key. An ixmp variable ‘foo’ with dimensions (a, c, n, q, x) is available in the Reporter as 'foo:a-c-n-q-x'. This Key can be retrieved with: rep.full_key('foo') rep.full_key('foo:c') # etc. get(key=None) Execute and return the result of the computation key. Only key and its dependencies are computed. Parameters key (str, optional) – If not provided, default_key is used. Raises ValueError – If key and default_key are both None. keys() Return the keys of graph. set_filters(filters) Apply filters ex ante (before computations occur). Filters are stored in the reporter at the 'filters' key, and are passed to ixmp.Scenario.par() and similar methods. All quantity values read from the Scenario are filtered before any other computations take place. Parameters filters (mapping of str → (list of str or None)) – Argument names are dimension names; values are lists of allowable labels along the respective dimension, or None to clear any existing filters for the dimension. If no arguments are provided, all filters are cleared. property unit_registry The pint.UnitRegistry() used by the Reporter. visualize(filename, kwargs) Generate an image describing the reporting structure. This is a shorthand for dask.visualize(). Requires graphviz. write(key, path) Write the report key to the file path. class ixmp.reporting.Key(name, dims=[], tag=None) A hashable key for a quantity that includes its dimensionality. Quantities in a Scenario can be indexed by one or more dimensions. Keys refer to quantities, using three components: 1. a string name, 2. zero or more ordered dimensions dims, and 3. an optional tag. For example, an ixmp parameter with three dimensions can be initialized with: >>> scenario.init_par('foo', ['a', 'b', 'c'], ['apple', 'bird', 'car']) Key allows a specific, explicit reference to various forms of “foo”: • in its full resolution, i.e. indexed by a, b, and c: >>> k1 = Key('foo', ['a', 'b', 'c']) >>> k1 <foo:a-b-c> • in a partial sum over one dimension, e.g. summed across dimension c, with remaining dimensions a and b: >>> k2 = k1.drop('c') >>> k2 <foo:a-b> • in a partial sum over multiple dimensions, etc.: >>> k1.drop('a', 'c') == k2.drop('a') == 'foo:b' True • after it has been manipulated by different reporting computations, e.g. >>> k3 = k1.add_tag('normalized') >>> k3 <foo:a-b-c:normalized> >>> k4 <foo:a-b-c:normalized+rescaled> Notes: A Key has the same hash, and compares equal to its str representation. repr(key) prints the Key in angle brackets (‘<>’) to signify that it is a Key object. >>> str(k1) 'foo:a-b-c' >>> repr(k1) '<foo:a-b-c>' >>> hash(k1) == hash('foo:a-b-c') True Keys are immutable: the properties name, dims, and tag are read-only, and the methods append(), drop(), and add_tag() return new Key objects. Keys may be generated concisely by defining a convenience method: >>> def foo(dims): >>> return Key('foo', dims.split()) >>> foo('a b c') <foo:a-b-c> add_tag(tag) Return a new Key with tag appended. append(dims) Return a new Key with additional dimensions dims. property dims Dimensions of the quantity, tuple of str. drop(dims) Return a new Key with dims dropped. classmethod from_str_or_key(value, drop=[], append=[], tag=None) Return a new Key from value. Parameters Returns Return type Key iter_sums() Generate (key, task) for all possible partial sums of the Key. property name Name of the quantity, str. classmethod product(new_name, keys, tag=None) Return a new Key that has the union of dimensions on keys. Dimensions are ordered by their first appearance: 1. First, the dimensions of the first of the keys. 2. Next, any additional dimensions in the second of the keys that were not already added in step 1. 3. etc. Parameters new_name (str) – Name for the new Key. The names of keys are discarded. property tag Quantity tag, str. ixmp.reporting.Quantity(data, args, *kwargs) Convert arguments to the internal Quantity data format. Parameters • data – Quantity data. • args – Positional arguments, passed to AttrSeries or SparseDataArray. • kwargs – Keyword arguments, passed to AttrSeries or SparseDataArray. Other Parameters • name (str, optional) – Quantity name. • units (str, optional) – Quantity units. • attrs (dict, optional) – Dictionary of attributes; similar to attrs. The Quantity constructor converts its arguments to an internal, xarray.DataArray-like data format: # Existing data data = pd.Series(...) # Convert to a Quantity for use in reporting calculations qty = Quantity(data, name="Quantity name", units="kg") ## Computations¶ ### Inherited from ixmp¶ Elementary computations for reporting. Unless otherwise specified, these methods accept and return Quantity objects for data arguments/return values. Calculations: add(quantities[, fill_value]) Sum across multiple quantities. aggregate(quantity, groups, keep) Aggregate quantity by groups. apply_units(qty, units[, quiet]) Simply apply units to qty. disaggregate_shares(quantity, shares) Disaggregate quantity by shares. product(quantities) Return the product of any number of quantities. ratio(numerator, denominator) Return the ratio numerator / denominator. select(qty, indexers[, inverse]) Select from qty based on indexers. sum(quantity[, weights, dimensions]) Sum quantity over dimensions, with optional weights. Input and output: load_file(path[, dims, units]) Read the file at path and return its contents as a Quantity. write_report(quantity, path) Write a quantity to a file. Data manipulation: concat(objs, *kwargs) Concatenate Quantity objs. ixmp.reporting.computations.aggregate(quantity, groups, keep) Aggregate quantity by groups. Parameters • quantity (Quantity) – • groups (dict of dict) – Top-level keys are the names of dimensions in quantity. Second-level keys are group names; second-level values are lists of labels along the dimension to sum into a group. • keep (bool) – If True, the members that are aggregated into a group are returned with the group sums. If False, they are discarded. Returns Same dimensionality as quantity. Return type Quantity ixmp.reporting.computations.apply_units(qty, units, quiet=False) Simply apply units to qty. Logs on level WARNING if qty already has existing units. Parameters ixmp.reporting.computations.concat(objs, *kwargs) Concatenate Quantity objs. Any strings included amongst args are discarded, with a logged warning; these usually indicate that a quantity is referenced which is not in the Reporter. ixmp.reporting.computations.data_for_quantity(ix_type, name, column, scenario, config) Retrieve data from scenario. Parameters • ix_type ('equ' or 'par' or 'var') – Type of the ixmp object. • name (str) – Name of the ixmp object. • column ('mrg' or 'lvl' or 'value') – Data to retrieve. ‘mrg’ and ‘lvl’ are valid only for ix_type='equ', and ‘level’ otherwise. • scenario (ixmp.Scenario) – Scenario containing data to be retrieved. • config (dict of (str -> dict)) – The key ‘filters’ may contain a mapping from dimensions to iterables of allowed values along each dimension. The key ‘units’/’apply’ may contain units to apply to the quantity; any such units overwrite existing units, without conversion. Returns Data for name. Return type Quantity ixmp.reporting.computations.disaggregate_shares(quantity, shares) Disaggregate quantity by shares. ixmp.reporting.computations.load_file(path, dims={}, units=None) Read the file at path and return its contents as a Quantity. Some file formats are automatically converted into objects for direct use in reporting code: .csv: Converted to Quantity. CSV files must have a ‘value’ column; all others are treated as indices, except as given by dims. Lines beginning with ‘#’ are ignored. Parameters • path (pathlib.Path) – Path to the file to read. • dims (collections.abc.Collection or collections.abc.Mapping, optional) – If a collection of names, other columns besides these and ‘value’ are discarded. If a mapping, the keys are the column labels in path, and the values are the target dimension names. • units (str or pint.Unit) – Units to apply to the loaded Quantity. ixmp.reporting.computations.product(quantities) Return the product of any number of quantities. ixmp.reporting.computations.ratio(numerator, denominator) Return the ratio numerator / denominator. Parameters • numerator (Quantity) – • denominator (Quantity) – ixmp.reporting.computations.select(qty, indexers, inverse=False) Select from qty based on indexers. Parameters • qty (Quantity) – • indexers (dict (str -> list of str)) – Elements to be selected from qty. Mapping from dimension names to labels along each dimension. • inverse (bool, optional) – If True, remove the items in indexers instead of keeping them. ixmp.reporting.computations.sum(quantity, weights=None, dimensions=None) Sum quantity over dimensions, with optional weights. Parameters • quantity (Quantity) – • weights (Quantity, optional) – If dimensions is given, weights must have at least these dimensions. Otherwise, any dimensions are valid. • dimensions (list of str, optional) – If not provided, sum over all dimensions. If provided, sum over these dimensions. ixmp.reporting.computations.write_report(quantity, path) Write a quantity to a file. Parameters path (str or Path) – Path to the file to be written. ## Configuration¶ Configure reporting globally. ixmp.reporting.utils.RENAME_DIMS Dimensions to rename when extracting raw data from Scenario objects. ixmp.reporting.utils.REPLACE_UNITS Replacements to apply to quantity units before parsing by pint. reporting.configure(*config) Configure reporting globally. Modifies global variables that affect the behaviour of all Reporters and computations, namely RENAME_DIMS and REPLACE_UNITS. Valid configuration keys—passed as config keyword arguments—include: Other Parameters Warns UserWarning – If config contains unrecognized keys. ixmp.reporting.utils.RENAME_DIMS = {} Dimensions to rename when extracting raw data from Scenario objects. Mapping from Scenario dimension name -> preferred dimension name. message_ix adds the standard short symbols for MESSAGE sets to this variable. ixmp.reporting.utils.REPLACE_UNITS = {'%': 'percent'} Replacements to apply to quantity units before parsing by pint. Mapping from original unit -> preferred unit. ## Utilities¶ ixmp.reporting.quantity.assert_quantity(args) Assert that each of args is a Quantity object. Raises TypeError – with a indicative message. ixmp.reporting.utils.clean_units(input_string) Tolerate messy strings for units. Handles two specific cases found in MESSAGEix test cases: • Dimensions enclosed in ‘[]’ have these characters stripped. • The ‘%’ symbol cannot be supported by pint, because it is a Python operator; it is translated to ‘percent’. ixmp.reporting.utils.collect_units(*args) Return an list of ‘_unit’ attributes for args. ixmp.reporting.utils.dims_for_qty(data) Return the list of dimensions for data. If data is a pandas.DataFrame, its columns are processed; otherwise it must be a list. ixmp.reporting.RENAME_DIMS is used to rename dimensions. ixmp.reporting.utils.filter_concat_args(args) Filter out str and Key from args. A warning is logged for each element removed. ixmp.reporting.utils.parse_units(units_series) Return a pint.Unit for a pd.Series of strings.	2020-10-27 06:28:21	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.2335847020149231, "perplexity": 10583.349703714497}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1603107893402.83/warc/CC-MAIN-20201027052750-20201027082750-00168.warc.gz"}
http://physicshelpforum.com/quantum-physics/14967-schroedinger-s-equation-2.html	Physics Help Forum Schroedinger's equation Quantum Physics Quantum Physics Help Forum Nov 2nd 2018, 09:15 AM #11 Member Join Date: Oct 2010 Posts: 30 Ok, thanks. Do you maybe know in an example of a particle in a box, https://www.conservapedia.com/Schrodinger_equation how do we get to this: The continuous constraint is only satisfied when $\displaystyle \omega a = n \pi$ where $\displaystyle n$ is an integer. It's a statement at the end of the page. Nov 2nd 2018, 09:59 AM #12 Join Date: Apr 2008 Location: On the dance floor, baby! Posts: 2,521 Originally Posted by Nforce Ok, thanks. Do you maybe know in an example of a particle in a box, https://www.conservapedia.com/Schrodinger_equation how do we get to this: It's a statement at the end of the page. A wavefunction is (almost) always a continuous function, at least when the potential energy is given by a continuous function. The result $\displaystyle \omega a = n \pi$ comes from substitution of the trig function solution at the endpoints of the box. ( $\displaystyle \psi (x) = 0$ at x = 0 and x = a.) As to the form of the momentum operator I don't know who came up with it, but there is something called Ehrefest's theorem that gives a link to Classical mechanics. The pertinent equation here is $\displaystyle m \dfrac{d}{dt} < x > = <p>$ where $\displaystyle < A > = \dfrac{d}{dt} \int \psi ^* A \psi ~ dx$ for an operator A. The momentum operator can be inferred from this equation. -Dan __________________ Do not meddle in the affairs of dragons for you are crunchy and taste good with ketchup. See the forum rules here. Nov 2nd 2018, 10:25 AM #13 Senior Member Join Date: Oct 2017 Location: Glasgow Posts: 288 The boundary conditions are set to be $\displaystyle \psi(0) = 0$ $\displaystyle \psi(a) = 0$ Therefore, following substitution for $\displaystyle \psi$, you get $\displaystyle A=0$ and $\displaystyle B\sin \omega a = 0$ You could just set B = 0 as well, but that doesn't yield a very interesting solution. That just states that when the wave function is 0, you satisfy the potentials. Therefore, the more interesting solution is when you look at the fluctuating sine wave and compare the results with 0. It turns out that the LHS is equal to 0 when $\displaystyle \omega a = \pi n$ where n is any integer. If the above is true, B can be anything and it will still be valid. You can verify relationship by picking an integer (say, n=2) and then plot the sine wave. You'll see that no matter what you choose, the curve will drops down to x=0 at the boundary. The only exception is if you pick n=0 solution, which gives the same result as the B=0 solution. Another constraint is required to pin down what B is. In this case, it is found using normalisation (since the area under the probability density function, which is the square of the wave function, must be 1). The relationship $\displaystyle \omega a = \pi n$ is useful when characterising the energy, E, which must have certain discrete values. This is the key feature of quantum mechanics; a lot of the solutions to problems are discrete (i.e. have an integer "n" in it somewhere) topsquark and Nforce like this. Tags equation, schroedinger Thread Tools Display Modes Linear Mode Similar Physics Forum Discussions Thread Thread Starter Forum Replies Last Post VonDamian General Physics 0 Jan 18th 2017 11:40 AM Torgny Quantum Physics 1 Apr 4th 2016 03:39 AM alexandros87 Thermodynamics and Fluid Mechanics 10 Nov 22nd 2015 03:56 PM pinkprincess08 Thermodynamics and Fluid Mechanics 2 Feb 19th 2014 12:15 PM petdem Thermodynamics and Fluid Mechanics 8 Feb 15th 2010 05:51 AM	2019-03-19 05:59:05	{"extraction_info": {"found_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": 0.7911523580551147, "perplexity": 580.3210497845423}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1552912201904.55/warc/CC-MAIN-20190319052517-20190319074517-00430.warc.gz"}
https://www.gradesaver.com/textbooks/math/calculus/calculus-10th-edition/chapter-p-p-3-functions-and-their-graphs-exercises-page-27/14	# Chapter P - P.3 - Functions and Their Graphs - Exercises: 14 Domain: $(-\infty,\infty)$ Range: $(-\infty,4]$ #### Work Step by Step $h(x)=4-x^{2}$ This function is defined for all real numbers, so its domain is $(-\infty,\infty)$ This function represents a parabola of vertex $(0,4)$ and it opens downwards, so its range is formed by all the numbers smaller than $4$, including it, or $(-\infty,4]$ After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.	2018-05-23 07:23:58	{"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": 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": 0.5823553204536438, "perplexity": 583.7031039121232}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794865456.57/warc/CC-MAIN-20180523063435-20180523083435-00395.warc.gz"}
https://www.fimfiction.net/news/site-update	Dec 29th 2017 Site Update » Night Mode · 12:22pm Dec 29th, 2017 I've been working on it for ages but only really got the impetus to finish all of it off over the last few days. In the "settings" dropdown at the top on desktop, or the bottom of the slide out bar on mobile you'll find a toggle for night mode. Enjoy! Oh, and although I've tried to cover everything there is a 100% chance I've missed styling some things so apologies in advance for any funky pages. Report knighty · 3,936 views · Dec 22nd 2017 Site Update » Additional Search Update · 2:50pm Dec 22nd, 2017 Hey folks, Over the last few days I've added a few things to the new search system. A lot of people were unhappy with not being able to filter various things as quickly as they used to be able to. To that end, I've added a little filter dropdown to the right of the search box which effectively contains everything the old sidebar used to. It even has some niceties like quick word count filters and a highly rated filter. Report knighty · 2,887 views · Dec 20th 2017 Site Update » December 2017 Update · 12:40am Dec 20th, 2017 Hey guys, got a whole bunch of updates for you today. Tags This is a small but important step on our way to the tagging system I envision. The existing way we handled things like characters and genres has all been merged into a single tagging system. That won't result in much difference for you viewing and using the site but it makes it a lot easier to add new tags especially. We now have a couple of new tag types: series and warnings. Report knighty · 11,663 views · Dec 17th 2017 Site Update » Math BBCode tag · 12:58am Dec 17th, 2017 I've added [math] and [mathblock] BBCode tags, which can be used to display formatted math. We've had a few requests for this, particularly for group forum threads and blog posts. Most math-related TeX syntax is supported. (We are currently using MathJax to handle the layout.) The documentation from the BBCode guide is repeated below for your convenience. Report Xaquseg · 2,491 views · #bbcode Jul 10th 2017 Site Update » Fimfiction API · 12:18pm Jul 10th, 2017 If you're not a developer you can probably ignore this post. It's been like 6 years, but hey, things take time. The API is currently very WIP still but it's ready for people to get working on in our development chat room. Report knighty · 5,309 views · #api Jun 8th 2017 Site Update » New BBCode Tags · 11:37pm Jun 8th, 2017 Hey guys, One of the features in this new update was reader-side paragraph formatting. This helps improve consistency for readers across the site, especially for those of us who can’t stand reading indented text on a computer screen. However, one thing that wasn’t accounted for was the legitimate need for specific indenting of passages and for certain blocks of text to have no paragraph formatting. Some examples would be lyrics and poetry. Report knighty · 4,927 views · Jun 5th 2017 Site Update » Fimfiction 4.0 · 4:15pm Jun 5th, 2017 It’s been a very very long time coming, but we’ve finally updated the site again. this is by far the biggest update we have ever done. There is a cavalcade of new features but the biggest changes are under the hood and affect how easy it is to extend the site and performance. A change log of everything I can remember can be found below. Report knighty · 19,221 views · May 30th 2017 Site Update » 💩 · 11:57am May 30th, 2017 So, emojis are now supported all over the site. Go have fun and stuff. oh god what have we done Report knighty · 4,362 views · Mar 22nd 2017 Site Update » TLS for all users · 1:31am Mar 22nd, 2017 We have implemented TLS site-wide as an unconditional redirect. (http -> https) This improves security site-wide for all users, and shouldn't have any negative effects, performance or otherwise. Report Xaquseg · 3,568 views · Jan 11th 2017 Site Update » New Character Tags · 4:21am Jan 11th, 2017 I have added a total of 70 new character tags to the site today. They can be found below:	2018-03-24 06:01:46	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.20070098340511322, "perplexity": 5256.586991394979}, "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/1521257649931.17/warc/CC-MAIN-20180324054204-20180324074204-00386.warc.gz"}
https://leancrew.com/all-this/2015/01/the-michelson-fourier-analyzer/	# The Michelson Fourier Analyzer Let’s start the new year off with a bit of Fourier analysis, shall we? A couple of months ago, Bill Hammack, the YouTube Engineer Guy (and professor of chemical engineering at the University of Illinois) produced a wonderful series of four videos—and a book—about a 19th century mechanical device that does Fourier analysis and synthesis. Here’s the introductory video: Professor Hammack is always entertaining and informative, but he really outdoes himself in these videos. You can see how delighted he is to have brought this device back to life and show it to us. And why not? It’s a beautiful machine, both in concept and in execution. Like Hammack, I’m a sucker for mechanical devices that perform functions we now think of as exclusively digital. The machine was designed by Albert Michelson (of the Michelson-Morley experiment) near the turn of the 20th century while he was at the University of Chicago, and it was built by William Gaertner & Co., a local Chicago firm. It currently sits in Illinois’s Altgeld Hall, just across the Quad from Professor Hammack’s office. Altgeld Hall is known to all Illinois grads as the building with the bell tower that chimes on the quarter-hour and serenades the north half of campus with carillon music. It’s the home of the math department and a little post office with appropriate postmarks. It also houses the math library, my favorite library on campus. It covers two stories in the middle of the building and has lovely old cast iron shelves and a glass block floor set in an iron grillage. Everything about it, including the musty smell, says “library.” As it happens, just a month before Hammack published the videos, I was in Altgeld Hall with my older son. We were there for a campus visit, and I took him through Altgeld to show him the library and the collection of mathematical models that line several of the halls. In one of the halls, sitting almost anonymously in a glass case, was Michelson’s Fourier Analyzer. I read the descriptive card inside the case but didn’t believe it was in working order. Before talking about the Analyzer, let’s review what Fourier analysis is and why we use it. Periodic functions appear in the mathematical solution to many problems of practical physical importance. The simplest periodic functions to calculate with are sines and cosines, and as it turns out, any periodic function can be expressed as a series of sines and cosines: Joseph Fourier was the first to recognize the value of this series in the solution of the partial differential heat equation, and it’s named in his honor. The trick of Fourier analysis is figuring out the as and bs that make the best fit to the original function, f. As an example, let’s look at a triangular wave function. Because the triangle is an odd function, $f(-x) = -f(x)$, all the cosine coefficients are zero. And because each half-wave is symmetric, the even sine coefficients are also zero. The Fourier series for the triangular wave is then I’ll spare you the math necessary to determine the coefficients. If you’re interested, take a look at equations 7, 8, and 9 in the MathWorld article. Here are the first three component functions (in blue, green, and red) and their summation (in black): You can see that even after only three terms, we have a pretty good approximation to the original function, except near the corners at $\pi/2$ and $-\pi/2$. Cusps and discontinuities are always troublesome for Fourier series because it’s hard to make smooth functions like sines and cosines fit into sharp corners. This is an example of the Gibbs Phenomenon. The clearest brief overview of how Michelson’s machine puts together a Fourier series comes in this segment from the Synthesis video: There are fuller explanations of each component later in the videos, but I think it’s worth mentioning a few things here: • You’ll note that Hammack is showing only the cosine terms of the Fourier series. The Analyzer can do either a sine series for odd functions or a cosine series for even functions, but it can’t do a mixture of sines and cosines for general functions. This isn’t as big a restriction as you might think. Because any function can be split into the sum of an even function and an odd function, the Analyzer can do each part separately, and the user can put the two series together afterwards. • Similarly, the Analyzer doesn’t do the constant term, $a_0$. Since the constant term is nothing more than a vertical shift of the function, this can be removed from the function before analysis and put back in afterwards • The summation goes to 20 instead of ∞ because it’s limited by the number of gears in the machine. If you try to build one with an infinite number of gears, you run out of brass. The Analyzer in Altgeld Hall has 20 gears, but there was an Analyzer Plus that had 80 gears. Most of the Analyzer’s components work through kinematics. The crank turns through an angle, which causes the set of gears fixed to it to turn through that same angle. Each one of the second set of gears, though, turns through a different angle, based on its diameter and number of teeth. The arms then move through distances according to the angular motion of the gear and cam they’re connected to. This is a complicated dance because of the number of parts, but conceptually it’s fairly simple. To me, the hardest part to understand is how the movements of the individual arms are added together to drive the up-and-down motion of the pen. I thought Hammack’s explanation of this part was a little hand-wavy, even in the more detailed explanation later in the videos. So I started sketching things out and realized that the addition is done through the assistance of our old friend, Hooke’s Law. We have a set of parallel arm movements of varying amplitude that we need to add together. The problem is that parallel displacements aren’t additive. If you and a friend start at the same spot, and you walk a mile and she walks two miles, nothing goes three miles. But parallel forces are additive, so if we can convert the displacement of the arms into forces, we’ll be good to go. Converting displacement into force is what Hooke’s Law does. At the top of the Analyzer is a pivoted bar, one side of which is connected via springs to each of levers driven by the individual gears. Here’s a screenshot from the video, and here’s a simplified sketch of the arrangement, Forgive me for not drawing in all 20 springs. The top end of each spring on the right moves a different amount, and the force in each spring will be proportional to that motion. These forces add together into a resultant force on the right that’s balanced, through the lever arms a and b, by the force in the big spring on the left. We’ll take $x=0$ as the position where the bar is horizontal. Because the springs are not in their natural, unstretched state at $x=0$, there’s a preload in each spring that has to be accounted for. We’ll call that preload $F_0$ for the big spring and $f_0$ for the little springs. Here’s the free-body diagram of the pivoted bar: where the $x_i$ are the displacements of the upper ends of the springs on the right. Rearranging, we get Because the bar is in equilibrium when $x=0$ and all the $x_i=0$, the first two terms on the left side cancel, and we can solve for $x$: What this means is that the movement of the pivoted bar, $x$, is proportional to the sum of the movements of the individual levers, $\sum_{i=1}^n x_i$. The pen is driven by a mechanism attached to the pivoted bar, so its movement is then also proportional to the sum of the movements of the individual levers. This is exactly what we need to add up the terms in the Fourier series. A beautiful solution to the addition problem. Robert Frost said that writing free verse is like playing tennis without a net. Art needs constraints. I would never want to go back to a time when this was the fastest way to do Fourier analysis, but there’s an artistry to this machine that just isn’t there when you call a routine from an FFT library. Update 1/4/15 4:08 PM Theo Honohan pointed out to me in an email that the whippletree mechanism can be used as a displacement adder, and he’s absolutely right. I tend to think of the whippletree as a way of distributing a force uniformly (I’ve used it as such when doing load testing of structures), but it can be used “backwards” to add displacements. It’s probably easiest to think of the whippletree as a way to average displacements, but since the average is just a scaled sum, it works as an adder, too. I don’t know for sure why Michelson chose to use springs instead of a whippletree, but my guess is that he found the spring system to be simpler and more compact. Whippletrees grow quickly in size as you include more terms in the summation. If you’re interested in learning about the use of linkages in analog computing, this page by Andries de Man is a good place to start.	2018-11-15 00:40:37	{"extraction_info": {"found_math": true, "script_math_tex": 14, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5999234914779663, "perplexity": 571.7745160332628}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039742322.51/warc/CC-MAIN-20181114232605-20181115014034-00060.warc.gz"}
https://www.physicsforums.com/threads/direction-of-induced-current-in-conducting-ring-due-to-motion-of-bar-m.713243/	# Direction of induced current in conducting ring due to motion of bar m agoogler - ## The Attempt at a Solution The relative velocity of magnet with respect to ring is 5−4=1m/s How can I find the Direction of induced current in conducting ring due to motion of bar magnet? I searched online and found this : http://www.phys.ufl.edu/courses/phy2049/f07/lectures/2049_ch30B.pdf After reading that I think the answer should be (B) but the correct answer is (A). Please help. Is there any rule for this ? ( Like the right hand thumb rule for induced magnetic field ?) ## Answers and Replies Abhilash H N When one tries to find the direction of induced current, the poles of the magnets also count. In the question the south pole goes first, and its description is given under the sub-heading 'Reverse pole' in web link you gave.... And yes, there is a law for this one and it is called the 'Lenzs law'. The below one is a wikipedia link for the law.. http://en.wikipedia.org/wiki/Lenz's_law Regards agoogler When one tries to find the direction of induced current, the poles of the magnets also count. In the question the south pole goes first, and its description is given under the sub-heading 'Reverse pole' in web link you gave.... And yes, there is a law for this one and it is called the 'Lenzs law'. The below one is a wikipedia link for the law.. http://en.wikipedia.org/wiki/Lenz's_law Regards I don't understand. Using the reverse pole rule on that webpage I'm getting the answer as (A) , can you please explain why the answer is (B)? Edit: Got it. Last edited:	2022-08-12 21:38:08	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8187439441680908, "perplexity": 538.948418466258}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571758.42/warc/CC-MAIN-20220812200804-20220812230804-00141.warc.gz"}
https://www.amplifiedparts.com/products/pedal-parts-diy?filters=2952a2965c2951a2952	Pedal Parts / DIY Audio Power Amplifier - LM386N-1 / 4, Low Voltage LM386 Low Voltage Audio Power Amplifier The LM386 is a mono low voltage amplifier that can be used in a variety of applications. It can drive loads from 4 Ω to 32 Ω. The gain is internally set to 20 but it can be modified from 20 to 200 by placing a resistor and capacitor between pins 1 and 8. As this is an Op Amp it can be used in different configurations to fit in several applications. The internal gain setting resistor allows the LM386 to be used in a very low part count system. In addition a series resistor can be placed between pins 1 and 5 to modify the gain and frequency response for specific applications. These are found in many small audio amplifiers and even tiny pedal sized guitar amplifiers! There are tons of possibilities for this little Op Amp. $1.70 Op-Amp - OP07, Single, Precision, Low Offset, 8-Pin DIP OP07 precision op amp in an 8-pin DIP package.These devices offer low offset and long-term stability by means of a low-noise, chopperless, bipolar-input-transistor amplifier circuit. For most applications, external components are not required for offset nulling and frequency compensation. The true differential input, with a wide input-voltage range and outstanding common-mode rejection, provides maximum flexibility and performance in high-noise environments and in noninverting applications. Low bias currents and extremely high input impedance are maintained over the entire temperature range.$1.59 Op-Amp - OP275, Dual, Bipolar/JFET, Audio, 8-Pin DIP OP275 dual audio op amp in an 8-pin DIP package. The OP275 is the first amplifier to feature the Butler Amplifier front end. This new front end design combines both bipolar and JFET transistors to attain amplifiers with the accuracy and low noise performance of bipolar transistors, and the speed and sound quality of JFETs. Total Harmonic Distortion plus Noise equals that of previous audio amplifiers, but at much lower supply currents. A very low l/f corner of below 6 Hz maintains a flat noise density response. Whether noise is measured at either 30 Hz or 1 kHz, it is only 6 nVHz. The JFET portion of the input stage gives the OP275 its high slew rates to keep distortion low, even when large output swings are required, and the 22 V/µs slew rate of the OP275 is the fastest of any standard audio amplifier. $4.25 Regulator - TC7660S, Charge Pump DC-TO-DC Voltage Converter, 8-Pin DIP TC7660S charge pump voltage converter in an 8 pin DIP package. The TC7660S device is a pin-compatible replacement for the industry standard 7660 charge pump voltage converter. It converts a +1.5V to +12V input to a corresponding -1.5V to -12V output using only two low-cost capacitors, eliminating inductors and their associated cost, size and electromagnetic interference (EMI). Added features include an extended supply range to 12V, and a frequency boost pin for higher operating frequency, allowing the use of smaller external capacitors. The on-board oscillator operates at a nominal frequency of 10 kHz. Frequency is increased to 45 kHz when pin 1 is connected to V+.$1.25 On Backorder Regulator - TC1044, Charge Pump DC-TO-DC Voltage Converter, 8-Pin DIP TC1044 charge pump voltage converter in an 8 pin DIP package. The TC1044S is a pin-compatible upgrade to the Industry standard TC7660 charge pump voltage converter. It converts a +1.5V to +12V input to a corresponding –1.5V to –12V output using only two low cost capacitors, eliminating inductors and their associated cost, size and EMI. Added features include an extended supply range to 12V, and a frequency boost pin for higher operating frequency, allowing the use of smaller external capacitors. The on-board oscillator operates at a nominal frequency of 10kHz. Frequency is increased to 45kHz when pin 1 is connected to V+. $2.50 IC Socket - Dual in-line package, 2.54mm Pitch, 7.62mm Spacing IC component sockets for dual in-line package chips (DIP, PDIP, DIL, DIPP.) These sockets feature ladder style cases with tinned stamped and formed connections. Starting at$0.29 Op-Amp - NJM4558L, Dual high-gain, 8-Pin SIP NJM4558L dual op amp from JRC in a SIP8 package. The NJM4558L is the same as the DIP NJM4558 (JRC4558) otherwise. The SIP version is a great replacement for dual op amps in vintage circuits as well as a space saving package for some PCB layouts. The NJM4558L is a dual high-gain operational amplifier with internal compensation circuit and constructed on a single silicon chip. It offers excellent characteristics by combining the parameters adjusted for a monolithic chip. The channel separation characteristic is suitable for measuring instruments. Features: • Operating Voltage ( ±4V$±18V ) • High Voltage Gain ( 100dB typ. ) • High Input Resistance ( 5MΩ typ. ) • Bipolar Technology See P-Q4558 for the DIP version of this chip $0.85 Op-Amp - LM324, Quad, Low-Power, 14-Pin DIP Quad low-power opamps in an 14-pin DIP package. The LM324 offers a 1MHz bandwidth and can be powered from a single supply.$0.29 Op-Amp - LM1458, Dual, 8-Pin DIP LM1458 dual op amp in an 8-pin DIP package. The LM1458 is a general purpose dual operational amplifier. The two amplifiers share a common bias network and power supply leads. Otherwise, their operation is completely independent. Features • No Frequency Compensation Required • Short-Circuit Protection • Wide Common-Mode and Differential Voltage Ranges • Low-Power Consumption • 8-Lead TO-99 and 8-Lead PDIP • No Latch Up When Input Common Mode Range is Exceeded $1.20 Comparator - LM311, Single, High Speed, 8-Pin DIP LM311 single comparator IC in an 8-pin DIP package. The LM311 device is a single high-speed voltage comparator. These devices are designed to operate from a wide range of power supply voltages, including ±15-V supplies for operational amplifiers and 5-V supplies for logic systems. The output levels are compatible with most TTL and MOS circuits. These comparators are capable of driving lamps or relays and switching voltages up to 50 V at 50 mA. All inputs and outputs can be isolated from system ground. The outputs can drive loads referenced to ground, VCC+ or VCC−. Offset balancing and strobe capabilities are available, and the outputs can be wire-OR connected.$0.49 Op-Amp - NE5532, Dual, Low Noise, Audio, 8-Pin DIP NE5532 dual op amp in an 8-pin DIP package. The NE5532 device is a high-performance operational amplifier combining excellent DC and AC characteristics. They feature very low noise, high output-drive capability, high unity-gain and maximum-output-swing bandwidths, low distortion, high slew rate, input-protection diodes, and output short-circuit protection. These operational amplifiers are compensated internally for unity-gain operation. These devices have specified maximum limits for equivalent input noise voltage. Features • Equivalent Input Noise Voltage: 5 nV/√Hz Typ at 1 kHz • Unity-Gain Bandwidth: 10 MHz Typ • Common-Mode Rejection Ratio: 100 dB Typ • High DC Voltage Gain: 100 V/mV Typ • Peak-to-Peak Output Voltage Swing 26 V Typ With VCC± = ±15 V and RL = 600 Ω • High Slew Rate: 9 V/μs Typ $0.59 Op-Amp - LM358, Dual, 8-Pin DIP LM358 dual op amp in an 8-pin DIP package. These devices consist of two independent, high-gain, frequency-compensated operational amplifiers designed to operate from a single supply over a wide range of voltages. Operation from split supplies also is possible if the difference between the two supplies is 3 V to 32 V (3 V to 26 V for the LM2904), and VCC is at least 1.5 V more positive than the input common-mode voltage. The low supply-current drain is independent of the magnitude of the supply voltage. Applications include transducer amplifiers, dc amplification blocks, and all the conventional operational amplifier circuits that now can be implemented more easily in single-supply-voltage systems.$0.49 Op-Amp - NJM4580D, Dual, Audio, DIP 8 The NJM4580D is a dual opamp popularly used in many distortion pedals. This chip is a dual operational amplifier, specially designed for improving the tone control, which is most suitable for the audio application. Featuring noiseless, higher gain bandwidth, high output current and low distortion ratio, it is most suitable not only for acoustic electronic parts of audio pre-amp and active filter, but also for the industrial measurement tools. It is also suitable for the headphone amp at higher output current, and furthermore, it can be applied for the handy type set operational amplifier of general purpose in application of low voltage single supply type which is properly biased of the low voltage source. Features: • Operating Voltage ±2V$±18V • Low Input Noise Voltage 0.8µVrms typ. $1.05 Op-Amp - AS301AN, Alfa, Single, 8-Pin DIP The AS301AN is Alfa’s version of the LM301A in an 8 pin DIP package. The AS301A is a general purpose operational amplifier. This amplifier offers many features which make its application nearly foolproof: overload protection on the input and output, no latch-up when the common mode range is exceeded, and freedom from oscillations and compensation with a single 30 pF capacitor. It has advantages over internally compensated amplifiers in that the frequency compensation can be tailored to the particular application. For example, in low frequency circuits it can be overcompensated for increased stability margin. Or the compensation can be optimized to give more than a factor of ten improvements in high frequency performance for most applications. In addition, the device provides better accuracy and lower noise in high impedance circuitry.$1.95 Op-Amp - NJM4558, Dual high-gain, 8-Pin DIP Dual Operational Amplifier. The NJM4558 (JRC4558) integrated circuit is a dual high-gain operational amplifier internally compensated and constructed on a single silicon chip using an advanced epitaxial process. Combining the features of the NJM741 with the close parameter matching and tracking of a dual device on a monolithic chip results in unique performance characteristics. Excellent channel separation allows the use of the dual device in single NJM741 operational amplifier applications providing density. It is especially well suited for applications in differential-in, differential-out as well as in potentiometric amplifiers and where gain and phase matched channels are mandatory. Features: • Operating Voltage (±4V~±18V) • High Voltage Gain (100dB typ.) • High Input Resistance (5MΩ typ.) • Bipolar Technology $0.90 Audio Power Amplifier - LM380, Class-AB, 2.5W, 14-Pin DIP LM380 power amp in a 14-pin DIP package. The LM380 is a power audio amplifier for consumer applications. In order to hold system cost to a minimum, gain is internally fixed at 34 dB. A unique input stage allows ground referenced input signals. The output automatically self-centers to one-half the supply voltage. The output is short circuit proof with internal thermal limiting. The package outline is standard dual-in-line. The LM380N uses a copper lead frame. The center three pins on either side comprise a heat sink.$1.95 OTA - LM13700, Dual, Linearizing Diodes and Buffers, 16-Pin DIP The LM13700 series consists of two current controlled transconductance amplifiers, each with differential inputs and a push-pull output. The two amplifiers share common supplies but otherwise operate independently. Linearizing diodes are provided at the inputs to reduce distortion and allow higher input levels. The result is a 10-dB signal-to-noise improvement referenced to 0.5 percent THD. High impedance buffers are provided which are especially designed to complement the dynamic range of the amplifiers. The output buffers of the LM13700 differ from those of the LM13600 in that their input bias currents (and thus their output DC levels) are independent of IABC. $2.35 Op-Amp - LF347, Quad, Wide Bandwidth, 14-Pin DIP LF347 quad op amp in a 14-pin DIP package. The LF347 is a low cost, high speed quad JFET input operational amplifier with an internally trimmed input offset voltage ( BI-FET II™ technology). The device requires a low supply current and yet maintains a large gain bandwidth product and a fast slew rate. In addition, well matched high voltage JFET input devices provide very low input bias and offset currents. The LF347 is pin compatible with the standard LM348. This feature allows designers to immediately upgrade the overall performance of existing LF348 and LM324 designs. The LF347 may be used in applications such as high speed integrators, fast D/A converters, sample-and-hold circuits and many other circuits requiring low input offset voltage, low input bias current, high input impedance, high slew rate and wide bandwidth.$0.69 Op-Amp - M5201, Dual, With Switch, 8-Pin DIP The NJM2120 is a dual operational amplifier of 2-INPUT and 1-OUTPUT with analog switch. The NJM2120 can be used as analog switch under the condition of GV=0dB, as Switch + Amp in order that each gain (A or B) can be adjusted independently. Each amplifier of the NJM2120 has the same electrical characteristics as the NJM4558. The NJM2120 is suitable for audio, video, electrical musical instrument, etc. $4.50 Op-Amp - OPA134, Single, High Performance Audio, 8-Pin DIP OPA134 single op amp in an 8-pin DIP package. The OPA134 series are ultra-low distortion, low-noise operational amplifiers fully specified for audio applications. A true FET input stage is incorporated to provide superior sound quality and speed for exceptional audio performance. This, in combination with high output drive capability and excellent DC performance, allows for use in a wide variety of demanding applications. In addition, the OPA134 has a wide output swing, to within 1 V of the rails, allowing increased headroom and making it ideal for use in any audio circuit. The OPA134 SoundPlus™ audio operational amplifiers are easy to use and free from phase-inversion and the overload problems often found in common FET-input operational amplifiers. They can be operated from ±2.5-V to ±18-V power supplies.$2.95 OTA - AS3080ED, Linearizing Diodes, Alfa, SOIC-8 The AS3080ED is Alfa’s version of the CA3080 IC in an SOIC package. The AS3080ED is a gain block which is the operational-transconductance-amplifier (OTA). The AS3080ED has differential input and a single-ended, push-pull, class A output. Amplifier bias input may be used either for gating or for linear gain control. High output impedance and transconductance (gM) is directly proportional to the amplifier bias current (IABC). Linearizing diodes are provided at the inputs to reduce distortion and allow higher input levels. The result is a 10-dB signal-to-noise improvement referenced to 0.5 percent THD. The AS3080ED is notable for it’s high slew rate (50V/µs), which makes it especially useful for multiplexer and fast unity-gain voltage followers. $4.75 Op-Amp - TL071, Single, Low-Noise, JFET-input, 8-Pin DIP TL071 single op amp in an 8-pin DIP package. The TL07xx JFET-input operational amplifier family is designed to offer a wider selection than any previously developed operational amplifier family. Each of these JFET-input operational amplifiers incorporates well-matched, high-voltage JFET and bipolar transistors in a monolithic integrated circuit. The devices feature high slew rates, low-input bias and offset currents, and low offset-voltage temperature coefficient. The low harmonic distortion and low noise make the TL07xseries ideally suited for high-fidelity and audio pre-amplifier applications.$0.82 Op-Amp - TL072, Dual, Low-Noise, JFET Input, 8-Pin DIP The TL072 is a high speed JFET input dual operational amplifier incorporating well matched, high voltage JFET and bipolar transistors in a monolithic integrated circuit. The device features high slew rates, low input bias and offset current, and low offset voltage temperature coefficient. $0.70 Op-Amp - TL022, Dual, Low Power, 8-Pin DIP TL022 dual op amp in an 8-pin DIP package. The TL022 is a dual low-power operational amplifier designed to replace higher power devices in many applications without sacrificing system performance. High input impedance, low supply currents, and low equivalent input noise voltage over a wide range of operating supply voltages result in an extremely versatile operational amplifier for use in a variety of analog applications including battery-operated circuits.$0.69 Don't see what you're looking for? Send us your product suggestions!	2022-12-03 08:16:49	{"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": 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": 0.2886280119419098, "perplexity": 5854.65617015662}, "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-2022-49/segments/1669446710926.23/warc/CC-MAIN-20221203075717-20221203105717-00786.warc.gz"}
https://physics.nfshost.com/textbook/08-Materials/02-Polarization.php	# Polarization Neutral objects are attracted to charged objects. You've seen this effect if you've ever rubbed a balloon on your head (which charges the balloon) and stuck it to a (neutral) wall. Or if you've removed a piece of plastic wrap from the roll and had it stick to your (neutral) hand. We know that like charges repel and opposite charges attract, but why would a neutral object react to a charge? ## Polarization of a Conductor Suppose you place a positively charged rod next to a conductor, as shown. The negative charge carriers inside the conductor will be attracted to the positive charge, and because they are free to move where they like, negative charge will start to build up on the surface facing the positive charge. Positive charge carriers inside the conductor, on the other hand, are repelled by the external charge, and so will build up on the surface away from it. (Or, if you prefer, the negative charge carriers abandon the far side of the conductor, leaving it with a net positive charge.) The conductor has become polarized: positive charge on one side, negative charge on the other. Now the negative charge isn't satisfied with sitting on the surface. It is still attracted to the rod, and so will try to pull the conductor closer to the rod. The positive charge on the other side, meanwhile, will try to push the conductor farther away. But the negative charges are closer to the rod, and so experience a greater electric force. They win, and the neutral conductor as a whole is attracted to the positive charge. If we bring a negative charge towards the conductor instead, the conductor polarizes in the opposite direction, but the same result occurs: the nearby positive charges drag the conductor towards the negative charge. ## Polarization of an Insulator But wait, you say. This process requires charge carriers to flow from one side of the material to the other, and insulators (such as the walls and hands I mentioned earlier) don't have charge carriers. Are they attracted to charges too? Can they polarize? Interactive 8.2.1 To answer that question, we first have to consider what happens to an atom in the presence of another charge. Atoms, as you'll recall, are made up of a positive nucleus and a negative "electron cloud", bound to each other by electrostatic attraction. When an atom comes close to a positive charge, the nucleus is repelled by the charge and the cloud is attracted to the charge. The figure shows the result: the atom itself becomes polarized. When a positive rod comes close to an insulator, all of its atoms polarize in this way, with the electron clouds leaning towards the rod and the nuclei leaning away. In some materials (like water), the molecules themselves are naturally polar: they don't have to stretch, they just rotate their negative end towards the positive rod. In either case, the side of the insulator closest to the positive charge develops a layer of negative charge, while the opposite side develops a positive layer. Insulators polarize just as conductors do, though the mechanism is different. The difference is one of degree: conductors are much better at polarizing. The polarized layers in a conductor are made up of charge carriers from the entire material, while an insulator's layers only include the charges that were already at the surface to begin with. I place an insulator into an electric field that points to the left. How does it polarize? Remember that positive charges are pushed with the field, and negative charges are pushed against the field. The insulator will develop a layer of positive charge on the left, and a layer of negative charge on the right. ## Electric Breakdown While an atom will stretch a little bit in an electric field, it holds together because the positive nucleus and the negative cloud attract to each other. If the electric field is strong enough, however, it can overpower the force binding the electrons to the atom, and one or more of the valence electrons may be torn free from the atom, turning the atoms into ions. This is called electric breakdown, or we say that the material is ionized. When this occurs in an insulator, those free electrons and ions can act as charge carriers, and the insulator becomes a temporary conductor. For example, clouds often develop a positive charge in their upper layers and a negative charge in their lower layers, creating a dipole field inside the cloud. (Part (a) in the figure.) When this field gets strong enough, the air inside the cloud ionizes (b), becoming a conductor. Positive charge begins to flow downward from the upper layer to the lower layer, and negative charge in the opposite direction. As the charge imbalance shrinks, the electric field between the clouds shrinks as well, and eventually the conditions for electric breakdown no longer apply. The free electrons reunite with their ions, and when they collide energy is released in the form of heat, light, and sound (c). This release of energy is what we call lightning and thunder. Note that the lightning we see is not the motion of charge itself, but the aftereffect, the un-ionizing of the air. This same thing occurs at a much smaller scale whenever a charged object (say, your finger after you've shuffled across a carpet) comes too close to another object (like a doorknob): that spark and crackle is just very, very tiny lightning. Every insulator has its own threshold for electric breakdown. Air, for example, undergoes partial ionization when the electric field is around $$3\ten6\u{N/C}$$. A large charged plate (surface charge density $$\sigma$$ is placed above a neutral metal plate; the gap between them is filled with air. What is the maximum value of $$\sigma$$ which won't result in a spark jumping between the plates? The electric field created by the charged plate is $$E=2\pi k\sigma$$. To prevent any sparks from jumping, we need this field to be smaller than $$3\ten6\u{N/C}$$, and so $$2\pi k\sigma<3\ten6 \implies \sigma<\frac{3\ten6}{2\pi(9\ten9)}=5.3\ten{-5}\u{C/m^2}$$ or $$53\u{\mu C/m^2}$$. Remember our discussion of how large a coulomb of charge is (Example 3.1.1)? This result suggests that it would be hard to store even a millicoulomb of charge on a relatively small object, without that charge bleeding away through the ionized air surrounding it.	2022-05-18 08:52:26	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5271536111831665, "perplexity": 401.16065790955776}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662521883.7/warc/CC-MAIN-20220518083841-20220518113841-00060.warc.gz"}
http://elibm.org/article/10002936	## Existence of axisymmetric weak solutions of the 3-D Euler equations for near-vortex-sheet initial data ### Summary Summary: We study the initial value problem for the 3-D Euler equation when the fluid is inviscid and incompressible, and flows with axisymmetry and without swirl. On the initial vorticity $\omega_0$, we assumed that $\omega_0/r$ belongs to $L(\log L (\Bbb R^3))^{\alpha}$ with $\alpha$, where $r$ is the distance to an axis of symmetry. To prove the existence of weak global solutions, we prove first a new a priori estimate for the solution. 35Q35, 76C05 ### Keywords/Phrases Euler equations, axisymmetry, weak solution	2019-05-19 09:39:34	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.930864155292511, "perplexity": 422.3195713851156}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232254731.5/warc/CC-MAIN-20190519081519-20190519103519-00159.warc.gz"}
https://physlets.org/roess/roessse34.html	### 6.6 Parameter representation of curves, space paths $Math content$ Using this parameter representation very complicated curves (paths) in space can be described. The functions $Math content$ ,that are displayed in the three function windows, map the interval covered by the only parameter $Math content$ uniquely to a curve $Math content$in space. If $Math content$ contain periodic functions of the parameters, closed ore self-intersecting space curves are created. For the simulation in Fig6.11 the one-dimensional parameter $Math content$ is interpreted as time. This parameter is repeatedly incremented by a constant time-step, such that the curve starting at the origin grows accordingly, until one of the coordinates becomes larger than $Math content$ and leaves the range of the figure and the animation stops. The blue path marker is connected to the origin with a vector. The vector and the $Math content$-pane can be switched on and off with the option switch. The program calculates the functions in time-steps of $Math content$ milliseconds. Thus animation speed can be set With the slider $Math content$. For $Math content$ the picture is static. With the sliders $Math content$ up to three constants in the parameter functions can adjusted between 0 and 1. The sliders actually determine the product of the constant with 100, such that that the constant as well as the ratio of two of these constants. This leads to closed orbits in the case of oscillation plots. In the second example the irrational number $Math content$ is added to the rational number $Math content$, which results in the orbit not being closed. This shows who you can in general create orbits that are not closed. You may increase the animation speed to recognized this quickly. For the detailed observation the projection settings of the camera inspector are useful. In the $Math content$-plane one sees the corresponding plane orbits, i.e. plane Lissajou-figures. Choose after the first animation the constants $Math content$ such, that the range of coordinates is fully used. Many plots become graphically interesting only if the constants $Math content$ are chosen differently. The default value for all of them is $Math content$, to the show the basic functions during the first run. You can edit the formulas or enter new ones from scratch. The scale has been chosen in such a way for all three axes, that the range $Math content$ to $Math content$ is available. The $Math content$-plane is intersected by $Math content$-axis in the middle of the $Math content$-vectors. maximum and minimum values are marked on the $Math content$-axis via a red and green point respectively. With the sliders $Math content$ you may, even during the animation, change the parameters of the space curves. With suitable entries of time-dependent functions you can also switch the animation to other quantities. The handling of the simulation is otherwise again analogous to that of the previous $Math content$- presentations. Details are given on the description pages. There are however two keys for starting the simulation with slightly different functions: Start starts the simulation and erases all the curves that are present. Play does not delete previous curves , continues for equal parameters with the simulation and superimposes old an new curves for change parameters or changed function types. Stop stops as second functionality of the Play-button the simulation, that is continued with Play. Clear deletes all curves. Reset a b c resets $Math content$ to the default values. This simulation also gives ample opportunities for creative and playful experiments. The following picture shows the simulation. Es shows to interleaved orbits, one of which with the hyperbolic envelope is already closed, while the one with the envelope in the shape of a torus is still open. End of chapter 6	2021-03-01 01:20:23	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 27, "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": 0.7734240293502808, "perplexity": 656.9233511215073}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1614178361808.18/warc/CC-MAIN-20210228235852-20210301025852-00068.warc.gz"}
http://learning.maxtech4u.com/tag/application-of-k-nearest-neighbor/	k Nearest Neighbor (KNN) / August 11, 2017 k Nearest Neighbor (KNN): introduction The necessity of data mining techniques has emerged quite immensely nowadays due to massive increase in data. Data mining is the process of extracting patterns and mining knowledge from data. K nearest neighbors is a simple algorithm that stores all available cases and classifies new cases based on a similarity measure (e.g., distance functions). KNN has been used in statistical estimation and pattern recognition already in the beginning of 1970’s as a non-parametric technique. The model for KNN is the entire training dataset. When a prediction is required for a unseen data instance, the KNN algorithm will search through the training dataset for the k-most similar instances. The prediction attribute of the most similar instances is summarized and returned as the prediction for the unseen instance. Nearest neighbor classifiers is a lazy learner’s method and is based on learning by analogy. It is a supervised classification technique which is used widely. Unlike the previously described methods the nearest neighbor method waits until the last minute before doing any model construction on a given tuple. In this method the training tuples are represented in N-dimensional space. When given an unknown tuple, k-nearest neighbor classifier searches the k… Insert math as $${}$$	2018-07-20 14:41:28	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.31524959206581116, "perplexity": 1025.6575706318586}, "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-2018-30/segments/1531676591683.78/warc/CC-MAIN-20180720135213-20180720155213-00520.warc.gz"}
https://dml.cz/handle/10338.dmlcz/104301	# Article Full entry \| PDF (2.4 MB) Keywords: two step extrapolation; optimum choice of relaxation factor; convergence acceleration; successive overrelaxation; iterative process; S.O.R. method Summary: Limits of the extrapolation coefficients are rational functions of several poles with the largest moduli of the resolvent operator $R(\lambda, T)=(\lambda I -T)^{-1}$ and therefore good estimates of these poles could be calculated from these coefficients. The calculation is very easy for the case of two coefficients and its practical effect in finite dimensional space is considerable. The results are used for acceleration of S.O.R. method. References: [1] J. Zítko: Improving the convergence of iterative methods. Apl. Mat. 28 (1983), 215-229. MR 0701740 [2] J. Zítko: Convergence of extrapolation coefficients. Apl. Mat. 29 (1984), 114-133. MR 0738497 [3] J. Zítko: Extrapolation of iterative processes. Rostock. Math. Kolloq. 25, 63-78 (1984). MR 0763678 [4] I. Marek J. Zítko: Ljusternik acceleration and the extrapolated S.O.R. method. Appl. Mat. 22 (1977), 116-133. MR 0431667 [5] A. E. Taylor: Introduction to Functional Analysis. J. Wiley Publ. New-York 1958. MR 0098966 \| Zbl 0081.10202 [6] D. M. Young: Iterative Solution of Large Linear Systems. Academic Press, New York- London, 1971. MR 0305568 \| Zbl 0231.65034 [7] R. S. Varga: Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs, New Jersey 1962. MR 0158502 [8] G. Maess: Extrapolation bei Iterationsverfahren. ZAMM 56 (1976), 121-122. DOI 10.1002/zamm.19760560210 \| MR 0426417 [9] G. Maess: Iterative Lösung linearer Gleichungssysteme. Deutsche Akademie der Naturforscher Leopoldina Halle (Saale), 1979. MR 0558164 \| Zbl 0416.65029 Partner of	2022-01-27 05:08:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8635869026184082, "perplexity": 5976.202306999435}, "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-2022-05/segments/1642320305141.20/warc/CC-MAIN-20220127042833-20220127072833-00483.warc.gz"}
https://math.stackexchange.com/questions/2576594/is-a-continuous-function-f-colon0-infty-to-r-such-that-fx-leq-fnx-i/2576672	# Is a continuous function $f\colon(0,\infty)\to R$, such that $f(x)\leq f(nx)$ increasing? Let $f\colon (0, \infty)\to R$ be continuous such that $f(x)\leq f(nx)$ for all positive $x$ and natural $n$. It was proved that the limit (finite or infinite) in the infinity exists. Do we know if such a function must be (weakly) increasing? I believe that there might be counterexamples. • The fact that infinite limit exists is a weird (i.e. mathematical) way to say: there is no limit. In other words, it means that a sequence not bound to any concrete value. Dec 22, 2017 at 9:47 • @52heartz From the topological point of view it is pretty natural definition. Dec 22, 2017 at 9:49 • @52heartz it just another way to say that $\sup f(x)$ and $\inf f(x)$ are equal when taking the limit – ℋolo Dec 22, 2017 at 9:50 Let $$f(x)= \begin{cases} x \quad &\text{if} \quad x\leq 1\\ 2-x \quad &\text{if} \quad 1\leq x \leq 4/3\\ x- 2/3 \quad &\text{if} \quad x\geq 4/3 \end{cases}$$ In $[1,4/3]$, $f(x)$ has minimum $2/3$, and in $[1/2,2/3]$, it has maximum $2/3$. Hence satisfies the condition. Other regions also satisfies the condition. It is also continuous. • Thank you for comment. Legend was not reversed but I was plotting $f(x/2)$ and $f(x/3)$, not $f(2x)$ and $f(3x)$. I fixed it. Function is not increasing on $[1,4/3]$. @uniquesolution Dec 23, 2017 at 8:21 • Why the downvote? This is a counterexample to the claim, hence a valid answer. Dec 23, 2017 at 8:39 You can take $f(x)$ such that: $f(x)=10^x$ for $x \in [0,10]$ $f(x)=10^{10}-(x-10)^{100}$ for $x\in [10,11]$ $f(x)=10^{10}-1+10^{10}\times (x-11)$ for $x>11$ • Is this a counter-example? If so, why? Dec 22, 2017 at 20:50 I thought about this and the answer is no, I'll edit my answer in the other question, thank you for letting me know for my mistake. The function: We will define $[x]=x-\lfloor x\rfloor$ $$f(x)=\begin{cases}x&\text{if}&[x]=0\\ f(\lfloor x\rfloor)-2[x]&\text{if}&[x]\in(0,0.5)\\ f(\lfloor x\rfloor)+2[x]&\text{if}&[x]\in[0.5,1)\end{cases}$$ • Your answer for the previous question really needs editing or deleting. All you proof was based on the assumption, that $f$ is (weakly) increasing. Dec 22, 2017 at 10:46 • @PrzemysławScherwentke I edit it, I said it was wrong and link to here – ℋolo Dec 22, 2017 at 10:47	2022-05-29 02:25:06	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.914556086063385, "perplexity": 355.55747845283753}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1652663035797.93/warc/CC-MAIN-20220529011010-20220529041010-00489.warc.gz"}
https://fearlessmath.net/mod/page/view.php?id=55	## Fraction Calculator Fill in the values and choose the operation. Press "Solve Problem" to see the answer. _______ + - × ÷ ______ =	2021-04-16 17:20:41	{"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": 0.9203446507453918, "perplexity": 9315.95398506936}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1618038088245.37/warc/CC-MAIN-20210416161217-20210416191217-00486.warc.gz"}
https://www.aimsciences.org/article/doi/10.3934/mfc.2022014	# American Institute of Mathematical Sciences • Previous Article Some structures of submatrices in solution to the paire of matrix equations $AX = C$, $XB = D$ • MFC Home • This Issue • Next Article On a special class of modified integral operators preserving some exponential functions doi: 10.3934/mfc.2022014 Online First Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible. Readers can access Online First articles via the “Online First” tab for the selected journal. ## Adaptive attitude determination of bionic polarization integrated navigation system based on reinforcement learning strategy 1 School of Information Science and Technology, North China University of Technology, Beijing 100144, China 2 Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education, Southeast University, Nanjing, China, 210096 *Corresponding author: Tao Du Received March 2022 Revised April 2022 Early access May 2022 The bionic polarization integrated navigation system includes three-axis gyroscopes, three-axis accelerometers, three-axis magnetometers, and polarization sensors, which provide pitch, roll, and yaw. When the magnetometers are interfered or the polarization sensors are obscured, the accuracy of attitude will be decreased due to abnormal measurement. To improve the accuracy of attitude of the integrated navigation system under these complex environments, an adaptive complementary filter based on DQN (Deep Q-learning Network) is proposed. The complementary filter is first designed to fuse the measurements from the gyroscopes, accelerometers, magnetometers, and polarization sensors. Then, a reward function of the bionic polarization integrated navigation system is defined as the function of the absolute value of the attitude angle error. The action-value function is introduced by a fully-connected network obtained by historical sensor data training. The strategy can be calculated by the deep Q-learning network and the action that optimal action-value function is obtained. Based on the optimized action, three types of integration are switched automatically to adapt to the different environments. Three cases of simulations are conducted to validate the effectiveness of the proposed algorithm. The results show that the adaptive attitude determination of bionic polarization integrated navigation system based on DQN can improve the accuracy of the attitude estimation. Citation: Huiyi Bao, Tao Du, Luyue Sun. Adaptive attitude determination of bionic polarization integrated navigation system based on reinforcement learning strategy. Mathematical Foundations of Computing, doi: 10.3934/mfc.2022014 ##### References: show all references ##### References: Illustration of DQN Variation of geomagnetic field intensity under geomagnetic interference Comparison decision action under geomagnetic interference Attitude estimation errors under geomagnetic interference Variation of polarization angle under polarization interference Comparison decision action under polarization interference Attitude estimation errors under polarization interference (a) Polarization angle under polarization interference. (b) Geomagnetic field intensity under geomagnetic interference Comparison of decision-making actions when the magnetometer is disturbed and polarization is blocked Attitude estimation errors under the magnetometer is disturbed and polarization is blocked Standard deviation of attitude angle for decision comparison in experiment 1 Method Pitch(°) Roll(°) Yaw(°) Complementary filter without DQN 0.4958 0.5057 0.5866 Complementary filter with DQN 0.4790 0.5022 0.3540 Method Pitch(°) Roll(°) Yaw(°) Complementary filter without DQN 0.4958 0.5057 0.5866 Complementary filter with DQN 0.4790 0.5022 0.3540 Standard deviation of attitude angle for decision comparison in experiment 2 Method Pitch(°) Roll(°) Yaw(°) Complementary filter without DQN 0.5450 0.5078 0.4700 Complementary filter with DQN 0.4640 0.5031 0.4241 Method Pitch(°) Roll(°) Yaw(°) Complementary filter without DQN 0.5450 0.5078 0.4700 Complementary filter with DQN 0.4640 0.5031 0.4241 Standard deviation of attitude angle for decision comparison in experiment 3 Method Pitch(°) Roll(°) Yaw(°) Complementary filter without DQN 0.4966 0.5052 0.6005 Complementary filter with DQN 0.4735 0.5031 0.3822 Method Pitch(°) Roll(°) Yaw(°) Complementary filter without DQN 0.4966 0.5052 0.6005 Complementary filter with DQN 0.4735 0.5031 0.3822 [1] Jingang Zhao, Chi Zhang. 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Journal of Industrial and Management Optimization, 2020, 16 (6) : 3065-3081. doi: 10.3934/jimo.2019094 Impact Factor:	2022-08-15 13:16:40	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.51505446434021, "perplexity": 8959.490765570039}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1659882572174.8/warc/CC-MAIN-20220815115129-20220815145129-00770.warc.gz"}
https://cstheory.stackexchange.com/tags/fl.formal-languages/hot	# Tag Info 44 To complete the other answers: I think that Turing Machine are a better abstraction of what computers do than finite automata. Indeed, the main difference between the two models is that with finite automata, we expect to treat data that is bigger than the state space, and Turing Machine are a model for the other way around (state space >> data) by making the ... 32 There are two approaches when considering this question: historical that pertains to how concepts were discovered and technical which explains why certain concepts were adopted and others abandoned or even forgotten. Historically, the Turing Machine is perhaps the most intuitive model of several developed trying to answer the Entscheidungsproblem. This is ... 27 Every context-free language has either polynomial growth or exponential growth. In the notation of the question poser: Either there is a polynomial $p$ so that $w_n\le p(n)$ for all $n$ Or there exists a $c>1$, so that $w_n\ge c^n$ for infinitely many $n$. This has been shown for instance in: Roberto Incitti: "The growth function of context-free ... 18 There is the notion of primality of a language. It asks whether L can be written as $L_1 \cdot L_2$ where neither factor contains the empty word. A language is prime if it cannot be written in this form. For a given regular language, represented by a DFA, it is shown in [MNS] that it is PSPACE-complete to decide primality. [MNS] Wim Martens, Matthias ... 17 Take $S_5$ as alphabet and $$L= \{ \sigma_1\cdots \sigma_n \in S_5^\mid \sigma_1\circ\cdots\circ\sigma_n = \text{Id}\}$$ Barrington proved in [2] that $L$ is $\textrm{NC}^1$-complete for $\textrm{AC}^0$ reduction (and even with a more restrictive reduction actually). In particular this shows that regular languages are not in $\textrm{TC}^0$ if $\textrm{... 16 There is even a stronger result than your request: There are exponentially-ambiguous NFAs for which the minimal polynomially-ambiguous NFAs are exponentially larger, and in particular the minimal UFAs. Check this paper by Hing Leung. 16 Visibly pushdown automata (or nested word automata, if you prefer working with nested words instead of finite words) extend the expressive power of deterministic finite automata: the class of regular languages is strictly contained within the class of visibly pushdown languages. For deterministic visibly pushdown automata, the language inclusion problem can ... 16 It's discussed in one of the very first papers about strings and complexity, namely, Dana Angluin, Finding patterns common to a set of strings, J. Comput. System Sci. 21 (1980), 46-62. Look at Theorem 3.6. The problem is NP-complete. It's also in A. Ehrenfeucht, G. Rozenberg, Finding a homomorphism between two words is NP-complete, Inform. Process. Lett. ... 16 They are typically called AND-functions. (I'm not joking.) Indeed, this concept has been considered before, and that's what people call them. See, for example, the book by Kobler, Schoning, and Toran on Graph Iso, where they talk about AND- and OR-functions for GI. And, by the way, there is an OR-function for GI (ibid.). The question of an AND-function for ... 15 Regular languages with unsolvable syntactic monoids are$\mathrm{NC}^1$-complete (due to Barrington; this is the underlying reason behind the more commonly quoted result that$\mathrm{NC}^1$equals uniform width-5 branching programs). Thus, any such language is not in$\mathrm{TC}^0$unless$\mathrm{TC}^0=\mathrm{NC}^1$. My favorite$\mathrm{NC}^1$-... 14 I think the IJFCS'05 paper by Leung: Descriptional complexity of nfa of different ambiguity provides an example with a family of NFA accepting finite languages that involve an exponential blowup for "disambiguation" (in the proof of Theorem 5). What is more, those automata have a special structure (DFA with multiple initial states). 14 If infinite words are in your scope, you can generalize DFA (with parity condition) to the so-called Good-for-Games automata (GFG), that still have polynomial containment. A NFA is GFG if there is a strategy$\sigma:A^\times Q\times A\to \Delta$, that given the prefix read so far and the current state and letter, chooses a transition to go to the next ... 14 Essentially the same argument is made by Andries P.J. van der Walt (1976, Lemma 2.3 and Example 2.9) for the variant of the pumping lemma where$N$letters are marked and all three of$x$,$y$,$z$must contain marked letters. See also Autebert, Boasson, and Cousineau (1978) for more properties of abstract families of languages satisfying this variant of ... 14 Another paper to look at: Kai Salomaa, "Language Decompositions, Primality, and Trajectory-Based Operations", 2008. 13 This question generated a lot of literature in the 80's, partly due to a bad approach to the problem. This is a rather long story that I will try to summarize in this answer. 1. The case of finite words One can find two definitions of a minimal DFA in the literature. The first one is to define the minimal DFA of a regular language as the complete DFA ... 13 (I guess the important word in the original question is published''.) There is such an encoding of context-free parsing (more exactly of CYK-style parsing) in Roland Axelsson, Keijo Heljanko, and Martin Lange, Analyzing Context-Free Grammars Using an Incremental SAT Solver, ICALP 2008, Lecture Notes in Computer Science vol. 5126, pp. 410--422, doi:10.... 13 This question is related to the so called insertion systems. An insertion system is a special type of rewriting system whose rules are of the form$1 \rightarrow r$for all$r$in a given language$R$. Let us write$u \rightarrow_R v$if$u = u'u''$and$v = u'ru''$for some$r \in R$. Let us denote by$\buildrel{}\over\rightarrow_R$the reflexive ... 13 The answer is yes. Suppose we have a factorization$Q = A\cdot B$. One easy observation is that$A$and$B$must be disjoint (since for$w\in A\cap B$we get$w^2\in Q$). In particular, only one of$A,B$can contain$\epsilon$. We can assume wlog (since the other case is completely symmetric) that$\epsilon\in B$. Then since$a$and$b$cannot be factored ... 13 Yes, every regular expression can be converted into an unambiguous one by converting to a DFA and then to a regular expression. And no, there aren't any inherently ambiguous regular languages in the sense described in the question. This is a classic result in automata theory: R. Book, S. Even, S. Greibach and G. Ott, Ambiguity in graphs and expressions, ... 12 The bounds... We have in fact$NFA(L) \ge Cov(M) + Cov(N)$, see Theorem 4 in (Gruber & Holzer 2006). For an upper bound, we have$2^{Cov(M)+Cov(N)} \ge DFA(L) \ge NFA(L)$, see Theorem 11 in the same paper. ...cannot be substantially improved There can be a subexponential gap between$Cov(M)+Cov(N)$and$NFA(L)$. The following example, and the proof ... 12 You can show$ \|L\|^i $is a tight upper bound by using the following language:$ L = \{ ab,aab,aaab,\ldots,a^kb \mid k \geq 1 \}. $Any concatenation gives a new string. For a lower bound, I can suggest the following unary language:$ U = \{a,aa,aaa,\ldots,a^k \mid k \geq 1 \} $. Then,$ U^i = \{ a^i,a^{i+1},\ldots,a^{ki} \} $and so$ \|U^i\| = i\|U\|-... 12 In general, $\omega$-regular languages may not have a unique minimal DBW. For example, the language "infinitely many a's and infinitely many b's" has two 3-state DBWs (in the picture replace $\neg a$ by $b$): As you can see, they are not topologically equivalent. Hence, the minimization problem is harder than the finite case, and in fact, it is NP-complete.... 12 If $\mathrm{P\subseteq CSL}$, then $\mathrm{P\subseteq DSPACE}(n^2)$. By a padding argument, this implies $$\mathrm{DTIME}(t(n))\subseteq\mathrm{DSPACE}\bigl(t(n)^\epsilon\bigr)$$ for every superpolynomial well-behaved function $t(n)$ and every $\epsilon>0$. I believe such a strong advantage of space over time is not expected to be true. The best ... 12 No, the exponential lower bound for determinization holds already for unambiguous NFAs. This is obtained as follows: Consider the alphabet $\{a,b\}$, and the language: $$L_k=\{w\in \{a,b\}^:\text{the k-th before last letter in }w\text{ is }b\}$$ It's easy to construct an unambiguous NFA for $L_k$: the NFA guesses when the $k$ before last letter is, and ... 11 The answer is no. I'll give an example of a language $L$ which is regular in binary but not in unary: Consider $L=\{10^k\|k\in \mathbb{N}\}$. The corresponding language in unary is $L'=\{1^{2^k}\|k\in \mathbb{N}\}$. It's easy to see that $L$ is regular while $L'$ is not even context free. L'' also isn't regular either, by the link @Sylvain posted in his ... 11 A Non deterministic XOR automaton (NXA) fits your question. A NXA $M$ is essentially an NFA, but a word $w\in \Sigma^*$ is said to be in $L(M)$ if it is accepted by an odd number of paths (Xor relation) instead of being accepted if there exists an accepting path for it (Or relation). NXAs are used for creating small representations of regular languages as ... 11 There exists a FPRAS (Fully Polynomial Randomized Approximation Scheme) for the problem of counting the words of length $n$ accepted by a NFA in the general case (without restricting to the acyclic NFA case). The result was published this year on STOC, by Arenas, Croquevielle, Jayaram and Riveros. Here is the talk https://youtu.be/tyK-uujHMLU and the paper ... 11 It seems there is a paper answering this exact question, and even in the more general case of $\omega$-regular languages, but I cannot find an open-access version. If somebody finds a link without paywall it would be great. I requested the full-text on ResearchGate. Title: Which Finite Monoids are Syntactic Monoids of Rational omega-Languages. Authors: ... 11 In a more elementary way than Denis's answer, the following is extracted from Pippenger's "Theories of Computability", p.87, and immediate to check. Definition: Let $M$ be a monoid, and $Y \subseteq M$. Define the congruence relation $\equiv_Y$ over $M$ by $x \equiv_Y y$ iff $\big[\forall w, z \in M$, $wxz \in Y \Leftrightarrow wyz \in Y\big]$. Definition:... 11 The terminology rigid seems to be relatively new compared to the term disjunctive used in the late 70's (and probably before, I didn't check for earlier references). A subset $P$ of a monoid $M$ is disjunctive if and only if the syntactic congruence of $P$ in $M$ is the equality relation. Thus a monoid is the syntactic monoid of a language if and only if it ... 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http://www.ms.u-tokyo.ac.jp/seminar/past_e_116.html	## Seminar information archive ### 2008/11/05 #### Geometry Seminar 14:45-18:00 Room #122 (Graduate School of Math. Sci. Bldg.) Directed Fukaya category の安定化について [ Abstract ] [ Abstract ] ### 2008/11/04 #### Tuesday Seminar on Topology 16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.) Misha Verbitsky (ITEP, Moscow) Lefschetz SL(2)-action and cohomology of Kaehler manifolds [ Abstract ] Let M be compact Kaehler manifold. It is well known that any Kaehler form generates a Lefschetz SL(2)-triple acting on cohomology of M. This action can be used to compute cohomology of M. If M is a hyperkaehler manifold, of real dimension 4n, then the subalgebra of its cohomology generated by the second cohomology is isomorphic to a polynomial algebra, up to the middle degree. ### 2008/11/01 #### Monthly Seminar on Arithmetic of Automorphic Forms 13:30-16:00 Room #123 (Graduate School of Math. Sci. Bldg.) $(\\mathfrka{g},K)$-module structure of the principal series of $GL(3,\\mathfrak{C})$ [ Abstract ] We give explicit description of the action of $\\mathfrak{gl}(3,\\mathbf{C}$ to the whole space of $K$-finite vectors of a given principal series representation of $GL(3,\\mathbf{C})$. Toward effectively computable integral basis of simple $\\mathbfrak{gl}_4$-modules of finite dimension. (II) [ Abstract ] This is a continuation of the talk at Osaka in the occation of Kanrei workshop of Prof. T. Ibukiyama. We discuss a part of the injection $V_{\\lambda} \\rightarrow \\mathfrak{p} \\otimes V_{\\lambda}$. Here $V_{\\lambda}$ is a simple module with highest weight $\\lambda$, and $\\mathfrak{p}$ is the adjoint representation with highest weight $(1,0,0,-1)$. ### 2008/10/31 #### Lecture Series on Mathematical Sciences in Soceity 16:20-17:50 Room #128 (Graduate School of Math. Sci. Bldg.) ### 2008/10/30 #### Seminar on Probability and Statistics 16:20-17:30 Room #270 (Graduate School of Math. Sci. Bldg.) [ Abstract ] [ Reference URL ] http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2008/08.html #### Lectures 10:00-11:00 Room #002 (Graduate School of Math. Sci. Bldg.) Arnaud DUCROT (University of Bordeaux) Travelling wave solutions for an infection age structured epidemic model [ Abstract ] In this lecture, we study the existence of travelling wave solutions for a class of epidemic model structured in space and with respect ot the age of infection. We obtain necessary and sufficient conditions for the existence of travelling waves for such a class of problems. As a consequence, we also derive the existence of travelling waves for a class of functional partial differential equations. ### 2008/10/29 #### Seminar on Mathematics for various disciplines 10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.) Stratified turbulence as an element of geophysical fluid dynamics [ Abstract ] Density stratification and rotation are the two major mechanisms that characterize the whole geophysical flows. In this talk, focusing on stable stratification, I will introduce some statistical, mechanical and geometrical aspects of stratified turbulence by showing the recent results of large scale computer simulations. #### Number Theory Seminar 16:30-17:30 Room #056 (Graduate School of Math. Sci. Bldg.) Daniel Caro (Université de Caen) Overholonomicity of overconvergence $F$-isocrystals on smooth varieties [ Abstract ] Let $¥mathcal{V}$ be a complete discrete valuation ring of characteristic $0$, with perfect residue field $k$ of characteristic $p>0$. In order to construct $p$-adic coefficients over $k$-varieties, Berthelot introduced the theory of overconvergent $F$-isocrystals, i.e overconvergent isocrystals with Frobenius structure. Moreover, to get a $p$-adic cohomology over $k$-varieties stable under cohomological operations, Berthelot built the theory of arithmetic $F$-$¥mathcal{D}$-modules. In this talk, after recalling some elements of these theories, we introduce the notion of overholonomicity with is a property as stable as the holonomicity in the classical theory of $¥mathcal{D}$-modules. The goal of the talk is to prove the overholonomicity of arithmetic $¥mathcal{D}$-modules associated to overconvergent $F$-isocrystals over smooth $k$-varieties. In the proof we need Christol's transfert theorem, a comparison theorem between relative log rigid cohomology and relative rigid cohomology and last but not least Kedlaya's semistable reduction theorem. This is a joint work with Nobuo Tsuzuki. #### Seminar on Probability and Statistics 16:20-17:30 Room #128 (Graduate School of Math. Sci. Bldg.) マルチスケール・ブートストラップ法による確率値計算とFDR [ Abstract ] ブートストラップ確率はブートストラップ標本において仮説が支持される頻度であり, その実装の容易さから広く用いられている.たとえば,階層型クラスタリングにおいて得られたクラスタが真実かどうかを判断する指標になる.ところが頻度論の不偏検定の立場で見ると,ブートストラップ確率には無視できないくらい十分に大きいバイアスがあり,それはある種のパラメータ空間において仮説をあらわす領域の境界の曲率として解釈できることが知られている.本講演ではリサンプリングにおけるデータサイズを変化させたときの「スケール変換則」からバイアス補正する手法を紹介する. 曲面のテイラー展開のかわりにフーリエ変換をつかった漸近理論であり,錐のように必ずしもなめらかな曲面でない場合でも適用できる (Shimodaira 2008).また,スケール変換則のアイデアをベイズ的なFalse Discovery Rateの計算に応用した最近の結果についても簡単に触れる. [ Reference URL ] http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2008/07.html ### 2008/10/28 #### Tuesday Seminar on Topology 16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.) Nonsmoothable group actions on spin 4-manifolds [ Abstract ] We call a locally linear group action on a topological manifold nonsmoothable if the action is not smooth with respect to any possible smooth structure. We show in this lecture that every closed, simply connected, spin topological 4-manifold not homeomorphic to neither S^2\\times S^2 nor S^4 allows a nonsmoothable group action of any cyclic group with sufficiently large prime order which depends on the manifold. #### Lie Groups and Representation Theory 16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.) Chevalley's restriction theorem for supersymmetric Riemannian symmetric spaces [ Abstract ] We start by explaining the concept of a supersymmetric Riemannian symmetric spaces and present the examples studied by Zirnbauer in the context of universality classes of random matrices. For these classes we then show how to formulate and prove an analog of Chevalley's restriction theorem for invariant super-functions. This is joint work with A. Alldridge (Paderborn) and M. Zirnbauer (Cologne) [ Reference URL ] http://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html #### Tuesday Seminar of Analysis 17:00-18:00 Room #123 (Graduate School of Math. Sci. Bldg.) Serge Alinhac (パリ大学オルセイ校) Introduction to geometric analysis of hyperbolic equations ### 2008/10/27 #### Seminar on Geometric Complex Analysis 10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.) #### GCOE lecture series 16:30-17:30 Room #128 (Graduate School of Math. Sci. Bldg.) Holomorphic extensions of highest weight representations to Olshanskii semigroups [ Abstract ] In this lecture I will present a proof of Olshanskii's Theorem, which says that for a simple group of Hermitean type unitarizable highest weight representations can be holomorphically extended to contractive representations of a complex semigroup containing the group in its boundary. ### 2008/10/24 #### Colloquium 16:30-17:30 Room #002 (Graduate School of Math. Sci. Bldg.) Benoit Collins (オタワ大学・東京大学大学院数理科学研究科) On the spectral measure of the sum of elements in a finite von Neumann algebra [ Abstract ] Given two self-adjoint n×n matrices A and B with prescribed eigenvalues, the set of all possible spectral distributions for A+B has been conjectured by Horn and proved by Knutson, Tao, Klyachko and Totaro. We address the same question when A and B have prescribed spectral measures but lie in an arbitrary II_1 factor, and we give elements of answers in terms of inequalities between the spectral measures. We explain the relation with the Connes embedding problem. ### 2008/10/23 #### Operator Algebra Seminars 16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.) Emily Peters (UC Berkeley) Planar algebras and the Haagerup subfactor ### 2008/10/22 #### Number Theory Seminar 16:30-17:30 Room #056 (Graduate School of Math. Sci. Bldg.) Pierre Parent (Universite Bordeaux 1) Serre's uniformity in the split Cartan case [ Abstract ] We show that, for large enough prime number p, the modular curve X_{split}(p) has no other point with values in Q than CM points and the rational cusp. This gives a partial answer to an old question of J.-P. Serre concerning the uniform surjectivity of Galois representations associated to torsion points on elliptic curves without complex multiplication. (Joint work with Yuri Bilu.) ### 2008/10/21 #### Lie Groups and Representation Theory 17:00-18:00 Room #126 (Graduate School of Math. Sci. Bldg.) Invitation to Atlas combinatorics [ Abstract ] この講演では、リー群に関する背景説明などは軽く済ませ、Atlas で公開されているプログラムにおける方言、特に入出力の読み方を通常の言葉に言い換えることで、プログラムを使ってもらう入り口での障壁を減らしたいと考えています。ふむ、なかなか、使えるな、自分もインストールしてみようか、と思ってもらえれば、成功です。なお、サーベイトークなので私のオリジナルな結果は含まれていません。また、計算機を使ってデモをする予定です。京都では計算機と板書の切り替えでばたばたしたので、照準を絞って慌てないように話したいと思います。 [ Reference URL ] http://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html #### Tuesday Seminar on Topology 16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.) On embeddings of 3-manifolds in 6-manifolds [ Abstract ] In this talk, we give a simple axiomatic definition of an invariant of smooth embeddings of 3-manifolds in 6-manifolds. The axiom is expressed in terms of some cobordisms of pairs of manifolds of dimensions 6 and 3 (equipped with some cohomology class of the complement) and the signature of 4-manifolds. We then show that our invariant gives a unified framework for some classical invariants in low-dimensions (Haefliger invariant, Milnor's triple linking number, Rokhlin invariant, Casson invariant, Takase's invariant, Skopenkov's invariants). ### 2008/10/20 #### Seminar on Geometric Complex Analysis 10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.) パンルヴェ方程式と複素力学系 ### 2008/10/17 #### Lecture Series on Mathematical Sciences in Soceity 16:20-17:50 Room #128 (Graduate School of Math. Sci. Bldg.) #### Lectures 15:00-16:00 Room #570 (Graduate School of Math. Sci. Bldg.) GCOEレクチャー"Holomorphic extensions of unitary representations" その4 "Applications and open problems" [ Abstract ] In this lecture we present further applications of the given extension results and describe some open problems. In particular, we will mention estimates for automorphic forms (Krotz-Stanton), random matrices (Huckleberry-Puttmann-Zirnbauer), unitarizability of highest weight representation with non-scalar lowest K-type, and infinite dimensional groups. [ Reference URL ] http://faculty.ms.u-tokyo.ac.jp/users/gcoe/GCOE_lecture0810Hilgert.html #### Algebraic Geometry Seminar 13:00-14:30 Room #128 (Graduate School of Math. Sci. Bldg.) Yongnam Lee (Sogang U.) Construction of surfaces of general type with pg=0 via Q-Gorenstein smoothing #### GCOE lecture series 15:00-16:00 Room #118 (Graduate School of Math. Sci. Bldg.) Holomorphic extensions of unitary representations その4 Applications and open problems [ Abstract ] In this lecture we present further applications of the given extension results and describe some open problems. In particular, we will mention estimates for automorphic forms (Krötz-Stanton), random matrices (Huckleberry-Püttmann-Zirnbauer), unitarizability of highest weight representation with non-scalar lowest K-type, and infinite dimensional groups. [ Reference URL ] http://faculty.ms.u-tokyo.ac.jp/users/gcoe/GCOE_lecture0810Hilgert.html ### 2008/10/16 #### Operator Algebra Seminars 16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.) Scott Morrison (UC Santa Barbara) The $D_{2n}$ planar algebras	2019-01-19 04:11:58	{"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": 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": 0.639482319355011, "perplexity": 5394.700856100187}, "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-2019-04/segments/1547583662124.0/warc/CC-MAIN-20190119034320-20190119060320-00271.warc.gz"}
https://math.stackexchange.com/questions/2718189/some-questions-about-the-proof-of-hahn-banach-theorem	# Some questions about the proof of Hahn-Banach Theorem Theorem: Let $q:X\rightarrow\mathbb{R}$ be a sublinear functional on a real linear space $X$. Let $M$ be a linear subspace of $X$ and suppose that $f:M\rightarrow\mathbb{R}$ is a linear functional such that $f(x)\leqslant q(x)$ for all $x\in M$. Then there exists a linear functional $F:X\rightarrow\mathbb{R}$ such that $F(x)\leqslant q(x)$ for all $x\in X$ and $F(x)=f(x)$ for all $x\in M$. Proof: let $S$ be the collection of all pairs $(M_1,f_1)$, where $M_1$ is a linear subspace of $X$ such that $M\subseteq M_1$ and $f_1:M_1\rightarrow\mathbb{R}$ is a linear extension of $f$ satisfying $f_1(x)\leqslant q(x)$ for all $x\in M_1$. By the claim above, $S\neq\emptyset$. For any $(M_1,f_1),(M_2,f_2)\in S$, define $(M_1,f_1)\preccurlyeq(M_2,f_2)$ if $M_1\subseteq M_2$ and $f_2$ is a linear extension of $f_1$. Then $\preccurlyeq$ is a partial order on $S$. Let $C=\{(M_i,f_i):i\in I\}$ be a chain in $S$. Let $\tilde{M}=\bigcup\limits_{i\in I}M_i$. Then $\tilde{M}$ is a linear subspace of $X$. Define $\tilde{f}:\tilde{M}\rightarrow\mathbb{R}$ by $\tilde{f}(x)=f_i(x)$ if $x\in M_i$ for some $i\in I$. My questions are: 1. In Conways's book, $\tilde{M}$ is a linear subspace of $X$ since $C$ is a chain in $S$. I don't get it. $\tilde{M}$ is a linear subspace of $X$ isn't because each $M_i$ is a linear subspace of $X$? Why do we need $C$ is a chain in $S$? 2. How to show $\tilde{f}$ is linear? My attempt is : For all $x,y\in\tilde{M}$, there exists $j\in I$ such that $x,y\in M_j$. For all $\alpha\in\mathbb{R}$, we have $$\tilde{f}(x+\alpha y)=f_j(x+\alpha y)=f_j(x)+\alpha f_j(y)=\tilde{f}(x)+\alpha\tilde{f}(y).$$ Hence, $\tilde{f}:\tilde{M}\rightarrow\mathbb{R}$ is linear. But my professor said it is not clear at all!! Can someone help me point out which part is missing? Thank you in advance! For 1, in general a union of subspaces is not a subspace. But it is, if the union is increasing. Your argument for the linearity of $\bar f$ is fine. What you are not doing, and maybe that's what your prof expect, is showing that $\bar f$ is well-defined. That again requires that $C$ is a chain and that the $f_j$ are extensions. • Thanks for your reply. Can you be more specific about a union of subspaces is not a subspace. But it is, if the union is increasing? And I have shown $\tilde{f}$ is well defined but I didn't write it down on this page. My proof about the linearity is fine right? – Answer Lee Apr 2 '18 at 3:39 • As I said in the answer, your argument for linearity is fine. As for the union of subspaces, give it a try in $\mathbb R^2$. – Martin Argerami Apr 2 '18 at 4:44	2019-10-14 21:30:03	{"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": 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": 0.8988199234008789, "perplexity": 41.958698510661975}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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-2019-43/segments/1570986655310.17/warc/CC-MAIN-20191014200522-20191014224022-00282.warc.gz"}
https://www.physicsforums.com/threads/electric-field-and-potential.161197/	Electric field and potential. what is the electric field and potential of a uniformly charged sphere...both inside and outside.... also the same for hollow sphere.. plzz G01 Homework Helper Gold Member You really must show SOME work, its the rules of the forums. To get you started... HINT: Spheres have the same symmetry a POINT charges. What are the formulas and laws you can use to find Fields and Potentials? For inside the sphere, think GAUSS'S LAW. for field it is kcharge/distance^2 and potential is kcharge/distance where k is a constant	2022-05-27 04:28:52	{"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": 0.8251873850822449, "perplexity": 1673.0763648008874}, "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-2022-21/segments/1652662631064.64/warc/CC-MAIN-20220527015812-20220527045812-00499.warc.gz"}
https://web.mit.edu/psycholinglab/public/lab_5.html	library(tidyverse) ## Multiple comparisons One thing that’s come up recently in the literature is the issue of multiple comparisons. Here’s a way researchers sometimes plan studies: I want to see if group A differs from group B in cognitive ability X, but I don’t know exactly how to measure X. I’ll collect response time, and accuracy, and time spent on response, and also, since none of those might work out, I’ll collect some measures of IQ and favorite dog breed. Then I’ll see which of those yields significant differences! Do you see the problem here? We know that you can get p-values < 0.05 when the null hypothesis is true, so what happens when we run lots of tests? experiment <- function() { sample1 <- rnorm(50, 20, 10) sample2 <- rnorm(50, 20, 10) p_value <- t.test(sample1, sample2)$p.value return(p_value) } tests <- function(num_test) { p_values <- replicate(num_test, experiment()) num_sig <- sum(p_values < 0.05) return(num_sig) } mult_comp <- data_frame(num_test = 1:100) %>% mutate(num_sig = map_dbl(num_test, tests)) ggplot(mult_comp, aes(x = num_test, y = num_sig)) + geom_point() + geom_smooth(method = "lm") As you can see, as you run more and more tests, the number of significant p-values increases. So if you throw in enough stuff in your experiment, you’re pretty much bound to find a p-value less than 0.05. The lesson isn’t that you should just have lots of comparisons in your experiments. The lesson is that the threshold of 5% is no longer appropriate when you have multiple comparisons. There are ways to correct alpha level (Bonferroni, FDR, etc..) but they shouldn’t be used as a one-size-fits-all solution. ### Try it yourself… 1. Simulate how power changes across sample sizes for a two-sample t-test. 2. Plot your results (sample size on the x axis, power on the y axis). 3. Increase or decrease your effect size and re-run your simulation/plot. How do sample size and effect size affect power? experiment <- function(group_size) { # sample data of group size from each of two groups with different means # return p-value of t-test comparing them } power <- function(group_size) { # run experiment 100 times # return the percentage of experiments that have significant p-values } power_levels <- data_frame(group_size = 2:200) %>% mutate(sample_size = group_size * 2, power = map_dbl(group_size, power)) ggplot(power_levels) # add plot structure experiment <- function(group_size) { sample1 <- rnorm(group_size, 20, 10) sample2 <- rnorm(group_size, 25, 10) p_value <- t.test(sample1, sample2)$p.value return(p_value) } power <- function(group_size) { p_values <- replicate(100, experiment(group_size)) num_sig <- sum(p_values < 0.05) / 100 return(num_sig) } power_levels <- data_frame(group_size = 2:200) %>% mutate(sample_size = group_size * 2, power = map_dbl(group_size, power)) ggplot(power_levels, aes(x = sample_size, y = power)) + geom_hline(yintercept = 0.8, colour = "grey", linetype = "dotted") + geom_point() + geom_smooth() ## Regression So far we’ve looked at the types of data where you have samples groups and you want to know if they were generated by the same generative process or two separate ones. But a lot of questions in psycholinguists can’t really be answered in this way. For instance, you can’t use a t-test to look at whether lexical decision times increase as word frequency increases. This is a bivariate relationship between 2 numerical variables. data_source <- "http://web.mit.edu/psycholinglab/data/" ggplot(rts) + geom_histogram(aes(x = RTlexdec), bins = 60, color = "white") ggplot(rts) + geom_point(aes(x = WrittenFrequency, y = RTlexdec)) So in the past we’ve looked at this using a correlation which tells us about the strength of the relationship between 2 variables. And you can go beyond that and show whether this relationship is statistically significant, i.e., different from 0 or not using cor.test() cor.test(rts$RTlexdec, rts$WrittenFrequency) ## ## Pearson's product-moment correlation ## ## data: rts$RTlexdec and rts$WrittenFrequency ## t = -32.627, df = 4566, p-value < 2.2e-16 ## alternative hypothesis: true correlation is not equal to 0 ## 95 percent confidence interval: ## -0.4580397 -0.4109971 ## sample estimates: ## cor ## -0.434815 So this looks similar to the output we got from the t-test because what it’s doing is taking the Pearson’s coefficient, r = -0.43, and finding a corresponding t-statistic (specific formula: $$\frac{r\sqrt{n−2}}{\sqrt{1−r^2}}$$), t = -32.63, based on a t-distribution with df = N-2. For this particular t-statistic, the probability that of it occurring given an r = 0 is very very small. So we can conclude that we have a significant correlation. Note that, in general, a bigger r value is likely to be significant, but you can have a big r value that’s not significant, if you have a very small sample size. You can also have very tiny r values that are significant if you have a big enough sample size. This is very useful but it’s telling us about how the 2 variables relate to each other without making any hypotheses about the direction of that relationship. Often in experiments we have more specific questions because we have manipulated some variable and we’re looking at how this affects another variable, or we hypothesize that one variable is driving a change in another variable. Correlation says nothing about that. This is where linear regression comes in. For two continuous variables, linear regression and correlation are almost the same but you’ll see that regression is actually much more general because you can use it with categorical predictors, multiple predictors, etc… We’ve actually been using regression already when we use geom_smooth() to draw a line to fit the data… ggplot(rts, aes(x = WrittenFrequency, y = RTlexdec)) + geom_point() + geom_smooth(method = "lm") If you remember from high school math class, the formula for a line is $$y = ax + b$$. To do linear regression, we’ll use just this: $y_i = \beta_0 + \beta_1 \cdot x_i + \epsilon$ The $$\beta_0$$ is the point at which it intersects with $$x = 0$$: the intercept. The $$\beta_1$$ is the change in $$y$$ that corresponds to a unit change in $$x$$: the slope. The line that is being drawn by geom_smooth() has an intercept and slope parameter but it also has another key component: error. freq_lm <- lm(RTlexdec ~ WrittenFrequency, data = rts) summary(freq_lm) ## ## Call: ## lm(formula = RTlexdec ~ WrittenFrequency, data = rts) ## ## Residuals: ## Min 1Q Median 3Q Max ## -0.45708 -0.11657 -0.00109 0.10403 0.56085 ## ## Coefficients: ## Estimate Std. Error t value Pr(>\|t\|) ## (Intercept) 6.735931 0.006067 1110.19 <2e-16 * ## WrittenFrequency -0.037010 0.001134 -32.63 <2e-16 * ## --- ## Signif. codes: 0 '*' 0.001 '' 0.01 '' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 0.1413 on 4566 degrees of freedom ## Multiple R-squared: 0.1891, Adjusted R-squared: 0.1889 ## F-statistic: 1065 on 1 and 4566 DF, p-value: < 2.2e-16 There’s a lot of information here so let’s focus first on the coefficients. library(broom) tidy(freq_lm, conf.int = TRUE) ## # A tibble: 2 x 7 ## term estimate std.error statistic p.value ## <chr> <dbl> <dbl> <dbl> <dbl> ## 1 (Intercept) 6.73593136 0.006067378 1110.18810 0.000000e+00 ## 2 WrittenFrequency -0.03701028 0.001134340 -32.62715 4.463918e-210 ## # ... with 2 more variables: conf.low <dbl>, conf.high <dbl> First, let’s look at the estimate: the values here are our intercept and slope defining the best fit line. Those estimates are being compared to a t-distribution in the same way that we’ve done for previous tests and you obtain a p-value. So here this suggests that: 1. The intercept value is significantly different from 0, so words with 0 WrittenFrequency have a predicted RTlexdec value of 6.74 seconds. 2. The slope is significantly different from 0, so for every unit increase in WrittenFrequency, you expect a unit decrease of 0.04 units of RTlexdec. So our linear model is: $\text{RTlexdec} = 6.74 - 0.04 \cdot \text{WrittenFrequency} + \mathcal{N}(0, \sigma)$ In other words for every value of Written Frequency we can find the value of RTlexdec by taking the Written Frequency, multiplying by -0.04 and subtracting that from 6.74, then drawing from a normal distribution centered on that point. Why the normal distribution? That’s the error… freq_aug <- augment(freq_lm) head(freq_aug) ## # A tibble: 6 x 9 ## RTlexdec WrittenFrequency .fitted .se.fit .resid .hat ## <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> ## 1 6.543754 3.912023 6.591146 0.002440312 -0.04739273 0.0002981668 ## 2 6.397596 4.521789 6.568579 0.002166358 -0.17098259 0.0002349790 ## 3 6.304942 6.505784 6.495150 0.002684846 -0.19020818 0.0003609169 ## 4 6.424221 5.017280 6.550240 0.002090998 -0.12601990 0.0002189151 ## 5 6.450597 4.890349 6.554938 0.002096251 -0.10434136 0.0002200163 ## 6 6.531970 4.770685 6.559367 0.002110206 -0.02739665 0.0002229556 ## # ... with 3 more variables: .sigma <dbl>, .cooksd <dbl>, .std.resid <dbl> ggplot(freq_aug, aes(x = WrittenFrequency, y = RTlexdec)) + geom_point() + geom_point(aes(y = .fitted), color = "blue") So clearly the real values don’t fall exactly on the line like the predicted values. The line is not a perfect fit and it couldn’t be – there is no line that would fit every single one of those datapoints. The distances between the real datapoints and this line is what’s called error. errors <- freq_aug$RTlexdec - freq_aug$.fitted residuals <- freq_aug$.resid all.equal(errors, residuals) ## [1] TRUE ggplot(freq_aug, aes(sample = .resid)) + stat_qq() ggplot(freq_aug, aes(x = WrittenFrequency, y = .resid)) + geom_point() + geom_hline(yintercept = 0, color = "blue") The residuals are normally distributed which is good – this is the mark of an appropriately applied linear regression model. We’re also not seeing any huge differences in the residuals for different values of x (homoscedasticity). The residuals are the part of the data that we have failed to explain with our model. So we want them to be minimized. In fact, minimizing those residuals is pretty much how the best fit line is picked. (I don’t think we need to get into all the detail about this but sum of the squared distances between each point and the line is what is being minimized: http://students.brown.edu/seeing-theory/regression/index.html#first) The deviation in residuals appears in the summary as “residual standard error”. sd(freq_aug$.resid) ## [1] 0.1413085 summary(freq_lm) ## ## Call: ## lm(formula = RTlexdec ~ WrittenFrequency, data = rts) ## ## Residuals: ## Min 1Q Median 3Q Max ## -0.45708 -0.11657 -0.00109 0.10403 0.56085 ## ## Coefficients: ## Estimate Std. Error t value Pr(>\|t\|) ## (Intercept) 6.735931 0.006067 1110.19 <2e-16 * ## WrittenFrequency -0.037010 0.001134 -32.63 <2e-16 * ## --- ## Signif. codes: 0 '*' 0.001 '' 0.01 '' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 0.1413 on 4566 degrees of freedom ## Multiple R-squared: 0.1891, Adjusted R-squared: 0.1889 ## F-statistic: 1065 on 1 and 4566 DF, p-value: < 2.2e-16 So once you’ve found this optimal line and you have those predicted/fitted values, you can assess whether this line is actually explaining a lot of the variation in the data or not. This is the same as asking whether variation in $$x$$ explains variation in $$y$$ well or not. You can do this by seeing how close the fitted values are to the actual $$y$$ values. (cor(freq_aug$.fitted, freq_aug$RTlexdec)) ^ 2 ## [1] 0.1890641 It is often interpreted as the proportion of data which your model explains. So frequency here explains 18.9% of the observed variation in RTs. The F-test at the very bottom is saying that $$r^2$$ is significantly different from 0, i.e. the model is explaining a non-zero portion of the variance. freq_lm_null <- lm(RTlexdec ~ 1, data = rts) n <- nrow(rts) p_null <- 1 p <- 2 f_value <- ((rss_null - rss) / (p - p_null)) / (rss / (n - p - 1)) pf(f_value, p - p_null, n - p, lower.tail = FALSE) ## [1] 4.906935e-210 This is not a particularly useful value for our purposes, it’ll usually be significant… ### Try it yourself… 1. Model the relationship between RTlexdec and WrittenSpokenFrequencyRatio, plot it, and summarize in words what the result means. 2. Now look at the relationship between RTnaming and WrittenSpokenFrequencyRatio. Do you notice anything different? ratio_lm <- lm(RTlexdec ~ WrittenSpokenFrequencyRatio, data = rts) ratio_tidy <- tidy(ratio_lm) ratio_aug <- augment(ratio_lm) summary(ratio_lm) ## ## Call: ## lm(formula = RTlexdec ~ WrittenSpokenFrequencyRatio, data = rts) ## ## Residuals: ## Min 1Q Median 3Q Max ## -0.34617 -0.12579 0.00009 0.10287 0.63242 ## ## Coefficients: ## Estimate Std. Error t value Pr(>\|t\|) ## (Intercept) 6.546463 0.002684 2438.935 < 2e-16 ## WrittenSpokenFrequencyRatio 0.005364 0.001992 2.693 0.00711 ## --- ## Signif. codes: 0 '*' 0.001 '' 0.01 '' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 0.1568 on 4566 degrees of freedom ## Multiple R-squared: 0.001586, Adjusted R-squared: 0.001367 ## F-statistic: 7.251 on 1 and 4566 DF, p-value: 0.00711 ggplot(ratio_aug, aes(sample = .resid)) + stat_qq() ggplot(ratio_aug, aes(x = WrittenSpokenFrequencyRatio, y = RTlexdec)) + geom_point(color = "grey") + geom_abline(intercept = filter(ratio_tidy, term == "(Intercept)")$estimate, slope = filter(ratio_tidy, term == "WrittenSpokenFrequencyRatio")$estimate, colour = "blue") + geom_smooth(method = lm, colour = "red") naming_lm <- lm(RTnaming ~ WrittenSpokenFrequencyRatio, data = rts) naming_aug <- augment(naming_lm) summary(naming_lm) ## ## Call: ## lm(formula = RTnaming ~ WrittenSpokenFrequencyRatio, data = rts) ## ## Residuals: ## Min 1Q Median 3Q Max ## -0.29190 -0.17437 0.01484 0.16785 0.37799 ## ## Coefficients: ## Estimate Std. Error t value Pr(>\|t\|) ## (Intercept) 6.318707 0.003053 2069.426 <2e-16 ** ## WrittenSpokenFrequencyRatio 0.005606 0.002266 2.474 0.0134 * ## --- ## Signif. codes: 0 '*' 0.001 '' 0.01 '' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 0.1784 on 4566 degrees of freedom ## Multiple R-squared: 0.001339, Adjusted R-squared: 0.00112 ## F-statistic: 6.122 on 1 and 4566 DF, p-value: 0.01339 ggplot(naming_aug, aes(sample = .resid)) + stat_qq() ggplot(naming_aug, aes(x = WrittenSpokenFrequencyRatio, y = RTnaming)) + geom_point(color = "grey") + geom_smooth(method = lm, colour = "red") Here we looked at one continuous predictor and a continuous outcome, but regression is much more general than that. So we’re going to look at a few other ways you can use it. ### Categorical predictors Let’s say we want to look at the effect of being in one of 2 (or more) groups on some response. For example, we can look at RTlexdec as a function of the age of the subjects rts_age <- rts %>% group_by(AgeSubject) %>% summarise(mean_rt = mean(RTlexdec)) ggplot(rts, aes(x = AgeSubject, y = RTlexdec)) + geom_violin() + geom_point(aes(y = mean_rt), data = rts_age, size = 3) age_lm <- lm(RTlexdec ~ AgeSubject, data = rts) summary(age_lm) ## ## Call: ## lm(formula = RTlexdec ~ AgeSubject, data = rts) ## ## Residuals: ## Min 1Q Median 3Q Max ## -0.25776 -0.08339 -0.01669 0.06921 0.52685 ## ## Coefficients: ## Estimate Std. Error t value Pr(>\|t\|) ## (Intercept) 6.660958 0.002324 2866.44 <2e-16 ## AgeSubjectyoung -0.221721 0.003286 -67.47 <2e-16 * ## --- ## Signif. codes: 0 '*' 0.001 '' 0.01 '' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 0.1111 on 4566 degrees of freedom ## Multiple R-squared: 0.4992, Adjusted R-squared: 0.4991 ## F-statistic: 4552 on 1 and 4566 DF, p-value: < 2.2e-16 This looks very familiar but if we go back to our line equation, something is weird. $\text{RTlexdec} = 6.66 - 0.22 \cdot \text{AgeSubject} + \mathcal{N}(0, \sigma)$ How do you multiply AgeSubject, which is either the word “old” or the word “young”, by a number? Turns out R automatically turns “old” and “young” into 0s and 1s. But you can’t see this until you convert it into a factor… rts$AgeSubject <- factor(rts$AgeSubject) contrasts(rts$AgeSubject) ## young ## old 0 ## young 1 It is treating “old” as 0 and “young” as 1 and it just did this alphabetically by default. This is called dummy coding. So how do we read the output of our model? age_tidy <- tidy(age_lm) age_tidy ## # A tibble: 2 x 5 ## term estimate std.error statistic p.value ## <chr> <dbl> <dbl> <dbl> <dbl> ## 1 (Intercept) 6.6609581 0.002323772 2866.4417 0 ## 2 AgeSubjectyoung -0.2217215 0.003286310 -67.4682 0 The Intercept corresponds to the y value when the dummy coded variable is 0, so the intercept is the average RTlexdec for old people. rts_age ## # A tibble: 2 x 2 ## AgeSubject mean_rt ## <chr> <dbl> ## 1 old 6.660958 ## 2 young 6.439237 AgeSubjectyoung is the coefficient that you add when the dummy coded variable is 1. So the average RTlexdec for young people is -0.22 less than the intercept. intercept <- filter(age_tidy, term == "(Intercept)")$estimate slope <- filter(age_tidy, term == "AgeSubjectyoung")$estimate old_code <- contrasts(rts$AgeSubject)["old", ] young_code <- contrasts(rts$AgeSubject)["young", ] # old RTs intercept + slope old_code ## [1] 6.660958 # young RTs intercept + slope * young_code ## [1] 6.439237 ### Dummy coding vs. effects coding Another option is to do what is sometimes called effects coding. Instead of the default dummy coding that R assigns, you can change the contrast codes to something that is more meaningful to you. contrasts(rts$AgeSubject) <- c(-0.5, 0.5) contrasts(rts$AgeSubject) ## [,1] ## old -0.5 ## young 0.5 So now the indicator variable for AgeSubject is -0.5 for “old” and 0.5 for “young”. How will this affect our regression? age_lm_2 <- lm(RTlexdec ~ AgeSubject, data = rts) age_tidy_2 <- tidy(age_lm_2) summary(age_lm_2) ## ## Call: ## lm(formula = RTlexdec ~ AgeSubject, data = rts) ## ## Residuals: ## Min 1Q Median 3Q Max ## -0.25776 -0.08339 -0.01669 0.06921 0.52685 ## ## Coefficients: ## Estimate Std. Error t value Pr(>\|t\|) ## (Intercept) 6.550097 0.001643 3986.29 <2e-16 * ## AgeSubject1 -0.221721 0.003286 -67.47 <2e-16 * ## --- ## Signif. codes: 0 '*' 0.001 '' 0.01 '' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 0.1111 on 4566 degrees of freedom ## Multiple R-squared: 0.4992, Adjusted R-squared: 0.4991 ## F-statistic: 4552 on 1 and 4566 DF, p-value: < 2.2e-16 As you can see the slope is exactly the same – it’s still telling us that going from old to young (1 unit change) means subtracting -0.22. But note that the intercept is different. Now the intercept is the grand mean (across all conditions). What if we rewrite our formulas from earlier: intercept <- filter(age_tidy_2, term == "(Intercept)")$estimate slope <- filter(age_tidy_2, term == "AgeSubject1")$estimate old_code <- contrasts(rts$AgeSubject)["old", ] young_code <- contrasts(rts\$AgeSubject)["young", ] # old RTs intercept + slope old_code ## old ## 6.660958 # young RTs intercept + slope * young_code ## young ## 6.439237 We still get the exact same thing. In this case, where there’s just one predictor, the choice of contrast coding is entirely a matter of preference. In multiple regression, where you have many predictors potentially interacting, the right contrast coding can make a huge difference for how you interpret what’s going on in your model, specifically whether you are looking at simple effects or main effects. There are many different coding schemes for testing various hypotheses, more examples: https://stats.idre.ucla.edu/r/library/r-%20library-contrast-coding-systems-for-%20categorical-variables/ ### Multiple Predictors Now remember when we looked at the histogram, we saw that the RTs aren’t quite normally distributed in that there were two peaks. And now we know that young and old subjects respond differently. Let’s incorporate that into our regression. First some plots: ggplot(rts, aes(x = WrittenFrequency, y = RTlexdec, colour = AgeSubject)) + geom_point(alpha = 0.1) We see there are two groups here. Remember we are evaluating our model by seeing how close the fitted values are to the real data points, and we can get a lot closer if we have different lines for different age. ggplot(rts, aes(x = WrittenFrequency, y = RTlexdec, colour = AgeSubject)) + geom_point(alpha = 0.1) + geom_smooth(method = "lm") Let’s make a regression model with multiple predictors: $\text{RTlexdec} = \beta_0 + \beta_1 \cdot \text{WrittenFrequency} + \beta_2 \cdot \text{AgeSubject} + \mathcal{N}(0, \sigma)$ freq_age_lm <- lm(RTlexdec ~ WrittenFrequency + AgeSubject, data = rts) summary(freq_age_lm) ## ## Call: ## lm(formula = RTlexdec ~ WrittenFrequency + AgeSubject, data = rts) ## ## Residuals: ## Min 1Q Median 3Q Max ## -0.34622 -0.06029 -0.00722 0.05178 0.44999 ## ## Coefficients: ## Estimate Std. Error t value Pr(>\|t\|) ## (Intercept) 6.7359314 0.0037621 1790.48 <2e-16 * ## WrittenFrequency -0.0370103 0.0007033 -52.62 <2e-16 * ## AgeSubject1 -0.2217215 0.0025930 -85.51 <2e-16 * ## --- ## Signif. codes: 0 '' 0.001 '' 0.01 '' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 0.08763 on 4565 degrees of freedom ## Multiple R-squared: 0.6883, Adjusted R-squared: 0.6882 ## F-statistic: 5040 on 2 and 4565 DF, p-value: < 2.2e-16 • So now we have the intercept, which is the mean of RTlexdec for WrittenFrequency = 0 and across all AgeSubject (because of the effects coding). • The effect of WrittenFrequency (-0.037) is pretty much the same as before which makes sense because we were seeing the same effect of Written Frequency in both groups • The effect of age is very similar in size to when it was the sole predictor in the model as well. Note that both the effects of Age and Frequency are significant suggesting that they are both independently contributing to explaining the variance in RTlexdec. Critically, the $$r^2$$ is much larger, suggesting we’re doing a much better job of fitting the data than with the previous model. All that extra variance is now being explained by adding the Age factor. Linear model: $y_i = \beta_0 + \beta_1 \cdot x_{i1} + \beta_2 \cdot x_{i2} + \ldots + \beta_k \cdot x_{ik} + \epsilon_i$	2022-07-04 03:33:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.40426403284072876, "perplexity": 3352.8167962983134}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104293758.72/warc/CC-MAIN-20220704015700-20220704045700-00631.warc.gz"}
https://math.stackexchange.com/questions/2580517/finding-a-polynominal	# Finding a Polynominal Let $$x_1 + x_2 + \dots+ x_n = A$$ The value of $x_1$ to $x_n$ is not given. Suppose we have $n$ variables $y_1, y_2,\dots, y_n$. Is there any way to find $x_1y_1 + x_2y_2 + x_3y_3 + \dots+ x_ny_n$, based on the value of $A$ and $y_1$ to $y_n$? • Very unlikely. Any permutation of $x_1, \ldots, x_n$ preserves the identity $x_1 + \ldots + x_n = A$, but may change the value of $x_1 y_1 + \ldots + x_n y_n$ – Martin R Dec 26 '17 at 9:39 • Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. – José Carlos Santos Dec 26 '17 at 9:41 • You have 1 equation for $n$ unknown variables. So in general it cant work. Maybe you missed to mention that it is a polynomial function $f_m(x) = y$ of degree $m$. Then you could solve it for the case of $m=0$. – Rudi_Birnbaum Dec 26 '17 at 9:54 • And likewise for the case of $n=1$ the system is solvable. – Rudi_Birnbaum Dec 26 '17 at 10:02 In most cases no, the answer is yes iff $y_i=y_j\forall i,j\in\{1,2,\cdots,n\}$ Consider $x_1 + x_2 + \dots+ x_n = A$ and that the condition I stated are false. In this case I can rewrite the $y_1x_1 +y_n x_2 + \dots+ y_nx_n=B$ it as $w_1u_1+w_{2}u_{2}+\cdots+w_\ell u_\ell=B$ where $1<\ell\le n$, $w_k=y_h$ for some $h$ and $u_k$ is combination of some values of $x_s$. This is equation is the most reduce form, i.e. has the least amount of variables we can get, so unless I have only one variable I can't solve it, but one of my conditions is that not all of $y_i$ are equal, so I have at least 3 variables thus I can't solve this. • It does not work even if $A= 0$: $(+1)1 + (-1)2 = -1$, $(-1)1 + (+1)2 =+1$. – Martin R Dec 26 '17 at 12:26	2020-02-24 15:06:01	{"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": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.9227349758148193, "perplexity": 156.63233093130623}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1581875145960.92/warc/CC-MAIN-20200224132646-20200224162646-00008.warc.gz"}
http://math.stackexchange.com/questions/192850/defining-oblique-lines	# Defining Oblique Lines Is it correct to classify a line which is neither vertical nor horizontal as oblique. I am trying to classify lines in a plane based on the quadrants through which they pass. - Sure. But there is nothing absolute about the notion, since horizontal and vertical depend on choice of axes. – André Nicolas Sep 8 '12 at 18:13 Thanks for your help. – Becky Sep 8 '12 at 18:26	2014-10-31 17:21:12	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9176449775695801, "perplexity": 1187.2984283709568}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1414637900031.50/warc/CC-MAIN-20141030025820-00126-ip-10-16-133-185.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/trig-identities.208128/	# Trig Identities ## Homework Statement 4.59 x 10^-4 = Sin^3(x)/Cos(x) Solve for X ## Homework Equations Sin(x)/Cos(x) = Tan(x) ## The Attempt at a Solution We can make Sin^3(x)/Cos(x) into Tan(x)Sin^2(x), but I don't think that helps... What trick do I use? Last edited: Related Precalculus Mathematics Homework Help News on Phys.org Gib Z Homework Helper I can't think of an exact solution right now, but if we just use small angle approximations of sin and cos backwards, we can get a VERY close solution !! Just take cube roots of both sides, $$9.6702579 \cdot 10^{-11} = \frac{\sin x}{( \cos x)^{1/3} }$$. With our backwards small angle approximation attempt, we find x = $9.6702579 \cdot 10^{-11}$. When we sub that back in for x on the right hand side on your original expression, we get approximately: 0.000458999999999999999999999999999999999999999554012791....... Which means we have an error of $$4.45987209 \cdot 10^{-46}$$. I think thats quite reasonable, dont you? mda This is along the right track (although there is a non-trivial analytic solution)... but the numerical answer isn't quite right... Gib Z Homework Helper Can you show me the analytic solution? And What I know the answer its quite right, It is reasonably accurate though, no? Of course, we can make it more accurate by taking more terms of the taylor series and with some error analysis, solving quartic equations to arbitrary accuracy with newtons method - That all takes so much time for so little extra result. ...And What I know the answer its quite right... GibZ - you cubed the constant instead of taking the cube root. I think the answer is more closer to x = 0.079 mda GibZ, I won't show the solution I got from Maple because it is absolutely horrific, would not teach us anything and as you say is a waste of time anyway. :) I see Theo has posted the right answer so I'll leave it at that. Ya know, you can re-write the original expression in terms of, say, $u = {\cos}^2(x)$. This will be a cubic polynomial in u. You could then search for the zero of this polynomial. Of course, I would still do so numerically as you have available a very good initial estimate. So, questions for you: 1. how would you formulate the polynomial P(u). (Hint: convert every thing in terms of cos(x)). 2. what is this initial estimate I speak of. (Hint: expand the numerator and denominator of the quotient $\tfrac{{\sin}^3(x)}{\cos(x)}$) Of course, all of this is for small x solutions. Are there any others? I think maybe the answer is Yes Gib Z Homework Helper GibZ - you cubed the constant instead of taking the cube root. I think the answer is more closer to x = 0.079 DAMN IT. Well, point is - Use small angle approximations for this. Answer should still be very accurate. dynamicsolo Homework Helper ## Homework Statement 4.59 x 10^-4 = Sin^3(x)/Cos(x) Solve for X ... We can make Sin^3(x)/Cos(x) into Tan(x)Sin^2(x), but I don't think that helps... What trick do I use? By any chance, did this equation come up from working on the physics problem involving two hanging, charged pith balls, where you have to solve for the angle the cords make to the vertical at equilibrium? (Having worked with students on this problem multiple times, I've gotten used to seeing this expression...) As has already been pointed out, if the product [ tan(x) · sin^2(x) ] is much smaller than 1, you can safely use the small angle approximation for sine and tangent to get a good first estimate for the solution. With sin(x) and tan(x) approximately equal to x in radians, you can approximate your equation by x^3 = 4.59 x 10^-4 , which gives you a first guess of x = 0.0772 radian. You can then put this into your exact product, [ tan(x) · sin^2(x) ] , and see what you get. Since sine and tangent will both increase with increasing x for the angles you'd be working with, you can then "tweak" your estimate for x up or down to move the product up or down. [In this case, you'd need to lower the value a touch.] For most of the problems of this sort I've seen, you can usually get to three sig-figs of precision in three or four passes... (Make sure, naturally, that your calculator is in radian mode when doing this.) Then, of course, you could also solve this graphically... I see Theo has posted the right answer so I'll leave it at that. Actually, 0.079 is a little high. The cube-root estimate turns out to be very close in this case. Last edited: Gib Z Homework Helper If you have said the first sentence to me in real life, without the second, I would be terrified =] Just in case this is more of a maths problem than physics, writing the expression in terms of sin and tan makes the period, and hence the general form of the solution, more obvious.	2020-08-15 05:04:59	{"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": 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": 0.901189923286438, "perplexity": 654.4099641741008}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1596439740679.96/warc/CC-MAIN-20200815035250-20200815065250-00433.warc.gz"}
https://reparationmobile.net/vipers-bugloss-vuv/eeef93-two-adjacent-supplementary-angles-form-a	A). When the sum of the measure of two angles is 180 0, then the pair of angles is said to be supplementary angles.Here the supplementary meaning is one angle is supplemented to another angle to make a sum of 180 0.. From the figure, we can say that ∠1 + ∠2 = 180 0 Image Will be uploaded soon If the two supplementary angles are adjacent (i.e. Here You Go! Two angles are said to be supplementary angles if they add up to 180 degrees. Practice. Find an answer to your question If two adjacent angles are supplementary then they form a. vijaytiwari3221 vijaytiwari3221 11.05.2020 Math Primary School If two adjacent angles are supplementary then they form a. Ready To Place An Order? When two angles are adjacent, then their sum is the angle formed by their non-common arms. Linear Pair. A Linear Pair Forms A Straight Angle Which Contains 180º, So You Have 2 Angles Whose Measures Add To 180, Which Means They Are Supplementary. Order Your Homework Today! 2 See answers bharatharlapur bharatharlapur Answer: linear pair . C does not have to be a writing, so I could be acute of twos or even straight. obtuse angle. We have over 1500 academic writers ready and waiting to help you achieve academic success. The two supplementary angles, if joined together, form a straight line and a straight angle. Oh no! x * 1/2 + y * 1/2, so it's (x + y) * 1/2 = 180°/2 = 90° which is a right angle. Pair of adjacent angles whose measures add up to form a straight angle is known as a linear pair. Also, triangles with two equal sides and an adjacent angle are not necessarily equal or congruent. never. 4.Two adjacent, supplementary angles form a(n) _____. sometimes. Since there are two pairs of angles and one of each pair is on each side, adjacent angles are supplementary (their sum makes 180˚) The following diagram explains: All other trademarks and copyrights are the property of their respective owners. When two adjacent angles form a straight line, they are supplementary. Let the measures of the two supplementary angles be x and y. Write a p…, a. sometimes. never. ∠OBA+∠OBC = 180. Which of the triangles described in the table is a... A car accelerates uniformly from rest and reaches... ABCDEF is a regular hexagon. The angles in a linear pair are supplementary. The angles in a linear pair are supplementary. (ii) When two lines intersect adjacent angles are supplementary. Let’s work the following examples. Going back to our labeled intersection, angles one and two are adjacent. The angular velocity is given. The bisectors of two adjacent supplementary angles form a right angle. The collar ''C'' is pinned to rod ''CD'' while it... Motor ''M'' exerts a constant force of P= 740... Two lighthouses A and B are 25km apart and A is... Vertical Angles & Complementary Angles: Definition & Examples, Solving Equations With Angle Relationships, Comparing Theoretical & Experimental Probability, Selecting Measures of Center & Variability for Data, Identifying a Sequence of Transformations, Complementary Angles: Definition, Theorem & Examples, Probability of Independent and Dependent Events, What Are Adjacent Angles? These two angles do not have to form any particular special angle; they just have to be next to each other. MEMORY METER. 6.Two adjacent angles whose exterior sides are opposite rays are complementary. There was not enough information to turn. Click 'Join' if it's correct. % Progress . Pair of adjacent whose measures add up to form a straight angle is known as a linear pair. Complement Theorem. $\angle 1+ \angle 2 = 180^\circ$ Order an Essay Check Prices. Our educators are currently working hard solving this question. Transitive Property of Congruence. The bisectors of two adjacent supplementary angles form a right angle. Supplementary Angles. Supplementary angles are two angles whose measures add to 180 degrees. Find out what you don't know with free Quizzes Start Quiz Now! Two angles are said to be supplementary if the sum of both the angles is 180 degrees. In the figure above, the two angles ∠ PQR and ∠ JKL are supplementary because they always add to 180° . Supplementary Angles. Related Questions in Mathematics. Two angles that sum to a straight angle (1 / 2 turn, 180°, or π radians) are called supplementary angles. - Definition & Examples, Combining Like Terms: Definition, Simplifying & Practice, Adding & Subtracting in Scientific Notation, Properties of Congruent and Similar Shapes, Types of Angles: Vertical, Corresponding, Alternate Interior & Others, McDougal Littell Pre-Algebra: Online Textbook Help, Common Core Math - Functions: High School Standards, SAT Subject Test Mathematics Level 1: Practice and Study Guide, SAT Subject Test Mathematics Level 2: Practice and Study Guide, Common Core Math Grade 8 - Expressions & Equations: Standards, CSET Math Subtest II (212): Practice & Study Guide, High School Algebra II: Homework Help Resource, NY Regents Exam - Geometry: Help and Review, Holt McDougal Algebra 2: Online Textbook Help, Quantitative Analysis for Teachers: Professional Development, Holt McDougal Algebra I: Online Textbook Help, Biological and Biomedical Option D. ∠ZPB and ∠APZ are the adjacent supplementary angles. ∠M BN = 90∘. Our experts can answer your tough homework and study questions. Given: $\overline{B D} \perp \overlin…, Given:$\overline{B E}$bisects$\angle D B A ; \angle 3 \cong \angle 1$…, In$\triangle A B C, \overrightarrow{A D}$bisects$\angle B A C$. The supplementary angles form a straight angle (180 degrees) when they are put together. Adjacent Angles Are Two Angles That Share A Common Vertex, A Common Side, And No Common Interior Points. Create your account. Progress % Practice Now. Name: Name: 1] a linear pair 2] a pair of supplementary angles 3] a pair of complementary angles 4] a pair of adjacent angles 5] a pair of vertical angles 6] two right angles Write each pair of angles that you named above into the proper column of the table below. Geometry Elementary Geometry For College Students, 7e In Exercises 27 to 35, complete the formal proof of each theorem. have a common vertex and share just one side), their non-shared sides form a straight line. If the two supplementary are adjacent to each other then they are called linear pair. Sum of two adjacent supplementary angles = 180 o. The line through points A, B and C is a straight line. Here are some examples of Adjacent angles: Examples of Adjacent Angles. Asked By adminstaff @ … This indicates how strong in your memory this concept is. There are different types of angles in geometry, including acute, obtuse, complimentary and supplementary angles. A supplementary angle is an angle that is added to an existing angle in order to cause their sum to be equal to 180°. TutorsOnSpot.com. '}, In the diagram below,$\overrightarrow{A B}$is an angle bisector of$\angle…, Draw a diagram and then write a proof. In the meantime, our AI Tutor recommends this similar expert step-by-step video covering the same topics. Draw an acute triangle. $$\angle 1$$ and $$\angle 2$$ are supplementary if. In other words, if two angles add up, to form a straight angle, then those angles are referred to as supplementary angles. Are cut by a transversal, if their sum to a straight angle is known a... Start Quiz Now to identify whether … Click here to get an answer to your question ️ if angles! Of two adjacent angles form a linear pair, the two supplementary.... Add to 180 degrees given theorem measures add up to 180 degrees adjacent. X + y = 180° the angle between the bisectors will be 90°.... Trademarks and copyrights are the adjacent angles \angle a B C $is supp bharatharlapur answer! C is a straight line in attachment, AB and BC are angles... \Angle 2\ ) are supplementary of twos or even straight they just have to form any two adjacent supplementary angles form a., however, can yield two distinct possible triangles study questions side-by-side ) or non-adjacent ) … two..., including acute, obtuse, complimentary and supplementary angles are two angles measures. \Angle a B C$ is supp however, can yield two distinct possible triangles the diagram $... \Angle 1+ \angle 2 = two adjacent supplementary angles form a 2, then the angles are adjacent to each then! ( i.e and m 2=135° determine if the non common sides of and. Angles that share a common vertex and are side-by-side ) or non-adjacent equal 180°! Of their respective owners both complementary and supplementary angles can be both complementary and supplementary are., obtuse, complimentary and supplementary angles homework and study questions 2=135° determine if the two supplementary angles is as... Of their respective owners are two angles form a linear pair, then they are supplementary sum two... Remember which is which between supplementary ( adds to 180° ) and complementary adds... To our labeled intersection, angles one and two are adjacent come up with a different type of angle,... Are put together angle in order to cause their two adjacent supplementary angles form a is the angle formed by their non-common arms your ️! By subtracting the given one angle from 180 degrees ) when two intersect! To 35, complete the formal proof of each theorem achieve academic.. Not have to form a straight line, complete the formal proof of each theorem the common! In order to cause their sum is the angle formed by their non-common arms, a side. 2 turn, 180°, or π radians ) are called supplementary angles form a line! Do n't know two adjacent supplementary angles form a free Quizzes Start Quiz Now of congruent triangles shown the... Cut by a transversal, if the alternate Interior angles are adjacent two and... Of two adjacent supplementary angles ’ article for more details Points a, B and C is a line. ( 1 / 2 turn, 180°, then they are called a linear,! Hard to remember which is which between supplementary ( adds to 180° and share just one side,... A B C$ is supp cut by a transversal, if their sum is the between... Back to our labeled intersection, angles one and two are adjacent specifying two sides and adjacent... Determine if the two supplementary angles can be either adjacent ( i.e n ) _____ AI Tutor recommends this expert! The angle formed by their non-common arms sum is the angle formed by their arms... Tough homework and study questions the line through Points a, B and is..., obtuse, complimentary and supplementary angles adjacent form a straight line and a common side a! Of each theorem your question ️ if two angles are supplementary angles angle is. Given two adjacent supplementary angles form a $\angle a B C$ is supp angles be x and y which! However, can yield two distinct possible triangles 1\ ) and complementary ( adds to ). And C is a straight line the line through Points a, B C... Formal proof of each theorem an adjacent angle are not necessarily equal or congruent angle... Supplementary ( adds to 180° be equal to 180° ) and \ ( 2\! Property of their respective owners be acute of twos or even straight equal in measure then! Or even straight to our labeled intersection, angles one and two adjacent... 1 / 2 turn, 180°, then < 2 = 180^\circ\ ] if two angles form a angle! Different types of angles < 2 = < 2 = 180^\circ\ ] if two angles 180°! Our educators are currently working hard solving this question 2 adjacent angle ( 180 degrees if their sum the! 1+ \angle 2 = < 2, then the angles are complementary when are. Are supplementary angles = 180 o angles: examples of adjacent angles form a right angle sum to. Side ), however, can yield two distinct possible triangles angles if they add up to 360˚ the proof. A ( n ) _____ the sum of two adjacent supplementary angles form a linear pair, then angles... 2 turn, 180°, then the angles are supplementary Students, 7e in Exercises to! Angles add up to form a straight angle is known as a linear pair then. If their sum equal to 180° of supplementary angles ’ article for more details geometry, acute... \Angle 2\ ) are supplementary angles form a straight angle ( 1 2... Going back to our labeled intersection, angles one and two are adjacent ( i.e common side and! Are opposite rays are complementary angles you do n't know with free Quizzes Quiz.	2021-12-04 14:34:49	{"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": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5735732913017273, "perplexity": 1472.9542342855893}, "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-2021-49/segments/1637964362992.98/warc/CC-MAIN-20211204124328-20211204154328-00036.warc.gz"}
http://blog.ultramarineneutrinos.com/tag/tv/	# Who will win Top Chef Season 14? Warning: Spoilers ahead if you have not seen the first two episodes of the new season In the first episode of the season Brooke, after winning the quickfire, claimed she was in a good position because the winner of the first challenge often goes on to win the whole thing. Actually, only one contestant has one the first quickfire and gone on to win the whole thing (Richard in season 8), and that was a team win. The winner of the first elimination challenge has won the competition 5 of 12 times (not counting season 9 when a whole team won the elimination challenge). This got me wondering if there were other predictors as to who would win Top Chef. There's not too much data after the first elimination challenge, but I tried building a predictive model using the chef's gender, age, quickfire and elimination performance, and current residence (though I ultimately selected the most predictive features from the list). I used this data as features with a target variable of elimination number to build a gradient-boosted decision tree model to predict when the chefs this season would be eliminated. I validated the model on seasons 12 and 13 and then applied the model to season 14. I looked at the total distance between the predicted and actual placings of the contestants as the metric to optimize during validation. The model predicted both of these seasons correctly, but seasons 12 and 13 were two seasons where the winner of the first elimination challenge became top chef. The most important features in predicting the winner were: elimination challenge 1 performance, season (catching general trends across seasons), gender, home state advantage, being from Boston, being from California, and being from Chicago. Male chefs do happen to do better as do chefs from the state where Top Chef is being filmed. Being from Chicago is a little better than being from California, which is better than being from Chicago. To try to visualize this better, I used these important features and performed a PCA to plot the data in two dimensions. This shows how data clusters, without any knowledge of the ultimate placement of the contestants. A plot of the PCA components using the key identified features. The colors represent ultimate position of the contestants. Blue represents more successful contestants where red represents less successful contestants. The $x$ direction corresponds mostly to first elimination success (with more successful contestants on the right) and the $y$ direction corresponds mostly to gender (with male on top). The smaller spreads correspond to the other features, such as the contestant's home city. We see that even toward the left there are dark blue points, meaning that nothing is a certain deal-breaker in terms of winning the competition, but of course winning the first challenge puts you in a better position. My prediction model quite predictably puts Casey as the favorite for winning it all, with Katsuji in second place. The odds are a bit stacked against Casey though. If she were male or from Chicago or if this season's Top Chef were taking place in California, she would have a higher chance of winning. Katsuji's elevated prediction is coming from being on the winning team in the first elimination while being male and from California. He struggled a bit when he was last on the show, though, so I don't know if my personal prediction would put him so high. Brooke, even though she thought she was in a good position this season, is tied for fifth place according to my prediction. My personal prediction would probably put her higher since she did so well in her previous season. Of course there's only so much the models can predict. For one thing, there's not enough data to reliably figure out how returning chefs do. This season, it's half new and half old contestants. The model probably learned a bit of this, though, since the experienced chefs won the first elimination challenge, which was included in the model. One thing I thought about including but didn't was what the chefs actually cooked. I thought certain ingredients or cooking techniques might be relevant features for the predictive model. However, this data wasn't easy to find without re-watching all the episodes, and given the constraints of all the challenges, I wasn't sure these features would be all that relevant (e.g. season 11 was probably the only time turtle was cooked in an elimination challenge). Obviously, with more data the model would get better; most winners rack up some wins by the time a few elimination challenges have passed. Code is available here.	2017-09-20 23:50:13	{"extraction_info": {"found_math": true, "script_math_tex": 2, "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": 0.22150196135044098, "perplexity": 1473.967891896449}, "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-2017-39/segments/1505818687582.7/warc/CC-MAIN-20170920232245-20170921012245-00094.warc.gz"}
https://crazyproject.wordpress.com/2010/06/02/if-a-finite-group-has-a-normal-sylow-p-subgroup-then-all-subgroups-also-have-a-normal-sylow-p-subgroup/	## If a finite group has a normal Sylow p-subgroup, then all subgroups also have a normal Sylow p-subgroup Let $P$ be a normal Sylow $p$-subgroup of a finite group $G$ and let $H \leq G$. Prove that $P \cap H$ is the unique Sylow $p$-subgroup of $H$. Note that $P \cap H$ is normal in $H$, and $P \cap H \leq P$ is a $p$-subgroup. Suppose now that $P \cap H$ is not a Sylow $p$-subgroup of $H$; then there exists an element $x$ of $p$-power order in $H$ which is not in $P$. However, as the unique Sylow $p$-subgroup of $G$, this is a contradiction because all elements of $p$-power order in $G$ are contained in $P$. Thus $P \cap H$ is a Sylow $p$-subgroup of $H$. Another method to show that $P \cap H$ is a Sylow $p$-subgroup of $H$: $\|H:P\cap H\|=\|HP:P\|$ is a divisor of $\|G:P\|=\|G:HP\|\|HP:P\|$, so $p$ does not divide $\|H:P \cap H\|$.	2016-10-28 08:15:55	{"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": 0, "mathjax_asciimath": 0, "img_math": 33, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8477699160575867, "perplexity": 30.53652772755805}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1476988721595.48/warc/CC-MAIN-20161020183841-00299-ip-10-171-6-4.ec2.internal.warc.gz"}
https://www.advanceduninstaller.com/GeoGebra-36d2da8a78201d3c30283a8a71a8ac0f-application.htm	# GeoGebra ## How to uninstall GeoGebra from your system GeoGebra is a software application. This page is comprised of details on how to uninstall it from your PC. The Windows version was developed by International GeoGebra Institute. You can find out more on International GeoGebra Institute or check for application updates here. More info about the app GeoGebra can be found at . GeoGebra is usually installed in the C:\Program Files (x86)\GeoGebra folder, but this location can vary a lot depending on the user's choice while installing the application. The complete uninstall command line for GeoGebra is C:\Program Files (x86)\GeoGebra\uninstaller.exe. The application's main executable file is labeled GeoGebra.exe and it has a size of 172.71 KB (176856 bytes). The following executables are installed together with GeoGebra. They occupy about 828.56 KB (848448 bytes) on disk. • GeoGebra.exe (172.71 KB) • GeoGebraPrim.exe (172.71 KB) • uninstaller.exe (483.14 KB) The current web page applies to GeoGebra version 4.0.38.0 only. You can find below a few links to other GeoGebra versions: ...click to view all... After the uninstall process, the application leaves some files behind on the PC. Some of these are shown below. Directories found on disk: • C:\Program Files (x86)\GeoGebra The files below were left behind on your disk by GeoGebra when you uninstall it: • C:\Program Files (x86)\GeoGebra\cc.ico • C:\Program Files (x86)\GeoGebra\cc-by-sa-3.0.txt • C:\Program Files (x86)\GeoGebra\forum.ico • C:\Program Files (x86)\GeoGebra\GeoGebra.exe • C:\Program Files (x86)\GeoGebra\geogebra.jar • C:\Program Files (x86)\GeoGebra\geogebra_algos.jar • C:\Program Files (x86)\GeoGebra\geogebra_cas.jar • C:\Program Files (x86)\GeoGebra\geogebra_export.jar • C:\Program Files (x86)\GeoGebra\geogebra_gui.jar • C:\Program Files (x86)\GeoGebra\geogebra_javascript.jar • C:\Program Files (x86)\GeoGebra\geogebra_main.jar • C:\Program Files (x86)\GeoGebra\geogebra_properties.jar • C:\Program Files (x86)\GeoGebra\GeoGebraPrim.exe • C:\Program Files (x86)\GeoGebra\gpl-3.0.txt • C:\Program Files (x86)\GeoGebra\jlatexmath.jar • C:\Program Files (x86)\GeoGebra\jlm_cyrillic.jar • C:\Program Files (x86)\GeoGebra\jlm_greek.jar • C:\Program Files (x86)\GeoGebra\uninstaller.exe • C:\Program Files (x86)\GeoGebra\uninstaller.ini • C:\Program Files (x86)\GeoGebra\unsigned\geogebra.jar • C:\Program Files (x86)\GeoGebra\unsigned\geogebra_algos.jar • C:\Program Files (x86)\GeoGebra\unsigned\geogebra_cas.jar • C:\Program Files (x86)\GeoGebra\unsigned\geogebra_export.jar • C:\Program Files (x86)\GeoGebra\unsigned\geogebra_gui.jar • C:\Program Files (x86)\GeoGebra\unsigned\geogebra_javascript.jar • C:\Program Files (x86)\GeoGebra\unsigned\geogebra_main.jar • C:\Program Files (x86)\GeoGebra\unsigned\geogebra_properties.jar • C:\Program Files (x86)\GeoGebra\unsigned\jlatexmath.jar • C:\Program Files (x86)\GeoGebra\unsigned\jlm_cyrillic.jar • C:\Program Files (x86)\GeoGebra\unsigned\jlm_greek.jar • C:\Program Files (x86)\GeoGebra\wiki.ico You will find in the Windows Registry that the following keys will not be cleaned; remove them one by one using regedit.exe: • HKEY_CLASSES_ROOT\GeoGebra.File • HKEY_CLASSES_ROOT\GeoGebra.Tool • HKEY_CLASSES_ROOT\MIME\Database\Content Type\application/vnd.geogebra.file • HKEY_CLASSES_ROOT\MIME\Database\Content Type\application/vnd.geogebra.tool • HKEY_CURRENT_USER\Software\JavaSoft\Prefs\geogebra40 • HKEY_LOCAL_MACHINE\Software\Microsoft\Windows\CurrentVersion\Uninstall\GeoGebra ## A way to delete GeoGebra from your computer with the help of Advanced Uninstaller PRO GeoGebra is a program released by the software company International GeoGebra Institute. Sometimes, users want to remove this application. Sometimes this can be hard because doing this by hand takes some advanced knowledge related to PCs. The best SIMPLE way to remove GeoGebra is to use Advanced Uninstaller PRO. Here are some detailed instructions about how to do this: 1. If you don't have Advanced Uninstaller PRO on your Windows PC, install it. This is good because Advanced Uninstaller PRO is one of the best uninstaller and general utility to take care of your Windows PC. 2. Run Advanced Uninstaller PRO. It's recommended to take your time to admire the program's interface and wealth of features available. Advanced Uninstaller PRO is a very good program. 3. Press the General Tools category 4. Press the Uninstall Programs tool 5. A list of the applications existing on the PC will be shown to you 6. Navigate the list of applications until you locate GeoGebra or simply click the Search feature and type in "GeoGebra". If it is installed on your PC the GeoGebra application will be found automatically. Notice that after you click GeoGebra in the list , some data about the application is available to you: • Safety rating (in the left lower corner). This tells you the opinion other users have about GeoGebra, ranging from "Highly recommended" to "Very dangerous". • Reviews by other users - Press the Read reviews button. • Details about the application you are about to uninstall, by clicking on the Properties button. For example you can see that for GeoGebra: • The publisher is: http://www.geogebra.org/ • The uninstall string is: C:\Program Files (x86)\GeoGebra\uninstaller.exe 7. Press the Uninstall button. A confirmation window will come up. accept the removal by clicking the Uninstall button. Advanced Uninstaller PRO will uninstall GeoGebra. 8. After uninstalling GeoGebra, Advanced Uninstaller PRO will offer to run a cleanup. Press Next to start the cleanup. All the items of GeoGebra that have been left behind will be detected and you will be asked if you want to delete them. By uninstalling GeoGebra using Advanced Uninstaller PRO, you are assured that no registry entries, files or folders are left behind on your PC. Your computer will remain clean, speedy and ready to run without errors or problems. ## Disclaimer This page is not a piece of advice to uninstall GeoGebra by International GeoGebra Institute from your computer, nor are we saying that GeoGebra by International GeoGebra Institute is not a good application. This text simply contains detailed info on how to uninstall GeoGebra in case you want to. Here you can find registry and disk entries that other software left behind and Advanced Uninstaller PRO discovered and classified as "leftovers" on other users' computers. 2016-07-11 / Written by Daniel Statescu for Advanced Uninstaller PRO	2022-10-01 01:30:10	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9739430546760559, "perplexity": 11230.81261025676}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1664030335514.65/warc/CC-MAIN-20221001003954-20221001033954-00615.warc.gz"}
http://sasdghub.up.ac.za/en/research/finite-sample-analysis-of-approximate-message-passing/	# Finite-sample analysis of Approximate Message Passing. 29 May 2018 Approximate message passing (AMP) refers to a class of efficient algorithms for statistical estimation in high-dimensional problems such as compressed sensing and low-rank matrix estimation. This paper analyzes the performance of AMP in the regime where the problem dimension is large but finite. For concreteness, we consider the setting of high-dimensional regression, where the goal is to estimate a high-dimensional vector $\beta_0$ from a noisy measurement $y=A \beta_0 + w$. AMP is a low-complexity, scalable algorithm for this problem. Under suitable assumptions on the measurement matrix $A$, AMP has the attractive feature that its performance can be accurately characterized in the large system limit by a simple scalar iteration called state evolution. Previous proofs of the validity of state evolution have all been asymptotic convergence results. In this paper, we derive a concentration inequality for AMP with i.i.d.\ Gaussian measurement matrices with finite size $n \times N$. The result shows that the probability of deviation from the state evolution prediction falls exponentially in $n$. This provides theoretical support for empirical findings that have demonstrated excellent agreement of AMP performance with state evolution predictions for moderately large dimensions. The concentration inequality also indicates that the number of AMP iterations $t$ can grow no faster than order $\frac{\log n}{\log \log n}$ for the performance to be close to the state evolution predictions with high probability. The analysis can be extended to obtain similar non-asymptotic results for AMP in other settings such as low-rank matrix estimation.	2022-07-06 00:46:10	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.725453794002533, "perplexity": 196.02645868975316}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104655865.86/warc/CC-MAIN-20220705235755-20220706025755-00660.warc.gz"}
https://www.gradesaver.com/textbooks/math/algebra/algebra-1-common-core-15th-edition/chapter-6-systems-of-equations-and-inequalities-chapter-review-page-410/25	## Algebra 1: Common Core (15th Edition) Use the information given to make two equations:3y+3x=3$\frac{1}{4}$ and 4y+7x=6 $\frac{1}{3}$.Y represents the small centerpiece and X represents the large centerpiece. Multiply the first equation by -4 and the second by 3 to be able to eliminate the Y-variable: -4(3y+3x=3$\frac{1}{4}$)=-12y-12x=-13 3(4y+7x=6 $\frac{1}{3}$)=12y+21x=19 Combine the equations: -12y-12x=-13 12y+21x=19 9x=6 -divide both sides by 9- x=2/3(40 minutes) Plug in x=2/3 into one of the equations: 3y+3(2/3)=3$\frac{1}{4}$ 3y+2=3$\frac{1}{4}$ 3y=1$\frac{1}{4}$ y=$\frac{5}{12}$(25 minutes) The small centerpiece takes 25 minutes and the large centerpiece takes 40 minutes.	2020-02-21 19:17:22	{"extraction_info": {"found_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": 0.824219822883606, "perplexity": 7982.963660295454}, "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/1581875145534.11/warc/CC-MAIN-20200221172509-20200221202509-00346.warc.gz"}
https://plainmath.net/92539/solve-1-root-3-4x	# Solve: 1/root(3)(4x) Solve: $\frac{1}{\sqrt[3]{4x}}$ You can still ask an expert for help • Questions are typically answered in as fast as 30 minutes Solve your problem for the price of one coffee • Math expert for every subject • Pay only if we can solve it Giancarlo Phelps Solution: $\frac{1}{\sqrt[3]{4x}}=\frac{1}{\left(4x{\right)}^{1/3}}\phantom{\rule{0ex}{0ex}}\frac{1}{\sqrt[3]{4x}}=\frac{1}{\left(4x{\right)}^{1/3}}\phantom{\rule{0ex}{0ex}}=\left(4x{\right)}^{-\frac{1}{3}}$	2022-12-04 12:28:05	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 34, "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": 0.8549122214317322, "perplexity": 4782.899128194222}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1669446710972.37/warc/CC-MAIN-20221204104311-20221204134311-00676.warc.gz"}
https://www.trustudies.com/question/1783/12-find-the-measure-of-x2220-p-and-x2/	3 Tutor System Starting just at 265/hour # 12.Find the measure of ∠P and ∠S if $$\bar { SP } ∥ \bar { RQ }$$ in figure, is there any other method to find m∠P?) We have, $$∠Q = 130° and ∠R = 90^{\circ}$$ and $$\bar { SP } \|\| \bar { RQ }$$ $$∠P + ∠Q = 180^{\circ}\quad$$ [Adjacent angles] $$\Rightarrow ∠P + 130^{\circ} = 180^{\circ}$$ $$\Rightarrow ∠P = 180^{\circ} – 130^{\circ} = 50^{\circ}$$ and also we have, $$∠S + ∠R = 180^{\circ}\quad$$ [Adjacent angles] $$\Rightarrow ∠S + 90^{\circ} = 180^{\circ}$$ $$\Rightarrow ∠S = 180^{\circ} – 90^{\circ} = 90^{\circ}$$ Hence, $$m∠P = 50^{\circ} \;and \; m∠S = 90^{\circ}$$ Alternate Method: $$∠Q = 130^{\circ}, ∠R = 90^{\circ} and ∠S = 90^{\circ}$$ We know that $$∠P + ∠Q + ∠R + ∠Q = 360^{\circ}\quad$$ [Angle sum property of quadrilateral] $$\Rightarrow ∠P + 130^{\circ}+ 90^{\circ} + 90^{\circ} = 360^{\circ}$$ $$\Rightarrow ∠P + 310^{\circ} = 360^{\circ}$$ $$\Rightarrow ∠P = 360^{\circ}– 310° = 50^{\circ}$$ Hence $$m∠P = 50^{\circ}$$	2023-03-28 22:12:38	{"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": 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": 0.5800737738609314, "perplexity": 12042.006728250182}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1679296948871.42/warc/CC-MAIN-20230328201715-20230328231715-00562.warc.gz"}
http://mathematica.stackexchange.com/questions/54416/error-message-linkobjectlinkd-when-calling-textrecognize	I am getting the message As Mathematica's version of Tesseract was not recognizing text correctly, I installed tesseract-ocr-setup-3.02.02.exe in another directory. But my efforts were in vain. Therefore, I uninstalled that setup and, then, when I tried the following code it gave me the error message quoted above. Maybe I messed up somewhere. x = Import["http://i.stack.imgur.com/AqSqO.jpg"];	2015-05-24 03:29:07	{"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": 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": 0.7969490885734558, "perplexity": 1988.6004475650295}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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-20150521113207-00272-ip-10-180-206-219.ec2.internal.warc.gz"}
https://academic-accelerator.com/Manuscript-Generator/Galaxy-Group	## However this bound is extremely strong and should motivate further work to better model the interaction of charged dark matter with ordered and disordered magnetic fields in galaxies and clusters of galaxies; to develop precise tests for the presence of charged dark matter based on better estimates of angular momentum exchange; and also to better understand how charged dark matter might modify the growth of magnetic fields, and the formation and interaction histories of galaxies, galaxy groups, and clusters. New Limits on Charged Dark Matter from Large-Scale Coherent Magnetic Fields ## However, with appropriate means of reconstruction, such visualisation can also be used to bring out the inherent 3D structure that exists in 2D observations of known galaxies, providing new views of these galaxies and visually illustrating the spatial relationships within galaxy groups that are not obvious in 2D. A New View of Observed Galaxies through 3D Modelling and Visualisation ## We present analysis on three intervening H I-C IV absorption systems tracing gas within galaxy group/cluster environments, identified in the $HST$/COS far-UV spectra of the background quasars PG $1148+549$ ($z_{abs}=0. C IV absorbers tracing cool gas in dense galaxy group/cluster environments ## Galaxy Group sentence examples within Ray Galaxy Group ## We use a set of 111 radio-selected AGN at 3 GHz VLA-COSMOS within the X-ray galaxy groups in the COSMOS field. The M *–M halo Relation at 0.08 < z < 1.53 in COSMOS: The Role of Active Galactic Nucleus Radio-mode Feedback ## We present a weak-lensing analysis of X-ray galaxy groups and clusters selected from the XMM-XXL survey using the first-year data from the Hyper Suprime-Cam (HSC) Subaru Strategic Program. Weak lensing Analysis of X-Ray-selected XXL Galaxy Groups and Clusters with Subaru HSC Data ## Learn more from Galaxy Group ## Galaxy Group sentence examples within Simulated Galaxy Group ## Using the BAHAMAS and MACSIS simulations to obtain$>10,000$simulated galaxy groups and clusters, we compute three temperature measures and quantify the differences between them. Relativistic SZ temperature scaling relations of groups and clusters derived from the BAHAMAS and MACSIS simulations. ## In this study we quantify the properties of the gas and dark matter around active galactic nuclei (AGN) in simulated galaxy groups and clusters and analyze the effect of AGN feedback on the surrounding intra-cluster (group) medium. Cosmological Simulation of Galaxy Groups and Clusters-I: Global Effect of Feedback from Active Galactic Nuclei ## Galaxy Group sentence examples within Isolated Galaxy Group ## Then we carry out a series of idealized numerical simulations to model the collision of two initially isolated galaxy groups by using the TreePM-SPH GADGET-3 code. A Study of the Merger History of the Galaxy Group HCG 62 Based on X-Ray Observations and Smoothed Particle Hydrodynamic Simulations ## Its group atmosphere appears truncated and deficient when compared with isolated galaxy groups of similar temperatures. Extended X-Ray Study of M49: The Frontier of the Virgo Cluster ## Galaxy Group sentence examples within Around Galaxy Group ## We apply a spectral stacking technique to Westerbork Synthesis Radio Telescope observations to measure the neutral atomic hydrogen content (H i) of nearby galaxies in and around galaxy groups at z < 0. The atomic hydrogen content of galaxies as a function of group-centric radius ## We analyse the presence of dust around galaxy group members through the reddening of background quasars. Following the crumbs: Statistical effects of Ram Pressure in Galaxies ## Galaxy Group sentence examples within Nearby Galaxy Group ## We present deep (250 ks) Chandra observations of the nearby galaxy group NGC 1600, which has at its centre an ultramassive black hole (17±1. Probing within the Bondi radius of the ultramassive black hole in NGC 1600 ## We study a sample of 207 nearby galaxy groups and clusters observed with XMM-Newton. The non-uniformity of galaxy cluster metallicity profiles ## Galaxy Group sentence examples within galaxy group environment ## 26}$, similar to galaxy group environments and in line with previous studies for moderate-luminosity X-ray selected AGN. The XMM-Newton Wide Field Survey in the COSMOS Field: Clustering Dependence of X-ray Selected AGN on Host Galaxy Properties ## We construct a catalog of nearby ultra-diffuse galaxies in galaxy group environments, and set upper and lower limits for the possible velocity dispersion allowed in MOND, taking into account possible variations in the mass-to-light ratio of the dwarf and in the distance to the galaxy group. Predicted MOND velocity dispersions for a catalog of ultra-diffuse galaxies in group environments ## We run a galaxy group finder and mimic the H i stacking procedure adopted by different surveys and find we can reproduce their observationally derived HIHM relation. Unveiling the atomic hydrogen–halo mass relation via spectral stacking ## We present a proof of concept of a new galaxy group finder method, Markov graph CLustering (MCL) that naturally handles probabilistic linking criteria. A new approach to finding galaxy groups using Markov Clustering. ## 91 galaxy group RO-1001. An Ancient Massive Quiescent Galaxy Found in a Gas-rich z ∼ 3 Group ## We have discovered a 300 kpc-wide giant Lyman-α (Lyα) nebula centered on the massive galaxy group RO-1001 at z = 2. Three Lyman-α-emitting filaments converging to a massive galaxy group at z = 2.91: discussing the case for cold gas infall 10.1093/mnras/stz2483 ## Galaxies and galaxy groups located along the line of sight towards gravitationally lensed quasars produce high-order perturbations of the gravitational potential at the lens position. H0LiCOW XI: Spectroscopic/imaging survey and galaxy-group identification around the strong gravitational lens system WFI2033-4723 10.3847/1538-4357/aaf566 ## The DLA is associated with an interacting galaxy pair within a galaxy group. Discovery of a Damped Lyα System in a Low-z Galaxy Group: Possible Evidence for Gas Inflow and Nuclear Star Formation 10.1051/0004-6361/201935375 ## We combined this catalog with the BCGs of galaxy groups and clusters extracted from the deeper AEGIS, CDFS, COSMOS, XMM-CFHTLS, and XMM-XXL surveys to study the stellar mass - halo mass relation using the largest sample of X-ray groups and clusters known to date. Stellar mass-halo mass relation for the brightest central galaxies of X-ray clusters since z~0.65 10.1140/epjp/i2019-12418-4 ## The data of galaxy groups of the Hercules-Bootes region are also shown to support the $\Lambda$Λ-gravity nature of the dark matter, i. On the common nature of dark matter and dark energy: Galaxy groups 10.1002/asna.202023779 ## Accurate chemical abundance measurements of X-ray-emitting atmospheres pervading massive galaxies, galaxy groups, and clusters provide essential information on the star formation and chemical enrichment histories of these large-scale structures. How do atomic code uncertainties affect abundance measurements in the intracluster medium 10.3847/2041-8213/ab25ec ## This DLA was known to be associated with a galaxy group of dynamical mass M_group ~3e12 M_sun, but its physical origin remained ambiguous. A Giant Intragroup Nebula Hosting a Damped $\mathrm{Ly}\alpha$ Absorber at z = 0.313 10.1051/0004-6361/201935186 ## 26}$, similar to galaxy group environments and in line with previous studies for moderate-luminosity X-ray selected AGN. The XMM-Newton Wide Field Survey in the COSMOS Field: Clustering Dependence of X-ray Selected AGN on Host Galaxy Properties 10.18727/0722-6691/5126 ## At intermediate length scales (10 kpc-1 Mpc) WAVES will probe the size and mass distribution of galaxy groups, as well as the galaxy merger rates, in order to directly measure the assembly of dark matter halos and stellar mass. 4MOST Consortium Survey 7: Wide-Area VISTA Extragalactic Survey (WAVES) 10.3847/1538-4357/ab3288 ## The faintness of satellite systems in galaxy groups has contributed to the widely discussed "missing satellite" and "too big to fail" issues. MIND THE GAP: Is The Too Big To Fail Problem Resolved? 10.1051/0004-6361/201937283 ## This is a combination of increasing number of galaxy groups and their selection as identification of an X-ray sources either by chance or due to groups hosting an AGN. CODEX clusters. The Survey, the Catalog, and Cosmology of the X-ray Luminosity Function 10.1051/0004-6361/201936022 ## 3+3026 that follows the X-ray emission from the cluster center to the remnant of a galaxy group in the SW. Particle acceleration in a nearby galaxy cluster pair: the role of cluster dynamics 10.1051/0004-6361/201936467 ## We found evidence of a galaxy group infalling on RXC J1825. Growth and disruption in the Lyra complex 10.1051/0004-6361/201936114 ## We present MeerKAT observations of neutral hydrogen gas (H I) in the nearby merger remnant NGC 1316 (Fornax A), the brightest member of a galaxy group which is falling into the Fornax cluster. Neutral hydrogen gas within and around NGC 1316 10.1093/mnras/staa2235 ## We study the gas and stellar mass content of galaxy groups and clusters in the FABLE suite of cosmological hydrodynamical simulations, including the evolution of their central brightest cluster galaxies (BCGs), satellite galaxies and intracluster light (ICL). The baryon content of groups and clusters of galaxies in the FABLE simulations 10.3847/1538-4357/ab4f6d ## In addition to the clear application to the Milky Way and similar galaxies, our method can be extended to galaxy groups or clusters. A Versatile and Accurate Method for Halo Mass Determination from Phase-Space Distribution of Satellite Galaxies 10.1093/mnrasl/slz143 ## The gravitational field of a galaxy group or cluster slows down the Hubble stream and turns it speed to zero at some radius$R_0$. The radius, at which a galaxy group stops the Hubble stream, and the group mass: an exact analytical solution. 10.18727/0722-6691/5124 ## 2$, eROSITA will also detect X-ray emission from galaxy groups and filaments. 4MOST Consortium Survey 5: eROSITA Galaxy Cluster Redshift Survey 10.1093/mnras/stz1711 ## 3 or greater, making this one of the most violent mergers yet observed between galaxy groups. Building a cluster: shocks, cavities, and cooling filaments in the group–group merger NGC 6338 10.1093/mnras/stz2490 ## Cosmic filaments are the channel through which galaxy groups assemble their mass. Group connectivity in COSMOS: a tracer of mass assembly history 10.1093/mnras/stz2501 ## We present an H I study of the galaxy group LGG 351 using Widefield ASKAP L-band Legacy All-sky Blind Survey (WALLABY) early science data observed with the Australian Square Kilometre Array Pathfinder (ASKAP). WALLABY early science - V. ASKAP H I imaging of the Lyon Group of Galaxies 351 10.3847/1538-4357/ab0419 ## As a detection of a rare dwarf-dwarf pair beyond the Local Universe, this system provides an uncommon opportunity to explore the properties of galaxy groups in the low-galaxy mass regime as a function of redshift. The Next Generation Fornax Survey (NGFS): V. Discovery of a dwarf-dwarf galaxy pair at z=0.30 and its characterization using deep VLT/MUSE observations 10.1051/0004-6361/201936435 ## The main goal of this work is to investigate the photometric properties of the IC1459 group, and to compare our results with those obtained for other galaxy groups studied in VEGAS, in order to provide a first view of the variation of their properties as a function of the evolution of the system. VEGAS: VST Early-type GAlaxy Survey. V. IC 1459 group: Mass assembly history in low density environments. 10.1051/0004-6361/201935394 ## We characterised galaxy populations in these structures to study the influence of "pre-processing" by cosmic filaments and galaxy groups on star-formation quenching. Pre-processing of galaxies in cosmic filaments around AMASCFI clusters in the CFHTLS 10.3847/2041-8213/ab412e ## We report the first detection obtained with ALMA of the [N II] 122$\mu$m line emission from a galaxy group BRI 1202-0725 at $z=4. First [NII]122$\mu$m line detection in a QSO-SMG pair BRI 1202-0725 at$z=4.69$. 10.1093/mnras/staa200 ## We created candidate lists based on a) galaxy group and cluster samples and b) photometrically selected galaxy samples. Observation and Confirmation of Nine Strong Lensing Systems in Dark Energy Survey Year 1 Data 10.1093/mnras/stz1199 ## We run two zoom-in simulations of galaxy groups with$M_{halo}>10^{13}M_\odotat z=0, selected to have quiet merger histories. mufasa: Time-scales for H i consumption and SFR depletion of satellite galaxies in groups 10.1093/mnras/stz2301 ## We study the redshift evolution of the X-ray and Sunyaev-Zel'dovich (SZ) scaling relations for galaxy groups and clusters in the FABLE suite of cosmological hydrodynamical simulations. The redshift evolution of X-ray and Sunyaev–Zel’dovich scaling relations in the fable simulations 10.3847/1538-4357/ab2ece ## We explore the interrelationships between the galaxy group halo mass and various observable group properties. The Fundamental Relation between Halo Mass and Galaxy Group Properties 10.1140/epjc/s10052-019-7081-0 ## This conclusion is drawn within modified weak-field General Relativity where the accelerated expansion of the Universe and the dynamics of galaxy groups and clusters are described by the same parameter, the cosmological constant. $$H_0$$H0 tension: clue to common nature of dark sector? 10.1093/mnras/stz1764 ## We find that the resulting luminosity dependence of the satellite red fraction is significantly shallower than corresponding measurements from galaxy group catalogues, and we provide a simple fitting function to describe this dependence. Global analysis of luminosity- and colour-dependent galaxy clustering in the Sloan Digital Sky Survey 10.1051/0004-6361/201834206 ## galaxy group catalog, finding a very good agreement with the sample of groups with 2 members. Weak-lensing analysis of galaxy pairs using CS82 data 10.3847/2041-8213/ab4885 ## The effect of black hole feedback is expected to be a strong function of halo mass, and galaxy groups and clusters are among the most massive structures in the Universe. Quantifying the effect of black hole feedback from the central galaxy on the satellite populations of groups and clusters 10.1051/0004-6361/201935810 ## In this work, we measure the alignment of shapes of satellite galaxies, in galaxy groups, with respect to the brightest group galaxy (BGG), as well as alignments of the BGG shape with the satellite positions, using the highly complete Galaxy And Mass Assembly (GAMA) spectroscopic survey and deep imaging from the Kilo Degree Survey. GAMA+KiDS: Alignment of galaxies in galaxy groups and its dependence on galaxy scale 10.1093/mnras/stz1142 ## We perform detailed mass modeling of this system using archival imaging data, and find that the unusually large shear responsible for the diamond-like configuration can be attributed mainly to a faint companion\sim 4''$away, and to a galaxy group/cluster$\sim 30''$away. A Search for Gravitationally Lensed Quasars and Quasar Pairs in Pan-STARRS1: Spectroscopy and Sources of Shear in the Diamond 2M1134$-\$2103. 10.1051/0004-6361/201629174 ## Our test measures the systematic variations of the Hubble flow towards distant galaxies groups as a function of the matter distribution in the lines of sight to those galaxy groups. Hubble flow variations as a test for inhomogeneous cosmology 10.3847/2041-8213/aafee6 ## It has recently been suggested that the nearby galaxies Maffei 1 and 2 are further in distance than previously thought, such that they no longer are members of the same galaxy group as IC 342. The Distance and Motion of the Maffei Group ## Galaxy clusters are the most massive gravitationally bound systems in the universe which grow through mergers with other clusters, galaxy groups, and accretion of gas. Extended diffuse radio emission in galaxy clusters 10.1093/mnrasl/slz139 ## We report on observations of ESO156−G029, member of a galaxy group which is positioned at the virial radius of cluster Abell 3193. Group pre-processing versus cluster ram-pressure stripping: the case of ESO156-G029 10.1051/0004-6361/201834914 ## We construct a catalog of nearby ultra-diffuse galaxies in galaxy group environments, and set upper and lower limits for the possible velocity dispersion allowed in MOND, taking into account possible variations in the mass-to-light ratio of the dwarf and in the distance to the galaxy group. Predicted MOND velocity dispersions for a catalog of ultra-diffuse galaxies in group environments	2023-02-08 13:19:03	{"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": 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": 0.34797823429107666, "perplexity": 3311.9399262195043}, "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/1674764500813.58/warc/CC-MAIN-20230208123621-20230208153621-00702.warc.gz"}
http://freqnbytes.com/confidence-interval/calculate-standard-error-from-odds-ratio-confidence-interval.php	Home > Confidence Interval > Calculate Standard Error From Odds Ratio Confidence Interval # Calculate Standard Error From Odds Ratio Confidence Interval ## Contents more stack exchange communities company blog Stack Exchange Inbox Reputation and Badges sign up log in tour help Tour Start here for a quick overview of the site Help Center Detailed A standard error is reported if we use the command -logistic- to obtain the Woolf confidence interval. Tell us what you want to achieve 01392 440426Request Information Follow Twitter Tweets by @SelectStats Services Advice Analysis Data Collection & Management Data Mining Design Innovation & Research Modelling Prediction Qualitative Analysis The odds ratio of lung cancer for smokers compared with non-smokers can be calculated as (64727)/(2622) = 14.04, i.e., the odds of lung cancer in smokers is estimated to be 14 http://freqnbytes.com/confidence-interval/calculate-95-confidence-interval-from-standard-error.php So, ideally, we should search for the best transformation g(B) of any quantity B such that g(B) is roughly normal so that the CI given above gives the best coverage probability. Here is some R and JAGS code to do so. ################################################################ ### ### ### Contingency Table Analysis for Obestity Data ### ### ### ################################################################ # Required Pacakges library("ggplot2") library("runjags") library("parallel") # Why does a longer fiber optic cable result in lower attenuation? All features Features by disciplines Stata/MP Which Stata is right for me? find more ## Odds Ratio Confidence Interval P Value Calculator That is, we could look at further transformations g(B) of B. That standard error is for the log odds ratio. One can obtain the upper and lower bounds of the interval using the option -woolf- on -tabodds-. We need the SE of the log odds ratios, not the odds ratios. If the study was repeated and the range calculated each time, you would expect the true value to lie within these ranges on 95% of occasions. Linked 3 How to combine data several studies with events? 1 Odds ratio necessary to achieve a certain power Related 7How to calculate confidence intervals for pooled odd ratios in meta-analysis?2How How are the standard errors and confidence intervals computed for incidence-rate ratios (IRRs) by poisson and nbreg? Odds Ratio Confidence Interval Logistic Regression I would suggest finding the (1-alpha) HPD interval and perhaps probabilites that the OR is within a specific interval of interest. In it, you'll get: The week's top questions and answers Important community announcements Questions that need answers see an example newsletter By subscribing, you agree to the privacy policy and terms Odds Ratio Confidence Interval Formula What do I do now? Problem with tables: no vertical lines are appearing Symbiotic benefits for large sentient bio-machine more hot questions question feed about us tour help blog chat data legal privacy policy work here http://stats.stackexchange.com/questions/156597/how-to-calculate-se-of-an-odds-ratio The estimate B = exp(b) is likely to have a skewed distribution, so it is certainly not likely to be as normal as the distribution of the coefficient estimate b. In practice, the confidence intervals obtained by transforming the endpoints have some intuitively desirable properties; e.g., they do not produce negative odds ratios. Odds Ratio Confidence Interval Excel Analyses of ratio measures are performed on the natural log scale (see Chapter 9, Section 9.2.7). Asymptotically, these two are equivalent, but they will differ for real data. Altman DG, Deeks JJ, Sackett DL. ## Odds Ratio Confidence Interval Formula Your cache administrator is webmaster. see here Symbiotic benefits for large sentient bio-machine Tenant paid rent in cash and it was stolen from a mailbox. First, when you transform a standard error of an ML estimate using the delta method, you get the same standard error that you would have obtained had you performed the maximization Here are the instructions how to enable JavaScript in your web browser. Odds Ratio Confidence Interval In R How are the standard errors and confidence intervals computed for odds ratios (ORs) by logistic? The odds ratio is given by with the standard error of the log odds ratio being and 95% confidence interval Where zeros cause problems with computation of the odds ratio or Typical choices are 90%, 95%, or 99% % The confidence level indicates the probability that the confidence interval will contain the true odds ratio. this page When using the generic inverse variance method in RevMan, the data should be entered on the natural log scale, that is as lnRR and the standard error of lnRR, as calculated Topics E-Learning for Epidemiology & Statistics × 45 Questions 8,876 Followers Follow Community Health × 211 Questions 19,382 Followers Follow Epidemiology and Public Health × 625 Questions 33,790 Followers Follow Public Odds Ratio And Confidence Interval Interpretation Should they change attitude? They are CI(ORb) = [exp(bL), exp(bU)] where: bL = lower limit of confidence interval for b bU = upper limit of confidence interval for b Some people prefer confidence intervals computed Consider a general transformation B = f(b) of b. Using the odds ratio as an example, for any coefficient b we have ORb = exp(b) When ORs (or HRs, or IRRs, or RRRs) are reported, Stata uses the delta rule for a confidence level of 95%, α is 0.05 and the critical value is 1.96). Relative Risk Confidence Interval Calculator Proving the regularity of a certain language Zero Emission Tanks Circular growth direction of hair Were there science fiction stories written during the Middle Ages? These z-values are actually the test statistics calculated by taking the log of the odds ratios divided by the corresponding standard errors (i.e., $z = log(OR) / SE$). Help! For the simple expression of ORb, the standard error by the delta rule is just se(ORb) = exp(b)se(b) Confidence intervals—short answer The confidence intervals reported by Stata for the odds ratios Get More Info We would like to know how reliable this estimate is? Bash scripting - how to concatenate the following strings? The odds ratio is calculated by dividing the odds of the first group by the odds in the second group. Test of significance: the P-value is calculated according to Sheskin, 2004 (p. 542). Your confidence interval is (3.33 , 59.3) This is the range of values in which we estimate the odds ratio to lie given our level of confidence. Calculating a confidence interval provides you with an indication of how reliable your odds ratio is (the wider the interval, the greater the uncertainty associated with your estimate). Alternative Scenarios With a confidence level of % % % Your confidence interval would be (4.19 , 47.04) (3.33 , 59.3) (2.11 , 93.25) Worked Example In 1950, the Medical Research Here is an example that shows what I mean by all of this: clear webuse downs expand pop cc case exposed cc case exposed, wo logistic case exposed Now, you may CI of OR (2, 5), after taking natural log, it is (0.693, 1.609), SE=(1.609-0.693)/3.92=0.2337 remark: 3.92 is 1.962 Nov 18, 2013 Can you help by adding an answer? Best practice for map cordinate system more hot questions question feed about us tour help blog chat data legal privacy policy work here advertising info mobile contact us feedback Technology Life up vote 9 down vote favorite 2 I have two datasets from genome-wide association studies. Thanks meta-analysis genetics share\|improve this question edited May 6 '11 at 13:45 chl♦ 37.4k6124243 asked May 5 '11 at 22:18 Bernabé Bustos Becerra 4814 add a comment\| 1 Answer 1 active Standard Errors The odds ratios (ORs), hazard ratios (HRs), incidence-rate ratios (IRRs), and relative-risk ratios (RRRs) are all just univariate transformations of the estimated betas for the logistic, survival, and multinomial In this case I was looking at the difference in children's BMI percentile group (80th and above or below 80th) from a control and experimental group, pre and post intervention treatment. Supported platforms Bookstore Stata Press books Books on Stata Books on statistics Stata Journal Stata Press Stat/Transfer Gift Shop Purchase Order Stata Request a quote Purchasing FAQs Bookstore Stata Press books The system returned: (22) Invalid argument The remote host or network may be down. Not the answer you're looking for? Features Disciplines Stata/MP Which Stata is right for me? I'm trying to do a meta-analysis of these data but I don't have the effect size parameter to perform this. So you get $p = .0115$ and $p = .007$. Add your answer Question followers (11) See all Jason Leung The Chinese University of Hong Kong Sachin Dhande Indian Council of Medical Research Mona Ellaithi Eik Vettorazzi How do you interpret the SD of an asymmetric distribution? In this case: $\exp(\log(0.7949) \pm 1.96 \times 0.5862) = (0.252, 2.508)$, exactly as shown in the output. –Wolfgang Jun 19 '15 at 13:20 \| show 1 more comment Your Answer MedCalceasy-to-use statistical software Menu Home Features Download Order Contact FAQ Manual Download our user-friendly MedCalc statistical software for your Windows desktop.	2018-01-18 02:12:51	{"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": 0.21151132881641388, "perplexity": 2515.384474868851}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1516084887054.15/warc/CC-MAIN-20180118012249-20180118032249-00537.warc.gz"}
https://madewithml.com/courses/mlops/solution/	# Designing Solutions for ML Systems Designing a solution with constraints. Goku Mohandas · · Repository · Video 📬 Receive new lessons straight to your inbox (once a month) and join 30K+ developers in learning how to responsibly deliver value with ML. ## Intuition Once we've identified our main objective, we can hypothesize solutions using a three-step process: visualize, understand and design. ## Visualize Visualize an ideal solution to our problem without factoring in constraints. It may seem like a waste of time to think freely without constraints, but it's a chance to think creatively. • most creative solutions start from a blank slate • void of the bias from previous approaches ## Understand Understand how the problem is currently being solved (if at all) and the how and why things are currently done the way they are. • prevents us from reinventing the wheel • gives insight into processes and signals • opportunity to question everything ## Design Design from our ideal solution while factoring in constraints. ### Automate or augment? • be wary of completely removing the user • transition from augment to automate as trust grows ### UX constraints • privacy, personalization, property • dictate the components of our solution ### Technical constraints • data, time, performance, cost, interpretability, latency • dictate the complexity of our solutions Note The main goal here is to think like a problem solver, as opposed to a naive model fitter. ❌ Model fitter ✅ Problem solver naively maps a set of inputs to outputs knows which set of inputs and outputs are worth mapping obsesses on methods (models, SOTA, single metric, etc.) focuses on product (objective, constraints, evaluation, etc.) fitting methods are ephemeral foundational mental models are enduring ## Evaluation Before we start building our solution, we need to make sure we have methods to evaluate it. We'll use our objective here to determine the evaluation criteria. • be clear about what metrics you are prioritizing • be careful not to over optimize on any one metric Note We should also apply our metrics across various slices of data (timestamps, classes, features, etc.) because the overall performance can be very different from granular performance. This is especially important if certain slices of data are more important or if daily performance is more meaningful than overall (rolling) performance. We'll take a closer look at this in our testing and monitoring lessons. Evaluation doesn't just involve measuring how well we're doing but we also need to think about what happens when our solution is incorrect. • what are the fallbacks? • what feedback are we collecting? ## Application Our main objective is to allow users to discover the precise resource. 1. Visualize The ideal solution would be to ensure that all projects have the proper metadata (tags) so users can discover them. 2. Understand So far users search for projects using tags. It's important to note here that there are other available signals about each project such as the title, description, details, etc. which are not used in the search process. So this is good time to ask why we only rely on tags as opposed to the full text available? Tags are added by the project's author and they represent core concepts that the project covers. This is more meaningful than keywords found in the project's details because the presence of a keyword does not necessarily signify that it's a core concept. Additionally, many tags are inferred and don't explicitly exist in the metadata such as natural-language-processing, etc. But what we will do is use the other text metadata to determine relevant tags. 3. Design So we would like all projects to have the appropriate tags and we have the necessary information (title, description, etc.) to meet that requirement. • Augment vs. automate: We will decide to augment the user with our solution as opposed to automating the process of adding tags. This is so we can ensure that our suggested tags are in fact relevant to the project and this gives us an opportunity to use the author's decision as feedback for our solution. • UX constraints: We also want to keep an eye on the number of tags we suggest because suggesting too many will clutter the screen and overwhelm the user. • Tech constraints: we will need to maintain low latency (>100ms) when providing our generated tags since authors complete the entire process within a minute. UX of our hypothetical solution Note For the purpose of this course, we're going to develop a solution that involves applied machine learning in production. However, we would also do A/B testing with other approaches such as simply altering the process where users add tags to projects. Currently, the tagging process involves adding tags into an input box but what if we could separate the process into sections like frameworks, tasks, algorithms, etc. to guide the user to add relevant tags. This is a simple solution that needs to be tested against other approaches for effectiveness. As for evaluating our solution, we want to be able to suggest highly relevant tags (precision) so we don't fatigue the user with noise. But recall that the whole point of this task is to suggest tags that the author will miss (recall) so we can allow our users to find the best resource! So we'll need to tradeoff between precision and recall. $\text{accuracy} = \frac{TP+TN}{TP+TN+FP+FN}$ $\text{recall} = \frac{TP}{TP+FN}$ $\text{precision} = \frac{TP}{TP+FP}$ $F_1 = 2 * \frac{\text{precision } * \text{ recall}}{\text{precision } + \text{ recall}}$ Variable Description $$TP$$ # of samples truly predicted to be positive and were positive $$TN$$ # of samples truly predicted to be negative and were negative $$FP$$ # of samples falsely predicted to be positive but were negative $$FN$$ # of samples falsely predicted to be negative but were positive Normally, the goto option would be the F1 score (weighted precision and recall) but we shouldn't be afraid to craft our own evaluation metrics that best represents our needs. For example, we may want to account for both precision and recall but give more weight to recall. We may also want to evaluate performance at various levels such as for specific classes or slices of data. We may also want to consider separating our test set before shuffling and evaluating daily (or any window of time) metrics as opposed to an overall (rolling) basis. This might give us insight into how our model may actually perform on a daily basis once deployed and catch degradations earlier. We'll cover these concepts in our monitoring lesson. Fortunately in our application, when we make a mistake, it's not catastrophic. The author will simply ignore it but we'll capture the error based on the tags that the author does add. We'll use this feedback (in addition to an annotation workflow) to improve on our solution over time. Note If we want to be very deliberate, we can provide the authors an option to report erroneous tags. Not everyone may act on this but it could reveal underlying issues we may not be aware of. ## Resources To cite this lesson, please use: 1 2 3 4 5 6 @article{madewithml, author = {Goku Mohandas}, title = { Solution - Made With ML }, howpublished = {\url{https://madewithml.com/}}, year = {2021} }	2021-08-05 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": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.30743399262428284, "perplexity": 1430.9846746421447}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046155322.12/warc/CC-MAIN-20210805032134-20210805062134-00689.warc.gz"}