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https://nforum.ncatlab.org/discussion/4445/nonabelian-stokes-theorem/	# Start a new discussion ## Not signed in Want to take part in these discussions? Sign in if you have an account, or apply for one below ## Discussion Tag Cloud Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support. • CommentRowNumber1. • CommentAuthorUrs • CommentTimeOct 26th 2012 a little remark at nonabelian Stokes theorem, still to be expanded • CommentRowNumber2. • CommentAuthorzskoda • CommentTimeOct 26th 2012 • (edited Oct 26th 2012) Maybe some basic references would be helpful for orientation. I knew of some references about some generalizations of Stokes to some nonabelian setups, but I am not sure how many of them where indeed about what is described in this entry, i.e. the case of Lie algebras and parallel transport (at least one or two were about that case, this I remember). • CommentRowNumber3. • CommentAuthorUrs • CommentTimeOct 26th 2012 • (edited Oct 26th 2012) Okay, I have added a pointer to • CommentRowNumber4. • CommentAuthorEric • CommentTimeOct 28th 2012 I made a minor formatting change to the sidebar. I noticed when I was revising geometry pf physics that the Contexts would display all listed contexts together when you hover over it. I thought it would be better to display only the specific item you hovered over. Now, the “Differential geometry” and “$\infty$-Lie theory” can display independently. This is a change I also made to geometry of physics that was lost when my changes were rolled back. • CommentRowNumber5. • CommentAuthorEric • CommentTimeOct 28th 2012 • (edited Oct 28th 2012) Is there a higher version of Stokes theorem involving higher parallel transport? I mean something that might relate $(n+1)$-transport to $n$-transport on a boundary for $0\le n\le D-1$. Something like that… • CommentRowNumber6. • CommentAuthorDavidRoberts • CommentTimeOct 28th 2012 • (edited Oct 28th 2012) Never mind. • CommentRowNumber7. • CommentAuthorUrs • CommentTimeOct 29th 2012 • (edited Oct 29th 2012) I made a minor formatting change to the sidebar. I noticed when I was revising geometry pf physics that the Contexts would display all listed contexts together when you hover over it. I thought it would be better to display only the specific item you hovered over. Ah, thanks, that’s indeed better. I should remember to use this code next time. Is there a higher version of Stokes theorem involving higher parallel transport? Yes, there are higher analogs in arbitrary degree. As I briefly indicate in the entry, the nonabelian Stokes theorem may be regarded as part of the $n$-functoriality of an $n$-functor from a path n-groupoid to the delooping smooth n-groupoid of a smooth n-group. It gets increasingly tedious to write it out explicitly, though. Partial formulas for $n = 3$ are in the literature. For instance in our appendix we do part of it when we discuss 3-form curvature.	2018-12-10 12:45:13	{"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": 0.8286164999008179, "perplexity": 3094.633941698239}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376823339.35/warc/CC-MAIN-20181210123246-20181210144746-00079.warc.gz"}
https://gmatclub.com/forum/is-x-positive-1-x-2-10-x-0-2-3-x-252389.html	GMAT Question of the Day - Daily to your Mailbox; hard ones only It is currently 20 Feb 2019, 11:35 ### 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 ## Events & Promotions ###### Events & Promotions in February PrevNext SuMoTuWeThFrSa 272829303112 3456789 10111213141516 17181920212223 242526272812 Open Detailed Calendar • ### Free GMAT Prep Hour February 20, 2019 February 20, 2019 08:00 PM EST 09:00 PM EST Strategies and techniques for approaching featured GMAT topics. Wednesday, February 20th at 8 PM EST • ### Online GMAT boot camp for FREE February 21, 2019 February 21, 2019 10:00 PM PST 11:00 PM PST Kick off your 2019 GMAT prep with a free 7-day boot camp that includes free online lessons, webinars, and a full GMAT course access. Limited for the first 99 registrants! Feb. 21st until the 27th. # Is x positive? (1) x^2 + 10 x > 0 (2) 3^x > 1 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics Author Message TAGS: ### Hide Tags Math Expert Joined: 02 Sep 2009 Posts: 53020 Is x positive? (1) x^2 + 10 x > 0 (2) 3^x > 1 [#permalink] ### Show Tags 29 Oct 2017, 00:41 00:00 Difficulty: 25% (medium) Question Stats: 73% (01:14) correct 27% (01:17) wrong based on 137 sessions ### HideShow timer Statistics Is x positive? (1) x^2 + 10 x > 0 (2) 3^x > 1 _________________ Retired Moderator Joined: 25 Feb 2013 Posts: 1216 Location: India GPA: 3.82 Is x positive? (1) x^2 + 10 x > 0 (2) 3^x > 1 [#permalink] ### Show Tags 29 Oct 2017, 10:37 2 Bunuel wrote: Is x positive? (1) x^2 + 10 x > 0 (2) 3^x > 1 Statement 1: implies $$x(x+10)>0$$ $$=> x<-10$$ or $$x>0$$. clearly $$x$$ can be negative or positive. Hence insufficient Statement 2: $$3^x>3^0 => x>0$$. Sufficient Option B CEO Joined: 11 Sep 2015 Posts: 3440 Is x positive? (1) x^2 + 10 x > 0 (2) 3^x > 1 [#permalink] ### Show Tags 29 Nov 2018, 17:07 Top Contributor Bunuel wrote: Is x positive? (1) x² + 10x > 0 (2) 3^x > 1 Target question: Is x positive? Statement 1: x² + 10x > 0 Let's TEST some values There are several values of x that satisfy statement 1. Here are two: Case a: x = 1. In this case, the answer to the target question is YES, x IS positive Case b: x = -20. In this case, the answer to the target question is NO, x is NOT positive Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT Statement 2: 3^x > 1 We know that 3^0 = 1 If x < 0, then 3^x < 1 And if x > 0, then 3^x > 1 So, statement 2 indirectly tells us that x IS positive Since we can answer the target question with certainty, statement 2 is SUFFICIENT Cheers, Brent _________________ Test confidently with gmatprepnow.com Is x positive? (1) x^2 + 10 x > 0 (2) 3^x > 1 [#permalink] 29 Nov 2018, 17:07 Display posts from previous: Sort by # Is x positive? (1) x^2 + 10 x > 0 (2) 3^x > 1 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics Powered by phpBB © phpBB Group \| Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.	2019-02-20 19:35:47	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.42508959770202637, "perplexity": 10864.327949001074}, "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-2019-09/segments/1550247496080.57/warc/CC-MAIN-20190220190527-20190220212527-00221.warc.gz"}
https://www.vedantu.com/question-answer/isomerism-name-the-isomers-of-butane-class-11-chemistry-cbse-5f530dc0cb67df7e4399842a	Question # What is isomerism? Name the isomers of butane. Hint: When two compounds have similar molecular formulas but different structures, they are called isomers of each other. 4 Carbon atoms are present in butane which can be arranged in two different ways. Complete step by step answer: Isomerism is the phenomenon in which more than one compounds have the same chemical formula but different chemical structures. When the atoms and the functional groups are joined together in different ways but the molecular formula remains the same, the type of isomerism is called Structural isomerism. Butane is an alkane with molecular formula ${{\text{C}}_4}{{\text{H}}_{10}}$. Following types of structures are possible with 4 Carbon atoms. - n-butane or straight chain butane - 2-methyl propane or isobutane In which there are 3 carbon atoms in the parent chain. Additional information: There are a few types of structural isomerism: - Chain Isomerism When the carbon molecules in the chain of an organic molecule are relocated but the molecular formulas remain the same, Chain isomerism is confirmed. This alters the chain of the organic compound. The number of possible structural isomers increases greatly with the number of available atoms. - Position isomerism When compounds having the same molecular formula, same structural formula but the position of functional groups between two identical organic compounds is different, the compounds are said to show positional isomerism. - Functional isomerism Organic compounds which have the same molecular formula but the functional group present is different are called functional isomers of each other and the phenomenon is called functional isomerism. Note: Isomers may or may not have the same chemical properties. For example - stereoisomerism like geometrical isomerism, the chemical properties of two isomers are different.	2020-10-27 18:48:47	{"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": 0.7340623140335083, "perplexity": 1691.1038672742466}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1603107894426.63/warc/CC-MAIN-20201027170516-20201027200516-00201.warc.gz"}
http://www.phy.ntnu.edu.tw/ntnujava/index.php?action=printpage;topic=1076.0	# NTNUJAVA Virtual Physics LaboratoryEnjoy the fun of physics with simulations! Backup site http://enjoy.phy.ntnu.edu.tw/ntnujava/ ## Easy Java Simulations (2001- ) => Kinematics => Topic started by: Fu-Kwun Hwang on April 04, 2009, 03:17:39 pm Title: X(t), V(t) and a(t) plots : drag x to view v(t)/a(t) or set constant a Post by: Fu-Kwun Hwang on April 04, 2009, 03:17:39 pm This applet hope to help you understand relation betwen x(t), v(t) and a(t).$v(t)=limit_{\Delta t\rightarrow 0}\frac{\Delta x}{\Delta t}= \frac{dx(t)}{dt}$ , and $a(t)=\frac{d v(t)}{dt}$The applet will draw x(t),v(t) and a(t) when you drag the tiger horizontally =>to change it's position.You can also set up acceleration or velocity with slider and click accelerate to star the acceleration.The step is for you to move one time step at a time.	2020-04-05 23:41:37	{"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": 0.6187894940376282, "perplexity": 9303.396308703299}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371611051.77/warc/CC-MAIN-20200405213008-20200406003508-00457.warc.gz"}
https://williamhaw.com/	# Using Mockito With Scala Value Classes 2 mins tl;dr: When mocking functions that have value classes as arguments, use Mockito matchers on the wrapped value instead of on the value class (i.e. MyId(any[Int]) instead of any[MyId]) or else NullPointerExceptions will be thrown at runtime. # How (Not) To Thread State Through a JavaScript Promise Chain 6 mins Last weekend I created a GitHub Action and submitted it to the GitHub Hackathon. It was a relatively simple workflow that brings up a WireMock API for tests to run against. It was pretty cool as I hadn’t had a chance to write a Node application before. (The other way to write an Action is to build a Docker image, in which case any language can be used.) In this post, I’m going to share my thought process bemoan my follies as I went about writing the Action. Action repository: https://github.com/williamhaw/setup-wiremock-action Tl;dr In JavaScript, use async await to deal with intermediate state, much simpler. Also, just having documentation from types make TypeScript awesome, even if you don’t write your application in TypeScript. # Data Processing with Bash 7 mins Inspired by this post, I built a pipeline to extract facilities from ~700K hotels, combine all repeated facilities and rank by number of occurences, all in Bash. TL;DR commands: echo "supplier_id,supplier_value,mapping_type" > header.csv cat hotel-facility-dump \| \ #read from file rg "facility:" \| \ #find relevant logs awk -F'facility: ' '{print $2}' \| \ #extract content in the form of 'facility name,facility code' sort \| \ #sort for the next step uniq -c \| \ #get all unique entries with count sort --numeric-sort --reverse \| \ #get entries with count in descending order head -n 500 \| \ #take top 500 entries awk -F ' "' '{print 12345",""\42"$2}' > \ #remove count and put back double quote hotel-processed #write to output file # Everything I Know About Development in 2019 13 mins I’m starting this series so that I can track how far I’ve gone in my career as an engineer. In true Dan Abramov style I’ll be listing what I don’t know as well, so that I can concentrate on learning those things in the future. # SBT Tricks 3 mins I was recently upgrading a library at work from using Scala 2.11 to 2.12. Here are some sbt tricks that I picked up while trying to perform the migration.	2022-06-27 09:46:35	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.25786092877388, "perplexity": 4885.27029870309}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1656103329963.19/warc/CC-MAIN-20220627073417-20220627103417-00337.warc.gz"}
https://fabienpesquerel.github.io/posts/2021/09/mcmc-sampler/	# MCMC Sampler Published: The github repository is available here. ## Description MCMC is a python implementation of some MCMC sampler algorithms that can leverage pyTorch’s automatic differentiation to sample from a given (unormalized or not) density distribution. For instance, if we were to sample from a Gaussian distribution using MALA algorithm, one can do like in the mcmc_test.py file and get the following kernel density estimation: A sample from the MCMC trajectory is: Within the mala_test.py file, one can also find a small function to compute autocorrelation and check that the Markov Chain “mixes” well: To help tuning parameters of the MALA algorithm, it is possible to print the acceptation ratio. A good empirical fit for this is around 63%. It means that the chain is exploring enough so that around 63% of the propositions are accepted as the next step. One can also find a small tutorial in the tutorial.ipynb file. ## Practical use A useful example can be found in mala_test.py as well as in the tutorial.ipynb. The repository is organized around two files: • mcmc.py which contains several classes associated to MCMC sampler algorithms. Since we need a density function and a gradient in Langevin methods, one can find an important class in the next file. • function.py which contains one class, MODEL, that is used by the MCMC classes. MODEL should contains all the information one have about the distribution one whish to sample from. The bare minimum is either an (unormalized) density or log-density. This function should be written using pyTorch only function so that one can use autodifferentiation to compute gradient of the log-density with respect to the input. However, if the gradient is known, you can still use it in the MODEL. ## Example Assume that one have an implementation of the density function we whish ta sample from. Then a minimal working example would be like: from mcmc import MALA from function import MODEL DIMENSION_INPUT_SPACE = int nbr RANDOM_STEPSIZE = float nbr INITIAL_POINT = torch.randn(DIMENSION_INPUT_SPACE) NBR_SAMPLES = int nbr def density_to_sample_from(x): # ... # Some pyTorch thingsgummy # ... return density_at_x # Create probabilistical model probabilistical_model = MODEL(density=density_to_sample_from) # Create MCMC method mala_mcmc = MALA(probabilistical_model, metric=torch.eye(DIMENSION_INPUT_SPACE),	2022-01-21 11:55:21	{"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": 0.5204915404319763, "perplexity": 1844.056043849666}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320303356.40/warc/CC-MAIN-20220121101528-20220121131528-00311.warc.gz"}
https://community.wolfram.com/groups/-/m/t/1362176	# Avoid problems with converting dimensionless units? Posted 10 months ago 1057 Views \| 2 Replies \| 0 Total Likes \| Relatively new to Mathematica so I've been playing around with it trying to learn how to use it and came across an unusual 'bug' (most likely user error). A simplified example: eq1 = Pn /. { P -> Quantity[P, "Kilopascals"], n -> Quantity[n, "DimensionlessUnit"] }; UnitConvert[eq1, "Atmospheres"] gives "(40 n P)/4053atm." On the other hand... eq2 = Pn/(n + 1) /. { P -> Quantity[P, "Kilopascals"], n -> Quantity[n, "DimensionlessUnit"] }; UnitConvert[eq2, "Atmospheres" ] gives two identical errors, "Quantity::unkunit: Unable to interpret unit specification {DimensionlessUnit,Kilopascals}."Any ideas as to what's going on and/or how to fix it? Removing the line which substitutes n as a dimensionless unit seems to help in this particular example; however, I still get the same error with my more complicated equation despite the general form being the same. 2 Replies Sort By: Posted 10 months ago re-check "DimensionlessUnit", if n->n it worksi cannot find "DimensionlessUnit" on my system, are you sure it's (still) on yours? Yes, "DimensionlessUnit" seems to be on my system; however, it does not autocomplete when you start to type it. I found it referenced in a forum here. I've verified by using QuantityQ[Quantity[n, "DimensionlessUnit"]] which outputs True. I thought that another potential unitless value may be "PureUnities" which does autocomplete, but this seems to have the same problem...as does using QuantityUnit[n] such as eq3 = P*n/(n + 1) /. {P -> Quantity[P, "Kilopascals"], n -> QuantityUnit[n]}; UnitConvert[eq3, "Atmospheres"]	2019-04-25 16:38:57	{"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": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.24931934475898743, "perplexity": 4543.622639040113}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578727587.83/warc/CC-MAIN-20190425154024-20190425180024-00381.warc.gz"}
https://www.physicsforums.com/threads/potential-to-be-real.151665/	# Potential to be real? 1. Jan 16, 2007 ### emanaly If we have a potential with two fields psi and phi with a phase Exp(itheta) , how can we get rid of the field in order for the potential to be real? 2. Jan 16, 2007 ### Staff: Mentor Your question is not really clear. The notation $$e^{i \theta}$$ represents a field that is oscillating at some frequency and has a phase offset from some reference phase. What do you mean to make the potential "real"? You want to make it DC instead of AC? 3. Jan 17, 2007 ### emanaly Sorry , may be I was not clear enough, but I mean by the potential, the one which appear in quantum field theory 4. Jan 17, 2007 ### Staff: Mentor Oopsies, sorry for my misinterpretation. I'm of no help on QFT. 5. Jan 17, 2007 ### Staff: Mentor I had someone who does understand QFT look at your question, and he doesn't understand what you are asking. Could you please provide more of a context for the question, and more details about the question? 6. Jan 17, 2007 ### emanaly Thank you for your effort bereman The potential is as follow:V=Psi Phi Exp(itheta) in order for the potential to be real, we should get rid of the imaginery part , how can we do that? Thanks again 7. Jan 17, 2007 ### Staff: Mentor I'm glad to try to help out. But what you've posted there is pretty much just a repeat of your original post. Could you please provide more context? What is the course, what is the textbook, what similar questions are from the same section of this course? Can you provide a physical example where this question would apply? 8. Jan 18, 2007 ### emanaly Thank you berkeman for all what have you done. The problem by the grace of God had been solved	2017-10-18 06:15:14	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.45552220940589905, "perplexity": 1038.7994670520713}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187822747.22/warc/CC-MAIN-20171018051631-20171018071631-00478.warc.gz"}
https://mathzsolution.com/how-many-7-note-musical-scales-are-possible-within-the-12-note-system/	# How many 7-note musical scales are possible within the 12-note system? This combinatorial question has a musical motivation, which I provide below using as little musical jargon as I can. But first, I’ll present a purely mathematical formulation for those not interested in the motivation: Define a signature as a 7-tuple over the set $\{1,2,3\}$ such that the sum of the elements in the tuple is $12$. Two signatures are said to be equivalent if they are either identical or there is a circular shift relation between them (i.e. one can be circularly shifted between 1 and 6 times to obtain the other). How many unique signatures are there? Most modern, Western music is based on the equal temperament system, which divides the octave logarithmically equally into 12 notes. Let’s refer to 2 adjacent notes from these 12 as being “1 step” apart, two notes with one skipped note in between them as being “2 steps” apart, and so on. The (arguably) most natural scale is the major scale, which uses 7 notes from these 12, and has the following signature: This means that we can construct a major scale as follows: given any note to start from, the second note is 2 steps away from first note, the third note is 1 step away from the 2nd note, and so on according to the above signature. Now this signature really has not just one, but seven scales embedded in it. This is because we can circularly shift the signature, effectively meaning that we are picking a different degree of the major scale to serve as our home note (these seven ‘permutations’ are called the modes of the signature). Thus, for example, $(1,2,2,2,1,2,2)$ is not a new signature, but just the major signature circularly shifted left twice (and is called the Phrygian mode). My question is: how many unique 7-note signatures are there under the restriction that any signature must not contain an interval greater than 3 steps (this is to respect the fact that any 7-note scale in common use uses only 1, 2 and 3 step intervals, to the best of my knowledge). Some signatures in common use are: The number of possible 7-note scales within the 12-note system is simply given by multiplying the number of unique signatures by 7. Since $7$ is prime and there is no $7$-note signature that sums to $12$ with all $7$ steps identical, we don’t have to worry about periodicity; we can just divide by $7$ in the end. Thus, we just have to count the number of ways of distributing $12-7=5$ balls into $7$ bins with capacity $3-1=2$. There are $\binom75=21$ ways to have $5$ steps of $2$, $\binom7{1,3,3}=140$ ways to have $3$ steps of $2$ and $1$ step of $3$, and $\binom7{2,1,4}=105$ ways to have $1$ step of $2$ and $2$ steps of $3$, for a total of $21+140+105=266$ scales in $266/7=38$ cyclically inequivalent types. In the present case, inclusion-exclusion would be a bit of an overkill, but since you said you’d like a method that generalises to any number of notes with any number of maximum steps, let’s generalise: For $k$ notes with a maximum of $m$ steps that sum to $12$, we want to distribute $12-k$ balls into $k$ bins with capacity $m-1$. As explained at Balls In Bins With Limited Capacity, inclusion-exclusion yields a count of where, contrary to convention, binomial coefficients with negative upper index are taken to be zero. For the present case of $k=7$, $m=3$, this again yields signatures. If $k$ isn’t prime, or if it divides $12$, then you have to do a bit more to deal with periodicity; otherwise, you can just divide the above result by $k$.	2023-02-06 06:38:29	{"extraction_info": {"found_math": true, "script_math_tex": 37, "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": 0.7412589192390442, "perplexity": 290.3123575983368}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1674764500304.90/warc/CC-MAIN-20230206051215-20230206081215-00420.warc.gz"}
https://zbmath.org/?q=an:06587960	Equivariant Pieri rules for isotropic Grassmannians.(English)Zbl 1339.14030 Let $$V$$ be an $$N$$-dimensional complex vector space equipped with a symmetric or skew-symmetric bilinear form $$\omega$$, which can be either trivial or non-degenerate. The Grassmannians $$I G_{\omega}(m, N)$$ of classical Lie type parameterize $$m$$-dimensional isotropic vector subspaces of $$V$$. The cohomology ring of an isotropic Grassmannian $$X =I G_{\omega}(m, N)$$, or more generally of a homogeneous variety, has an additive basis of Schubert classes represented by Schubert subvarieties $$X_{\lambda}$$. One of the central problems of Schubert calculus is to find a manifestly positive formula for the structure constants of the cup product of two Schubert cohomology classes, or equivalently, for the triple intersection numbers of three Schubert subvarieties in general position. Such a positive formula, called a Littlewood-Richardson rule, has deep connections to various subjects, including geometry, combinatorics and representation theory. An isotropic Grassmannian $$X$$ can be written as a quotient of a classical complex simple Lie group $$G$$ by a maximal parabolic subgroup $$P$$ (with two notable exceptions of Lie type $$D_n$$). Fix a choice of maximal complex torus $$T$$ and a Borel subgroup $$B$$ with $$T \subset B \subset P$$. The Schubert varieties $$X_{\lambda}$$ are closures of $$B$$-orbits, and hence are $$T$$-stable. They give a basis $$[X_{\lambda}]^T$$ for the $$T$$-equivariant cohomology $${H^}_T (X)$$ as a $${H^}_T (pt)$$-module. The structure coefficients $${N^{\nu}}_{\lambda,\mu}$$ in the equivariant product, $[X_{\lambda}]^T \cdot [X_{\mu}]^T=\sum\limits_{\nu}{N^{\nu}}_{\lambda,\mu} [X_{\nu}]^T,$ are homogeneous polynomials which satisfy a positivity condition conjectured by D. Peterson [Lectures on quantum cohomology of $$G/B$$, MIT (1996)] and proved by W. Graham [Duke Math. J. 109, No. 3, 599–614 (2001; Zbl 1069.14055)]. In particular, they are Graham-positive, meaning they are polynomials in the negative simple roots, with non-negative integer coefficients. These equivariant structure coefficients carry much more information than the triple intersection numbers of Schubert varieties, and are more challenging to study. In the present paper, the authors give for the first time an equivariant Pieri rule for Grassmannians of Lie types $$B, C$$, and $$D$$, as well as a new proof of the Pieri rule in type $$A$$. Such a rule concerns products with the special Schubert classes $$[X_{p}]^T$$ , which are related to the equivariant Chern classes of the tautological quotient bundle, and generate the $$T$$ -equivariant cohomology ring. Using geometric methods, they give a manifestly positive formula for the structure coefficients $${N^{\mu}}_{\lambda,p}$$ of the equivariant multiplication $$[X_{\lambda}]^T \cdot [X_{p}]^T$$. Reviewer: Cenap Özel (Bolu) MSC: 14M15 Grassmannians, Schubert varieties, flag manifolds 14N15 Classical problems, Schubert calculus 55N91 Equivariant homology and cohomology in algebraic topology Zbl 1069.14055 Software: Quantum Calculator Full Text: References: [1] Andersen, HH; Jantzen, JC; Soergel, W, Representations of quantum groups at a pth root of unity and of semisimple groups in characteristic p: independence of p, Astérisque, 220, 297-302, (1994) · Zbl 0802.17009 [2] Anderson, D.: Introduction to equivariant cohomology in algebraic geometry. In: Contributions to Algebraic Geometry, pp. 71-92. EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich (2012) · Zbl 1262.14019 [3] Arabia, A, Cohomologie T-équivariante de la variété de drapeaux d’un groupe de Kac-Moody, Bull. Soc. Math. 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(to appear). arXiv:1109.6669 [math.AG] · Zbl 1342.14108 [11] Buch, A.S.: Equivariant quantum calculator, a free software package for Maple. http://math.rutgers.edu/ asbuch/equivcalc/. Accessed on 6 July 2015 · Zbl 1205.05244 [12] Buch, AS; Kresch, A; Tamvakis, H, Quantum Pieri rules for isotropic Grassmannians, Invent. Math., 178, 345-405, (2009) · Zbl 1193.14071 [13] Buch, AS; Mihalcea, LC, Quantum K-theory of Grassmannians, Duke Math. J., 156, 501-538, (2011) · Zbl 1213.14103 [14] Buch, AS; Ravikumar, V, Pieri rules for the K-theory of cominuscule Grassmannians, J. Reine Angew. Math., 668, 109-132, (2012) · Zbl 1298.14059 [15] Buch, AS; Rimanyi, R, Specializations of Grothendieck polynomials, C. R. Math. Acad. Sci. Paris, 339, 1-4, (2004) · Zbl 1051.14062 [16] Fulton, W.: Equivariant cohomology in algebraic geometry. In: Eilenberg lectures. Columbia University, Springer, Berlin (2007) (Notes by D. Anderson) · Zbl 1361.14029 [17] Fulton, W.: Young tableaux. In: London Mathematical Society Student Texts, vol. 35. Cambridge University Press, Cambridge (1997) · Zbl 0878.14034 [18] Fun, A.: Raising operators and the Littlewood-Richardson polynomials. arXiv:1203.4729 [math.CO] · Zbl 0972.05053 [19] Gatto, L; Santiago, T, Equivariant Schubert calculus, Ark. Mat., 48, 41-55, (2010) · Zbl 1188.14033 [20] Graham, W, Positivity in equivariant Schubert calculus, Duke Math. J., 109, 599-614, (2001) · Zbl 1069.14055 [21] Huang, Y., Li, C.: On equivariant quantum Schubert calculus for $$G/P$$. J. Algebra (to appear). arXiv:1506.00872 [math.AG] · Zbl 1188.14033 [22] Ikeda, T, Schubert classes in the equivariant cohomology of the Lagrangian Grassmannian, Adv. Math., 215, 1-23, (2007) · Zbl 1126.14060 [23] Ikeda, T; Mihalcea, L; Naruse, H, Double Schubert polynomials for the classical groups, Adv. Math., 226, 840-886, (2011) · Zbl 1291.05222 [24] Ikeda, T; Matsumura, T, Pfaffian sum formula for the symplectic Grassmannians, Math. Zeit., 280, 269-306, (2015) · Zbl 1361.14029 [25] Ikeda, T; Naruse, H, Excited Young diagrams and equivariant Schubert calculus, Trans. Am. Math. Soc., 361, 5193-5221, (2009) · Zbl 1229.05287 [26] Knutson, A; Tao, T, Puzzles and (equivariant) cohomology of Grassmannians, Duke Math. J., 119, 221-260, (2003) · Zbl 1064.14063 [27] Kostant, B; Kumar, S, The nil Hecke ring and the cohomology of $$G/P$$ for a Kac-Moody group $$G$$, Adv. Math., 62, 187-237, (1986) · Zbl 0641.17008 [28] Kreiman, V, Equivariant Littlewood-Richardson skew tableaux, Trans. Am. Math. Soc., 362, 2589-2617, (2010) · Zbl 1205.05244 [29] Kumar, S.: Kac-Moody groups, their flag varieties and representation theory. In: Progress in Mathematics, vol. 204. Birhäuser, Boston (2002) · Zbl 1026.17030 [30] Lakshmibai, V., Raghavan, K.N., Sankaran, P.: Equivariant Giambelli and determinantal restriction formulas for the Grassmannian. Pure Appl. Math. Q. 2(3) (2006) (Special Issue: In honor of Robert D. MacPherson. 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In: Proceedings of the 9-th Conference on Factorial Power Series and Algebraic Combinatorics, Vienna, vol. 3, pp. 517-523 (1997) · Zbl 1229.05287 [38] Molev, AI, Littlewood-Richardson polynomials, J. Algebra, 321, 3450-3468, (2009) · Zbl 1169.05050 [39] Molev, AI; Sagan, BE, A Littlewood-Richardson rule for factorial Schur functions, Trans. Am. Math. Soc., 351, 4429-4443, (1999) · Zbl 0972.05053 [40] Peterson D.: Lectures on quantum cohomology of G/P, Unpublished (1997) [41] Ravikumar, V.: Triple intersection formulas for isotropic Grassmannians. Algebra Num. Theory (to appear, preprint). arXiv:1403.1741 [math.AG] · Zbl 1342.14109 [42] Robinson, S, A Pieri-type formula for $$H_T^*(SL_n(\mathbb{C})/B)$$, J. Algebra, 249, 38-58, (2002) · Zbl 1061.14060 [43] Sottile, F, Pieri-type formulas for maximal isotropic Grassmannians via triple intersections, Colloq. Math., 82, 49-63, (1999) · Zbl 0977.14023 [44] Tamvakis, H., Wilson, E.: Double theta polynomials and equivariant Giambelli formulas (preprint). arXiv:1410.8329 [math.AG] · Zbl 1371.14058 [45] Tamvakis, H.: Giambelli and degeneracy locus formulas for classical $$G/P$$ spaces (preprint). arXiv:1305.3543 [math.AG] · Zbl 1379.14026 [46] Thomas, H., Yong, A.: Equivariant Schubert calculus and jeu de taquin. Ann. l’Inst. Four. (to appear, preprint). arXiv:1207.3209 [math.CO] · Zbl 0977.14023 [47] Wilson, E.V.: Equivariant Giambelli Formulae for Grassmannians, Ph.D. Thesis, University of Maryland, College Park (2010) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.	2022-06-30 05:03:02	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8822922706604004, "perplexity": 3159.3542296061437}, "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-2022-27/segments/1656103661137.41/warc/CC-MAIN-20220630031950-20220630061950-00352.warc.gz"}
http://www.theinfolist.com/php/SummaryGet.php?FindGo=group_homomorphism	group homomorphism TheInfoList Image:Group homomorphism ver.2.svg, 250px, Image of a group homomorphism (h) from G (left) to H (right). The smaller oval inside H is the image of h. N is the Kernel_(algebra)#Group_homomorphisms, kernel of h and aN is a coset of N. In mathematics, given two group (mathematics), groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ... ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that :$h\left(uv\right) = h\left(u\right) \cdot h\left(v\right)$ where the group operation on the left side of the equation is that of ''G'' and on the right side that of ''H''. From this property, one can deduce that ''h'' maps the identity element In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ... ''eG'' of ''G'' to the identity element ''eH'' of ''H'', :$h\left(e_G\right) = e_H$ and it also maps inverses to inverses in the sense that :$h\left\left(u^\right\right) = h\left(u\right)^. \,$ Hence one can say that ''h'' "is compatible with the group structure". Older notations for the homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homom ... ''h''(''x'') may be ''x''''h'' or ''x''''h'', though this may be confused as an index or a general subscript. In automata theory Automata theory is the study of abstract machines and automata, as well as the computational problem In theoretical computer science An artistic representation of a Turing machine. Turing machines are used to model general computing devices. T ... , sometimes homomorphisms are written to the right of their arguments without parentheses, so that ''h''(''x'') becomes simply ''x h''. In areas of mathematics where one considers groups endowed with additional structure, a ''homomorphism'' sometimes means a map which respects not only the group structure (as above) but also the extra structure. For example, a homomorphism of topological group 350px, The real numbers form a topological group under addition ">addition.html" ;"title="real numbers form a topological group under addition">real numbers form a topological group under addition In mathematics, a topological group is a group ... s is often required to be continuous. # Intuition The purpose of defining a group homomorphism is to create functions that preserve the algebraic structure. An equivalent definition of group homomorphism is: The function ''h'' : ''G'' → ''H'' is a group homomorphism if whenever : ''a'' ∗ ''b'' = ''c'' we have ''h''(''a'') ⋅ ''h''(''b'') = ''h''(''c''). In other words, the group ''H'' in some sense has a similar algebraic structure as ''G'' and the homomorphism ''h'' preserves that. # Types ; Monomorphism 220px In the context of abstract algebra or universal algebra, a monomorphism is an Injective function, injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of catego ... : A group homomorphism that is injective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ... (or, one-to-one); i.e., preserves distinctness. ; Epimorphism 220px In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labe ... : A group homomorphism that is surjective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ... (or, onto); i.e., reaches every point in the codomain. ; Isomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ... : A group homomorphism that is bijective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ... ; i.e., injective and surjective. Its inverse is also a group homomorphism. In this case, the groups ''G'' and ''H'' are called ''isomorphic''; they differ only in the notation of their elements and are identical for all practical purposes. ; Endomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ... : A homomorphism, ''h'': ''G'' → ''G''; the domain and codomain are the same. Also called an endomorphism of ''G''. ; Automorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ... : An endomorphism that is bijective, and hence an isomorphism. The set of all automorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ... s of a group ''G'', with functional composition as operation, forms itself a group, the ''automorphism group'' of ''G''. It is denoted by Aut(''G''). As an example, the automorphism group of (Z, +) contains only two elements, the identity transformation and multiplication with −1; it is isomorphic to Z/2Z. # Image and kernel We define the '' kernel Kernel may refer to: Computing Kernel (operating system) The kernel is a computer program at the core of a computer's operating system that has complete control over everything in the system. It is the "portion of the operating system co ... of h'' to be the set of elements in ''G'' which are mapped to the identity in ''H'' : $\operatorname\left(h\right) \equiv \left\.$ and the '' image File:TEIDE.JPG, An Synthetic aperture radar, SAR radar imaging, radar image acquired by the SIR-C/X-SAR radar on board the Space Shuttle Endeavour shows the Teide volcano. The city of Santa Cruz de Tenerife is visible as the purple and white a ... of h'' to be : $\operatorname\left(h\right) \equiv h\left(G\right) \equiv \left\.$ The kernel and image of a homomorphism can be interpreted as measuring how close it is to being an isomorphism. The first isomorphism theorem In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for ... states that the image of a group homomorphism, ''h''(''G'') is isomorphic to the quotient group ''G''/ker ''h''. The kernel of h is a normal subgroup In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ... of ''G'' and the image of h is a subgroup In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by ... of ''H'': : $\begin h\left\left(g^ \circ u \circ g\right\right) &= h\left(g\right)^ \cdot h\left(u\right) \cdot h\left(g\right) \\ &= h\left(g\right)^ \cdot e_H \cdot h\left(g\right) \\ &= h\left(g\right)^ \cdot h\left(g\right) = e_H. \end$ If and only if , the homomorphism, ''h'', is a ''group monomorphism''; i.e., ''h'' is injective (one-to-one). Injection directly gives that there is a unique element in the kernel, and a unique element in the kernel gives injection: :$\begin && h\left(g_1\right) &= h\left(g_2\right) \\ \Leftrightarrow && h\left(g_1\right) \cdot h\left(g_2\right)^ &= e_H \\ \Leftrightarrow && h\left\left(g_1 \circ g_2^\right\right) &= e_H,\ \operatorname\left(h\right) = \ \\ \Rightarrow && g_1 \circ g_2^ &= e_G \\ \Leftrightarrow && g_1 &= g_2 \end$ # Examples * Consider the cyclic group In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ... Z/3Z = and the group of integers Z with addition. The map ''h'' : Z → Z/3Z with ''h''(''u'') = ''u'' mod 3 is a group homomorphism. It is surjective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ... and its kernel consists of all integers which are divisible by 3. * The yields a group homomorphism from the group of real number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ... s R with addition to the group of non-zero real numbers R* with multiplication. The kernel is and the image consists of the positive real numbers. * The exponential map also yields a group homomorphism from the group of complex number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ... s C with addition to the group of non-zero complex numbers C* with multiplication. This map is surjective and has the kernel , as can be seen from Euler's formula Euler's formula, named after Leonhard Euler, is a mathematics, mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex number, complex exponential function. Euler's ... . Fields like R and C that have homomorphisms from their additive group to their multiplicative group are thus called exponential fields. # The category of groups If and are group homomorphisms, then so is . This shows that the class of all groups, together with group homomorphisms as morphisms, forms a category theory, category. # Homomorphisms of abelian groups If ''G'' and ''H'' are abelian group, abelian (i.e., commutative) groups, then the set of all group homomorphisms from ''G'' to ''H'' is itself an abelian group: the sum of two homomorphisms is defined by :(''h'' + ''k'')(''u'') = ''h''(''u'') + ''k''(''u'') for all ''u'' in ''G''. The commutativity of ''H'' is needed to prove that is again a group homomorphism. The addition of homomorphisms is compatible with the composition of homomorphisms in the following sense: if ''f'' is in , ''h'', ''k'' are elements of , and ''g'' is in , then : and . Since the composition is associative, this shows that the set End(''G'') of all endomorphisms of an abelian group forms a ring (algebra), ring, the ''endomorphism ring'' of ''G''. For example, the endomorphism ring of the abelian group consisting of the Direct sum of groups, direct sum of ''m'' copies of Z/''n''Z is isomorphic to the ring of ''m''-by-''m'' matrix (mathematics), matrices with entries in Z/''n''Z. The above compatibility also shows that the category of all abelian groups with group homomorphisms forms a preadditive category; the existence of direct sums and well-behaved kernels makes this category the prototypical example of an abelian category.	2022-01-20 13:09:30	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 7, "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": 0.8599743247032166, "perplexity": 1891.073630604115}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1642320301863.7/warc/CC-MAIN-20220120130236-20220120160236-00281.warc.gz"}
https://www.europeanpharmaceuticalreview.com/news/99541/global-lc-ms-market-to-exhibit-cagr-of-6-9-percent/	news # Global LC-MS market to exhibit CAGR of 6.9 percent 0 SHARES Share via Research has found that the worldwide LC-MS market will be worth $2750 million by the end of the forecast period in 2024. A study has found that the global liquid chromatography-mass spectrometry (LC-MS) market will increase at a CAGR of 6.9 percent in the forecast period 2019 to 2024. According to Market Study Report, the worldwide market is currently valued at$1840 million but will grow to \$2750 million by 2024. The report found that the top three producers for the LC-MS industry account for approximately 67.34 percent of the revenue market. LC-MS is a technique used in separation, general detection and identification of chemicals. It can also be used for mass-directed purification of substances and to quantify product quality characteristics, within processes such as multi-attribute method (MAM) workflows. Other findings from the report include the US being the largest manufacturing region and consumer of LC-MS products. The prices of LC-MS products are predicted by the report to decrease, due to an intensified competition. The key players in the industry include Thermo Fisher Scientific, Agilent Technologies and SCIEX. ### Related organisations This site uses Akismet to reduce spam. Learn how your comment data is processed.	2022-11-29 07:47:16	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.27681419253349304, "perplexity": 7055.646013982962}, "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-2022-49/segments/1669446710690.85/warc/CC-MAIN-20221129064123-20221129094123-00217.warc.gz"}
https://sbebo.github.io/posts/2014/04/10/minlp/	Lately, I’ve had to deal with mixed-integer nonlinear programming (MINLP) models. I’ll share a couple of considerations – and show one weird trick to reduce your optimization time by 99.96% adding just a single, simple cut! You can thank me later. I’ve experimented with some global solvers (SCIP, BARON, Couenne) on a nonconvex MINLP problem. Bringing together two inherently difficult aspects as nonconvexity and integrality sure is an arduous task. I would say that MINLP solvers (and theory, and algorithms?) are less mature than their linear counterparts, although it is a very active field of research. What I’ve seen is that even small details in the formulation are crucial (no big surprise, I guess): 1. Scaling – Scaling is important. At one point I had an instance whose variables could assume values over 10^6. I thought that wasn’t ideal, but hopefully it wouldn’t cause too many problems. It did. Those variables were squared in a constraint, and the solvers would either a) fail b) issue some cryptic warning c) say nothing, and return a plainly wrong result (actually, an infeasible solution). Manual scaling was necessary. 2. Bounds – If you don’t provide any bounds, you’d better be a really patient guy. Even what look like trivial bounds are essential. 3. Cutting planes – This is what inspired me to write this post. MINLP solvers apply manipulations and reformulations of the nonlinear expressions. However, my feeling is that, in doing so, some important structure of the problem might be missed. Sometimes even the simplest valid inequalities are helpful in cutting down the solving time. As an example for point 3, I solve a small instance of a nonlinear multi-item production planning problem with 168 variables (24 binary) and 240 constraints. I’ll describe the model later. Solver Time (s) Nodes Gap% SCIP 3.1 49.8 94882 0.00 BARON 10.2 1563.4 156520 1.00 This small problem seems already to be nontrivial for state-of-the-art global solvers (to be fair, that is not the most recent version of Baron). Is it really hard already, or are the solvers missing something? Can we help them in any way? Adding a single, very simple cutting plane (details in the next paragraph, pictures included), you get these results: Solver Time (s) Nodes Gap% SCIP 3.1 0.02 1 0.00 BARON 10.2 0.18 1 0.00 Now both solvers breeze through it, solving the problem in the root node. It is pretty remarkable how effective the cutting plane was. This valid inequality doesn’t always make such an impact – hey, it’s just one cutting plane, what do you expect? But in quite a few of my instances, it seems to be very useful. Why is that the case? #The gory details Here is a basic formulation of the nonlinear production planning problem mentioned above. Simple AMPL and GAMS implementations can be found here. We have one production plant that, given an input quantity $x_t$, provides two commodities $q_t$ and $u_t$. We want to decide when and how much to produce in order to satisfy the demands $d^q_t$ and $d^u_t$ in each hour, minimizing the total costs. The production is nonlinear and is a function of the variable $x_t$ via $f(\cdot)$ and $g(\cdot)$, if the plant is on (4–5). Constraints (3) impose technical bounds on the input quantity $x_t$ (i.e., minimum and maximum lot size). The plant produces the commodities simultaneously, but according to different “performance curves”. Constraints (1) and (2) are simple balance constraints. This is not extremely important here, but the items we produce are managed differently. Commodity $q_t$ can be sold, if in excess, and bought (back-ordered), if the production is not sufficient (1). Commodity $u_t$ can be stocked for the following time slot. Storing is free, but we have an upper bound $S$ on the quantity in stock (2). Quantity in excess can be discarded with no additional costs. If $f$ and $g$ are both concave, this problem is (or can be reformulated as) a convex MINLP. However, it may well be the case that $f$ and $g$ are not concave. Assume that, for commodity $u_t$, the plant is more efficient at lower loads, “saturating” as $x$ grows ($g$ is concave); while the production of $q_t$ increases its efficiency as $x_t$ grows ($f$ is convex). This leads to a nasty nonconvex MINLP. If you simply approximate the nonlinear functions with piecewise linear ones (and use a big-M term) you can solve the resulting MILP with your favorite solver in a matter of seconds (or less), even for quite large-size instances. Going back to the original MINLP, let’s plot the feasible region defined by the constraint $q_t \le z_tf(x_t)$: Points in the optimal solution are going to lie either in the origin ($z_t$ is 0, so production is off) or on the graph of $f$ in the feasible domain $[X^{min},X^{max}]$. The simplest (upper) convex hull approximation you can take is the line connecting the extreme point $(X^{max}, f(X^{max}))$ and the origin (0,0) – assuming, as it is the case, that $(X^{min}, f(X^{min}))$ lies under the line. [Otherwise, you can have a cutting plane just taking the line through $(X^{min}, f(X^{min}))$ and $(X^{max}, f(X^{max}))$. I think you can call that a secant approximation of $f$.] And this is the glorious cutting plane that achieves the remarkable speed-up reported above. Note that the cut is so effective probably because the curve, in that instance, is actually quite close to being linear. But, are solvers really missing that cutting plane? It looks to me that would be one of the easiest thing you could come up with. Is perhaps my initial formulation not good for the solvers – maybe it’s not what they expect at all? I’ve tried quite a few equivalent formulations (e.g., replacing the product in (4–5) with big-M terms, or projecting out variables $x_t$), but I’ve seen no big difference. Still, I’m not really used to working with MINLP solvers, so that just might be the case. Anyway, I sure wish it happened more often that such an easy trick could be this effective.	2020-08-03 18:06:34	{"extraction_info": {"found_math": true, "script_math_tex": 32, "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": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7308513522148132, "perplexity": 799.1235366812843}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1596439735823.29/warc/CC-MAIN-20200803170210-20200803200210-00047.warc.gz"}
http://mathhelpforum.com/calculus/154720-limit-derivative-problem.html	1. ## Limit/Derivative Problem I'm having trouble getting started with a calculus problem, can anyone help me? The limit $\lim_{x \to 4} \frac{\sqrt{x}-2}{x-4}$ represents the derivative of a function taken at a point. What function? What point? What is the limit? I need help getting started with this problem... 2. Well as $x\to 4$ the point is $x=4$ I get $\displaystyle \lim_{x \to 4} \frac{\sqrt{x}-2}{x-4}= \frac{1}{4}$ What methods for taking limits have you been taught? 3. Sorry, I didn't really explain this problem well. I knew the answer to the acutal limit, and the point, but I'm having trouble figuring out what function this represents the derivative taken at a point of. If this limit represents the numerical derivative of 1/4 taken at x=4, what would the original function that I'm taking the derivative of be? Thanks? 4. $\displaystyle f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x-a}$ 5. Makes perfect sense now! Thanks so much!	2013-12-12 20:00:11	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8779503107070923, "perplexity": 367.83590904177174}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386164695251/warc/CC-MAIN-20131204134455-00018-ip-10-33-133-15.ec2.internal.warc.gz"}
http://math.biu.ac.il/en/node/2147	# A two-phase mother body and a Muskat problem Mon, 20/06/2016 - 15:05 Speaker: Seminar: Place: Abstract: A Muskat problem describes an evolution of the interface $\Gamma (t)\subset{\mathbb R}^{2}$ between two immiscible fluids, occupying regions $\Omega _1$ and $\Omega _2$ in a Hele-Shaw cell. The interface evolves due to the presence of sinks and sources located in $\Omega _j$, $j=1,2$. The case where one of the fluids is effectively inviscid, that is, it has a constant pressure, is called one-phase problem. This case has been studied extensively. Much less progress has been made for the two-phase problem, the Muskat problem. The main difficulty of the two-phase problem is the fact that the pressure on the interface, separating the fluids, is unknown. In this talk we introduce a notion of a two-phase mother body (the terminology comes from the potential theory) as a union of two distributions $\mu _j$ with integrable densities of sinks and sources, allowing to control the evolution of the interface, such that $\rm{supp}\, \mu _j \subset\Omega _j$. We use the Schwarz function approach and the introduced two-phase mother body to find the evolution of the curve $\Gamma (t)$ as well as two harmonic functions $p_j$, the pressures, defined almost everywhere in $\Omega_j$ and satisfied prescribed boundary conditions on $\Gamma (t)$.	2018-06-25 02:08:21	{"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": 0.7804173231124878, "perplexity": 879.4188035463847}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267867364.94/warc/CC-MAIN-20180625014226-20180625034226-00414.warc.gz"}
https://math.hawaii.edu/wordpress/calendar/action~agenda/exact_date~1565344800/cat_ids~59/	# Calendar Sep 4 Wed Colloquium: Stuart White (U. of Oxford) @ Keller 401 Sep 4 @ 3:30 pm – 4:30 pm Speaker: Stuart White (U. of Oxford) Title: Amenable Operator Algebras Abstract: Operator algebras arise as suitably closed subalgebras of the bounded operators on a Hilbert space. They come in two distinct types: von Neumann algebras which have the flavour of measure theory, and C-algebras which have the flavour of topology. In the 1970’s Alain Connes obtained a deep structural theorem for amenable von Neumann algebras, leading to a complete classification of these objects. For the last 25 years the Elliott classification programme has been seeking a corresponding result for simple amenable C-algebras, and now, though the efforts of numerous researchers worldwide, we have a definitive classification theorem. In this talk, I’ll explain what this theorem says, and the analogies it makes to Connes work, using examples from groups and dynamics as motivation. I won’t assume any prior exposure to operator algebras or functional analysis. Sep 13 Fri Colloquium: Asaf Hadari (UHM) @ Keller 401 Sep 13 @ 3:30 pm – 4:30 pm Title: In search of a representation theory of mapping class groups. Abstract: Mapping class groups are nearly ubiquitous in low dimensional topology. They’ve been studied for over a century. Various results discovered during the past few decades it has become quite clear that there is much to gain by studying them via their linear representations. Somewhat surprisingly, many such representations are known. Unfortunately, until recently there was almost no representation theory, that is – no underlying structure that allows you to say anything about the class of representations as a whole. It is precisely such an understanding that is necessary for studying mapping class groups. In this talk I’ll talk about the major source of representations of mapping class groups, and talk about new results in their emerging representation theory. Oct 18 Fri Colloquium: Marion Campisi (SJSU) @ Keller 401 Oct 18 @ 3:30 pm – 4:30 pm Title: Analysis of partisan gerrymandering tools in advance of the US 2020 census Abstract: Over the last decade, mapmakers have precisely gerrymandered political districts for the benefit of their party. In response, political scientists and mathematicians have more extensively investigated tools to quantify and understand the mathematical structure of redistricting problems. Two primary tools for determining whether a particular redistricting plan is fair are partisan-gerrymandering metrics and stochastic sampling algorithms. In this work we explore the Declination, a new metric intended to detect partisan gerrymandering. Within out analyses, we show that Declination cannot detect all forms of packing and cracking, and we compare the Declination to the Efficiency Gap. We show that these two metrics can behave quite differently, and give explicit examples where that occurs. Oct 23 Wed Applied Math Seminar: Nicolette Meshkat (Santa Clara University) @ Keller 402 Oct 23 @ 3:30 pm – 4:30 pm Title: Structural Identifiability of Biological Models Abstract: Parameter identifiability analysis addresses the problem of which unknown parameters of a model can be determined from given input/output data. If all of the parameters of a model can be determined from data, the parameters and the model are called identifiable. However, if some subset of the parameters can not be determined from data, the model is called unidentifiable. We examine this problem for the case of perfect input/output data, i.e. absent of any experimental noise. This is called the structural identifiability problem. We show that, even in the ideal case of perfect input/output data, many biological models are structurally unidentifiable, meaning some subset of the parameters can take on an infinite number of values, yet yield the same input/output data. In this case, one attempts to reparametrize the model in terms of new parameters that can be determined from data. In this talk, we discuss the problem of finding an identifiable reparametrization and give necessary and sufficient conditions for a certain class of linear compartmental models to have an identifiable reparametrization. We also discuss finding classes of identifiable models and finding identifiable submodels of identifiable models. Our work uses graph theory and tools from computational algebra. This is joint work with Elizabeth Gross and Anne Shiu. Oct 29 Tue Colloquium: Marissa Loving (Georgia Tech) @ Keller Hall 401 Oct 29 @ 4:00 pm – 5:00 pm Title: Symmetries of Surfaces Abstract: There are many ways to study surfaces: topologically, geometrically, dynamically, algebraically, and combinatorially, just to name a few. We will touch on some of the motivation for studying surfaces and their associated mapping class groups, which is the collection of symmetries of a surface. We will also describe a few of the ways that these different perspectives for studying surfaces come together in beautiful and sometimes unexpected ways. Nov 6 Wed PhD defense for Don Krasky @ Keller 401 Nov 6 @ 3:00 pm – 5:00 pm Nov 27 Wed Colloquium: Elijah Liflyand (Bar-Ilan University) @ Keller 401 Nov 27 @ 3:30 pm – 4:30 pm Speaker: Elijah Liflyand (Bar-Ilan University) Title: A tale of two Hardy spaces Abstract: New relations between the Fourier transform of a function of bounded variation and the Hilbert transform of its derivative are revealed. If we do not distinguish between the cosine and sine transforms and consider the general Fourier transform of $f$, direct calculations give the belonging of the derivative $f’$ to the real Hardy space $H^1$ as a sufficient condition for the integrability of the Fourier transform. Our analysis is more delicate. The main result is an asymptotic formula for the {bf cosine} Fourier transform, while much earlier known results gives an asymptotic formula for the sine Fourier transform. The difference is achieved by assuming that the derivative belongs to different subspaces of $H^1$. However, this tale of each of the two subspaces were impossible if we would not have a new proof even for the old result. The known proofs used to give strong priority just to the sine transform. Interrelations of various function spaces are studied in this context, first of all of these two types of Hardy spaces. The obtained results are used for proving completely new results on the integrability of trigonometric series. Dec 6 Fri Colloquium: Nate Brown (Penn State) Dec 6 @ 3:30 pm – 4:30 pm Speaker: Nate Brown (Penn State) Title: Tomorrow’s STEM leaders are diverse Abstract: Thirty years ago a radical experiment began at the University of Maryland Baltimore County (UMBC). The aim was to prepare undergraduates from underrepresented groups to be successful graduate students in STEM fields. The pillars of the program were unorthodox and the results have been stunning. In this talk I will discuss the Driving Change Initiative, funded by the Howard Hughes Medical Institute, which aims to replicate UMBC’s experiment at research institutions across the country.	2023-03-23 03:53:14	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.47785940766334534, "perplexity": 1055.0818029755555}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296944996.49/warc/CC-MAIN-20230323034459-20230323064459-00093.warc.gz"}
https://ncatlab.org/nlab/show/Gelfand+duality	nLab Gelfand duality Context Algebra higher algebra universal algebra duality In QFT and String theory Noncommutative geometry noncommutative geometry (geometry $\leftarrow$ Isbell duality $\to$ algebra) Contents Idea Gelfand duality is a duality between spaces and their algebras of functions for the case of compact topological spaces and commutative C-star algebras: every (nonunital) $C^\ast$-algebra $A$ is equivalent to the $C^\ast$-algebra of continuous functions on the topological space called its Gelfand spectrum $sp(A)$. This theorem is the basis for regarding non-commutative $C^\ast$-algebras as formal duals to spaces in noncommutative geometry. Definitions The statement of Gelfand duality involves the following categories and functors. Definition Write • $C^\ast Alg$ for the category of C-star algebras; • $C^\ast Alg_{nu}$ for the category of non-unital $C^\ast$-algebras; • $C^\ast Alg_{com} \subset C^\ast Alg$ for the full subcategory of commutative $C^\ast$-algebras; • $C^\ast Alg_{com,nu} \subset C^\ast Alg_{nu}$ for the full subcategory of commutative non-unital $C^\ast$-algebras. And The duality itself is exhibited by the following functors: Definition Write $C \;\colon\; Top_{cpt} \to C^\ast Alg_{com}^{op}$ for the functor which sends a compact topological space $X$ to the algebra of continuous functions $C(X) = \{f : X \to \mathbb{C} \| f \; continuous\}$, equipped with the structure of a $C^\ast$-algebra in the evident way (…). Write $C_0 \;\colon\; /Top_{cpt} \to C^\ast Alg_{com,nu}$ for the functor that sends a pointed topological compact Hausdorff space $(X,x_0)$ to the algebra of continuous functions $f : X \to \mathbb{C}$ for which $f(x_0) = 0$. Definition Write $sp : C^\ast Alg_{com}^{op} \to Top_{cpt}$ for the Gelfand spectrum functor: it sends a commutative $C^\ast$-algebra $A$ to the set of characters – non-vanishing $C^\ast$-algebra homomorphisms $x : A \to \mathbb{C}$ – equipped with the spectral topology. Similarly write $sp : C^\ast Alg_{com,nu}^{op} \to Top_{lcpt} \,.$ Statement Theorem (Gelfand duality theorem) The pairs of functors $C^\ast Alg_{com}^{op} \stackrel{\overset{C}{\leftarrow}}{\underset{sp}{\to}} Top_{cpt}$ Here $C^\ast Alg^{op}_{\cdots}$ denotes the opposite category of $C^\ast Alg_{\cdots}$. Corollary On non-unital $C^\ast$-algebras the above induces an equivalence of categories $C^\ast Alg_{com,nu}^{op} \stackrel{\overset{C_0}{\leftarrow}}{\underset{sp}{\to}} /Top_{cpt} \,.$ Proof The operation of unitalization $(-)^+$ constitutes an equivalence of categories $C^\ast Alg_{nu} \stackrel{\overset{ker}{\leftarrow}}{\underset{(-)^+}{\to}} C^\ast Alg / \mathbb{C}$ between non-unital $C^\ast$-algebras and the over-category of $C^\ast$-algebras over the complex numbers $\mathbb{C}$. Composed with the equivalence of theorem 1 this yields $C^\ast Alg_{com,nu}^{op} \underoverset{\simeq}{(-)^+}{\to} (C^\ast Alg_{com}/\mathbb{C})^{op} \underoverset{\simeq}{C}{\to} * / Top_{cpt} \,.$ The weak inverse of this is the composite functor $C_0 : /Top_{cpt} \underoverset{\simeq}{sp}{\to} (C^\ast Alg_{com}/\mathbb{C})^{op} \underoverset{\simeq}{ker}{\to} C^\ast Alg_{com,nu}^{op}$ which sends $( \stackrel{x_0}{\to} X)$ to $ker(C(X) \stackrel{ev_{x_0}}{\to} \mathbb{C})$, hence to $\{f \in C(X) \| f(x_0) = 0\}$. This is indeed $C_0$ from def. 2. Remark Since locally compact Hausdorff spaces are equivalently open subspaces of compact Hausdorff spaces, via the construction that sends a locally compact Hausdorff space $X$ to its one-point compactification, and since a continuous function on the compact Hausdorff spce $X^\ast$ which vanishes at the extra point is equivalently a continuous function on $X$ which vanishes at infinity, the above induces an equivalence between locally compact Hausdorff spaces and $C^\ast$-algebras of functions that vanish at infinity. With due care on defining the right morphisms, the duality generalizes also to locally compact topological spaces. See for instance (Brandenburg 07). For an overview of other generalizations see also this MO discussion. Generalizations In constructive mathematics Gelfand duality makes sense in constructive mathematics hence internal to any topos: see constructive Gelfand duality theorem. By horizontal categorification Gelfand duality can be extended by horizontal categorification to define the notion of spaceoids as formal duals of commutative $C^*$-categories. References Textbook accounts include • N. P. Landsman, Mathematical topics between classical and quantum mechanics, Springer Monographs in Mathematics 1998. xx+529 pp. MR2000g:81081 doi • Gerald B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995. x+276 pp. gBooks An exposition that explicitly gives Gelfand duality as an equivalence of categories and introduces all the notions of category theory necessary for this statement is in • Ivo Dell’Ambrogio, Categories of $C^\ast$-algebras (pdf) Careful discussion of the duality for the more general case of locally compact topological spaces includes Some other generalized contexts for Gelfand duality: • Hans Porst, Manfred B. Wischnewsky, Every topological category is convenient for Gelfand duality, Manuscripta mathematica 25:2, (1978) pp 169-204 • H. Heunen, Klaas Landsman, Bas Spitters, S. Wolters, The Gelfand spectrum of a noncommutative $C^\ast$-algebra, J. Aust. Math. Soc. 90 (2011), 39–52 doi pdf • Christopher J. Mulvey, A generalisation of Gelfand duality, J. Algebra 56, n. 2, (1979) 499–505 doi Revised on June 3, 2017 06:25:25 by Urs Schreiber (178.6.236.87)	2018-03-20 00:22:50	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 57, "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": 0.8869212865829468, "perplexity": 761.7410791680405}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647244.44/warc/CC-MAIN-20180319234034-20180320014034-00320.warc.gz"}
https://docs.dgl.ai/en/latest/generated/dgl.adj_product_graph.html	dgl.adj_product_graph(A, B, weight_name, etype='_E')[source] Create a weighted graph whose adjacency matrix is the product of the adjacency matrices of the given two graphs. Namely, given two weighted graphs A and B, whose rows represent source nodes and columns represent destination nodes, this function returns a new graph whose weighted adjacency matrix is $$\mathrm{adj}(A) \times \mathrm{adj}(B)$$. The two graphs must be simple graphs, and must have only one edge type. Moreover, the number of nodes of the destination node type of A must be the same as the number of nodes of the source node type of B. The source node type of the returned graph will be the same as the source node type of graph A. The destination node type of the returned graph will be the same as the destination node type of graph B. If the two node types are the same, the returned graph will be homogeneous. Otherwise, it will be a bipartite graph. Unlike scipy, if an edge in the result graph has zero weight, it will not be removed from the graph. Notes This function works on both CPU and GPU. For GPU, the number of nodes and edges must be less than the maximum of int32 (i.e. 2 ** 31 - 1) due to restriction of cuSPARSE. The edge weights returned by this function is differentiable w.r.t. the input edge weights. If the graph format is restricted, both graphs must have CSR available. Parameters • A (DGLGraph) – The graph as left operand. • B (DGLGraph) – The graph as right operand. • weight_name (str) – The feature name of edge weight of both graphs. The corresponding edge feature must be scalar. • etype (str, optional) – The edge type of the returned graph. Returns The new graph. The edge weight of the returned graph will have the same feature name as weight_name. Return type DGLGraph Examples The following shows weighted adjacency matrix multiplication between two bipartite graphs. You can also perform this between two homogeneous graphs, or one homogeneous graph and one bipartite graph, as long as the numbers of nodes of the same type match. >>> A = dgl.heterograph({ ... ('A', 'AB', 'B'): ([2, 2, 0, 2, 0, 1], [2, 1, 0, 0, 2, 2])}, ... num_nodes_dict={'A': 3, 'B': 4}) >>> B = dgl.heterograph({ ... ('B', 'BA', 'A'): ([0, 3, 2, 1, 3, 3], [1, 2, 0, 2, 1, 0])}, ... num_nodes_dict={'A': 3, 'B': 4}) If your graph is a multigraph, you will need to call dgl.to_simple() to convert it into a simple graph first. >>> A = dgl.to_simple(A) >>> B = dgl.to_simple(B) Initialize learnable edge weights. >>> A.edata['w'] = torch.randn(6).requires_grad_() Take the product. >>> C = dgl.adj_product_graph(A, B, 'w') >>> C.edges() (tensor([0, 0, 1, 2, 2, 2]), tensor([0, 1, 0, 0, 2, 1])) >>> C.edata['w'] tensor([0.6906, 0.2002, 0.0591, 0.3672, 0.1066, 0.1328], Note that this function is differentiable: >>> C.edata['w'].sum().backward() tensor([0.7153, 0.2775, 0.7141, 0.7141, 0.7153, 0.7153]) >>> B.edata['w'].grad tensor([0.4664, 0.0000, 1.5614, 0.3840, 0.0000, 0.0000]) If the source node type of the left operand is the same as the destination node type of the right operand, this function returns a homogeneous graph: >>> C.ntypes ['A'] Otherwise, it returns a bipartite graph instead: >>> A = dgl.heterograph({ ... ('A', 'AB', 'B'): ([2, 2, 0, 2, 0, 1], [2, 1, 0, 0, 2, 2])}, ... num_nodes_dict={'A': 3, 'B': 4}) >>> B = dgl.heterograph({ ... ('B', 'BC', 'C'): ([0, 3, 2, 1, 3, 3], [1, 2, 0, 2, 1, 0])}, ... num_nodes_dict={'C': 3, 'B': 4})	2021-10-25 10:33:41	{"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": 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": 0.47169068455696106, "perplexity": 1142.8812381886414}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1634323587659.72/warc/CC-MAIN-20211025092203-20211025122203-00585.warc.gz"}
https://collegemathteaching.wordpress.com/category/integrals/	# College Math Teaching ## June 7, 2016 ### Infinite dimensional vector subspaces: an accessible example that W-perp-perp isn’t always W Filed under: integrals, linear albegra — Tags: , — collegemathteaching @ 9:02 pm This is based on a Mathematics Magazine article by Irving Katz: An Inequality of Orthogonal Complements found in Mathematics Magazine, Vol. 65, No. 4, October 1992 (258-259). In finite dimensional inner product spaces, we often prove that $(W^{\perp})^{\perp} = W$ My favorite way to do this: I introduce Grahm-Schmidt early and find an orthogonal basis for $W$ and then extend it to an orthogonal basis for the whole space; the basis elements that are not basis elements are automatically the basis for $W^{\perp}$. Then one easily deduces that $(W^{\perp})^{\perp} = W$ (and that any vector can easily be broken into a projection onto $W, W^{\perp}$, etc. But this sort of construction runs into difficulty when the space is infinite dimensional; one points out that the vector addition operation is defined only for the addition of a finite number of vectors. No, we don’t deal with Hilbert spaces in our first course. 🙂 So what is our example? I won’t belabor the details as they can make good exercises whose solution can be found in the paper I cited. So here goes: let $V$ be the vector space of all polynomials. Let $W_0$ the subspace of even polynomials (all terms have even degree), $W_1$ the subspace of odd polynomials, and note that $V = W_0 \oplus W_1$ Let the inner product be $\langle p(x), q(x) \rangle = \int^1_{-1}p(x)q(x) dx$. Now it isn’t hard to see that $(W_0)^{\perp} = W_1$ and $(W_1)^{\perp} = W_0$. Now let $U$ denote the subspace of polynomials whose terms all have degree that are multiples of 4 (e. g. $1 + 3x^4 - 2x^8$ and note that $U^{\perp} \subset W_1$. To see the reverse inclusion, note that if $p(x) \in U^{\perp}$, $p(x) = p_0 + p_1$ where $p_0 \in W_0, p_1 \in W_1$ and then $\int^1_{-1} (p_1(x))x^{4k} dx = 0$ for any $k \in \{1, 2, ... \}$. So we see that it must be the case that $\int^1_{-1} (p_0(x))x^{4k} dx = 0 = 2\int^1_0 (p_0(x))x^{4k} dx$ as well. Now we can write: $p_0(x) = c_0 + c_1 x^2 + ...c_n x^{2n}$ and therefore $\int^1_0 p_0(x) x^{4k} dx = c_0\frac{1}{4k+1} + c_1 \frac{1}{2 + 4k+1}...+c_n \frac{1}{2n + 4k+1} = 0$ for $k \in \{0, 1, 2, ...2n+1 \}$ Now I wish I had a more general proof of this. But these equations (for each $k$ leads a system of equations: $\left( \begin{array}{cccc} 1 & \frac{1}{3} & \frac{1}{5} & ...\frac{1}{2n+1} \\ \frac{1}{5} & \frac{1}{7} & \frac{1}{9}...&\frac{1}{2n+5} \\ ... & ... & ... & ... \\ \frac{1}{4k+1} & \frac{1}{4k+3} & ...& \frac{1}{10n+4} \end{array} \right) \left( \begin{array}{c} c_0 \\ c_1 \\ ... \\ c_n \end{array} \right) = \left( \begin{array}{c} 0 \\ 0 \\ ... \\ 0 \end{array} \right)$ It turns out that the given square matrix is non-singular (see page 92, no. 3 of Polya and Szego: Problems and Theorems in Analysis, Vol. 2, 1976) and so the $c_j = 0$. This means $p_0 = 0$ and so $U^{\perp} = W_1$ Anyway, the conclusion leaves me cold a bit. It seems as if I should be able to prove: let $f$ be some, say…$C^{\infty}$ function over $[0,1]$ where $\int^1_0 x^{2k} f(x) dx = 0$ for all $k \in \{0, 1, ....\}$ then $f = 0$. I haven’t found a proof as yet…perhaps it is false? ## May 20, 2016 ### Student integral tricks… Ok, classes ended last week and my brain is way out of math shape. Right now I am contemplating how to show that the complements of this object and of the complement of the object depicted in figure 3, are NOT homeomorphic. I can do this in this very specific case; I am interested in seeing what happens if the “tangle pattern” is changed. Are the complements of these two related objects always topologically different? I am reasonably sure yes, but my brain is rebelling at doing the hard work to nail it down. Anyhow, finals are graded and I am usually treated to one unusual student trick. Here is one for the semester: $\int x^2 \sqrt{x+1} dx =$ Now I was hoping that they would say $u = x +1 \rightarrow u-1 = x \rightarrow x^2 = u^2-2u+1$ at which case the integral is translated to: $\int u^{\frac{5}{2}} - 2u^{\frac{3}{2}} + u^{\frac{1}{2}} du$ which is easy to do. Now those wanting to do it a more difficult (but still sort of standard) way could do two repetitions of integration by parts with the first set up being $x^2 = u, \sqrt{x+1}dx =dv \rightarrow du = 2xdx, v = \frac{2}{3} (x+1)^{\frac{3}{2}}$ and that works just fine. But I did see this: $x =tan^2(u), dx = 2tan(u)sec^2(u)du, x+1 = tan^2(x)+1 = sec^2(u)$ (ok, there are some domain issues here but never mind that) and we end up with the transformed integral: $2\int tan^5(u)sec^3(u) du$ which can be transformed to $2\int (sec^6(u) - 2 sec^4(u) + sec^2(u)) tan(u)sec(u) du$ by elementary trig identities. And yes, that leads to an answer of $\frac{2}{7}sec^7(u) +\frac{4}{5}sec^5(u) + \frac{2}{3}sec^3(u) + C$ which, upon using the triangle Gives you an answer that is exactly in the same form as the desired “rationalization substitution” answer. Yeah, I gave full credit despite the “domain issues” (in the original integral, it is possible for $x \in (-1,0]$ ). What can I say? ## December 22, 2015 ### Multi leaf polar graphs and total area… Filed under: calculus, elementary mathematics, integrals — Tags: , — collegemathteaching @ 4:07 am I saw polar coordinate calculus for the first time in 1977. I’ve taught calculus as a TA and as a professor since 1987. And yet, I’ve never thought of this simple little fact. Consider $r(\theta) = sin(n \theta), 0 \theta \ 2 \pi$. Now it is well know that the area formula (area enclosed by a polar graph, assuming no “doubling”, self intersections, etc.) is $A = \frac{1}{2} \int^b_a (r(\theta))^2 d \theta$ Now the leaved roses have the following types of graphs: $n$ leaves if $n$ is odd, and $2n$ leaves if $n$ is even (in the odd case, the graph doubles itself). So here is the question: how much total area is covered by the graph (all the leaves put together, do NOT count “overlapping”)? Well, for $n$ an integer, the answer is: $\frac{\pi}{4}$ if $n$ is odd, and $\frac{\pi}{2}$ if $n$ is even! That’s it! Want to know why? Do the integral: if $n$ is odd, our total area is $\frac{n}{2}\int^{\frac{\pi}{n}}_0 (sin(n \theta)^2 d\theta = \frac{n}{2}\int^{\frac{\pi}{n}}_0 \frac{1}{2} + cos(2n\theta) d\theta =\frac{\pi}{4}$. If $n$ is even, we have the same integral but the outside coefficient is $\frac{2n}{2} = n$ which is the only difference. Aside from parity, the number of leaves does not matter as to the total area! Now the fun starts when one considers a fractional multiple of $\theta$ and I might ponder that some. ## May 11, 2015 ### The hypervolume of the n-ball enclosed by a standard n-1 sphere I am always looking for interesting calculus problems to demonstrate various concepts and perhaps generate some interest in pure mathematics. And yes, I like to “blow off some steam” by spending some time having some non-technical mathematical fun with elementary mathematics. This post uses only: 1. Integration by parts and basic reduction formulas. 2. Trig substitution. 3. Calculation of volumes (and hyper volumes) by the method of cross sections. 4. Induction 5. Elementary arithmetic involving factorials. The quest: find a formula that finds the (hyper)volume of the region $\{(x_1, x_2, x_3,....x_k) \| \sum_{i=1}^k x_i^2 \leq R^2 \} \subset R^k$ We will assume that the usual tools of calculus work as advertised. Start. If we done the (hyper)volume of the k-ball by $V_k$ we will start with the assumption that $V_1 = 2R$; that is, the distance between the endpoints of $[-R,R]$ is $2R$. Step 1: we show, via induction, that $V_k =c_kR^k$ where $c_k$ is a constant and $R$ is the radius. Our proof will be inefficient for instructional purposes. We know that $V_1 =2R$ hence the induction hypothesis holds for the first case and $c_1 = 2$. We now go to show the second case because, for the beginner, the technique will be easier to follow further along if we do the $k = 2$ case. Yes, I know that you know that $V_2 = \pi R^2$ and you’ve seen many demonstrations of this fact. Here is another: let’s calculate this using the method of “area by cross sections”. Here is $x^2 + y^2 = R^2$ with some $y = c$ cross sections drawn in. Now do the calculation by integrals: we will use symmetry and only do the upper half and multiply our result by 2. At each $y = y_c$ level, call the radius from the center line to the circle $R(y)$ so the total length of the “y is constant” level is $2R(y)$ and we “multiply by thickness “dy” to obtain $V_2 = 4 \int^{y=R}_{y=0} R(y) dy$. But remember that the curve in question is $x^2 + y^2 = R^2$ and so if we set $x = R(y)$ we have $R(y) = \sqrt{R^2 -y^2}$ and so our integral is $4 \int^{y=R}_{y=0}\sqrt{R^2 -y^2} dy$ Now this integral is no big deal. But HOW we solve it will help us down the road. So here, we use the change of variable (aka “trigonometric substitution”): $y = Rsin(t), dy =Rcos(t)$ to change the integral to: $4 \int^{\frac{\pi}{2}}_0 R^2 cos^2(t) dt = 4R^2 \int^{\frac{\pi}{2}}_0 cos^2(t) dt$ therefore $V_2 = c_2 R^2$ where: $c_2 = 4\int^{\frac{\pi}{2}}_0 cos^2(t)$ Yes, I know that this is an easy integral to solve, but I first presented the result this way in order to make a point. Of course, $c_2 = 4\int^{\frac{\pi}{2}}_0 cos^2(t) = 4\int^{\frac{\pi}{2}}_0 \frac{1}{2} + \frac{1}{2}cos(2t) dt = \pi$ Therefore, $V_2 =\pi R^2$ as expected. Exercise for those seeing this for the first time: compute $c_3$ and $V_3$ by using the above methods. Inductive step: Assume $V_k = c_kR^k$ Now calculate using the method of cross sections above (and here we move away from x-y coordinates to more general labeling): $V_{k+1} = 2\int^R_0 V_k dy = 2 \int^R_0 c_k (R(x_{k+1})^k dx_{k+1} =c_k 2\int^R_0 (R(x_{k+1}))^k dx_{k+1}$ Now we do the substitutions: first of all, we note that $x_1^2 + x_2^2 + ...x_{k}^2 + x_{k+1}^2 = R^2$ and so $x_1^2 + x_2^2 ....+x_k^2 = R^2 - x_{k+1}^2$. Now for the key observation: $x_1^2 + x_2^2 ..+x_k^2 =R^2(x_{k+1})$ and so $R(x_{k+1}) = \sqrt{R^2 - x_{k+1}^2}$ Now use the induction hypothesis to note: $V_{k+1} = c_k 2\int^R_0 (R^2 - x_{k+1}^2)^{\frac{k}{2}} dx_{k+1}$ Now do the substitution $x_{k+1} = Rsin(t), dx_{k+1} = Rcos(t)dt$ and the integral is now: $V_{k+1} = c_k 2\int^{\frac{\pi}{2}}_0 R^{k+1} cos^{k+1}(t) dt = c_k(2 \int^{\frac{\pi}{2}}_0 cos^{k+1}(t) dt)R^{k+1}$ which is what we needed to show. In fact, we have shown a bit more. We’ve shown that $c_1 = 2 =2 \int^{\frac{\pi}{2}}_0(cos(t))dt, c_2 = 2 \cdot 2\int^{\frac{\pi}{2}}_0 cos^2(t) dt = c_1 2\int^{\frac{\pi}{2}}_0 cos^2(t) dt$ and, in general, $c_{k+1} = c_{k}c_{k-1}c_{k-2} ....c_1(2 \int^{\frac{\pi}{2}}_0 cos^{k+1}(t) dt) = 2^{k+1} \int^{\frac{\pi}{2}}_0(cos^{k+1}(t))dt \int^{\frac{\pi}{2}}_0(cos^{k}(t))dt \int^{\frac{\pi}{2}}_0(cos^{k-1}(t))dt .....\int^{\frac{\pi}{2}}_0(cos(t))dt$ Finishing the formula We now need to calculate these easy calculus integrals: in this case the reduction formula: $\int cos^n(x) dx = \frac{1}{n}cos^{n-1}sin(x) + \frac{n-1}{n} \int cos^{n-2}(x) dx$ is useful (it is merely integration by parts). Now use the limits and elementary calculation to obtain: $\int^{\frac{\pi}{2}}_0 cos^n(x) dx = \frac{n-1}{n} \int^{\frac{\pi}{2}}_0 cos^{n-2}(x)dx$ to obtain: $\int^{\frac{\pi}{2}}_0 cos^n(x) dx = (\frac{n-1}{n})(\frac{n-3}{n-2})......(\frac{3}{4})\frac{\pi}{4}$ if $n$ is even and: $\int^{\frac{\pi}{2}}_0 cos^n(x) dx = (\frac{n-1}{n})(\frac{n-3}{n-2})......(\frac{4}{5})\frac{2}{3}$ if $n$ is odd. Now to come up with something resembling a closed formula let’s experiment and do some calculation: Note that $c_1 = 2, c_2 = \pi, c_3 = \frac{4 \pi}{3}, c_4 = \frac{(\pi)^2}{2}, c_5 = \frac{2^3 (\pi)^2)}{3 \cdot 5} = \frac{8 \pi^2}{15}, c_6 = \frac{\pi^3}{3 \cdot 2} = \frac{\pi^3}{6}$. So we can make the inductive conjecture that $c_{2k} = \frac{\pi^k}{k!}$ and see how it holds up: $c_{2k+2} = 2^2 \int^{\frac{\pi}{2}}_0(cos^{2k+2}(t))dt \int^{\frac{\pi}{2}}_0(cos^{2k+1}(t))dt \frac{\pi^k}{k!}$ $= 2^2 ((\frac{2k+1}{2k+2})(\frac{2k-1}{2k})......(\frac{3}{4})\frac{\pi}{4})((\frac{2k}{2k+1})(\frac{2k-2}{2k-1})......\frac{2}{3})\frac{\pi^k}{k!}$ Now notice the telescoping effect of the fractions from the $c_{2k+1}$ factor. All factors cancel except for the $(2k+2)$ in the first denominator and the 2 in the first numerator, as well as the $\frac{\pi}{4}$ factor. This leads to: $c_{2k+2} = 2^2(\frac{\pi}{4})\frac{2}{2k+2} \frac{\pi^k}{k!} = \frac{\pi^{k+1}}{(k+1)!}$ as required. Now we need to calculate $c_{2k+1} = 2\int^{\frac{\pi}{2}}_0(cos^{2k+1}(t))dt c_{2k} = 2\int^{\frac{\pi}{2}}_0(cos^{2k+1}(t))dt \frac{\pi^k}{k!}$ $= 2 (\frac{2k}{2k+1})(\frac{2k-2}{2k-1})......(\frac{4}{5})\frac{2}{3}\frac{\pi^k}{k!} = 2 (\frac{(2k)(2k-2)(2k-4)..(4)(2)}{(2k+1)(2k-1)...(5)(3)} \frac{\pi^k}{k!}$ To simplify this further: split up the factors of the $k!$ in the denominator and put one between each denominator factor: $= 2 (\frac{(2k)(2k-2)(2k-4)..(4)(2)}{(2k+1)(k)(2k-1)(k-1)...(3)(5)(2)(3)(1)} \pi^k$ Now multiply the denominator by $2^k$ and put one factor with each $k-m$ factor in the denominator; also multiply by $2^k$ in the numerator to obtain: $(2) 2^k (\frac{(2k)(2k-2)(2k-4)..(4)(2)}{(2k+1)(2k)(2k-1)(2k-2)...(6)(5)(4)(3)(2)} \pi^k$ Now gather each factor of 2 in the numerator product of the 2k, 2k-2… $= (2) 2^k 2^k \pi^k \frac{k!}{(2k+1)!} = 2 \frac{(4 \pi)^k k!}{(2k+1)!}$ which is the required formula. So to summarize: $V_{2k} = \frac{\pi^k}{k!} R^{2k}$ $V_{2k+1}= \frac{2 k! (4 \pi)^k}{(2k+1)!}R^{2k+1}$ Note the following: $lim_{k \rightarrow \infty} c_{k} = 0$. If this seems strange at first, think of it this way: imagine the n-ball being “inscribed” in an n-cube which has hyper volume $(2R)^n$. Then consider the ratio $\frac{2^n R^n}{c_n R^n} = 2^n \frac{1}{c_n}$; that is, the n-ball holds a smaller and smaller percentage of the hyper volume of the n-cube that it is inscribed in; note the $2^n$ corresponds to the number of corners in the n-cube. One might see that the rounding gets more severe as the number of dimensions increases. One also notes that for fixed radius R, $lim_{n \rightarrow \infty} V_n = 0$ as well. There are other interesting aspects to this limit: for what dimension $n$ does the maximum hypervolume occur? As you might expect: this depends on the radius involved; a quick glance at the hyper volume formulas will show why. For more on this topic, including an interesting discussion on this limit itself, see Dave Richardson’s blog Division by Zero. Note: his approach to finding the hyper volume formula is also elementary but uses polar coordinate integration as opposed to the method of cross sections. ## October 29, 2014 ### Hyperbolic Trig Functions and integration… In college calculus courses, I’ve always wrestled with “how much to cover in the hyperbolic trig functions” section. On one hand, the hyperbolic trig functions make some integrals much easer. On the other hand: well, it isn’t as if our classes are populated with the highest caliber student (I don’t teach at MIT); many struggle with the standard trig functions. There is only so much that the average young mind can absorb. In case your memory is rusty: $cosh(x) =\frac{e^x + e^{-x}}{2}, sinh(x) = \frac{e^x -e^{-x}}{2}$ and then it is immediate that the standard “half/double angle formulas hold; we do remember that $\frac{d}{dx}cosh(x) = sinh(x), \frac{d}{dx} = cosh(x)$. What is less immediate is the following: $sinh^{-1}(x) = ln(x+\sqrt{x^2+1}), cosh^{-1}(x) = ln(x + \sqrt{x^2 -1}) (x \ge 1)$. Exercise: prove these formulas. Hint: if $sinh(y) = x$ then $e^{y} - 2x- e^{-y} =0$ so multiply both sides by $e^{y}$ to obtain $e^{2y} -2x e^y - 1 =0$ now use the quadratic formula to solve for $e^y$ and keep in mind that $e^y$ is positive. For the other formula: same procedure, and remember that we are using the $x \ge 0$ branch of $cosh(x)$ and that $cosh(x) \ge 1$ The following follows easily: $\frac{d}{dx} sinh^{-1} (x) = \frac{1}{\sqrt{x^2 + 1}}$ (just set up $sinh(y) = x$ and use implicit differentiation followed by noting $cosh^2(x) -sinh^2(x) = 1$. ) and $\frac{d}{dx} cosh^{-1}(x) = \frac{1}{\sqrt{x^2-1}}$ (similar derivation). Now, we are off and running. Example: $\int \sqrt{x^2 + 1} dx =$ We can make the substitution $x =sinh(u), dx = cosh(u) du$ and obtain $\int cosh^2(u) du = \int \frac{1}{2} (cosh(2u) + 1)du = \frac{1}{4}sinh(2u) + \frac{1}{2} u + C$. Now use $sinh(2u) = 2 sinh(u)cosh(u)$ and we obtain: $\frac{1}{2}sinh(u)cosh(u) + \frac{u}{2} + C$. The back substitution isn’t that hard if we recognize $cosh(u) = \sqrt{sinh^2(u) + 1}$ so we have $\frac{1}{2} sinh(u) \sqrt{sinh^2(u) + 1} + \frac{u}{2} + C$. Back substitution is now easy: $\frac{1}{2} x \sqrt{x^2+1} + \frac{1}{2} ln(x + \sqrt{x^2 + 1}) + C$. No integration by parts is required and the dreaded $\int sec^3(x) dx$ integral is avoided. Ok, I was a bit loose about the domains here; we can make this valid for negative values of $x$ by using an absolute value with the $ln(x + \sqrt{x^2 + 1})$ term. ## August 31, 2014 ### The convolution integral: do some examples in Calculus III or not? For us, calculus III is the most rushed of the courses, especially if we start with polar coordinates. Getting to the “three integral theorems” is a real chore. (ok, Green’s, Divergence and Stoke’s theorem is really just $\int_{\Omega} d \sigma = \int_{\partial \Omega} \sigma$ but that is the subject of another post) But watching this lecture made me wonder: should I say a few words about how to calculate a convolution integral? Note: I’ve discussed a type of convolution integral with regards to solving differential equations here. In the context of Fourier Transforms, the convolution integral is defined as it was in analysis class: $fg = \int^{\infty}_{-\infty} f(x-t)g(t) dt$. Typically, we insist that the functions be, say, $L^1$ and note that it is a bit of a chore to show that the convolution of two $L^1$ functions is $L^1$; one proves this via the Fubini-Tonelli Theorem. (The straight out product of two $L^1$ functions need not be $L^1$; e.g, consider $f(x) = \frac {1}{\sqrt{x}}$ for $x \in (0,1]$ and zero elsewhere) So, assuming that the integral exists, how do we calculate it? Easy, you say? Well, it can be, after practice. But to test out your skills, let $f(x) = g(x)$ be the function that is $1$ for $x \in [\frac{-1}{2}, \frac{1}{2}]$ and zero elsewhere. So, what is $fg$??? So, it is easy to see that $f(x-t)g(t)$ only assumes the value of $1$ on a specific region of the $(x,t)$ plane and is zero elsewhere; this is just like doing an iterated integral of a two variable function; at least the first step. This is why it fits well into calculus III. $f(x-t)g(t) = 1$ for the following region: $(x,t), -\frac{1}{2} \le x-t \le \frac{1}{2}, -\frac{1}{2} \le t \le \frac{1}{2}$ This region is the parallelogram with vertices at $(-1, -\frac{1}{2}), (0, -\frac{1}{2}), (0 \frac{1}{2}), (1, \frac{1}{2})$. Now we see that we can’t do the integral in one step. So, the function we are integrating $f(x-t)f(t)$ has the following description: $f(x-t)f(t)=\left\{\begin{array}{c} 1,x \in [-1,0], -\frac{1}{2} t \le \frac{1}{2}+x \\ 1 ,x\in [0,1], -\frac{1}{2}+x \le t \le \frac{1}{2} \\ 0 \text{ elsewhere} \end{array}\right.$ So the convolution integral is $\int^{\frac{1}{2} + x}_{-\frac{1}{2}} dt = 1+x$ for $x \in [-1,0)$ and $\int^{\frac{1}{2}}_{-\frac{1}{2} + x} dt = 1-x$ for $x \in [0,1]$. That is, of course, the tent map that we described here. The graph is shown here: So, it would appear to me that a good time to do a convolution exercise is right when we study iterated integrals; just tell the students that this is a case where one “stops before doing the outside integral”. ## August 25, 2014 ### Fourier Transform of the “almost Gaussian” function with a residue integral This is based on the lectures on the Fourier Transform by Brad Osgood from Stanford: And here, $F(f)(s) = \int^{\infty}_{-\infty} e^{-2 \pi i st} f(t) dt$ provided the integral converges. The “almost Gaussian” integrand is $f(t) = e^{-\pi t^2}$; one can check that $\int^{\infty}_{-\infty} e^{-\pi t^2} dt = 1$. One way is to use the fact that $\int^{\infty}_{-\infty} e^{-x^2} dx = \sqrt{\pi}$ and do the substitution $x = \sqrt{\pi} t$; of course one should be able to demonstrate the fact to begin with. (side note: a non-standard way involving symmetries and volumes of revolution discovered by Alberto Delgado can be found here) So, during this lecture, Osgood shows that $F(e^{-\pi t^2}) = e^{-\pi s^2}$; that is, this modified Gaussian function is “its own Fourier transform”. I’ll sketch out what he did in the lecture at the end of this post. But just for fun (and to make a point) I’ll give a method that uses an elementary residue integral. Both methods start by using the definition: $F(s) = \int^{\infty}_{-\infty} e^{-2 \pi i ts} e^{-\pi t^2} dt$ Method 1: combine the exponential functions in the integrand: $\int^{\infty}_{-\infty} e^{-\pi(t^2 +2 i ts} dt$. Now complete the square to get: $\int^{\infty}_{-\infty} e^{-\pi(t^2 +2 i ts-s^2)-\pi s^2} dt$ Now factor out the factor involving $s$ alone and write as a square: $e^{-\pi s^2}\int^{\infty}_{-\infty} e^{-\pi(t+is)^2} dt$ Now, make the substitution $x = t+is, dx = dt$ to obtain: $e^{-\pi s^2}\int^{\infty+is}_{-\infty+is} e^{-\pi x^2} dx$ Now we show that the above integral is really equal to $e^{-\pi s^2}\int^{\infty}_{-\infty} e^{-\pi x^2} dx = e^{\pi s^2} (1) = e^{-\pi s^2}$ To show this, we perform $\int_{\gamma} e^{z^2} dz$ along the retangular path $\gamma$: $-x, x, x+is, -x+is$ and let $x \rightarrow \infty$ Now the integral around the contour is 0 because $e^{-z^2}$ is analytic. We wish to calculate the negative of the integral along the top boundary of the contour. Integrating along the bottom gives 1. As far as the sides: if we fix $s$ we note that $e^{-z^2} = e^{(s^2-x^2)+2si}$ and the magnitude goes to zero as $x \rightarrow \infty$ So the integral along the vertical paths approaches zero, therefore the integrals along the top and bottom contours agree in the limit and the result follows. Method 2: The method in the video This uses “differentiation under the integral sign”, which we talk about here. Stat with $F(s) = \int^{\infty}_{-\infty} e^{-2 \pi i ts} e^{-\pi t^2} dt$ and note $\frac{dF}{ds} = \int^{\infty}_{-\infty} (-2 \pi i t) e^{-2 \pi i ts} e^{-\pi t^2} dt$ Now we do integration by parts: $u = e^{-2 \pi i ts}, dv = (-2 \pi i t)e^{-\pi t^2} \rightarrow v = i e^{-\pi t^2}, du = (-2 \pi i s)e^{-2 \pi i ts}$ and the integral becomes: $(i e^{-\pi t^2} e^{-2 \pi i ts}\|^{\infty}_{-\infty} - (i)(-2 \pi i s) \int^{\infty}_{-\infty} e^{-2 \pi i ts} e^{-\pi t^2} dt$ Now the first term is zero for all values of $s$ as $t \rightarrow \infty$. The second term is merely: $-(2 \pi s) \int^{\infty}_{-\infty} e^{-2 \pi i ts} e^{-\pi t^2} dt = -(2 \pi s) F(s)$. So we have shown that $\frac{d F}{ds} = (-2 \pi s)F$ which is a differential equation in $s$ which has solution $F = F_0 e^{- \pi s^2}$ (a simple separation of variables calculation will verify this). Now to solve for the constant $F_0$ note that $F(0) = \int^{\infty}_{-\infty} e^{0} e^{-\pi t^2} dt = 1$. The result follows. Now: which method was easier? The second required differential equations and differentiating under the integral sign; the first required an easy residue integral. By the way: the video comes from an engineering class. Engineers need to know this stuff! ## August 21, 2014 ### Calculation of the Fourier Transform of a tent map, with a calculus tip…. I’ve been following these excellent lectures by Professor Brad Osgood of Stanford. As an aside: yes, he is dynamite in the classroom, but there is probably a reason that Stanford is featuring him. 🙂 And yes, his style is good for obtaining a feeling of comradery that is absent in my classroom; at least in the lower division “service” classes. This lecture takes us from Fourier Series to Fourier Transforms. Of course, he admits that the transition here is really a heuristic trick with symbolism; it isn’t a bad way to initiate an intuitive feel for the subject though. However, the point of this post is to offer a “algebra of calculus trick” for dealing with the sort of calculations that one might encounter. By the way, if you say “hey, just use a calculator” you will be BANNED from this blog!!!! (just kidding…sort of. 🙂 ) So here is the deal: let $f(x)$ represent the tent map: the support of $f$ is $[-1,1]$ and it has the following graph: The formula is: $f(x)=\left\{\begin{array}{c} x+1,x \in [-1,0) \\ 1-x ,x\in [0,1] \\ 0 \text{ elsewhere} \end{array}\right.$ So, the Fourier Transform is $F(f) = \int^{\infty}_{-\infty} e^{-2 \pi i st}f(t)dt = \int^0_{-1} e^{-2 \pi i st}(1+t)dt + \int^1_0e^{-2 \pi i st}(1-t)dt$ Now, this is an easy integral to do, conceptually, but there is the issue of carrying constants around and being tempted to make “on the fly” simplifications along the way, thereby leading to irritating algebraic errors. So my tip: just let $a = -2 \pi i s$ and do the integrals: $\int^0_{-1} e^{at}(1+t)dt + \int^1_0e^{at}(1-t)dt$ and substitute and simplify later: Now the integrals become: $\int^{1}_{-1} e^{at}dt + \int^0_{-1}te^{at}dt - \int^1_0 te^{at} dt.$ These are easy to do; the first is merely $\frac{1}{a}(e^a - e^{-a})$ and the next two have the same anti-derivative which can be obtained by a “integration by parts” calculation: $\frac{t}{a}e^{at} -\frac{1}{a^2}e^{at}$; evaluating the limits yields: $-\frac{1}{a^2}-(\frac{-1}{a}e^{-a} -\frac{1}{a^2}e^{-a}) - (\frac{1}{a}e^{a} -\frac{1}{a^2}e^a)+ (-\frac{1}{a^2})$ Add the first integral and simplify and we get: $-\frac{1}{a^2}(2 - (e^{-a} -e^{a})$. NOW use $a = -2\pi i s$ and we have the integral is $\frac{1}{4 \pi^2 s^2}(2 -(e^{2 \pi i s} -e^{-2 \pi i s}) = \frac{1}{4 \pi^2 s^2}(2 - cos(2 \pi s))$ by Euler’s formula. Now we need some trig to get this into a form that is “engineering/scientist” friendly; here we turn to the formula: $sin^2(x) = \frac{1}{2}(1-cos(2x))$ so $2 - cos(2 \pi s) = 4sin^2(\pi s)$ so our answer is $\frac{sin^2( \pi s)}{(\pi s)^2} = (\frac{sin(\pi s)}{\pi s})^2$ which is often denoted as $(sinc(s))^2$ as the “normalized” $sinc(x)$ function is given by $\frac{sinc(\pi x)}{\pi x}$ (as we want the function to have zeros at integers and to “equal” one at $x = 0$ (remember that famous limit!) So, the point is that using $a$ made the algebra a whole lot easier. Now, if you are shaking your head and muttering about how this calculation was crude that that one usually uses “convolution” instead: this post is probably too elementary for you. 🙂 ## August 6, 2014 ### Where “j” comes from I laughed at what was said from 30:30 to 31:05 or so: If you are wondering why your engineering students want to use $j = \sqrt{-1}$ is is because, in electrical engineering, $i$ usually stands for “current”. Though many of you know this, this lesson also gives an excellent reason to use the complex form of the Fourier series; e. g. if $f$ is piece wise smooth and has period 1, write $f(x) = \Sigma^{k = \infty}_{k=-\infty}c_k e^{i 2k\pi x}$ (usual abuse of the equals sign) rather than writing it out in sines and cosines. of course, $\overline{c_{-k}} = c_k$ if $f$ is real valued. How is this easier? Well, when you give a demonstration as to what the coefficients have to be (assuming that the series exists to begin with, the orthogonality condition is very easy to deal with. Calculate: $c_m= \int^1_0 e^{i 2k\pi t}e^{i 2m\pi x} dx$ for when $k \ne m$. There is nothing to it; easy integral. Of course, one has to demonstrate the validity of $e^{ix} = cos(x) + isin(x)$ and show that the usual differentiation rules work ahead of time, but you need to do that only once. ## July 31, 2014 ### Stupid question: why does it appear to us that differentiation is easier than anti-differentiation? Filed under: calculus, elliptic curves, integrals — Tags: , — collegemathteaching @ 8:05 pm This post is inspired by my rereading a favorite book of mine: Underwood Dudley’s Mathematical Cranks There was the chapter about the circumference of an ellipse. Now, given $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ it isn’t hard to see that $s^2 = {dx}^2 + {dy}^2$ and so going with the portion in the first quadrant: one can derive that the circumference is given by the elliptic integral of the second kind, which is one of those integrals that can NOT be solved in “closed form” by anti-differentiation of elementary functions. There are lots of integrals like this; e. g. $\int e^{x^2} dx$ is a very famous example. Here is a good, accessible paper on the subject of non-elementary integrals (by Marchisotto and Zakeri). So this gets me thinking: why is anti-differentiation so much harder than taking the derivative? Is this because of the functions that we’ve chosen to represent the “elementary anti-derivatives”? I know; this is not a well formulated question; but it has always bugged me. Oh yes, I am teaching two sections of first semester calculus this upcoming semester. 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https://doc.ecole.ai/master/reference/information.html	# Informations¶ ## Interface¶ class ecole.typing.InformationFunction(args, kwds)[source] Class repsonsible for extracting the the information dictionnary. Information functions are objects given to the Environment to extract the addtional information about the environment. A common pattern is use additional ecole.typing.RewardFunction and ecole.typing.ObservationFunction to easily create information functions. This class presents the interface expected to define a valid information function. It is not necessary to inherit from this class, as information functions are defined by structural subtyping. It is exists to support Python type hints. DataFunction Information function are a specific type of generic data function where the data extracted are dictionnary of string to any type. __init__(args, **kwargs) Initialize self. See help(type(self)) for accurate signature. before_reset(model: ecole.scip.Model) → None[source] Reset internal data at the start of episodes. The method is called on new episodes reset() right before the MDP is actually reset, that is right before the environment calls reset_dynamics(). It is usually used to reset the internal data. Parameters model – The Model, model defining the current state of the solver. extract(model: ecole.scip.Model, done: bool) → Dict[str, Information][source] Extract the information on the given state. Extract the information after transitionning on the new state given by model. The function is reponsible for keeping track of relevant information from previous states. This can safely be done in this method as it will only be called once per state i.e., this method is not a getter and can have side effects. Parameters • model – The Model, model defining the current state of the solver. • done – A flag indicating wether the state is terminal (as decided by the environment). Returns The return is passed to the user by the environment. ## Listing¶ The list of information functions relevant to users is given below. ### Nothing¶ class ecole.information.Nothing __init__(self: ecole.information.Nothing) → None before_reset(self: ecole.information.Nothing, model: ecole.scip.Model) → None Do nothing. extract(self: ecole.information.Nothing, model: ecole.scip.Model, done: bool) → Dict[str, None] Return an empty dictionnary.	2021-01-23 02:14:22	{"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": 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": 0.5053634643554688, "perplexity": 4192.8346051834715}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703531702.36/warc/CC-MAIN-20210123001629-20210123031629-00321.warc.gz"}
https://proxies123.com/tag/programming/	## Code for a programming of python lists In the python list, the name contains these names: Name =[“zam”,”tam”,”pam”] How to get an exit like this Zam Zam Zam Tam Tam Tam Pam Pam Pam And also like that Zam Tam Pam Zam Tam Pam … Without using multiple printing instructions? ## Merging two sequences ordered in linear programming. Given two sequences of integers. $$a_1, dots, a_t$$ Y $$b_1, dots, b_m$$ every $$> 0$$ where $$a_i Y $$b_i Is there a linear program to merge in orderly sequence without integer variables? Related: Classification as a linear program. ## programming challenge – Euler Project Problem # 14 largest Collatz sequence written in Python Problem Statement: The following iterative sequence is defined for the set of positive integers: n → n / 2 (n is even) n → 3n + 1 (n is odd) Using the previous rule and starting with 13, we generate the following sequence: 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1 You can see that this sequence (starting at 13 and ending at 1) contains 10 terms. Although it does not have still tested (Collatz problem), it is thought that all the initial numbers end in 1. What initial number, less than one million, produces the longest chain? NOTE: Once the string starts, the terms can go over a million. ``````from the time of import from the operator import itemgetter def collatz_count (n, count = {1: 1}): "" "use cache (account) to speed up the search, return the sequence length for the given number "" " try: return count[n] except KeyError: if n% 2 == 0: tell[n] = collatz_count (n / 2) + 1 plus: tell[n] = collatz_count (n * 3 + 1) + 1 return count[n] def run_test (): "" "uses the previous function, returns the number generated by the Major sequence "" " time1 = time () articles = ordered ([(x, collatz_count(x)) for x in range(1, 1000000)], key = itemgetter (1), reverse = True) maximum = articles[0] print (f & # 39; Initial number: {maximum[0]} n sequence length: {maximum[1]} n & # 39; f & # 39; Calculated in: {time () - time1} seconds. & # 39;) yes __name__ == & # 39; __ main __ & # 39; run_test () `````` ## programming challenge – Euler Project, problem # 9, Pythagoras triplet A triplet of Pythagoras is a set of three natural numbers, a <b <c, for which, a 2 + b 2 = c 2 For example, 3 2 + 4 2 = 9 + 16 = 25 = 52. There is exactly one triplet of Pythagoras for which a + b + c = 1000. Find the product abc. ``````def get_triplet (): for c in range (2, 1000): for b in rank (2, c): a = 1000 - c - b if a 2 + b 2 == c 2: returns & # 39; n1 =% s n n2 = % s n n3 =% s n product =% s & # 39;% (a, b, c, a * b * c) yes __name__ == & # 39; __ main __ & # 39; print (get_triplet ()) `````` ## Linear programming: Why does not my code correctly complete the augmented matrix? I'm trying to enter data from a training file. He's skipping the first row altogether. `````` double augmentMatrix (double matrix, int dim) { double identityMatrix = CreateIdentityMatrix (dim, dim); // create identity matrix double augmentedMatrix = (double *) malloc (sizeof (double ) * dim); for (int row = 0; row <tenue; row ++) { increased matrix[row] = (double ) malloc (sizeof (double) dim * 2); } // attribute space for (int row = 0; row <tenue; row ++) { for (int col = 0; col <dim; col ++) { increased matrix[row][col] = matrix[row][col]; // set the first part of the matrix augmented to the parameters of the matrix parameter } for (int col = dim; col <2 * dim; col ++) { increased matrix[row][col] = identityMatrix[row][col - dim]; // separates the identity matrix } } printMatrix (augmentedMatrix, dim, dim * 2); return augmentedMatrix; } `````` This is what is producing: `````` 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 26.2, 84.8, 61.9, 58900.0, 51789.8, 0.0, 1.0, 0.0, 0.0, 0.0, 6493.8, 19487.8, 12220.3, 9864262.5, 12861241.2, 0.0, 0.0, 1.0, 0.0, 0.0, 21384.0, 70762.0, 55533.2, 57206075.0, 42298973.0, 0.0, 0.0, 0.0, 1.0, 0.0, 19713.0, 61172.0, 41552.0, 39621035.0, 38865313.0, 0.0, 0.0, 0.0, 0.0, 1.0, `````` Here is the training data file: `````` 18.0, 55.0, 37.0, 33025.0, 35598.0, 26.2, 84.8, 61.9, 58900.0, 51789.8, 6493.8, 19487.8, 12220.3, 9864262.5, 12861241.2, 21384.0, 70762.0, 55533.2, 57206075.0, 42298973.0, 19713.0, 61172.0, 41552.0, 39621035.0, 38865313.0, `````` ## Programming: how to run a specific cell in a notebook from another notebook I have a long `Mathematics` Code in a notebook, denoted by NB1. Since it is a very long code, and I just need some of the results of NB1 to be used in another laptop indicated by NB2, I assigned `Mobile Tags` to those cells NB1 that generate the outputs required by NB2. I like to know how to remember the cells labeled in NB1 in NB2. I know how to run NB1 while it is in NB2, but I do not want to execute all the operations in NB1 since I only need a subset of the outputs of NB1. Here is an example of NB1: ``````Clear[Evaluate[Context[] <> ""]]; Clear[n, d, G, transfer, absorption, info, capParam, infoStock, edgeCapacityMat, infoStocks, system, reducedSystem, sa, wG]; SeedRandom[14]; n = 6; d = 0.3; G = RandomGraph[{Round[n], Round[n(n - 1)d]}, DirectedEdges-> True]; transfer = Table[t[i], {i, 1, n}]; ( transfer capacity ) absorption = table[a[i], {i, 1, n}]; ( Absorption capacity ) info = table[x[i], {i, 1, n}]; ( information stock capacity ) ( parameterization ) capParam = Table[{t[i] -> RandomReal[], a[i]-> RandomReal[]} {i, 1, n}]//Flatten; infoStock = Table[x[i] -> RandomInteger[{1, 10}], {i, 1, n}]//Flatten; edgeCapacityMat[transferCap_, absorptionCap_] : = (# - DiagonalMatrix[Diagonal[#]]) &[ KroneckerProduct[transferCap, absorptionCap]]; InfoStocks[stock_] : = DiagonalMatrix[stock]; system = infoStocks[info].edgeCapacityMat[transfer, absorption]; ( completely defined system ) ReducedSystem = system; ( the system associated with AdjacencyGraph "G" ) sa = SparseArray[reducedSystem]; wG = Graphic[sa["NonzeroPositions"], EdgeWeight-> sa["NonzeroValues"], DirectedEdges -> True, VertexLabels -> "Name"]; ( Answer by @ KGLR: use this to formulate "attention assignment problem in my work *) Clear all[edgeW]; edgeW = Module[{g = #, e = DirectedEdge @@@ Partition[#, 2, 1] & / @ FindPath[##, [Infinity], Everyone]} Transpose[{e, PropertyValue[{g, #}, EdgeWeight] & / @ # & / @ my}]]&; desiredOutput1 = edgeW[wG, 5, 2] desiredOutput2 = HighlightGraph[wGedgeW[wGedgeW[wGedgeW[wGedgeW[wG, 5, 2][[All, 1]]] `````` I like to use `desiredOutput1` Y `desiredOutput2` in NB2, therefore I have created two `cell tags` in NB1 it must be recovered from NB2, but I do not know how to recover only the two cell labels of NB2. ## design – How can you call other programming languages? This mechanism is called an external function call. Almost all major programming languages can do so, although the ease of use of an external function call differs between implementations of each language pair. Python itself, for example, has `ctypes` in the standard library and a third-party library `Cffi` that allows you to call any C code, and other languages that expose a foreign function interface similar to C (for which there is a lot). Java has JNI (Java Native Interface). .NET CLR and Java are somewhat different, since they expose an interface by which any language that is executed in the same virtual machine can call any other language in the virtual machine, so the foreign function call from the same time of execution is generally quite simple. This is possible because those languages already have to be assigned in the construction of the virtual machine anyway. ## design – What are some programming languages that are compiled with others? So today I found about IronPython: IronPython is an open source implementation of the Python programming language that is tightly integrated with the .NET Framework. This was my first experience with the use of multiple programming languages in a single program. I thought it was really cool to be able to call my Python script as a container for the compiled dll from C #. But this made me think … how many more programming languages could be attached to this? Is there a limit? What are some other programming languages that have a similar feature? ## c # – book to learn object oriented programming in C # _ Thanks for contributing in StackOverflow in Spanish with an answer! But avoid • Make statements based on opinions; Be sure to back them up with references or your own personal experience. ## Functional programming: refactoring an if statement into a higher order function in Swift Using Swift, I am sorting some objects first by date order, then second by title if the date is the same, according to the following extract: ``````.sorted { yes \$ 0.date! = \$ 1.date { return \$ 0. date <\$ 1 date // object date } else { return \$ 0.title <\$ 1.title // String } } `````` This brief si / else is used several times, and as such I would like to extract this functionality and convert it into a reusable component. Any ideas or suggestions on how this can be achieved, given that the return value could be a date or a chain?	2019-07-15 22:19:34	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "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": 0.21189501881599426, "perplexity": 5745.276649535544}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1563195524254.28/warc/CC-MAIN-20190715215144-20190716001144-00433.warc.gz"}
http://bjsm.bmj.com/content/37/3/212	Article Text Four weeks of androstenedione supplementation diminishes the treatment response in middle aged men 1. S G Beckham1, 2. C P Earnest2 1. 1University of Texas at Arlington, USA 2. 2Cooper Institute for Aerobics Research, Texas, USA 1. Correspondence to:  Assistant Professor Beckham, Department of Kinesiology, University of Texas at Arlington, Box 19259, Arlington, TX 76019, USA;   sbeckham{at}worldnet.att.net ## Abstract Objectives: To examine baseline hormonal concentrations and the pharmacokinetic response on day 0 and day 28 of 28 days of androstenedione supplementation. Methods: Eight men (mean (SD) age 44.1 (3.0) years (range 40–48), weight 76.3 (9.4) kg, and percentage body fat 20.6 (6.7)) participated in a randomised, double blind, cross over, 2 × 28 day placebo controlled study. Subjects were tested on day 0 and 28 days after receiving 200 mg/day oral androstenedione and a placebo treatment with a 28 day washout period between treatments. Serum hormone concentrations were examined at baseline (time 0) and then at 30 minute intervals for 180 minutes to measure day 0 and day 28 pharmacokinetic responses. Analytes included androstenedione, total testosterone, dehydroepiandrosterone sulfate (DHEAS), oestradiol, and sex hormone binding globulin (SHBG). Lipid concentrations, weight, body composition, resting heart rate, and blood pressure were also measured. Results: Analysis of integrated area under the curve (AUC) and time 0 hormonal concentrations by repeated measures multivariate analysis of variance (p<0.05) and Fisher’s post hoc analysis showed a significant increase in AUC for serum androstenedione at day 0 (108.3 (27.6) nmol/l) in the supplemented condition as compared with day 28 (43.4 (13.1) nmol/l) and placebo (2.1 (0.8) nmol/l) conditions. No other significant AUC changes were noted. After 28 days of supplementation, DHEAS levels were significantly elevated (p = 0.00002) at time 0 (12.9 (1.3) μmol/l) compared with placebo (7.0 (0.8) μmol/l) with a trend (p = 0.08) toward elevation of time 0 androstenedione concentrations (16.4 (7.0) nmol/l) compared with placebo (5.6 (0.4) nmol/l). No changes were found for lipids, resting heart rate, or blood pressure, weight, or percentage body fat. Conclusion: Although supplementation with 200 mg/day androstenedione increases AUC for serum androstenedione in the day 0 condition, continued supplementation is characterised by a diminished treatment response, coupled with time 0 increases in testosterone precursors but not testosterone. • dehydroepiandrosterone sulfate • sex hormone binding globulin • testosterone • DHEA, dehydroepiandrosterone • DHEAS, dehydroepiandrosterone sulfate • SHBG, sex hormone binding globulin • AUC, area under the curve • LDL, low density lipoprotein • HDL, high density lipoprotein • VLDL, very low density lipoprotein • POMS, profile of mood state ## Statistics from Altmetric.com Androstenedione is marketed to middle aged (⩾ 40 years) men to ameliorate the effect of aging on the hypothalamic pituitary gonadal axis. Some studies of male androgenic function report no decline in serum total testosterone concentration with age,1 whereas others report decreasing concentration at age 45–50 years that may be within the normal range for young men.2–4 As androstenedione is a testosterone precursor,5 and supplementation has been shown to raise serum androstenedione concentration,6–12 it is plausible that elevated androstenedione concentrations could result in increased conversion of androstenedione into testosterone, especially in an aging population. Several studies have reported that supplementation with 200–300 mg/day androstenedione can produce acute9,11 and chronic8 increases in total and free testosterone concentrations, whereas others have reported no changes6–8,10 in younger (<40 years) and middle aged (⩾40 years) men. These studies used different dosages, dosing patterns, lengths of supplementation, as well as, a variety of age groups, which may contribute to these inconsistent findings. Of perhaps greater concern is the potential for alterations in hormonal balance with prolonged supplementation. Strong regulatory mechanisms exist that alter the production and conversion of hormones after prolonged administration of androgens.5 The purpose of this study was to determine if baseline (time 0) and pharmacokinetic hormonal responses to androstenedione supplementation are altered after four weeks of supplementation with 200 mg/day androstenedione in middle aged men. ## METHODS ### Subjects Eight healthy men volunteered to participate in this study which was approved by the Institution’s human research review committee. Before the study, each subject completed and signed a medical history and consent form that described the study protocol and potential risks and discomforts. Baseline characteristics were mean (SD) age 44.1 (3.0) years (range 40–48), body mass 76.3 (9.4) kg, and height 175 (5.3) cm. Only two subjects participated in a resistance training programme; both had participated for five years or more. For the duration of the trial, all subjects agreed not to change the frequency, intensity, or volume of their exercise programme, or their dietary habits. ### Study design A randomised, double blind, cross over, placebo controlled design was used to assess subject responses to supplemented and placebo treatment conditions. A cross over design was chosen to control for initial differences in hormonal concentrations and the effects this may have on individual responses to supplementation. Each subject was tested at the beginning (day 0) and end of a 28 day supplementation period with 200 mg/day androstenedione and a placebo. Figure 1 shows a schematic of the testing protocol. A 28 day washout period was used between the two supplementation periods. During both treatments, subjects were examined to determine hormonal concentrations at time 0, as well as a 180 minute pharmacokinetic response to each treatment. Figure 1 Schematic of the testing protocol. ### Treatment The treatments consisted of 200 mg/day androstenedione (two 100 mg tablets) or two tablets of a rice powder placebo, administered each morning. Each treatment was distributed in bottles that were individually coded with a random number sequence so that, in the case of an adverse event, the code could be broken without sacrificing the integrity of the remaining cohort. All supplements were of the same lot and provided by Metabolic Response Modifiers (Newport Beach, California, USA). A certificate of analysis from the company reported the product to be 95% pure. ### Cardiovascular and anthropometric testing Subjects reported to the laboratory after a 12 hour fast and abstinence from exercise for 24 hours. Data were collected at about the same time each visit to control for diurnal variations in certain hormones. Resting heart rate and blood pressure measurements were taken at the beginning of each testing session. Subjects were then weighed for total body mass before determination of body density by hydrostatic weighing. The three highest underwater weights at each visit were averaged to obtain underwater weight. Underwater weights were corrected for residual volume using an automated multiple breath oxygen dilution procedure13 with a Vmax series 229 metabolic analyser (SensorMedics, Yorba Linda, California, USA). The Siri equation14 was used to convert body density into percentage body fat. Next, waist and hip circumferences were measured using a Gulick tape. Lastly, subjects completed a questionnaire designed by the researchers at each visit that enquired about androgenic effects typically associated with steroid use. ### Blood lipid assessment Blood lipid indices were measured before the pharmacokinetic assessment from a finger stick (Cholestech LDX System; Cholestech LDX, Hayward, California, USA) to determine total, low density lipoprotein (LDL), high density lipoprotein (HDL), and very low density lipoprotein (VLDL) cholesterol, total cholesterol/HDL cholesterol ratio, and triglycerides. The intra-assay and interassay coefficients of variation for the analysis were 2.5% and 3.1% for total cholesterol, 4.1% and 5.4% for HDL cholesterol, 2.6% and 2.8% for VLDL cholesterol, and 2.6% and 2.8% for triglycerides. ### Pharmacokinetic and baseline hormone assessments After a fasting assessment for lipid indices, the time 0 and pharmacokinetic response for the 180 minute period after ingestion of a placebo or androstenedione were assessed. As the pharmacokinetic assessment required an additional four hours to complete the data collection process, subjects were permitted to eat a snack. Meikle et al15 reported that ingestion of a high fat meal (57% fat) produced acute reductions in total and free testosterone, but ingestion of a low fat/mixed carbohydrate/protein meal did not; therefore subjects were permitted to ingest a low fat (<20% fat)/mixed carbohydrate/protein energy bar and consume a carbohydrate sports drink (Gatorade) after the cholesterol testing and before the pharmacokinetic assessment. Analysis of testosterone concentrations for the placebo trials by repeated measures analysis of variance (p<0.05) failed to show any significant differences across the 180 minute sampling period. This finding supports the assumption that the low fat/mixed carbohydrate/protein meal had no effect on testosterone levels. Before the pharmacokinetic trial, subjects were fitted with a 21 gauge catheter that was inserted into an antecubital vein. After a 30 minute habituation period, the first blood sample was collected at time 0. Supplements were then given orally, and blood samples were collected every 30 minutes for 180 minutes. Blood samples were collected in a plain tube, allowed to clot for 20 minutes, and spun in a centrifuge at 4°C. Serum was then pipetted into a storage tube and stored at −70°C. At a later date, serum androstenedione was analysed in duplicate in antibody coated tubes with radioimmunoassay kits (Diagnostic Products Corporation, Los Angeles, California, USA). Total testosterone, dehydroepiandrosterone sulfate (DHEAS), oestradiol 17 (oestradiol), and sex hormone binding globulin (SHBG) concentrations were assessed using a solid phase, chemiluminescent immunoassay (Immulite; Diagnostic Products Corporation). With this system, the lowest detectable oestradiol concentration is 73.4 pmol/l. Therefore oestradiol values <73.4 pmol/l were recorded as 73.4 pmol/l for the purpose of this study. The intra-assay and interassay coefficients of variation for the analysis were 5.7% and 8.4% for androstenedione, 8.2% and 12.0% for DHEAS, 7.4% and 9.8% for total testosterone, 9.3% and 10.6% for oestradiol, and 6.5 and 8.7% for SHBG. ### Psychometric assessment All subjects completed a profile of mood state (POMS) questionnaire at each visit. The POMS measures dimensions of affect or mood in six categories.16 A Wilcoxon signed rank test (p<0.05) was used to determine if total scores were significantly different between trials. Experiment wise error rate for the POMS was calculated for the six categories as 1−(1−0.05)6 = 0.264, where the iteration “6” is the number of categories in the POMS assessment. Subsequently, 0.264 was divided by six so that the α level for determining significance was p = 0.044 of self reported behaviours. ### Statistical analysis Integrated area under the curve (AUC) above time 0 concentrations and the associated metabolic clearance rate (litres/h) for androstenedione were determined by trapezoid calculation for the 180 minutes after supplementation for each analyte at each of the four testing conditions (PK Solutions Software 2.0, Montrose, Colorado, USA). The following equation was used to calculate systemic clearance rate (CL) for the observed AUC (0 t): $Math$ where F = fraction of dose absorbed and D = dose.17 A repeated measures multivariate analysis of variance (p<0.05) with a Fisher’s least squares difference post hoc analysis was used to compare AUC for the testing conditions (Statview, Cary, North Carolina, USA). As no statistical difference was observed between the two placebo conditions, results from the first placebo visit (day 0) were used in the analysis of all analytes. A repeated measures multivariate analysis of variance (p<0.05) with a Fisher’s least squares difference post hoc analysis was also used to compare time 0 hormonal concentrations measured at the beginning of each test session. A repeated measures analysis of variance (p<0.05) was used to assess concentration differences across time for analytes that showed significant changes for either time 0 or AUC values compared with placebo. In addition, a Pearson product-moment correlation was computed to assess the relations between AUC for various hormonal responses. A repeated measures multivariate analysis of variance (p<0.05) was also used to examine changes related to body composition (body mass, body fat, and waist/hip ratio), resting cardiovascular data (resting heart rate, systolic and diastolic blood pressure), and the lipid profile (total, LDL, HDL, and VLDL cholesterol, total cholesterol/HDL cholesterol ratio, and triglycerides). ## RESULTS ### Pharmacokinetics (AUC) Figures 2 and 3 depict integrated AUC concentrations for each analyte during the 180 minute period after administration of androstenedione or a placebo. The serum androstenedione concentration (AUC) was significantly higher after the first administration of 200 mg androstenedione (day 0) compared with both placebo and day 28 of supplementation. However, AUC concentrations were not significantly different between any of the treatments for other hormones. Furthermore, no differences were noted between androstenedione clearance rates for AUC at day 0 compared with day 28 of supplementation. Figure 2 Integrated area under the curve (AUC) concentrations for (A) androstenedione, (B) dehydroepiandrosterone sulfate (DHEAS), and (C) total testosterone for day 0, day 28, and placebo conditions. Values are mean (SE). Significantly different from placebo (p<0.0005); *significantly different from day 28 (p<0.05). Figure 3 Integrated area under the curve (AUC) concentrations for (A) oestradiol and (B) sex hormone binding globulin (SHBG) for day 0, day 28, and placebo conditions. Values are mean (SE). Correlations of AUC examining the relation between androstenedione and total testosterone were r = 0.66 (p = 0.08) and r = 0.79 (p = 0.01) for day 0 and day 28 responses respectively. AUC for androstenedione correlated strongly with DHEAS (r = 0.871, p<0.005) on day 0 of androstenedione supplementation but not on day 28 (r = 0.59, p = 0.1). In addition, AUC for total testosterone and DHEAS correlated significantly for both the day 0 (r = 0.89, p = 0.001) and day 28 (r = 0.87, p<0.005) supplemented conditions. ### Time 0 hormonal concentrations The repeated measures multivariate analysis of variance for time 0 hormonal concentrations showed significant differences (p = 0.01) between treatments (table 1). Time 0 androstenedione concentrations increased by 215% after 28 days of supplementation compared with placebo (p = 0.08). DHEAS (time 0) concentrations were significantly elevated after 28 days of supplementation (p = 0.0002). No significant differences in concentrations for total testosterone, oestradiol, or SHBG were found between treatments. Table 1 Baseline (time 0) concentrations for serum analytes for day 0, day 28, and placebo ### Time series Statistical analysis of androstenedione concentrations across time showed significant treatment (p<0.0001), time (p<0.0001), and time × treatment interaction (p<0.0001) effects (fig 4). Beginning at 90 minutes after administration, serum androstenedione concentrations were significantly higher for both the day 0 (p<0.05) and day 28 (p<0.05) supplemented trials compared with placebo. The only exception was minute 120 when only day 0 (p<0.005) but not day 28 (p = 0.07) concentrations were higher than placebo. For DHEAS, the repeated measures analysis of variance showed a significant treatment (p<0.005), time (p = 0.01), and treatment × time interaction (p = 0.001) effect. Figure 4 also depicts the DHEAS concentrations across time for all three conditions. As noted above, DHEAS concentrations were significantly higher at time 0 after 28 days of androstenedione supplementation compared with day 0 and placebo (p<0.001). This trend continued throughout the 180 minute sampling phase. DHEAS concentrations on day 0 were significantly higher than placebo (p = 0.009) only at 180 minutes after administration. Figure 4 Serum androstenedione (A) and dehydroepiandrosterone sulfate (DHEAS) (B) concentrations across time for day 0, day 28, and placebo trials. Values are mean (SE). a, Significantly different from placebo (p<0.05); b, significantly different from placebo (p<0.005); c, significantly different from placebo (p<0.0005); d, significantly different from day 0 (p<0.05); e, significantly different from day 0 (p<0.005); f, significantly different from day 0 (p<0.0005). ### Resting cardiovascular and anthropometric data, lipids, and POMS Body composition, body mass, and waist/hip ratio did not change significantly (p = 0.95) after four weeks of supplementation with androstenedione. Furthermore, no significant differences (p<0.05) were found for heart rate or systolic or diastolic blood pressure across the treatment conditions (p = 0.31). With regard to the lipid profile, concentrations across treatment conditions also did not change significantly (p = 0.80). Table 2 lists data for body composition, heart rate, blood pressure, and lipid profiles. Statistical analysis of total POMS scores showed no significant difference (p = 0.11) in total scores between day 0 (259 (19)) and day 28 (253 (16)) supplementation. Likewise, no significant differences were noted for any of the subcategories. Table 2 Day 0, day 28, and placebo values for anthropometric, cardiovascular, and lipid variables ## DISCUSSION Consistent with most studies,6,9–12 serum androstenedione concentrations increased significantly during the 180 minutes after ingestion of 200 mg androstenedione. An interesting finding from our study is that the day 0 pharmacokinetic trial yielded a greater AUC response than the day 28 and placebo conditions. Specifically, after 28 days of supplementation, the pharmacokinetic response (AUC) was 60% lower than on day 0 and no longer significantly greater than with placebo (p = 0.1). Androstenedione concentrations at time 0, however, were about threefold higher in the day 28 than day 0 (p = 0.07) and placebo (p = 0.08) conditions. This increase in baseline (time 0) androstenedione concentration was also consistent with other studies7,8,10,11 using doses of 200 and 300 mg/day. King et al,10 however, reported a decline in baseline androstenedione concentrations after 12 weeks of supplementation as compared with five weeks. In our study, the increase in time 0 androstenedione concentration, combined with the significant decline in the pharmacokinetic response after 28 days of supplementation suggests physiological compensatory mechanisms associated with the chronic administration of androstenedione. Potential sources of compensation include alterations in absorption, androgen production, receptor number, conversion, and clearance rate, which has been shown with testosterone.2,18,19 If the decrease in AUC for androstenedione was primarily a function of enhanced clearance, a greater clearance rate for integrated AUC would be expected after 28 days of supplementation. However, the clearance rate on day 28 (24.9 litres/h) was not significantly different from that on day 0 (26.1 litres/h). In addition, if a greater proportion of androstenedione in the day 28 condition were converted into other hormones, increased concentrations of testosterone and oestradiol would be expected. Although some studies reported conversion of androstenedione into total9,11 and free8,9 testosterone after administration of 200–300 mg/day androstenedione, others did not.6,10,11 It is difficult to interpret these findings because of differences in the age of subjects, length of supplementation, dose, dosing pattern, type of response measured (acute compared with chronic changes), and the specific method used to assess acute responses (AUC compared with change at different time points). These studies suggest that short term (0–7 days) supplementation with 200–300 mg/day androstenedione may produce acute increases in AUC for total and free testosterone in younger men,9,11 but not when analysed as changes across time points.6,9 Only one study reported a chronic increase in free testosterone concentration after four weeks of supplementation8 in younger men. With regard to middle aged men, 200–300 mg/day androstenedione administered for 4–8 weeks has been shown to temporarily elevate free and/or total testosterone levels.7,8 Broeder et al,7 however, reported that total and free testosterone concentrations returned to baseline after 12 weeks of supplementation. No other studies have examined the acute response to supplementation in older men. Our study used integrated AUC to assess the acute response to supplementation because it provides a better appreciation of the appearance, metabolism, and clearance of a hormone as contrasted with individual time points or total AUC. We found a slight (p = 0.23) increase in AUC for total testosterone in the day 0 condition, which was 143% and 148% greater than the placebo and day 28 trials respectively. In a similar pattern to androstenedione, the pharmacokinetic response for total testosterone in the day 28 condition decreased by 60% as compared with the day 0 condition. The strong correlations between androstenedione and total testosterone are consistent with this finding. Furthermore, time 0 values for total testosterone were not significantly elevated after four weeks of supplementation. Failure of androstenedione ingestion to produce a significant increase in total testosterone concentrations may be an integrated function of several metabolic regulatory areas. These areas include increased clearance rate for testosterone,2,3,18 declines in secretion of luteinising hormone,7 and conversion into other hormones such as oestradiol.5,20 In addition to changes in hormonal concentrations, androgen administration may alter the concentration of SHBG, the primary transport protein for testosterone, dihydrotestosterone, and to a lesser extent oestradiol.5 Physiological changes in the concentration of SHBG can alter clearance rate, and hence, the amount of free (non-protein) versus protein bound steroid.22 Plasma SHBG concentrations are sensitive to changes in the circulating androgen/oestrogen ratio and have been shown to increase with oestrogen and decrease with androgen administration.23–25 In our study, we found that acute ingestion of androstenedione decreased SHBG AUC on day 0 and day 28 of androstenedione supplementation by 58% and 50% respectively compared with placebo; however, neither of these changes were significant. Consistent with our findings, Wallace et al26 and Broeder et al7 did not report any significant changes in SHBG concentration. To the contrary, Leder et al11 reported a significant decrease in mean daily baseline SHBG concentrations after seven days of supplementation with both 100 and 300 mg/day. The physiological significance of changes in SHBG concentrations with supplementation is unclear. Studies suggest that non-protein bound testosterone may underrepresent the biologically active portion in plasma.27 Furthermore, Brown et al8 reported that supplementation can alter the concentration of dihydrotestosterone, which may have a higher binding affinity for SHBG than testosterone.28 In our study, plasma SHBG AUC did not correlate significantly with AUC for androgens such as androstenedione, total testosterone, and DHEAS, suggesting that changes in SHBG cannot be explained on the basis of the effect of these androgens. An interesting finding in our study was a change in hormonal concentrations upstream to androstenedione and testosterone. DHEAS concentrations increased by 84% at time 0 after 28 days of supplementation with androstenedione (p = 0.09). This pattern is consistent with those observed by Broeder et al.7 Also, as seen for androstenedione and total testosterone, the AUC response for DHEAS showed a tendency to decrease by the same proportion (58%) as observed for androstenedione and total testosterone after 28 days of supplementation. These changes in the relation between AUC for androstenedione and the DHEAS responses from day 0 to day 28 were significantly correlated (r = 0.86; p = 0.003). In addition, the strong correlations between AUC for total testosterone and DHEAS, as well as androstenedione and DHEAS, suggest that changes in these hormones are related. Most (90%) of the circulating DHEAS is secreted by the adrenal gland and then converted into more potent androgens.20 As dehydroepiandrosterone (DHEA) and DHEAS are produced upstream of androstenedione, changes in DHEAS concentrations may be related to alterations in adrenal secretion, metabolism, or excretion and/or enzyme activity. The addition of androstenedione, as well as testosterone, to tissue preparations from human testis has been shown to inhibit the Δ5,3β-hydroxysteroid dehydrogenase activity for dehydroepiandrosterone.29 Inhibition would tend to reduce the conversion of DHEA into androstenedione, thereby increasing serum concentrations of DHEA. As DHEA is readily and reversibly converted into its sulfate conjugate,20 DHEAS concentrations would also be expected to increase. Consistent with this finding, patients with Δ5,3β-hydroxysteroid dehydrogenase deficiency exhibit elevated plasma concentrations of DHEAS.20 ### Take home message Androstenedione supplementation in middle aged men did not produce significant increases in total testosterone but elevated the concentration of DHEAS, upstream of androstenedione. Furthermore, the acute pharmacokinetic response to supplementation was diminished after 28 days of supplementation. These findings raise health concerns associated with alteration of the endogenous hormonal profile. With regard to the effect of androstenedione supplementation on the lipid profile, no changes were noted during our study. However, a 13% reduction in HDL cholesterol (p = 0.21) was noted in our trial. Despite the observation that this was not “statistically” significant, similar reductions in HDL cholesterol have been reported in other studies in which 200–300 mg/day was administered.7,8,10 The dosing pattern (one dose of 200 mg/day v two doses of 100 mg) used in this study may not be adequate to produce significant changes in HDL cholesterol with only 28 days of supplementation. In addition, the statistical power (0.23) for this analysis was low, which increases the probability of a type II error. The lipid profile was a secondary outcome measure, while the hormonal responses to supplementation were the primary outcome measure. Thus future investigations using similar research designs should take this into account. Lastly, our study confirms those of others7,10,26 by showing that androstenedione does not affect anthropometric indices. These include percentage body fat, body mass, and waist/hip ratio. Neither does it appear to alter resting heart rate, blood pressure, or mood state. Two subjects did report an increase in mood swings and aggression/hostility and another reported an increase in hair growth after four weeks of androstenedione supplementation. Otherwise, there were no personal reports of other changes typically associated with steroid administration, such as acne or sex drive for either the androstenedione or placebo trials. Androgen/pro hormone supplements have not been shown to enhance performance, favourably alter body composition, or positively affect various parameters associated with good health in younger or middle aged men.6–8,10–12,26 Moreover, the potential for alterations in endogenous hormonal concentrations, including elevated concentrations of weaker androgens upstream of androstenedione is a concern as the long term effect in previously healthy men is unknown. Interestingly, the pharmacokinetic response to androstenedione supplementation is diminished after only 28 days of supplementation. Lastly, this study also provides no evidence that supplementation with 200 mg/day androstenedione will significantly elevate testosterone concentration. This study raises further concerns that androstenedione supplementation may lead to alterations in the hormonal profile that diminish the acute effects of supplementation and elevate baseline concentrations of hormones upstream of testosterone. View Abstract ## Request permissions If you wish to reuse any or all of this article please use the link below which will take you to the Copyright Clearance Center’s RightsLink service. You will be able to get a quick price and instant permission to reuse the content in many different ways.	2017-09-25 04:41:17	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.4224787652492523, "perplexity": 8297.663875842696}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1505818690318.83/warc/CC-MAIN-20170925040422-20170925060422-00394.warc.gz"}
https://www.gamedev.net/forums/topic/353383-which-scripting-language/	# Unity Which scripting language? This topic is 4645 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 looking into implementing a scripting language into fs2_open, which is the engine of the game Freespace 2 (now open-source.) FS2 uses table files like this: #Section $Name: Object name$Variable 1: 10 $Variable 2: moo ;;etc etc... #End Now, what I'd like to do is make it possible to use a scripting language, using one of those variables. EG with Python: #Section$Name: Object name $Variable 1: cmath.cos(35)$Trigger: { if conditional: do_something() } #End The preference of the modding community, however, is Lua. Which is where I run into a problem; Lua doesn't seem to let you precompile code chunks on runtime. Since some of the code chunks may be executed per frame, obviously I don't want it to be parsing the file or some text buffer in memory. In addition, I understand that Python has large memory requirements. So, wish list here for a scripting language (in no particular order): - Precompile code chunks - Allow for object-oriented programming - Small and fast - EASY TO LEARN for non-coders - Lets you use named arrays, eg Array['element'] - Lets you extend variables, functions, classes from C/++ - Is cross-platform (fs2_open runs on Win32, OSX, Linux, and Linux64) - JIT compile to machine code, as long as it doesn't interfere with cross-platformability Thanks. :) ##### Share on other sites Of course lua lets you precompile at runtime. Consider this code fragment: ontrigger = function() if active then print "I'm triggering" endendontrigger()ontrigger()ontrigger()ontrigger()ontrigger() In that fragment, the ontrigger command is compiled exactly once. If Lua doesn't seem to precompile stuff, it's only because the precompilation is completely transparent. ##### Share on other sites Indeed. Getting Lua to run a script happens in two stages. First, you load the script, at which point it will be compiled into a function, and second, you run that function. As long as you keep the function around to re-use, you won't have to compile it again. Secondly, for JIT compilation, you could take a look at LuaJIT. It may or may not speed things up though, depending on what type of code you're running (there are some statistics on the LuaJIT site - take a look and see if it's appropriate). For JIT compilation in Python, look at Psyco. I don't know how easy it is to set that up when you're got Python embedded in something though. Python is beefier than Lua, and includes more language features, eg support for classes and so on - you can write Object Oriented code in Lua, but you have a bit more framework to write yourself, since Lua doesn't provide 'classes' as such built-in; it just provides tables and metatables, and you build everything up from those. Lua and Python aren't the only options of course - you could check out Squirrel (a lot like Lua), AngelScript, Small, GameMonkey, or even something like Guile. John B ##### Share on other sites That's not exacly what I mean...I mean that, given a chunk of code, it may be executed at different places within the C code, or simply once a frame, or maybe multiple times per frame. AFAIK the Lua API only provides functions that will run script from a text file, or buffer. EG: $Name: Homing Rocket$Damage: 25 * current_speed Where "current_speed" is the speed that frame. It's not technically a function; but it still needs to be executed every frame that the weapon damage is needed. ##### Share on other sites Quote: Original post by WMCoolmonThat's not exacly what I mean...I mean that, given a chunk of code, it may be executed at different places within the C code, or simply once a frame, or maybe multiple times per frame.AFAIK the Lua API only provides functions that will run script from a text file, or buffer. No, as I said, the process happens in two stages. First you load from a text file or buffer (with lua_load), at which point it gets compiled into a function. Then, you call that function with lua_call (or one of the other call functions like lua_pcall). As long as you keep a copy of the function somewhere, you can keep calling it whenever you need it, without having to recompile everything. John B ##### Share on other sites In my inexperience with scripting languages, I see lua having the easiest syntax even for non coders. Squirrel, however, has native support for classes, and was made with lua's problems in minds. It, however, has a more C++'y syntax, though it's still easy to learn. Python is, like you've heard, very beefy with a lot of features, but very difficult to get going. In my opinion, squirrel is the best of both worlds. Like they said, you could check out other scripting languages, though I recommend squirrel. ##### Share on other sites See lua_loadfile() (loads chunk from disk as function onto the stack: can be set to a table and called later at will) and lua_dofile() (executes a chunk from disk (elements defined on root table will be available for later use)): See: Lua 5.0 Manual If your fellow programmers are set on using Lua, you should have no problem given your requirements (not clear whether Lua will work OK in a 64 bit environment (support appears to be coming in Lua 5.1)). However (IMO), Squirrel is easier to use (both script side and C/C++ side (binding/class handling)), and was developed as an improvement to Lua specifically for use in commercial game applications. Those with experience with Lua will have no problem using Squirrel. Lua compared to Squirrel. Both Squirrel and SqPlus compile and run with gcc. There is an ongoing effort to get Squirrel 100% compatible with 64 bit environments (see the Squirrel forum): it appears a few base typedefs must be changed and Squirrel and SqPlus will compile and run OK in 64 bit. [Edited by - John Schultz on October 29, 2005 1:12:22 PM] ##### Share on other sites (I apologize in advance if this post is hideous, I'm trying to finish quickly) I'm considering Lua 5.1 at the moment (just to learn, don't have any projects going). I liked Python a lot but, as you've definitely heard, it's beefy. But so far it seems that it's already "prettified" a lot of the solutions to problems I'm already starting to see in Lua (for instance, I much prefer Python strings to Lua strings, because there's quite a lot more functionality as soon as the builtins are imported). An advantage of Lua is that, if a feature you want doesn't exist (read: it exists, but you haven't found it yet), you can build it yourself. Python (I'll assume) is the same way, but is quite a bit more to read before you're really comfortable. I think Python looks prettier, as well, and, if you're trying to embed it, boost.python makes it much easier than with the native embedding API (?). Someone above mentioned LuaJIT, which I hadn't even heard of until a month or so ago, so I suppose you can look into that for JIT compiling. ...Did I mention that I think Lua is ugly? In a way, though, I like that Lua doesn't look too much like C/++ -- Python is a little harder to work with for me sometimes (but not by much) because of carry-over C-isms, which Lua's sometimes-foreign syntax avoids. Python has much better documentation, and it seems like builtin commands connotate well (the words that I would think to use for a situation often are, and I like that). Embedding python with boost.python is actually a little bit more enjoyable than embedding lua, because you avoid the "stack" idea. It's not a huge problem, though, because much of the work is done through the scripts regardless (both languages). And then, I never did find a good Python debugger. Admittedly, I didn't look very hard, and my code problems came from syntactical errors (again, the C-isms), which were easily solved with the interpreter's help. This post came out differently than I'd planned -- but I guess a personal comparison won't hurt too much. ##### Share on other sites To either of the two Johns: I understand the lua_load* functions, but how do I get something off the stack, or keep the stack from getting extremely large from loading minor chunks from everywhere? Edit: And the current status is that three of the principal coders are against it, but a number of modders seem interested, and the project head (who actually runs a game dev company) is definitely for it. The reasons being that there's a scripting language (SEXPs) implemented for mission scripting, which could conceivably be expanded to support floats and strings. But, it doesn't allow you to define new functions, doesn't support functions that both change something and return a value, doesn't allow classes, and doesn't support floating point values, as well as being a PITA to code for as it requires you to add code in 6 different places to add a function for it. It's also LISP-based, which at least one strongly feels is a better syntax than Lua. (I'd never heard of LISP until it was mentioned in passing ~3-4 months ago) But the biggest reason for their disagreement that they feel that any scripting language will be much slower and cause massive memory overhead or leaks than what reworking the SEXP system would do. The SEXP source code files can be found here (.cpp, .h): http://fs2source.warpcore.org/cgi-bin/cvsweb/cvsweb.cgi/fs2_open/code/parse/sexp.cpp?rev=2.179&content-type=text/x-cvsweb-markup http://fs2source.warpcore.org/cgi-bin/cvsweb/cvsweb.cgi/fs2_open/code/parse/sexp.h?rev=2.98&content-type=text/x-cvsweb-markup ##### Share on other sites Quote: Original post by WMCoolmonTo either of the two Johns: I understand the lua_load* functions, but how do I get something off the stack, or keep the stack from getting extremely large from loading minor chunks from everywhere? The entire contents of the file are returned as a single function (closure) on the stack. The closure (effectively a pointer; does not take up much "stack space"; the stack won't grow, as you'll store the result in a table somewhere, then pop the stack) can be called/saved, etc. I use this methodology in Squirrel to compile and load a script for later execution (and done in such a way to to put everything declared in the script file in a specific table (not the root table)): // Squirrel script:// ... try { game.init <- loadfile("levels/game.nut",true); } catch (e) { return 0; } // catch game.init(); // Everything in game.nut is defined in the table 'game'// ... game.update(); // Calls update(), defined in game.nut.// Or called from C/C++:// gameObject is a SquirrelObject (the 'game' table) res = SquirrelFunction<int>(gameObject,"update")(); You can do similar operations with Lua (see the Lua docs, perhaps google for Lua examples). 1. 1 2. 2 Rutin 22 3. 3 JoeJ 18 4. 4 5. 5 • 15 • 40 • 23 • 13 • 13 • ### Forum Statistics • Total Topics 631724 • Total Posts 3001903 ×	2018-07-17 16:00:09	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.26906776428222656, "perplexity": 2725.339096270922}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1531676589752.56/warc/CC-MAIN-20180717144908-20180717164908-00634.warc.gz"}
http://mathhelpforum.com/geometry/52613-please-solve-question.html	1. ## please solve this question please solve this question using heron's formula... The lengths of the sides of a triangle are 5cm, 12cm and 13cm.Find the length of perpendicular from the opposite vertex to the side whose length is 13cm. 2. Originally Posted by rickymylv please solve this question using heron's formula... The lengths of the sides of a triangle are 5cm, 12cm and 13cm.Find the length of perpendicular from the opposite vertex to the side whose length is 13cm. This is a right triangle since $13^2=5^2+12^2$ See diagram: The side of 5 is the geometric mean between x and 13. $\frac{x}{5}=\frac{5}{13}$ $13x=25$ $x=\frac{25}{13}$ The side 12 is the geometric mean between y and 13. $\frac{y}{12}=\frac{12}{13}$ $13y=144$ $x=\frac{144}{13}$ The length of the perpendicular (h) to the hypotenuse is the geometric mean between x and y. $\frac{\frac{25}{13}}{h}=\frac{h}{\frac{144}{13}}$ Can you finish up. Sorry, I misread your instructions. I did not use Heron's formula to solve this.....I'll see if someone else does. Otherwise, I'll be back. Okay, I'm back. I really don't know how Heron's formula fits into all this. Heron's formula is a formula for finding the area of any triangle knowing only the lengths of the sides. $A=\sqrt{s(s-a)(s-b)(s-c)}$, where a, b, and c are the sides and s is the semi-perimeter. That is, $s=\frac{a+b+c}{2}$ The perimeter of your triangle = 5 + 12 + 13 = 30. The semi-perimeter (s) = 15 Substituting into Heron's formula to find Area: $A=\sqrt{15(15-5)(15-12)(15-13)}=\sqrt{900}=30$ This could just as easily been found using $A=\frac{1}{2}bh$ using b = 12 and h = 5. But, I digress. Now, since you just found what the area was, you can use this to solve for your missing perpendicular. $A=\frac{1}{2}bh$ $30=\frac{1}{2}(13)h$ $60=13h$ $h=\frac{60}{13}$ All done!! , , , , , , , , , , , , , , # find length of perpendicular to vertex Click on a term to search for related topics.	2017-11-22 14:54:15	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 16, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8683457970619202, "perplexity": 517.5994985563261}, "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-2017-47/segments/1510934806609.33/warc/CC-MAIN-20171122141600-20171122161600-00540.warc.gz"}
https://www.zbmath.org/authors/?q=ai%3Azhang.yu.3	## Zhang, Yu Compute Distance To: Author ID: zhang.yu.3 Published as: Zhang, Yu; Zhang, Y.; zhang, Yu; Zhang, Yú Documents Indexed: 466 Publications since 1985, including 3 Books Co-Authors: 514 Co-Authors with 323 Joint Publications 15,428 Co-Co-Authors all top 5 ### Co-Authors 19 single-authored 16 Yang, Yidu 14 Zhang, Yanyan 12 Bi, Hai 9 Sarkar, Tapan Kumar 8 Zhang, Guanquan 7 Gui, Yuanxing 7 Hu, Hongchang 7 Jiang, Jack J. 7 Shen, Aiting 6 Tao, Chao 6 Wang, Jinhuan 5 Du, Gonghuan 5 Li, Shengjie 5 Yang, Hanchun 4 Cai, Jianfei 4 Duan, Pengfei 4 Geng, Jin-Ling 4 Hua, Changchun 4 Li, En-Kun 4 Li, Longsuo 4 Liu, Jianchang 4 McLeod, Angus Ian 4 Mei, Zicong 4 Nowak, David E. 4 Salazar-Palma, Magdalena 4 Ting, Sioweng 4 Wu, Jianxin 4 Xia, Tianyang 4 Zhao, Xunwang 3 Bleistein, Norman 3 Chen, Tao 3 Chen, Xiaowu 3 Cheng, Li 3 Fu, Ming 3 Jin, Hai 3 Jung, Baek Ho 3 Liao, Xiaofei 3 Liu, Xinsheng 3 Pang, Yicheng 3 Ren, Litong 3 Song, Yan 3 Tang, Jiafu 3 Teng, Zhi-dong 3 Tong, Minglei 3 Wang, Honghai 3 Wu, Lihua 3 Xie, Shousheng 3 Zhang, Ledi 2 Bai, Zhengyao 2 Bakr, Mohamed H. 2 Bose, Amitabha 2 Cai, Wenyu 2 Chen, Guanrong 2 Chioncel, L. 2 De, Arijit 2 Deng, Yuxin 2 Deng, Zhidong 2 Deng, Zichen 2 Do, Minh N. 2 Draayer, Jerry P. 2 Eisenbach, Markus 2 Er, X. Z. 2 Evans, David John 2 Fang, Xiangzhong 2 Feng, Xinyu 2 Fu, Xianlong 2 Gao, Hongya 2 Gao, Lingyun 2 Gao, Shujing 2 Garcia-Doñoro, Daniel 2 Govindarajan, Lakshmi Narasimhan 2 Guan, Xinping 2 Guo, Lei 2 Guo, Yuanbo 2 Guo, Zhichang 2 He, Jingsong 2 Hou, Chengmin 2 Hu, Jianping 2 Hu, Weipeng 2 Huang, Li 2 Ji, Zhong 2 Jin, Xinyu 2 Katebi, M. Reza 2 Khodadadi, J. M. 2 Li, Luyu 2 Li, Quanlin 2 Li, Songsong 2 Li, Yafeng 2 Li, Yanling 2 Li, Yin 2 Li, Yong 2 Lu, Jiangbo 2 Luo, Ziqiang 2 Ma, Xingpo 2 Moon, Hongsik 2 Nadim, Farzan 2 Ni, Xiaodong 2 Pan, Feng 2 Qi, Chuanda 2 Shen, Jiang ...and 605 more Co-Authors all top 5 ### Serials 8 Physics Letters. 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Natural Science Edition 2 Chinese Journal of Engineering Mathematics 2 Communication on Applied Mathematics and Computation 2 Fuzzy Systems and Mathematics 2 Open Mathematics 1 Acta Mechanica 1 Applicable Analysis 1 Astrophysics and Space Science 1 Biological Cybernetics ...and 155 more Serials all top 5 ### Fields 66 Computer science (68-XX) 64 Partial differential equations (35-XX) 63 Numerical analysis (65-XX) 53 Operations research, mathematical programming (90-XX) 48 Systems theory; control (93-XX) 44 Statistics (62-XX) 42 Fluid mechanics (76-XX) 37 Information and communication theory, circuits (94-XX) 35 Biology and other natural sciences (92-XX) 27 Ordinary differential equations (34-XX) 25 Mechanics of deformable solids (74-XX) 24 Relativity and gravitational theory (83-XX) 22 Optics, electromagnetic theory (78-XX) 22 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 19 Calculus of variations and optimal control; optimization (49-XX) 19 Probability theory and stochastic processes (60-XX) 16 Dynamical systems and ergodic theory (37-XX) 16 Quantum theory (81-XX) 14 Statistical mechanics, structure of matter (82-XX) 13 Mathematical logic and foundations (03-XX) 11 Operator theory (47-XX) 10 Classical thermodynamics, heat transfer (80-XX) 9 Mechanics of particles and systems (70-XX) 9 Astronomy and astrophysics (85-XX) 8 Geophysics (86-XX) 7 Difference and functional equations (39-XX) 6 Nonassociative rings and algebras (17-XX) 5 Real functions (26-XX) 4 Group theory and generalizations (20-XX) 4 Differential geometry (53-XX) 3 Order, lattices, ordered algebraic structures (06-XX) 3 Associative rings and algebras (16-XX) 3 Category theory; homological algebra (18-XX) 3 Functions of a complex variable (30-XX) 3 Potential theory (31-XX) 2 Topological groups, Lie groups (22-XX) 2 Several complex variables and analytic spaces (32-XX) 2 Special functions (33-XX) 2 Harmonic analysis on Euclidean spaces (42-XX) 2 Integral equations (45-XX) 2 Functional analysis (46-XX) 1 General and overarching topics; collections (00-XX) 1 Combinatorics (05-XX) 1 General algebraic systems (08-XX) 1 Approximations and expansions (41-XX) 1 Integral transforms, operational calculus (44-XX) 1 General topology (54-XX) 1 Algebraic topology (55-XX) 1 Global analysis, analysis on manifolds (58-XX) ### Citations contained in zbMATH Open 194 Publications have been cited 994 times in 561 Documents Cited by Year Adaptive output control of nonlinear systems with uncertain dead-zone nonlinearity. Zbl 1366.93306 Zhou, J.; Wen, C.; Zhang, Y. 2006 Removing the cell resonance error in the multiscale finite element method via a Petrov-Galerkin formulation. Zbl 1085.65109 Hou, Thomas Y.; Wu, Xiao-Hui; Zhang, Yu 2004 Stability of impulsive neural networks with time delays. Zbl 1195.93122 Zhang, Yu; Sun, Jitao 2005 Continuity of the solution mappings to parametric generalized strong vector equilibrium problems. Zbl 1290.90076 Li, S. J.; Liu, H. M.; Zhang, Y.; Fang, Z. M. 2013 Coexisting multiple attractors and riddled basins of a memristive system. Zbl 1390.34158 Wang, Guangyi; Yuan, Fang; Chen, Guanrong; Zhang, Yu 2018 Multi-exponential wave solutions to two extended Jimbo-Miwa equations and the resonance behavior. Zbl 1448.35459 Xu, Hao-Nan; Ruan, Wei-Yong; Zhang, Yu; Lü, Xing 2020 Quasinormal modes of gravitational perturbation around a Schwarzschild Black hole surrounded by quintessence. Zbl 1133.83367 Zhang, Yu; Gui, Y. X. 2006 Flux approximation to the isentropic relativistic Euler equations. Zbl 1398.35235 Yang, Hanchun; Zhang, Yu 2016 Generalized prolate spheroidal wave functions for offset linear canonical transform in Clifford analysis. Zbl 1269.30053 Kou, K.; Morais, J.; Zhang, Y. 2013 A quadratic $$C^0$$ interior penalty method for an elliptic optimal control problem with state constraints. Zbl 1282.65074 Brenner, S. C.; Sung, L.-Y.; Zhang, Y. 2014 Distributed filtering under false data injection attacks. Zbl 1415.93271 Yang, Wen; Zhang, Yu; Chen, Guanrong; Yang, Chao; Shi, Ling 2019 Flux-approximation limits of solutions to the relativistic Euler equations for polytropic gas. Zbl 1338.35347 Zhang, Yu; Yang, Hanchun 2016 A new ZK-ILW equation for algebraic gravity solitary waves in finite depth stratified atmosphere and the research of squall lines formation mechanism. Zbl 1416.86008 Guo, Min; Zhang, Yu; Wang, Man; Chen, Yaodeng; Yang, Hongwei 2018 Exact null controllability of non-autonomous functional evolution systems with nonlocal conditions. Zbl 1299.34255 Fu, Xianlong; Zhang, Yu 2013 On the nonlinear difference equation system $$x_{n+1}=A+y_{n - m}/x_n,y_{n+1}=A+x_{n - m}/y_n$$. Zbl 1152.39312 Zhang, Yu; Yang, Xiaofan; Evans, David J.; Zhu, Ce 2007 Single-machine ready times scheduling with group technology and proportional linear deterioration. Zbl 1427.90164 Xu, Yang-Tao; Zhang, Yu; Huang, Xue 2014 On the system of rational difference equations $$x_{n}=A+ \frac {1}{y_{n-p}}, y_{n}=A+ \frac {y_{n-1}}{x_{n-r}y_{n-s}}$$. Zbl 1096.39015 Zhang, Yu; Yang, Xiaofan; Megson, Graham M.; Evans, David J. 2006 Permanence criteria for general delayed discrete nonautonomous $$n$$-species Kolmogorov systems and its applications. Zbl 1189.39017 Teng, Zhidong; Zhang, Yu; Gao, Shujing 2010 Minimax problems for set-valued mappings. Zbl 1237.49010 Zhang, Y.; Li, S. J.; Zhu, S. K. 2012 A note on the lower semicontinuity of efficient solutions for parametric vector equilibrium problems. Zbl 1264.90156 Zhang, W. Y.; Fang, Z. M.; Zhang, Y. 2013 Yang-Mills condensate dark energy coupled with matter and radiation. Zbl 1126.83016 Zhang, Y.; Xia, T. Y.; Zhao, W. 2007 MHD instability in differentially-rotating cylindric flows. Zbl 1038.85003 Rüdiger, G.; Zhang, Y. 2001 Minimax theorems for scalar set-valued mappings with nonconvex domains and applications. Zbl 1281.49023 Zhang, Y.; Li, S. J. 2013 Affine-periodic solutions for dissipative systems. Zbl 1303.34033 Zhang, Yu; Yang, Xue; Li, Yong 2013 Routing optimization with time windows under uncertainty. Zbl 1412.90100 Zhang, Yu; Baldacci, Roberto; Sim, Melvyn; Tang, Jiafu 2019 Constant width simultaneous confidence bands in multiple linear regression with predictor variables constrained in intervals. Zbl 1074.62044 Liu, W.; Jamshidian, M.; Zhang, Y.; Bretz, F. 2005 Rational solutions for the Fokas system. Zbl 1330.35386 Rao, Ji-Guang; Wang, Li-Hong; Zhang, Yu; He, Jing-Song 2015 Quasinormal modes of a Schwarzschild black hole surrounded by free static spherically symmetric quintessence: electromagnetic perturbations. Zbl 1129.83312 Zhang, Yu; Gui, Yuanxing; Yu, Fei; Li, FengLin 2007 Attitude control for part actuator failure of agile small satellite. Zbl 1271.70078 Zhang, J. R.; Rachid, A.; Zhang, Y. 2008 $$H_ \infty$$ robust control design for dynamic ship positioning. Zbl 0875.93341 Katebi, M. R.; Grimble, M. J.; Zhang, Y. 1997 Existence of solutions for neutral integrodifferential equations with nonlocal conditions. Zbl 1259.34076 Fu, Xianlong; Gao, Yan; Zhang, Yu 2012 Breaking a chaotic image encryption algorithm based on perceptron model. Zbl 1254.68118 Zhang, Yu; Li, Chengqing; Li, Qin; Zhang, Dan; Shu, Shi 2012 Effects of buoyancy-driven convection on melting within spherical containers. Zbl 0992.76085 2001 On the numerical integration of a class of pressure-dependent plasticity models with mixed hardening. Zbl 0825.73820 Lee, J. H.; Zhang, Y. 1991 2-loop quantum Yang-Mills condensate as dark energy. Zbl 1246.83277 Xia, T. Y.; Zhang, Y. 2007 Determining the effects of surface elasticity and surface stress by measuring the shifts of resonant frequencies. Zbl 1348.74274 Zhang, Y.; Zhuo, L. J.; Zhao, H. S. 2013 Analysis of concrete fracture using a novel cohesive crack method. Zbl 1201.74288 Dong, Y.; Wu, S.; Xu, S. S.; Zhang, Y.; Fang, S. 2010 Exact solutions for a new coupled MKdV equations and a coupled KdV equations. Zbl 0994.35104 Cao, Dong Bo; Yan, Jia Ren; Zhang, Yu 2002 Green QFD-II: A life cycle approach for environmentally conscious manufacturing by integrating LCA and LCC into QFD matrices. Zbl 0951.90550 Zhang, Y.; Wang, H.-P.; Zhang, C. 1999 The efficiency of pooling mRNA in microarray experiments. Zbl 1154.62395 Kendziorski, C. M.; Zhang, Y.; Lan, H.; Attie, A. D. 2003 A limit theorem for matching random sequences allowing deletions. Zbl 0855.62012 Zhang, Yu 1995 Minimax inequalities for set-valued mappings. Zbl 1255.49014 Zhang, Y.; Li, S. J.; Li, M. H. 2012 Second order sufficient optimality conditions in vector optimization. Zbl 1283.90036 E., Ning; Song, Wen; Zhang, Yu 2012 A two-scale discretization scheme for mixed variational formulation of eigenvalue problems. Zbl 1246.65220 Yang, Yidu; Jiang, Wei; Zhang, Yu; Wang, Wenjun; Bi, Hai 2012 Asymptotic normality of DHD estimators in a partially linear model. Zbl 1349.62132 Hu, Hongchang; Zhang, Yu; Pan, Xiong 2016 Multi-symplectic method for peakon-antipeakon collision of quasi-Degasperis-Procesi equation. Zbl 1352.65648 Hu, Weipeng; Deng, Zichen; Zhang, Yu 2014 Partial autocorrelation parametrization for subset autoregression. Zbl 1111.62080 McLeod, A. I.; Zhang, Y. 2006 Residual power series method for time-fractional Schrödinger equations. Zbl 06816043 Zhang, Yu; Kumar, Amit; Kumar, Sunil; Baleanu, Dumitru; Yang, Xiao-Jun 2016 A type of adaptive $$C^0$$ non-conforming finite element method for the Helmholtz transmission eigenvalue problem. Zbl 1441.78026 Yang, Yidu; Zhang, Yu; Bi, Hai 2020 Pressure and flux-approximation to the isentropic relativistic Euler equations for modified Chaplygin gas. Zbl 1419.76727 Yang, Hanchun; Zhang, Yu 2019 On stochastic gradient Langevin dynamics with dependent data streams in the logconcave case. Zbl 1475.60156 Barkhagen, M.; Chau, N. H.; Moulines, É.; Rásonyi, M.; Sabanis, S.; Zhang, Y. 2021 The transient heat conduction MPM and GIMP applied to isotropic materials. Zbl 1402.80008 Gu, X. Y.; Dong, Chun Ying; Cheng, T.; Zhang, Y.; Bai, Yang 2016 An edge-based SIR model for sexually transmitted diseases on the contact network. Zbl 1397.92696 Yan, Shuixian; Zhang, Yu; Ma, Junling; Yuan, Sanling 2018 Analysis of tooth contact and load distribution of helical gears with crossed axes. Zbl 1049.70570 Zhang, Y.; Fang, Z. 1999 Decoding digital information from the cascaded heterogeneous chaotic systems. Zbl 1067.94503 Tao, Chao; Du, Gonghuan; Zhang, Yu 2003 Control of time-periodic systems via symbolic computation with application to chaos control. Zbl 1147.93336 Sinha, S. C.; Gourdon, E.; Zhang, Y. 2005 True amplitude wave equation migration arising from true amplitude one-way wave equations. Zbl 1048.35118 Zhang, Yu; Zhang, Guanquan; Bleistein, Norman 2003 Ky Fan minimax inequalities for set-valued mappings. Zbl 1273.49012 Zhang, Yu; Li, Sheng-Jie 2012 An exact analytic spectrum of relic gravitational waves in an accelerating universe. Zbl 1104.83022 Zhang, Y.; Er, X. Z.; Xia, T. Y.; Zhao, W.; Miao, H. X. 2006 Delta-shocks and vacuums in the relativistic Euler equations for isothermal fluids with the flux approximation. Zbl 1406.76093 Zhang, Yu; Zhang, Yanyan 2019 Riemann problems for a class of coupled hyperbolic systems of conservation laws with a source term. Zbl 1415.35197 Zhang, Yu; Zhang, Yanyan 2019 Local well-posedness of KP-I initial value problem on torus in the Besov space. Zbl 1342.35078 Zhang, Yu 2016 Image denoising using SVM classification in nonsubsampled contourlet transform domain. Zbl 1320.68220 Wang, Xiang-Yang; Yang, Hong-Ying; Zhang, Yu; Fu, Zhong-Kai 2013 Eigenanalysis and continuum modelling of pre-twisted repetitive beam-like structures. Zbl 1121.74382 Stephen, N. G.; Zhang, Y. 2006 Fitting model equations to time series using chaos synchronization. Zbl 1123.37325 Tao, Chao; Zhang, Yu; Du, Gonghuan; Jiang, Jack J. 2004 Computer algebra derivation of the bias of linear estimators of autoregressive models. Zbl 1115.62093 Zhang, Y.; McLeod, A. I. 2006 Consistent parameter estimation of systems disturbed by correlated noise. Zbl 0876.93098 Zhang, Y.; Lie, T. T.; Soh, C. B. 1997 Fitting MA($$q$$) models in the closed invertible region. Zbl 1094.62118 Zhang, Y.; Mcleod, A. I. 2006 Global minimization of non-smooth unconstrained problems with filled function. Zbl 1268.90056 Wang, W. X.; Shang, Y. L.; Zhang, L. S.; Zhang, Y. 2013 Neuro-sliding-mode control of flexible-link manipulators based on singularly perturbed model. Zbl 1212.93041 Zhang, Yu; Yang, Tangwen; Sun, Zengqi 2009 The computational SLR: a logic for reasoning about computational indistinguishability. Zbl 1200.94052 Zhang, Yu 2010 Determination of the unique orientation of two bodies connected by a ball-and-socket joint from four measured displacements. Zbl 0952.70005 Zhang, Yu; Crane, Carl D. III; Duffy, Joseph 1998 Studying vocal fold vibrations in Parkinson’s disease with a nonlinear model. Zbl 1144.37426 Zhang, Yu; Jiang, Jack; Rahn, Douglas A. III 2005 Wavelet method for boundary integral equations. Zbl 0952.65095 Zhang, Pinwen; Zhang, Yu 2000 Reasoning about optimistic concurrency using a program logic for history. Zbl 1287.03071 Fu, Ming; Li, Yong; Feng, Xinyu; Shao, Zhong; Zhang, Yu 2010 Fenchel-Lagrange duality for DC programs with composite functions. Zbl 1327.90223 Sun, Xiang-Kai; Guo, Xiao-Le; Zhang, Yu 2015 Dynamic data-driven finite element models for laser treatment of cancer. Zbl 1114.92043 Oden, J. T.; Diller, K. R.; Bajaj, C.; Browne, J. C.; Hazle, J.; Babuška, I.; Bass, J.; Biduat, L.; Demkowicz, L.; Elliott, A.; Feng, Y.; Fuentes, D.; Prudhomme, S.; Rylander, M. N.; Stafford, R. J.; Zhang, Y. 2007 Robust stability and $$H_{\infty }$$ control of discrete-time uncertain impulsive systems with time-varying delay. Zbl 1345.93130 Zhang, Yu 2016 Equilibrium pricing in an M/G/1 retrial queue with reserved idle time and setup time. Zbl 1480.90109 Zhang, Yu; Wang, Jinting 2017 Non-conforming Crouzeix-Raviart element approximation for Stekloff eigenvalues in inverse scattering. Zbl 1472.65137 Yang, Yidu; Zhang, Yu; Bi, Hai 2020 Construction and application of algebraic dual polynomial representations for finite element methods on quadrilateral and hexahedral meshes. Zbl 07362357 Jain, V.; Zhang, Y.; Palha, A.; Gerritsma, M. 2021 Two-grid discretizations and a local finite element scheme for a non-selfadjoint Stekloff eigenvalue problem. Zbl 1453.65402 Bi, Hai; Zhang, Yu; Yang, Yidu 2020 Stochastic stability of discrete-time Markovian jump delay neural networks with impulses and incomplete information on transition probability. Zbl 1296.93202 Zhang, Yu 2013 Vanishing viscosity limit for Riemann solutions to a class of non-strictly hyperbolic systems. Zbl 1395.35145 Zhang, Yanyan; Zhang, Yu 2018 Output feedback control for stochastic nonlinear time delay systems using dynamic gain technique. Zbl 1393.93115 Li, Yafeng; Zhang, Liuliu; Hua, Changchun; Zhang, Yu; Guan, Xinping 2018 Extended Farkas’s lemmas and strong dualities for conic programming involving composite functions. Zbl 1390.90440 Fang, Dong Hui; Zhang, Y. 2018 $$H_ \infty$$ control analysis and design for nonlinear systems. Zbl 0825.93223 Katebi, M. R.; Zhang, Y. 1995 Dirac quasinormal modes of Reissner-Nödstrom black hole surrounded by quintessence. Zbl 1222.81179 Wang, Chun-Yan; Zhang, Yu; Gui, Yuan-Xing; Lü, Jian-Bo 2010 Multigrid discretization and iterative algorithm for mixed variational formulation of the eigenvalue problem of electric field. Zbl 1256.78001 Yang, Yidu; Zhang, Yu; Bi, Hai 2012 Faster ARMA maximum likelihood estimation. Zbl 1452.62665 McLeod, A. I.; Zhang, Y. 2008 Large deflection behavior of simply supported laminated panels under in- plane loading. Zbl 0571.73071 Zhang, Y.; Matthews, F. L. 1985 Pseudosteady-state natural convection inside spherical containers partially filled with a porous medium. Zbl 0974.76082 1999 A calculus for game-based security proofs. Zbl 1287.68023 Nowak, David; Zhang, Yu 2010 Conic relaxations for power system state estimation with line measurements. Zbl 07044984 2018 An efficient analytical method for solving local fractional nonlinear PDEs arising in mathematical physics. Zbl 1446.35260 Zhang, Yu; Yang, Xiao-Jun 2016 Weakly supervised fine-grained categorization with part-based image representation. Zbl 1408.94828 Zhang, Yu; Wei, Xiu-Shen; Wu, Jianxin; Cai, Jianfei; Lu, Jiangbo; Nguyen, Viet-Anh; Do, Minh N. 2016 A new algorithm for contact detection between spherical particle and triangulated mesh boundary in discrete element method simulations. Zbl 1352.74364 Hu, Li; Hu, Guo Ming; Fang, Z. Q.; Zhang, Y. 2013 Score tests for both extra zeros and extra ones in binomial mixed regression models. Zbl 1332.62251 Deng, Dianliang; Zhang, Yu 2015 Fast and accurate approximation to significance tests in genome-wide association studies. Zbl 1229.62150 Zhang, Yu; Liu, Jun S. 2011 Generation and control of striped attractors of Rössler systems with feedback. Zbl 1129.93495 Chen, Qingfei; Zhang, Yu; Hong, Yiguang 2007 On stochastic gradient Langevin dynamics with dependent data streams in the logconcave case. Zbl 1475.60156 Barkhagen, M.; Chau, N. H.; Moulines, É.; Rásonyi, M.; Sabanis, S.; Zhang, Y. 2021 Construction and application of algebraic dual polynomial representations for finite element methods on quadrilateral and hexahedral meshes. Zbl 07362357 Jain, V.; Zhang, Y.; Palha, A.; Gerritsma, M. 2021 Multi-objective optimization based algorithms for solving mixed integer linear minimum multiplicative programs. Zbl 07350566 Mahmoodian, Vahid; Charkhgard, Hadi; Zhang, Yu 2021 Concentration and cavitation in the vanishing pressure limit of solutions to a simplified isentropic relativistic Euler equations. Zbl 1458.35322 Zhang, Yu; Pang, Yicheng 2021 Simultaneous inversion of the potential term and the fractional orders in a multi-term time-fractional diffusion equation. Zbl 1462.35469 Sun, L. L.; Li, Y. S.; Zhang, Y. 2021 Efficient reliability analysis with a CDA-based dimension-reduction model and polynomial chaos expansion. Zbl 07337751 Zhang, Yu; Xu, Jun 2021 Concentration in the zero-exponent limit of solutions to the isentropic Euler equations for extended Chaplygin gas. Zbl 07367958 Zhang, Yu; Zhang, Yanyan; Wang, Jinhuan 2021 Multi-exponential wave solutions to two extended Jimbo-Miwa equations and the resonance behavior. Zbl 1448.35459 Xu, Hao-Nan; Ruan, Wei-Yong; Zhang, Yu; Lü, Xing 2020 A type of adaptive $$C^0$$ non-conforming finite element method for the Helmholtz transmission eigenvalue problem. Zbl 1441.78026 Yang, Yidu; Zhang, Yu; Bi, Hai 2020 Non-conforming Crouzeix-Raviart element approximation for Stekloff eigenvalues in inverse scattering. Zbl 1472.65137 Yang, Yidu; Zhang, Yu; Bi, Hai 2020 Two-grid discretizations and a local finite element scheme for a non-selfadjoint Stekloff eigenvalue problem. Zbl 1453.65402 Bi, Hai; Zhang, Yu; Yang, Yidu 2020 Constructions of locally repairable codes with multiple recovering sets via rational function fields. Zbl 1433.94118 Jin, Lingfei; Kan, Haibin; Zhang, Yu 2020 Mixed methods for the elastic transmission eigenvalue problem. Zbl 1447.65166 Yang, Yidu; Han, Jiayu; Bi, Hai; Li, Hao; Zhang, Yu 2020 Exact solution of the extended dimer Bose-Hubbard model with multi-body interactions. 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Zbl 1411.35203 Zhang, Yu; Zhang, Yanyan; Wang, Jinhuan 2018 Viscous limits for a Riemannian problem to a class of systems of conservation laws. Zbl 1421.35215 Zhang, Yanyan; Zhang, Yu 2018 Finite-time stabilization for a class of nonlinear systems via optimal control. Zbl 07316178 2018 Structure of geodesics in the regular Hayward black hole space-time. Zbl 1398.83051 Hu, Jian-Ping; Zhang, Yu; Shi, Li-Li; Duan, Peng-Fei 2018 Security analysis of measurement-device-independent quantum key distribution in collective-rotation noisy environment. Zbl 1387.81188 Li, Na; Zhang, Yu; Wen, Shuang; Li, Lei-lei; Li, Jian 2018 A two-grid discretization scheme of non-conforming finite elements for transmission eigenvalues. Zbl 1409.65099 Wang, Shixi; Bi, Hai; Zhang, Yu; Yang, Yidu 2018 Carleson measures for the generalized Schrödinger operator. Zbl 1464.35073 Qi, S.; Liu, Y.; Zhang, Y. 2018 Equilibrium pricing in an M/G/1 retrial queue with reserved idle time and setup time. 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Zbl 1408.94828 Zhang, Yu; Wei, Xiu-Shen; Wu, Jianxin; Cai, Jianfei; Lu, Jiangbo; Nguyen, Viet-Anh; Do, Minh N. 2016 Unsupervised 3D shape segmentation and co-segmentation via deep learning. Zbl 1417.65094 Shu, Zhenyu; Qi, Chengwu; Xin, Shiqing; Hu, Chao; Wang, Li; Zhang, Yu; Liu, Ligang 2016 Rogue waves in a resonant erbium-doped fiber system with higher-order effects. Zbl 1410.78013 Zhang, Yu; Li, Chuanzhong; He, Jingsong 2016 Stability of numerical method for semi-linear stochastic pantograph differential equations. Zbl 1382.65025 Zhang, Yu; Li, Longsuo 2016 Multisymplectic method for the Camassa-Holm equation. Zbl 1419.35172 Zhang, Yu; Deng, Zi-Chen; Hu, Wei-Peng 2016 Equilibrium pricing strategies in retrial queueing systems with complementary services. Zbl 1465.90024 Zhang, Yu; Wang, Jinting; Wang, Fang 2016 Central limit theorems for AANA random sequences. Zbl 1363.60021 Zhang, Yu; Hu, Hongchang; Zeng, Zhen 2016 Analyzing gene expression time-courses based on multi-resolution shape mixture model. Zbl 1348.92110 Li, Ying; He, Ye; Zhang, Yu 2016 Strong consistency of $$M$$-estimation in linear regression model with AANA errors. Zbl 1363.62072 Hu, Hongchang; Zhang, Yu 2016 Rational solutions for the Fokas system. Zbl 1330.35386 Rao, Ji-Guang; Wang, Li-Hong; Zhang, Yu; He, Jing-Song 2015 Fenchel-Lagrange duality for DC programs with composite functions. Zbl 1327.90223 Sun, Xiang-Kai; Guo, Xiao-Le; Zhang, Yu 2015 Score tests for both extra zeros and extra ones in binomial mixed regression models. Zbl 1332.62251 Deng, Dianliang; Zhang, Yu 2015 New result on PID controller design of LTI systems via dominant eigenvalue assignment. Zbl 1329.93074 Liu, Jianchang; Wang, Honghai; Zhang, Yu 2015 Controller design for delay systems via eigenvalue assignment – on a new result in the distribution of quasi-polynomial roots. Zbl 1335.93054 Wang, Honghai; Liu, Jianchang; Yang, Feisheng; Zhang, Yu 2015 Semi-analytical finite strip transfer matrix method for buckling analysis of rectangular thin plates. Zbl 1394.74104 Yao, Li-Ke; He, Bin; Zhang, Yu; Zhou, Wei 2015 Computational fluid dynamics-discrete element method analysis of the onset of scour around subsea pipelines. Zbl 1443.76093 Zhang, Y.; Zhao, M.; Kwok, K. C. S.; Liu, M. M. 2015 An approximate framework for quantum transport calculation with model order reduction. Zbl 1351.78015 Chen, Quan; Li, Jun; Yam, Chiyung; Zhang, Yu; Wong, Ngai; Chen, Guanhua 2015 Formal security proofs with minimal fuss: implicit computational complexity at work. Zbl 1310.94164 Nowak, David; Zhang, Yu 2015 Generalized holographic Ricci dark energy and generalized second law of thermodynamics in Bianchi type I universe. Zbl 1329.83222 Li, En-Kun; Zhang, Yu; Geng, Jin-Ling; Duan, Peng-Fei 2015 A quadratic $$C^0$$ interior penalty method for an elliptic optimal control problem with state constraints. Zbl 1282.65074 Brenner, S. C.; Sung, L.-Y.; Zhang, Y. 2014 Single-machine ready times scheduling with group technology and proportional linear deterioration. Zbl 1427.90164 Xu, Yang-Tao; Zhang, Yu; Huang, Xue 2014 Multi-symplectic method for peakon-antipeakon collision of quasi-Degasperis-Procesi equation. Zbl 1352.65648 Hu, Weipeng; Deng, Zichen; Zhang, Yu 2014 An immersed boundary method for simulation of inviscid compressible flows. Zbl 1455.65187 Zhang, Y.; Zhou, C. H. 2014 Using pulsar timing arrays and the quantum normalization condition to constrain relic gravitational waves. Zbl 1470.83064 Tong, M. L.; Zhang, Y.; Zhao, W.; Liu, J. Z.; Zhao, C. S.; Yang, T. G. 2014 Orbital dynamics of the gravitational field of stringy black holes. Zbl 1297.83025 Zhang, Yu; Geng, Jin-Ling; Li, En-Kun 2014 Generalized Ky Fan minimax inequalities for set-valued mappings. Zbl 1301.49018 Zhang, Yu; Li, S. J. 2014 Stability of exponential Euler method for stochastic systems under Poisson white noise excitations. Zbl 1310.34106 Li, Longsuo; Zhang, Yu 2014 Minimax problems for set-valued mappings with set optimization. Zbl 1315.49004 Chen, Tao; Zhang, Yu 2014 Modelling rock fracturing caused by magma intrusion using the smoothed particle hydrodynamics method. Zbl 1392.86007 Das, R.; Zhang, Y.; Schaubs, P.; Cleary, P. W. 2014 On dispersive propagation of surface waves in patchy saturated porous media. Zbl 1456.76031 Zhang, Yu; Xu, Yixian; Xia, Jianghai; Ping, Ping; Zhang, Shuangxi 2014 An SIQR epidemic model with vertical transmission and impulsive vaccination. Zbl 1324.92060 Song, Yan; Liu, Wei; Zhang, Yu 2014 Quasinormal modes of massless scalar field perturbation of a Reissner-Nordström de Sitter black hole with a global monopole. Zbl 1291.83168 Zhang, Yu; Li, En-Kun; Geng, Jin-Ling 2014 Floyd-A algorithm solving the least-time itinerary planning problem in urban scheduled public transport network. Zbl 1407.90190 Zhang, Yu; Tang, Jiafu; Lv, Shimeng; Luo, Xinggang 2014 Continuity of the solution mappings to parametric generalized strong vector equilibrium problems. Zbl 1290.90076 Li, S. J.; Liu, H. M.; Zhang, Y.; Fang, Z. M. 2013 Generalized prolate spheroidal wave functions for offset linear canonical transform in Clifford analysis. Zbl 1269.30053 Kou, K.; Morais, J.; Zhang, Y. 2013 Exact null controllability of non-autonomous functional evolution systems with nonlocal conditions. Zbl 1299.34255 Fu, Xianlong; Zhang, Yu 2013 A note on the lower semicontinuity of efficient solutions for parametric vector equilibrium problems. Zbl 1264.90156 Zhang, W. Y.; Fang, Z. M.; Zhang, Y. 2013 Minimax theorems for scalar set-valued mappings with nonconvex domains and applications. Zbl 1281.49023 Zhang, Y.; Li, S. J. 2013 Affine-periodic solutions for dissipative systems. Zbl 1303.34033 Zhang, Yu; Yang, Xue; Li, Yong 2013 Determining the effects of surface elasticity and surface stress by measuring the shifts of resonant frequencies. Zbl 1348.74274 Zhang, Y.; Zhuo, L. J.; Zhao, H. S. 2013 Image denoising using SVM classification in nonsubsampled contourlet transform domain. Zbl 1320.68220 Wang, Xiang-Yang; Yang, Hong-Ying; Zhang, Yu; Fu, Zhong-Kai 2013 Global minimization of non-smooth unconstrained problems with filled function. Zbl 1268.90056 Wang, W. X.; Shang, Y. L.; Zhang, L. S.; Zhang, Y. 2013 Stochastic stability of discrete-time Markovian jump delay neural networks with impulses and incomplete information on transition probability. Zbl 1296.93202 Zhang, Yu 2013 ...and 94 more Documents all top 5 ### Cited by 1,077 Authors 48 Zhang, Yu 14 Yang, Yidu 12 Zhang, Yanyan 9 Bi, Hai 8 Li, Zhilin 7 Wang, Jinhuan 7 Yang, Xue 6 Chen, Guanrong 6 Ji, Haifeng 6 Li, Yuxia 6 Lü, Xing 6 Sun, Meina 5 Chen, Jinru 5 Chen, Pengyu 5 Chen, Sijia 5 Elsayed, Elsayed Mohammed 5 Fu, Xianlong 5 Han, Jiayu 5 He, Jingsong 5 Hou, Thomas Yizhao 5 Lin, Kai 5 Yang, Hongwei 4 Biswas, Anjan 4 Deng, Weibing 4 Deng, Yue 4 El-Dessoky, Mohamed M. 4 Li, Yong 4 Li, Yongxiang 4 Peng, Zhenhua 4 Rao, Jiguang 4 Sun, Ji Tao 4 Tao, Chao 4 Teng, Zhi-dong 4 Xu, Changjin 4 Yang, Wen 4 Zhang, Xuping 3 Alquran, Marwan Taiseer 3 Álvarez, Gonzalo A. 3 Anthoni, Selvaraj Marshal 3 Assari, Pouria 3 Cao, Jinde 3 Dehghan Takht Fooladi, Mehdi 3 Gloria, Antoine 3 Guo, Hailong 3 Han, Maoan 3 Henning, Patrick 3 Huang, Ting-Zhu 3 Jeon, Youngmok 3 Jiang, Haijun 3 Jiang, Jack J. 3 Kurbanli, Abdullah Selçuk 3 Legoll, Frédéric 3 Li, Jin 3 Li, Shujun 3 Li, Xiaodi 3 Liu, Wenjun J. 3 Lu, Lin 3 Ma, Wen-Xiu 3 Nowak, David E. 3 Ohlberger, Mario 3 Pang, Yicheng 3 Raja, Ramachandran 3 Shen, Chun 3 Shi, Ling 3 Song, Qiankun 3 Tang, Yang 3 Wang, Xuejun 3 Wu, Yusen 3 Xu, Yihong 3 Yan, Wei 3 Yang, Hanchun 3 Yang, Liangui 3 Yang, Xu 3 Yin, Xiaojun 3 Yuan, Fang 3 Zhang, Pengchuan 3 Zhang, Yunfeng 3 Zheng, Wei Xing 3 Zhou, Qin 3 Zhu, Jianbo 2 Abdurahman, Abdujelil 2 Adem, Abdullahi Rashid 2 Ahmed, Hamdy M. 2 Akhmet, Marat Ubaydulla 2 Aryati, Lina 2 Baghbani, R. 2 Bao, Bocheng 2 Bhaintwal, Maheshanand 2 Bo, Yuming 2 Bouetou Bouetou, Thomas 2 Cataldo, Edson 2 Chen, Fengde 2 Chen, Guangye 2 Chen, Tao 2 Cui, Li 2 Darvishi, Mohammad Taghi 2 Deng, Zichen 2 Di Gregorio, Raffaele 2 Ding, Wei 2 Dong, Huanhe ...and 977 more Authors all top 5 ### Cited in 206 Serials 22 Nonlinear Dynamics 20 Chaos, Solitons and Fractals 15 Applied Mathematics and Computation 14 Chaos 14 Communications in Nonlinear Science and Numerical Simulation 14 Advances in Difference Equations 13 Mathematical Problems in Engineering 11 Applied Mathematical Modelling 11 Abstract and Applied Analysis 10 Journal of the Franklin Institute 10 Automatica 10 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 9 General Relativity and Gravitation 9 Journal of Computational and Applied Mathematics 9 Discrete Dynamics in Nature and Society 8 Journal of Computational Physics 8 European Journal of Operational Research 7 Computers & Mathematics with Applications 7 Information Sciences 7 Journal of Inequalities and Applications 7 Multiscale Modeling & Simulation 6 International Journal of Theoretical Physics 6 Journal of Mathematical Physics 6 Nonlinear Analysis. Real World Applications 5 Astrophysics and Space Science 5 Wave Motion 5 Applied Numerical Mathematics 5 Journal of Scientific Computing 5 International Journal of Nonlinear Sciences and Numerical Simulation 4 Journal of Optimization Theory and Applications 4 Acta Applicandae Mathematicae 4 Applied Mathematics Letters 4 International Journal of Computer Mathematics 4 Journal of Applied Mathematics and Computing 4 Communications in Computational Physics 4 Analysis and Mathematical Physics 4 Open Mathematics 3 Mathematical Methods in the Applied Sciences 3 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 3 Operations Research 3 Acta Mathematicae Applicatae Sinica. English Series 3 Information and Computation 3 Neural Networks 3 Journal of Global Optimization 3 Complexity 3 Journal of Dynamical and Control Systems 3 Discrete and Continuous Dynamical Systems. Series B 3 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 3 Communications in Theoretical Physics 3 International Journal of Systems Science. Principles and Applications of Systems and Integration 3 Proceedings of the Royal Society of London. A. Mathematical, Physical and Engineering Sciences 2 Modern Physics Letters A 2 Computer Methods in Applied Mechanics and Engineering 2 Indian Journal of Pure & Applied Mathematics 2 Journal of Mathematical Analysis and Applications 2 Physics Letters. A 2 ZAMP. Zeitschrift für angewandte Mathematik und Physik 2 Mathematics of Computation 2 The Annals of Statistics 2 Journal of Differential Equations 2 Mathematics and Computers in Simulation 2 Numerical Functional Analysis and Optimization 2 Numerische Mathematik 2 Quaestiones Mathematicae 2 SIAM Journal on Control and Optimization 2 Theoretical Computer Science 2 Circuits, Systems, and Signal Processing 2 Chinese Annals of Mathematics. Series B 2 Bulletin of the Iranian Mathematical Society 2 Computer Aided Geometric Design 2 Computers & Operations Research 2 Asia-Pacific Journal of Operational Research 2 Mathematical and Computer Modelling 2 Journal of Robotic Systems 2 Journal of Statistical Computation and Simulation 2 SIAM Journal on Scientific Computing 2 International Journal of Modern Physics D 2 Computational and Applied Mathematics 2 Discrete and Continuous Dynamical Systems 2 International Journal of Applied Mathematics and Computer Science 2 Qualitative Theory of Dynamical Systems 2 Nonlinear Analysis. Modelling and Control 2 Journal of Systems Science and Complexity 2 Bulletin of the Malaysian Mathematical Sciences Society. Second Series 2 Communications on Pure and Applied Analysis 2 Mathematical Biosciences and Engineering 2 Boundary Value Problems 2 Journal of Industrial and Management Optimization 2 Journal of Biological Dynamics 2 Journal of Fixed Point Theory and Applications 2 Nonlinear Analysis. Hybrid Systems 2 Mathematical Modelling of Natural Phenomena 2 The Annals of Applied Statistics 2 International Journal of Biomathematics 2 Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM 2 Numerical Algebra, Control and Optimization 2 Journal of Applied Analysis and Computation 2 Journal of Mathematics 2 International Journal of Applied and Computational Mathematics 1 Modern Physics Letters B ...and 106 more Serials all top 5 ### Cited in 41 Fields 158 Partial differential equations (35-XX) 103 Numerical analysis (65-XX) 97 Ordinary differential equations (34-XX) 78 Systems theory; control (93-XX) 73 Biology and other natural sciences (92-XX) 60 Dynamical systems and ergodic theory (37-XX) 57 Operations research, mathematical programming (90-XX) 53 Fluid mechanics (76-XX) 47 Information and communication theory, circuits (94-XX) 33 Computer science (68-XX) 33 Relativity and gravitational theory (83-XX) 31 Probability theory and stochastic processes (60-XX) 29 Difference and functional equations (39-XX) 25 Statistics (62-XX) 21 Mechanics of deformable solids (74-XX) 18 Calculus of variations and optimal control; optimization (49-XX) 16 Operator theory (47-XX) 13 Optics, electromagnetic theory (78-XX) 13 Quantum theory (81-XX) 10 Integral equations (45-XX) 10 Statistical mechanics, structure of matter (82-XX) 9 Mechanics of particles and systems (70-XX) 7 Classical thermodynamics, heat transfer (80-XX) 5 Mathematical logic and foundations (03-XX) 5 Real functions (26-XX) 5 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 3 Differential geometry (53-XX) 3 Global analysis, analysis on manifolds (58-XX) 3 Geophysics (86-XX) 2 Linear and multilinear algebra; matrix theory (15-XX) 2 Special functions (33-XX) 2 Sequences, series, summability (40-XX) 2 Functional analysis (46-XX) 2 General topology (54-XX) 2 Astronomy and astrophysics (85-XX) 1 Nonassociative rings and algebras (17-XX) 1 Category theory; homological algebra (18-XX) 1 Topological groups, Lie groups (22-XX) 1 Functions of a complex variable (30-XX) 1 Potential theory (31-XX) 1 Harmonic analysis on Euclidean spaces (42-XX)	2022-05-23 22:59:36	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.7898044586181641, "perplexity": 10896.29029857355}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1652662562106.58/warc/CC-MAIN-20220523224456-20220524014456-00332.warc.gz"}
https://www.risk.net/journal-of-computational-finance/2160541/robust-numerical-methods-for-pde-models-of-asian-options	# Robust numerical methods for PDE models of Asian options ## R. Zvan, P. A. Forsyth, K. R. Vetzal #### Abstract ABSTRACT The pricing of Asian options is explored by numerically solving the associated partial differential equations. The authors demonstrate that numerical PDE techniques commonly used in finance for standard options are inaccurate in the case of Asian options and illustrate modifications which alleviate this problem. In particular, the usual methods generally produce solutions containing spurious oscillations. Flux-limiting techniques originally developed in the field of computational fluid dynamics are adapted in order to rapidly obtain accurate solutions. It is shown that flux-limiting methods are total variation diminishing (and hence free of spurious oscillations) for nonconservative PDEs such as those typically encountered in finance, for fully explicit, and fully and partially implicit, schemes. The van Leer flux limiter is also modified so that the second-order total variation diminishing property is preserved for nonuniform grid spacing.	2017-11-19 10:37:31	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8650372624397278, "perplexity": 1060.7454202030904}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934805541.30/warc/CC-MAIN-20171119095916-20171119115916-00242.warc.gz"}
https://socratic.org/questions/57310bbf7c01494799659c9c	What is the range of the function y = sqrt(x^2+1)-x ? May 9, 2016 Answer: Use an algebraic method to find range is $\left(0 , \infty\right)$ Explanation: If a particular value of $y$ is in the range, then we can solve for $x$ to give that value of $y$. Suppose: $y = \sqrt{{x}^{2} + 1} - x$ Adding $x$ to both sides we get: $y + x = \sqrt{{x}^{2} + 1}$ Squaring both sides (which may introduce spurious solutions) we get: ${y}^{2} + 2 x y + {x}^{2} = {x}^{2} + 1$ Subtract ${x}^{2} + {y}^{2}$ from both sides to get: $2 x y = 1 - {y}^{2}$ Note that we must have $y \ne 0$ since otherwise this equation would become $0 = 1$. So in particular $y = 0$ is not in the range. Divide both sides by $2 y$ to get: $x = \frac{1 - {y}^{2}}{2 y}$ So this is a requirement. To see if it is sufficient, substitute it back in the left hand side of the original equation: $\sqrt{{x}^{2} + 1} - x$ $= \sqrt{{\left(\frac{1 - {y}^{2}}{2 y}\right)}^{2} + 1} - \frac{1 - {y}^{2}}{2 y}$ $= \sqrt{\frac{\left(1 - 2 {y}^{2} + {y}^{4}\right) + 4 {y}^{2}}{4 {y}^{2}}} - \frac{1 - {y}^{2}}{2 y}$ $= \sqrt{{\left(1 + {y}^{2}\right)}^{2} / \left(4 {y}^{2}\right)} - \frac{1 - {y}^{2}}{2 y}$ $= \frac{1 + {y}^{2}}{2 \left\mid y \right\mid} - \frac{1 - {y}^{2}}{2 y}$ If $y > 0$ then $\left\mid y \right\mid = y$ and this becomes: $\frac{1 + {y}^{2}}{2 y} - \frac{1 - {y}^{2}}{2 y} = \frac{2 {y}^{2}}{2 y} = y$ If $y < 0$ then $\left\mid y \right\mid = - y$ and this becomes: $- \frac{1 + {y}^{2}}{2 y} - \frac{1 - {y}^{2}}{2 y} = - \frac{2}{2 y} = - \frac{1}{y} \ne y$ So there is a solution for $x$ if and only if $y > 0$. So the range is $\left(0 , \infty\right)$	2019-11-21 10:20:17	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 29, "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": 0.8640307784080505, "perplexity": 141.4037351125358}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496670770.21/warc/CC-MAIN-20191121101711-20191121125711-00241.warc.gz"}
https://www.gradesaver.com/textbooks/math/algebra/algebra-1-common-core-15th-edition/chapter-3-solving-inequalities-3-6-compound-inequalities-practice-and-problem-solving-exercises-page-204/18	## Algebra 1: Common Core (15th Edition) $m \gt -1$ or $m \lt -5$ Given: $5+m \gt 4$ or $7m \lt -35$ $m \gt -1$ or $m \lt -5$ The solutions as a single inequality are: $m \gt -1$ or $m \lt -5$	2022-05-27 15:13:54	{"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": 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": 0.5640210509300232, "perplexity": 1215.8621482356925}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662658761.95/warc/CC-MAIN-20220527142854-20220527172854-00693.warc.gz"}
https://friendica.cwiwie.org/display/3f585507-185e-e93c-1110-4e1296577556	Skip to main content With each additional training sample I grow more confident about the following predictive model about my health at day t: $$sick?(t) = moody?(t-2) \wedge moody?(t-1)$$	2020-08-07 15:07:40	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.24443092942237854, "perplexity": 4329.235910679241}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439737204.32/warc/CC-MAIN-20200807143225-20200807173225-00366.warc.gz"}
https://publons.com/publon/546458/#review-502770	5 pre-pub reviews 0 post-pub reviews Abstract Background: Collective animal behavior, such as the flocking of birds or the shoaling of fish, has inspired a class of algorithms designed to optimize distance-based clusters in various applications, including document analysis and DNA microarrays. In a flocking model, individual agents respond only to their immediate environment and move according to a few simple rules. After several iterations the agents self-organize, and clusters emerge without the need for partitional seeds. In addition to its unsupervised nature, flocking offers several computational advantages, including the potential to reduce the number of required comparisons.Findings: In the tool presented here, Clusterflock, we have implemented a flocking algorithm designed to locate groups (flocks) of orthologous gene families (OGFs) that share an evolutionary history. Pairwise distances that measure phylogenetic incongruence between OGFs guide flock formation. We tested this approach on several simulated datasets by varying the number of underlying topologies, the proportion of missing data, and evolutionary rates, and show that in datasets containing high levels of missing data and rate heterogeneity, Clusterflock outperforms other well-established clustering techniques. We also verified its utility on a known, large-scale recombination event in Staphylococcus aureus. By isolating sets of OGFs with divergent phylogenetic signals, we were able to pinpoint the recombined region without forcing a pre-determined number of groupings or defining a pre-determined incongruence threshold.Conclusions: Clusterflock is an open-source tool that can be used to discover horizontally transferred genes, recombined areas of chromosomes, and the phylogenetic 'core' of a genome. Although we used it here in an evolutionary context, it is generalizable to any clustering problem. Users can write extensions to calculate any distance metric on the unit interval, and can use these distances to 'flock' any type of data. Authors Narechania, Apurva; Baker, Richard; DeSalle, Rob; Mathema, Barun; Kolokotronis, Sergios-Orestis; Kreiswirth, Barry; Planet, Paul J. Publons users who've claimed - I am an author No Publons users have claimed this paper. • 3 reviewers • Review of Clusterflock: a flocking algorithm for isolating congruent phylogenomic datasets Reviewer #1 reported problems with using the Clusterflock tool due to the complexity with installing the software and its dependencies. In response, the authors of Clusterflock have provided a Docker container which ships all of the code and associated software libraries in a standalone package ready for use. I have tested the clusterflock-0.1 Docker container and can report that I have successfully executed the clusterflock.pl and clusterflock_simulations.pl scripts to completion using the instructions available from https://github.com/narechan/clusterflock/blob/master/MANUAL. This involved: 1. Deploying an Ubuntu-14.04 EC2 virtual server as a t2.medium instance on the AWS cloud and installing the Docker software on it. 3. The Clusterflock scripts can then be executed by running the clusterflock-0.1 Docker container with this command on the host server: $docker run -v /mount/path/on/host:/home/test -it narechan/clusterflock-0.1 The following two commands can then be executed using clusterflock-0.1 Docker image:$ clusterflock.pl -i test_data/4/fastas/ -c config.boids.simulations -l test_data/4/4.lds -s all -b 1 -d -x -o /home/test/4_out \$ clusterflock_simulations.pl -c config.boids.simulations -r 10 -p 10 -o /home/test/4_sim/ -i test_data/4/fastas/ -l test_data/4/4.lds -j /home/clusterflock/dependencies/elki-bundle- 0.6.5~20141030.jar -k 4 -f 500 > /home/test/4_sim.avg_jaccard Both of the above commands generated outputs as described in https://github.com/narechan/clusterflock/blob/master/MANUAL. Level of interest Please indicate how interesting you found the manuscript: An article whose findings are important to those with closely related research interests Quality of written English Please indicate the quality of language in the manuscript: Acceptable Declaration of competing interests Please complete a declaration of competing interests, considering the following questions: 1. Have you in the past five years received reimbursements, fees, funding, or salary from an organisation that may in any way gain or lose financially from the publication of this manuscript, either now or in the future? 2. Do you hold any stocks or shares in an organisation that may in any way gain or lose financially from the publication of this manuscript, either now or in the future? 3. Do you hold or are you currently applying for any patents relating to the content of the manuscript? 4. Have you received reimbursements, fees, funding, or salary from an organization that holds or has applied for patents relating to the content of the manuscript? 5. Do you have any other financial competing interests? 6. Do you have any non-financial competing interests in relation to this paper? If you can answer no to all of the above, write 'I declare that I have no competing interests' I declare that I have no competing interests. I agree to the open peer review policy of the journal. I understand that my name will be included on my report to the authors and, if the manuscript is accepted for publication, my named report including any attachments I upload will be posted on the website along with the authors' responses. I agree for my report to be made available under an Open Access Creative Commons which I do not wish to be included in my named report can be included as confidential comments to the editors, which will not be published. I agree to the open peer review policy of the journal. Published in Reviewed by • Review of Clusterflock: a flocking algorithm for isolating congruent phylogenomic datasets The authors propose a method that uses a modified flocking algorithm figure out how many trees are needed to represent a set of alignments of the same taxa. This is an interesting problem, and the proposed solution is a valuable contribution. Nevertheless, there were a number of places in which I thought the study could and perhaps should be improved. I split them into two below: those to do with the software, and those to do with the manuscript. 1. Only a passing mention is given to previous solutions to this problem. Given that there are various previous solutions, it would be useful for the reader to be given some comparison of the relative merits and shortcomings of the solutions, to motivate the current study. It would also be worth noting in the discussion whether and how this new method overcomes the limitations of previous methods. This seems like an important point for readers who areconsidering which tool to use. 2. There are no simulations. This seems like an important omission to me, because without simulations it's impossible to know when the method works well, and when it doesn't. Although the authors present an analysis of one empirical dataset in which the algorithm appears to do roughly what it should, this is not sufficient to make robust judgements as to the general performance of the algorithm. Thus, without simulations I would argue that the conclusions of the paper are not supported by the data, specifically it is not possible claim that 'we show that [clusterflock] is particularly well suited to isolating genes into discrete flocks that share a unique phylogenetic history'. Simulations could obviously take many forms, but a simple approach would be to consider 100 datasets with 1-100 trees underlying them. 100 loci could then be sampled across these trees, and fed into the algorithm. Repeating each simulation 10 times would require only 1000 analyses, and could give a quite detailed picture of the method's performance. Specific questions to ask would be: what is the false positive rate (i.e. how often do you detect more than one cluster when there is only a single underlying tree)? What is the false negative rate (i.e. how often do you cluster together genes with different underlying trees)? What are the detection limits (e.g. how much data and how different do two trees have to be before you can detect the differences)? What aspects of sequence evolution can mislead the algorithm (e.g. rates of evolution, see below)? How does the ratio of the number of loci to the number of trees affect performance (this seems like a particularly important point to address in a flocking algorithm - it's not obvious to me what will happen to trees that are represented by a single locus, and particularly in the case where most trees are represented by a very small number of loci)? 3. The design choices are described relatively thoroughly in the paper, but very few motivations for these choices are given. Thus, while I might be able to re-implement a similar algorithm by reading the paper, I have no idea why most of the choices were made. It would be nice to include the background to the decisions made when implementing the algorithms, because this would facilitate progress in this area. 4. The use of LD seems reasonable here, but it seems like it could also be misled by genes evolving at different rates. This is because higher rates will tend to exacerbate problems like long-branch attraction. Thus, under parsimony, a slow gene and a fast gene may have quite different most-parsimonious topologies. Given the vast differences in rates between many genes, this seems like a potential issue that could at the very least be explored with simulation, e.g. by simulating 100 genes on the same tree, where 50 evolve slowly and 50 evolve more quickly. By varying the rate ratio of the two genes, one could determine whether this is an issue, and at what kinds of scales it manifests itself. 5. A simple question - could the authors include some information on the relative proportion of the runtimes that are associated with different parts of the algorithm. I ask this because it's easy to think of other options (like calculating ML or NJ trees, and then using any of a number of metrics of tree distances) which might improve accuracy but increase runtimes. However, without knowing what the rate-limiting steps of the algorithm are, it's not possible to know whether such improvements are worth even thinking about. 6. Following from point 5: given that you have to run the algorithm 100 times to get some idea of the robustness of the flocking, how does the aggregated runtime compare to other approaches to this problem? E.g. what about software such as concaterpillar or conclustador? The latter states that it is specifically designed to solve the same problem as clusterflock, so it seems worth comparing the two here. Note that I don't think it's necessary to do better than any other software - this is a very interesting approach that should be described regardless of whether it's better on any particular metric - but it does seem important to make some attempt to compare performance in terms of accuracy and speed. 1. The way that github has been used is unconventional, and inconvenient. The only way I could download the software was to download a whole collection of other pieces of software along with it. Please give this software its own repository. This will also facilitate future collaboration and development, since github works fundamentally at the level of the single repository. 2. Please mint a DOI for the released version of the software with Zenodo or some other service. This ensures that the software will stay around if the github repo is deleted, and it also ensures that the ms refers to a persistent and tagged version of the software even if the repo stays around and the software continues to be developed. 3. There are no tests in the software. In this case, tests seem rather vital. The paper describes clusterflock 'an open source tool', so presumably the intention is that many others will use it. Simulations will form a useful set of tests on their own, and should be included in the repository with a script to run all tests and check that they produce the expected results. (note - the results don't have to be correct, but there should be some checking to make sure that they are expected). Given that the algorithm is stochastic, it might be useful to include an option to provide a random number seed in the code, in particular to facilitate testing. Unit tests would also be useful, to ensure that key functions are behaving as expected. As it stands, software with no tests does not inspire a great deal of confidence. 4. More documentation is needed. I suspect this is particularly the case here, since the vast majority of the end-users of the tool will not know Perl. It would be worth putting together a comprehensive manual, and in particular providing detailed installation instructions and a quickstart guide. For example, although I am quite proficient in a couple of languages I do not use Perl. Even if I had access to a linux machine to test the software (sadly, I don't, but I hope at least one reviewer does), I'm guessing that getting it up an running would have taken me some time. 5. I searched for a license, and found one in the script. But I am confused. The license states that the work is copyright of the AMNH, but also that it is released under the same terms as Perl itself. These seem incompatible, and also perhaps incompatible with the three dependencies that are packaged in the repo. Can the authors double check this, and when they are sure they have a valid license, include it somewhere obvious in the repository and the manual. 6. Just an observation: 'Clusterflock' is a very popular name for many things, and that makes this tool very hard to find on google. Even typing 'clusterflock phylogenetics github' does not produce a link to the tool. It might be worth considering a name that makes the tool easier to find. Level of interest Please indicate how interesting you found the manuscript: An article whose findings are important to those with closely related research interests Quality of written English Please indicate the quality of language in the manuscript: Acceptable Declaration of competing interests Please complete a declaration of competing interests, considering the following questions: 1. Have you in the past five years received reimbursements, fees, funding, or salary from an organisation that may in any way gain or lose financially from the publication of this manuscript, either now or in the future? 2. Do you hold any stocks or shares in an organisation that may in any way gain or lose financially from the publication of this manuscript, either now or in the future? 3. Do you hold or are you currently applying for any patents relating to the content of the manuscript? 4. Have you received reimbursements, fees, funding, or salary from an organization that holds or has applied for patents relating to the content of the manuscript? 5. Do you have any other financial competing interests? 6. Do you have any non-financial competing interests in relation to this paper? If you can answer no to all of the above, write 'I declare that I have no competing interests' None I agree to the open peer review policy of the journal. I understand that my name will be included on my report to the authors and, if the manuscript is accepted for publication, my named report including any attachments I upload will be posted on the website along with the authors' responses. I agree for my report to be made available under an Open Access Creative Commons which I do not wish to be included in my named report can be included as confidential comments to the editors, which will not be published. I agree to the open peer review policy of the journal. Authors' response to reviews: (https://static-content.springer.com/openpeerreview/art%3A10.1186%2Fs13742-016-0152-3/13742_2016_152_AuthorComment_V1.pdf) Published in Reviewed by • Review of Clusterflock: a flocking algorithm for isolating congruent phylogenomic datasets The manuscript present a method to identify phylogenetically-congruent genes through an agentbased modelling approach originally designed to model bird flocking. The method is concisely presented and applied to a set Staphylococcus aureus genomes known to have evolved via large hybridisation events. The problem is relevant and the idea has merit, but as I elaborate below, I am concerned that no efforts were done to compare the approach to standard clustering approaches or to existing methods for the same problem. I am also concerned that the authors only provide results for a single dataset. The minimum standard in the field is to validate one's approach on a variety of simulated data, showing that the method performs well under these ideal conditions at least. Major points: 1. Method only validated on a single problem instance. This is inadequate for a new method. Instead, the authors should at least show on simulated datasets covering a variety of scenarios that the algorithm is able to cluster the data correctly. 2. No comparison with other methods: As the authors correctly point out, their approach boils down to a clustering method. There are many such methods, so why should the proposed approach be preferred? Contrary to the claim two paragraphs prior to the conclusions (please number your ms pages), there are other clustering methods that do not require specifying the number of clusters. Even for those that do, there are heuristics available (elbow, silhouette, etc.). At the very least, it seems that embedding the genes in a space using a standard multidimensional scaling procedure followed by clustering (e.g. using the OPTICS algorithm used by the authors) would provide a reasonable baseline to gauge how useful the flocking approach is. Minor Points: 3. What genomes were used as input? (accession number/date) 4. How were the orthologous groups computed? 5. How were the single-gene trees computed? 5. How were the single-gene trees computed? 6. Given that orthologous groups were inferred, why did the authors need to map genes to USA300/TCH1516 via profile HMM? In any case, this needs to be described. 7. Paragraph right before conclusions: "The LDs of these genes with respect...". The authors probably mean ILD here. In the context of recombination, LD usually means linkage disequilibrium, which could be confusing. 8. Same sentence: the conjecture that genes that are both in the "core cluster" and hybridisation region could have reverted back to the core phylogeny seems highly improbable to me. Assuming these indeed follow the core phylogeny, it seems more likely that they were translocated to that region after the hybridisation event. 9. The labels on Fig. 3 are illegible. Level of interest Please indicate how interesting you found the manuscript: An article of limited interest Quality of written English Please indicate the quality of language in the manuscript: Acceptable Declaration of competing interests Please complete a declaration of competing interests, considering the following questions: 1. Have you in the past five years received reimbursements, fees, funding, or salary from an organisation that may in any way gain or lose financially from the publication of this manuscript, either now or in the future? 2. Do you hold any stocks or shares in an organisation that may in any way gain or lose financially from the publication of this manuscript, either now or in the future? 3. Do you hold or are you currently applying for any patents relating to the content of the manuscript? 4. Have you received reimbursements, fees, funding, or salary from an organization that holds or has applied for patents relating to the content of the manuscript? 5. Do you have any other financial competing interests? 6. Do you have any non-financial competing interests in relation to this paper? If you can answer no to all of the above, write 'I declare that I have no competing interests' By way of full disclosure, I am the senior author of a loosely related manuscript submitted to another journal. However, the two manuscripts use different approaches, and have very different focuses, so they are not in competition. I agree to the open peer review policy of the journal. I understand that my name will be included on my report to the authors and, if the manuscript is accepted for publication, my named report including any attachments I upload will be posted on the website along with the authors' responses. I agree for my report to be made available under an Open Access Creative Commons which I do not wish to be included in my named report can be included as confidential comments to the editors, which will not be published. I agree to the open peer review policy of the journal. Authors' response to reviews: (https://static-content.springer.com/openpeerreview/art%3A10.1186%2Fs13742-016-0152-3/13742_2016_152_AuthorComment_V1.pdf) Published in Reviewed by • Review of Clusterflock: a flocking algorithm for isolating congruent phylogenomic datasets I have a few remaining comments: 1. Usability While I think the method is interesting, the implementation remains very difficult to use. After two hours of attempting to install and run the software (I am a proficient programmer in python and R, but have zero Perl experience) I gave up. The installation remains complex for non-perl experts, and the sparsity of the documentation does not help (the documentation has been expanded somewhat, but it remains far too sparse to be useful to non-perl programmers). Because of that, the utility of the tool for the end-users (presumably, biologists with multi-locus datasets) is questionable. This is not something I see as a barrier to publication of the method - which itself is interesting - but since the primary focus of this paper is the software itself, this does seem to me to be an issue. 2. DOI The authors provide no cogent reason not to provide a DOI for their software. I don't know what the issue is here. By not providing a DOI (e.g. through Zenodo), there is no guarantee that the software will stay around. This is a problem for reproducibility and for the general utility of the work. Given that link rot, and lost/broken software in general is such a huge problem in our field, and given that the primary focus of this paper is the provision of 'an open-source tool' I think it's important to properly archive a version of the software with a DOI here. Neither tagging versions in github nor making a copy of the repo on bitbucket guarantees persistence. But the ~10 minutes it takes to provide a DOI through Zenodo does guarantee persistence. It means that, no matter what the authors decide to do with their github repository, the copy of the code used for this ms will be around and will be discoverable from the manuscript itself. A side note: the authors state that they have tagged the current version of the software as 0.1. However, there are no tags or releases on their github repository. Tags and releases are specific things designed to help people get to particular versions of software: https://help.github.com/articles/creating-releases/ . Minting a DOI with Zenodo would solve this problem too - Zenodo works with tagged versions of the repository only. 3. Simulations Can the authors please provide data (in a figure) on the number of clusters returned by clusterflock in each of the simulated datasets, versus the number of underlying topologies that were simulated. It's not possible to get this from the currently-presented data, and this is an important part of assessing the accuracy of the algorithm on the simulated datasets. 4. Data availability Please provide the output data from the simulations: specifically, the data that could be used to recalculate figures 3 and 4 on the identity of the simulated topology versus the topology to which clusterflock assigned that locus. 5. Discussion of performance Figures 3 and 4 would benefit from having the expected jacard index with random assignment of trees to loci plotted. This way we could see which methods do no better than randomly assigning trees to groups. As far as I can tell, clusterflock with 50% missing data tracks the random expectation very closely (JI = 0.5 with 2 trees; 0.1 with 10 trees; 0.04 with 25 trees). This in itself is interesting - even with data for 50% of the species, clusterflock does not appear to gain any benefit over randomly assigning trees to groups. Can comment on this particular case? It seems counterintuitive to me that with data for 50% of the species at each locus, the method gains no benefit over randomly assigning trees. More generally, can the authors comment on the meaning (for biologists) of the fact that clusterflock gets a JI of ~0.4 when there are 25 simulated topologies. If the algorithm correctly assigns loci to topologies less than half of the time in these simulations, what does this mean for biological inferences from the data? For example, it seems from the simulated and empirical data that while clusterflock might be useful when the number of clusters is very small (e.g. <10) it might be much less useful with >10 clusters. For example, while the empirical test presented in the paper is compelling, it seems likely that the algorithm may be much less useful if there had been a lot of recombination events (as might be the case in many empirical datasets, such as the analysis of whole-bird genomes from across the avian tree of life). As above, some comparison with existing approaches to this problem is warranted here: if clusterflock does better than existing approaches (i.e. Concaterpillar, conclustador, etc), then that's great even if the absolute performance remains less than ideal. In this case, biologists should prefer clusterflock because it makes the best inferences. However, if clusterflock is consistently worse than other methods, then we know that it is a neat method that requires additional development before it is useful. In my opinion, knowing which of these situations is the case would vastly strengthen the paper. Level of interest Please indicate how interesting you found the manuscript: An article whose findings are important to those with closely related research interests Quality of written English Please indicate the quality of language in the manuscript: Acceptable Declaration of competing interests Please complete a declaration of competing interests, considering the following questions: 1. Have you in the past five years received reimbursements, fees, funding, or salary from an organisation that may in any way gain or lose financially from the publication of this manuscript, either now or in the future? 2. Do you hold any stocks or shares in an organisation that may in any way gain or lose financially from the publication of this manuscript, either now or in the future? 3. Do you hold or are you currently applying for any patents relating to the content of the manuscript? 4. Have you received reimbursements, fees, funding, or salary from an organization that holds or has applied for patents relating to the content of the manuscript? 5. Do you have any other financial competing interests? 6. Do you have any non-financial competing interests in relation to this paper? If you can answer no to all of the above, write 'I declare that I have no competing interests' None. I agree to the open peer review policy of the journal. I understand that my name will be included on my report to the authors and, if the manuscript is accepted for publication, my named report including any attachments I upload will be posted on the website along with the authors' responses. I agree for my report to be made available under an Open Access Creative Commons which I do not wish to be included in my named report can be included as confidential comments to the editors, which will not be published. I agree to the open peer review policy of the journal. Authors' response to reviews: (https://static-content.springer.com/openpeerreview/art%3A10.1186%2Fs13742-016-0152-3/13742_2016_152_AuthorComment_V2.pdf) Published in Reviewed by • Review of Clusterflock: a flocking algorithm for isolating congruent phylogenomic datasets My authors have satisfactorily addressed my remarks. I only have two small comments on the new analyses: 1) The comparison with other clustering methods is good addition. The fragility of hierarchical methods and "partitioning around medoid" with respect to missing data is surprising. The authors should make data and scripts available. 2) The legend of new figures 3 and 4 should be clearer. As it stands, one needs to read the main text to understand that "zero, ten, twenty" refers to percentages of missing data. Level of interest Please indicate how interesting you found the manuscript: An article of limited interest Quality of written English Please indicate the quality of language in the manuscript: Acceptable Declaration of competing interests Please complete a declaration of competing interests, considering the following questions: 1. Have you in the past five years received reimbursements, fees, funding, or salary from an organisation that may in any way gain or lose financially from the publication of this manuscript, either now or in the future? 2. Do you hold any stocks or shares in an organisation that may in any way gain or lose financially from the publication of this manuscript, either now or in the future? 3. Do you hold or are you currently applying for any patents relating to the content of the manuscript? 4. Have you received reimbursements, fees, funding, or salary from an organization that holds or has applied for patents relating to the content of the manuscript? 5. Do you have any other financial competing interests? 6. Do you have any non-financial competing interests in relation to this paper? If you can answer no to all of the above, write 'I declare that I have no competing interests' I declare that I have no competing interests. I agree to the open peer review policy of the journal. I understand that my name will be included on my report to the authors and, if the manuscript is accepted for publication, my named report including any attachments I upload will be posted on the website along with the authors' responses. I agree for my report to be made available under an Open Access Creative Commons	2021-09-28 23:16:31	{"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": 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": 0.4289627969264984, "perplexity": 1263.831828182424}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780060908.47/warc/CC-MAIN-20210928214438-20210929004438-00432.warc.gz"}
https://www.shaalaa.com/question-bank-solutions/division-fractions-division-whole-number-fraction-find-4by9-division-5_17463	Share # Find:4by9 Division 5 - CBSE Class 7 - Mathematics ConceptDivision of Fractions Division of Whole Number by a Fraction #### Question Find:4/9 ÷ 5 #### Solution 4/9 ÷ 5 = 4/9 xx 1/5 = 4/45 Is there an error in this question or solution? #### APPEARS IN NCERT Solution for Mathematics for Class 7 (2018 to Current) Chapter 2: Fractions and Decimals Ex. 2.40 \| Q: 3.2 \| Page no. 46 Solution Find:4by9 Division 5 Concept: Division of Fractions - Division of Whole Number by a Fraction. S	2019-12-07 06:07:52	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.21376366913318634, "perplexity": 8864.350023554796}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540496492.8/warc/CC-MAIN-20191207055244-20191207083244-00424.warc.gz"}
http://mathhelpforum.com/discrete-math/87838-looking-algorithm-generate-occupancy-numbers-n-balls-into-k-cells.html	# Thread: Looking for algorithm to generate occupancy numbers of n balls into k cells 1. ## Looking for algorithm to generate occupancy numbers of n balls into k cells Hey everyone, So I need an algorithm that will generate all the occupancy numbers for n (indistinguishable) balls into k (distinguishable) cells, where you are allowed to have empty cells. I know the total number of such ways to do so is $C(k+n-1,n)=\binom{k+n-1}{n}$ But for a code I am writing I need to be able to generate all of them. Unfortunately, since I will be doing this for fairly large numbers (16 balls in 3-10 cells, so we're talking 10^2 - 10^9 possibilities), I don't want to include a routine to check each permutation, I want to just generate them straight away. In case anyone isn't following, say I want to put 3 balls into 3 cells. Then the possible ways of doing this are 3 0 0 2 1 0 2 0 1 0 3 0 1 2 0 0 2 1 0 0 3 1 0 2 0 1 2 1 1 1 ie There are 10. But I need to throw this in a code; anyone know an easy way to do this for general n and k? Thanks! 2. I think that you need to find a really good programmer. I don’t this is a mathematical problem. The best we can give is to tell the number of ways to partition integers into a certain number of summands. For example 16 can be partitioned into 10 or fewer summands in 212 ways. Example: 16=1+1+2+3+4+5 that is with six summands. But that does help list the different occupancy configurations 3. Hi, I think a slightly mathematically enclined casual programmer is enough. And I think that algorithmics is indeed maths (and informatics), even though there certainly are forums that are more relevant for such questions. About your problem. How would you procede "by hand"? Put a few balls in the first box (any number from 0 to n), then (among the remaining ones) put a few balls in the second ones, etc., until the last box where you put all the remaining balls. If you have 4 boxes, you can do this with three nested "for" loops: Code: ```for i_1=0 to n for i_2=n-i_1 to n for i_3=n-i_1-i_2 to n a solution is : i_1, i_2, i_3, n-i_1-i_2-i_3 end_for end_for end_for``` "Fine, you say, but what if I don't know k? How can I have my program do k-1 for-loops?". Answer (a classical one): to overcome this problem, use a recursive function. The idea is that you make a function that calls itself, according to the following scheme : Code: ```function balls_in_boxes(n,k) if k=1, put the n balls in the only box else, for i=0 to n, use balls_in_boxes(n-i,k-1) (it allows to put n-i balls in k-1 boxes) and put i balls in the last box end_for end_if end_function``` To make this more precise, you need the function to return the list of possible solutions. Depending on the programming language, this can be done in various ways. Here is a working program in C language that I just wrote. It writes the solutions on the screen. You'll have to make a few changes if you want to store the configurations. In my function, I chose to give as an argument the positions of the balls that have been put before, so I can print them at the end. I put comments in the code. Code: ```#include <stdio.h> #include <stdlib.h> // Lists on the screen the different Ways to put n balls in k(>=1) boxes (empty boxes allowed), followed by the "nb" numbers given by the array "start" (useful for recursion) ; returns the number of solutions // So if we only want the ways to put n balls in k boxes, we choose nb=0 and "start" is any pointer (for instance, the NULL pointer will do fine) int balls_in_boxes(int n,int k,int start,int nb) { int i,j,total; int start2; if(k==1) { // Just print the given starting numbers and put the balls in the only box i=0; while(i<nb) { printf("%d ",start[i]); i++; } printf("%d\n", n); return 1; } else { // Choose the number i (0<=i<=n) of balls to put in the first box, and call the function on the next k-1 boxes for n-i balls. total=0; for(i=0;i<=n;i++) { // Make an array of size nb+1 to store the position of balls start2=malloc((nb+1)*sizeof(int)); // Copy "start" into the first nb cells of "start2" for(j=0;j<nb;j++) start2[j]=start[j]; // Add a new cell with i balls start2[nb]=i; total += balls_in_boxes(n-i,k-1,start2,nb+1); // Print what we want, and count free(start2); } } } int main(void) { int n,k,total; n=3; k=3; total=balls_in_boxes(n,k,NULL,0); printf("\nNumber of solutions : %d\n",total); exit(0); }``` 4. I am sorry if people don't feel this is an appropriate place to post this question, but I categorize studying algorithms as part of descrete math...I was not actually expecting anyone to provide me with specific code. But since someone did... Great, thanks Laurent! My programming knowledge is pretty thin, so I might not have thought of just making a recursive function. In any case, it sounds like a fine solution for my problem. Thanks a bunch! EDIT: Laurent, I've checked your algorithm (by hand, not coding it) and I don't think it works. Basically, since you are starting with i_1=0 to n and i_2=n-i_1 to n, all the balls go into the first two cells. In other words, when I check your algorithm I get (for n=4) 0 4 0 0 1 3 0 0 2 2 0 0 3 1 0 0 4 0 0 0 I think you need to count upwards for each i, like 0 0 0 4, 0 0 1 3, 0 0 2 2 etc. In any case, the concept of the recursive function is probably good enough to get me going, which means I should have posted this in a coding forum anyway :-P Thanks all! 5. Originally Posted by cduston EDIT: Laurent, I've checked your algorithm (by hand, not coding it) and I don't think it works. Basically, since you are starting with i_1=0 to n and i_2=n-i_1 to n, all the balls go into the first two cells. In other words, when I check your algorithm I get (for n=4) 0 4 0 0 1 3 0 0 2 2 0 0 3 1 0 0 4 0 0 0 Yes indeed, it should have been "for i_2=0 to n-i_1" and the same for the others, I wrote this too quickly. In fact my point was that this method was not the good one, and I took greater care of the recursive algorithm, which works fine (the code runs well).	2013-12-10 00:56:50	{"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": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6468839049339294, "perplexity": 314.3112359981293}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386164003787/warc/CC-MAIN-20131204133323-00007-ip-10-33-133-15.ec2.internal.warc.gz"}
http://www.thulasidas.com/category/articles-and-essays/?lang=vi	# Category Archives: Articles and Essays This category archives my articles and essays published in peer reviewed journals or semi-academic magazines. These publications span a wide range from physics to spirituality, touching on neuroscience and philosophy. You will find articles that appeared in the International Journal of Modern Physics, IEEE Tran. Neural Systems and Rehab. Engineering, The Philosopher (the journal of the Philosophical Society of England), OmegaIndian Journal of Science and Religion, Galilean Electrodynamics and so on. # Ants and Grasshoppers The allegorical tale of Ants and Grasshoppers is often used to drive home the inevitable connection between handwork and success, as well as laziness and hardship. Or between talent and riches, indolence and penury. Here is another story that may run contrary to this message. Among the religious texts of Hinduism, the Bhagavad Gita is the most revered one. Literally presented as the word of God, the Bhagavad Gita enjoys a stature similar to the Bible or the Koran. Like all scriptures, the Bhagavad Gita also can be read, not merely as an act of devotion, but as a philosophical discourse as well. It presents a philosophical stance in understanding the world, which forms (for those from India) the basic and fundamental assumptions in dealing with life, and the unknowable reality around them. In fact, it is more than just assumptions and hypotheses; it is the basis of commonsense handed down from generation to generation. It is the foundations of intellect, which form the instinctive and emotional understanding of reality that is assimilated before logic and cannot be touched or analyzed with rationality. They are the mythos that trump logos every time. # Luddite Thoughts For all its pretentiousness, French cuisine is pretty amazing. Sure, I’m no degustation connoisseur, but the French really know how to eat well. It is little wonder that the finest restaurants in the world are mostly French. The most pivotal aspect of a French dish usually is its delicate sauce, along with choice cuts, and, of course, inspired presentation (AKA huge plates and minuscule servings). The chefs, those artists in their tall white hats, show off their talent primarily in the subtleties of the sauce, for which knowledgeable patrons happily hand over large sums of money in those establishments, half of which are calledCafe de Parisor have the wordpetitin their names. Seriously, sauce is king (to use Bollywood lingo) in French cuisine, so I found it shocking when I saw this on BBC that more and more French chefs were resorting to factory-manufactured sauces. Even the slices of boiled eggs garnishing their overpriced salads come in a cylindrical form wrapped in plastic. How could this be? How could they use mass-produced garbage and pretend to be serving up the finest gastronomical experiences? Sure, we can see corporate and personal greed driving the policies to cut corners and use the cheapest of ingredients. But there is a small technology success story here. A few years ago, I read in the newspaper that they found fake chicken eggs in some Chinese supermarkets. They werefresheggs, with shells, yolks, whites and everything. You could even make omelets with them. Imagine thata real chicken egg probably costs only a few cents to produce. But someone could set up a manufacturing process that could churn out fake eggs cheaper than that. You have to admire the ingenuity involvedunless, of course, you have to eat those eggs. The trouble with our times is that this unpalatable ingenuity is all pervasive. It is the norm, not the exception. We see it in tainted paints on toys, harmful garbage processed into fast food (or even fine-dining, apparently), poison in baby food, imaginative fine-print on financial papers andEULAs”, substandard components and shoddy workmanship in critical machineryon every facet of our modern life. Given such a backdrop, how do we know that theorganicproduce, though we pay four times as much for it, is any different from the normal produce? To put it all down to the faceless corporate greed, as most of us tend to do, is a bit simplistic. Going one step further to see our own collective greed in the corporate behavior (as I proudly did a couple of times) is also perhaps trivial. What are corporates these days, if not collections of people like you and me? There is something deeper and more troubling in all this. I have some disjointed thoughts, and will try to write it up in an ongoing series. I suspect these thoughts of mine are going to sound similar to the luddite ones un-popularized by the infamous Unabomber. His idea was that our normal animalistic instincts of the hunter-gatherer kind are being stifled by the modern societies we have developed into. And, in his view, this unwelcome transformation and the consequent tension and stress can be countered only by an anarchical destruction of the propagators of our so-called developmentnamely, universities and other technology generators. Hence the bombing of innocent professors and such. Clearly, I don’t agree with this luddite ideology, for if I did, I would have to first bomb myself! I’m nursing a far less destructive line of thought. Our technological advances and their unintended backlashes, with ever-increasing frequency and amplitude, remind me of something that fascinated my geeky mindthe phase transition between structured (laminar) and chaotic (turbulent) states in physical systems (when flow rates cross a certain threshold, for instance). Are we approaching such a threshold of phase transition in our social systems and societal structures? In my moody luddite moments, I feel certain that we are. # The Unreal Universe We know that our universe is a bit unreal. The stars we see in the night sky, for instance, are not really there. They may have moved or even died by the time we get to see them. It takes light time to travel from the distant stars and galaxies to reach us. We know of this delay. The sun that we see now is already eight minutes old by the time we see it, which is not a big deal. If we want to know what is going on at the sun right now, all we have to do is to wait for eight minutes. Nonetheless, we do have tocorrectfor the delay in our perception due to the finite speed of light before we can trust what we see. Now, this effect raises an interesting questionwhat is therealthing that we see? If seeing is believing, the stuff that we see should be the real thing. Then again, we know of the light travel time effect. So we should correct what we see before believing it. What then doesseeingmean? When we say we see something, what do we really mean? Seeing involves light, obviously. It is the finite (albeit very high) speed of light influences and distorts the way we see things, like the delay in seeing objects like stars. What is surprising (and seldom highlighted) is that when it comes to seeing moving objects, we cannot back-calculate the same way we take out the delay in seeing the sun. If we see a celestial body moving at an improbably high speed, we cannot figure out how fast and in what direction it isreallymoving without making further assumptions. One way of handling this difficulty is to ascribe the distortions in our perception to the fundamental properties of the arena of physicsspace and time. Another course of action is to accept the disconnection between our perception and the underlyingrealityand deal with it in some way. This disconnect between what we see and what is out there is not unknown to many philosophical schools of thought. Phenomenalism, for instance, holds the view that space and time are not objective realities. They are merely the medium of our perception. All the phenomena that happen in space and time are merely bundles of our perception. In other words, space and time are cognitive constructs arising from perception. Thus, all the physical properties that we ascribe to space and time can only apply to the phenomenal reality (the reality as we sense it). The noumenal reality (which holds the physical causes of our perception), by contrast, remains beyond our cognitive reach. One, almost accidental, difficulty in redefining the effects of the finite speed of light as the properties of space and time is that any effect that we do understand gets instantly relegated to the realm of optical illusions. For instance, the eight-minute delay in seeing the sun, because we can readily understand it and disassociate it from our perception using simple arithmetic, is considered a mere optical illusion. However, the distortions in our perception of fast moving objects, although originating from the same source are considered a property of space and time because they are more complex. At some point, we have to come to terms with the fact that when it comes to seeing the universe, there is no such thing as an optical illusion, which is probably what Goethe pointed out when he said, “Optical illusion is optical truth. The distinction (or lack thereof) between optical illusion and truth is one of the oldest debates in philosophy. After all, it is about the distinction between knowledge and reality. Knowledge is considered our view about something that, in reality, is “actually the case.” In other words, knowledge is a reflection, or a mental image of something external. In this picture, the external reality goes through a process of becoming our knowledge, which includes perception, cognitive activities, and the exercise of pure reason. This is the picture that physics has come to accept. While acknowledging that our perception may be imperfect, physics assumes that we can get closer and closer to the external reality through increasingly finer experimentation, and, more importantly, through better theorization. The Special and General Theories of Relativity are examples of brilliant applications of this view of reality where simple physical principles are relentlessly pursued using the formidable machine of pure reason to their logically inevitable conclusions. But there is another, competing view of knowledge and reality that has been around for a long time. This is the view that regards perceived reality as an internal cognitive representation of our sensory inputs. In this view, knowledge and perceived reality are both internal cognitive constructs, although we have come to think of them as separate. What is external is not the reality as we perceive it, but an unknowable entity giving rise to the physical causes behind sensory inputs. In this school of thought, we build our reality in two, often overlapping, steps. The first step consists of the process of sensing, and the second one is that of cognitive and logical reasoning. We can apply this view of reality and knowledge to science, but in order do so, we have to guess the nature of the absolute reality, unknowable as it is. The ramifications of these two different philosophical stances described above are tremendous. Since modern physics has embraced a non-phenomenalistic view of space and time, it finds itself at odds with that branch of philosophy. This chasm between philosophy and physics has grown to such a degree that the Nobel prize winning physicist, Steven Weinberg, wondered (in his bookDreams of a Final Theory”) why the contribution from philosophy to physics have been so surprisingly small. It also prompts philosophers to make statements like, “Whether ‘noumenal reality causes phenomenal realityor whether ‘noumenal reality is independent of our sensing itor whether ‘we sense noumenal reality,’ the problem remains that the concept of noumenal reality is a totally redundant concept for the analysis of science. From the perspective of cognitive neuroscience, everything we see, sense, feel and think is the result of the neuronal interconnections in our brain and the tiny electrical signals in them. This view must be right. What else is there? All our thoughts and worries, knowledge and beliefs, ego and reality, life and deatheverything is merely neuronal firings in the one and half kilograms of gooey, grey material that we call our brain. There is nothing else. Nothing! In fact, this view of reality in neuroscience is an exact echo of phenomenalism, which considers everything a bundle of perception or mental constructs. Space and time are also cognitive constructs in our brain, like everything else. They are mental pictures our brains concoct out of the sensory inputs that our senses receive. Generated from our sensory perception and fabricated by our cognitive process, the space-time continuum is the arena of physics. Of all our senses, sight is by far the dominant one. The sensory input to sight is light. In a space created by the brain out of the light falling on our retinas (or on the photo sensors of the Hubble telescope), is it a surprise that nothing can travel faster than light? This philosophical stance is the basis of my book, The Unreal Universe, which explores the common threads binding physics and philosophy. Such philosophical musings usually get a bad rap from us physicists. To physicists, philosophy is an entirely different field, another silo of knowledge, which holds no relevance to their endeavors. We need to change this belief and appreciate the overlap among different knowledge silos. It is in this overlap that we can expect to find great breakthroughs in human thought. The twist to this story of light and reality is that we seem to have known all this for a long time. Classical philosophical schools seem to have thought along lines very similar to Einstein’s reasonings. The role of light in creating our reality or universe is at the heart of Western religious thinking. A universe devoid of light is not simply a world where you have switched off the lights. It is indeed a universe devoid of itself, a universe that doesn’t exist. It is in this context that we have to understand the wisdom behind the statement thatthe earth was without form, and voiduntil God caused light to be, by sayingLet there be light. The Quran also says, “Allah is the light of the heavens and the earth,” which is mirrored in one of the ancient Hindu writings: “Lead me from darkness to light, lead me from the unreal to the real.The role of light in taking us from the unreal void (the nothingness) to a reality was indeed understood for a long, long time. Is it possible that the ancient saints and prophets knew things that we are only now beginning to uncover with all our supposed advances in knowledge? I know I may be rushing in where angels fear to tread, for reinterpreting the scriptures is a dangerous game. Such alien interpretations are seldom welcome in the theological circles. But I seek refuge in the fact that I am looking for concurrence in the metaphysical views of spiritual philosophies, without diminishing their mystical and theological value. The parallels between the noumenal-phenomenal distinction in phenomenalism and the Brahman-Maya distinction in Advaita are hard to ignore. This time-tested wisdom on the nature of reality from the repertoire of spirituality is now being reinvented in modern neuroscience, which treats reality as a cognitive representation created by the brain. The brain uses the sensory inputs, memory, consciousness, and even language as ingredients in concocting our sense of reality. This view of reality, however, is something physics is yet to come to terms with. But to the extent that its arena (space and time) is a part of reality, physics is not immune to philosophy. As we push the boundaries of our knowledge further and further, we are beginning to discover hitherto unsuspected and often surprising interconnections between different branches of human efforts. In the final analysis, how can the diverse domains of our knowledge be independent of each other when all our knowledge resides in our brain? Knowledge is a cognitive representation of our experiences. But then, so is reality; it is a cognitive representation of our sensory inputs. It is a fallacy to think that knowledge is our internal representation of an external reality, and therefore distinct from it. Knowledge and reality are both internal cognitive constructs, although we have come to think of them as separate. Recognizing and making use of the interconnections among the different domains of human endeavor may be the catalyst for the next breakthrough in our collective wisdom that we have been waiting for. # Food Prices and Terrible Choices Economists have too many hands. On the one hand, they may declare something good. On the other hand, they may say, “well, not so much.Some of them may have even a third or fourth hand. My ex-boss, an economist himself, once remarked that he wished he could chop off some of these hands. In the last couple of weeks, I plunged right into an ocean of economist hands as I sat down to do a minor research into this troubling phenomenon of skyrocketing food prices. The firsthandpointed out that the demand for food (and commodities in general) has surged due to the increase in the population and changing consumption patterns in the emerging giants of Asia. The well-known demand and supply paradigm explains the price surge, it would seem. Is it as simple as that? On the other hand, more and more food crops are being diverted into bio-fuel production. Is the bio-fuel demand the root cause? Bio-fuels are attractive because of the astronomical crude oil prices, which drive up the prices of everything. Is the recent OPEC windfall driving the price hikes? What about the food subsidies in wealthy nations that skew the market in their favour? Yet another economics hand puts the blame squarely on the supply side. It points an unwavering finger at the poor weather in food producing countries, and the panic measures imposed on the supply chain, such as export bans and smaller scale hoarding, that drive up the prices. I’m no economist, and I would like just one hand, one opinion, that I can count on. In my untrained view, I suspect that the speculation in commodities market may be driving the prices up. I felt vindicated in my suspicions when I read a recent US senate testimony where a well-known hedge fund manager, Michael Masters, shed light on the financial labyrinth of futures transactions and legal loopholes through which enormous profits were generated in commodity speculation. The real reasons behind the food crisis are likely to be a combination of all these factors. But the crisis itself is a silent tsunami sweeping the world, as the UN World Food Program puts it. Increase in the food prices, though unpleasant, is not such a big deal for a large number of Singaporeans. With our first world income, most of us spend about 20% of our salary on food. If it becomes 30% as a result of a 50% increase in the prices, we certainly won’t like it, but we won’t suffer that much. We may have to cut down on the taxi rides, or fine-dining, but it is not the end of our world. If we are in the top 10% of the households, we may not even notice the increase. The impact of the high food prices on our lifestyle will be minimal — say, a four-star holiday instead of a five-star one. It is a different story near the bottom. If we earn less than $1000 a month, and we are forced to spend$750 instead of \$500 on food, it may mean a choice between an MRT ride and legging it. At that level, the increase in food prices does hurt us as our grim choices become limited. But there are people in this world who face a much harsher reality as the prices shoot up with no end in sight. Their choices are often as terrible as Sophie’s choice. Which child goes to sleep hungry tonight? Medicine for the sick one or food for the rest? We are all powerless against the juggernaut of market forces creating the food crisis. Although we cannot realistically change the course of this silent tsunami, let’s at least try not to exacerbate the situation through waste. Buy only what you will use, and use only what you need to. Even if we cannot help those who will invariably go hungry, let’s not insult them by throwing away what they will die yearning for. Hunger is a terrible thing. If you don’t believe me, try fasting for a day. Well, try it even if you dofor it may help someone somewhere. # Light Travel Time Effects and Cosmological Features This unpublished article is a sequel to my earlier paper (also posted here asAre Radio Sources and Gamma Ray Bursts Luminal Booms?“). This blog version contains the abstract, introduction and conclusions. The full version of the article is available as a PDF file. . Abstract Light travel time effects (LTT) are an optical manifestation of the finite speed of light. They can also be considered perceptual constraints to the cognitive picture of space and time. Based on this interpretation of LTT effects, we recently presented a new hypothetical model for the temporal and spatial variation of the spectrum of Gamma Ray Bursts (GRB) and radio sources. In this article, we take the analysis further and show that LTT effects can provide a good framework to describe such cosmological features as the redshift observation of an expanding universe, and the cosmic microwave background radiation. The unification of these seemingly distinct phenomena at vastly different length and time scales, along with its conceptual simplicity, can be regarded as indicators of the curious usefulness of this framework, if not its validity. #### Introduction The finite speed of light plays an important part in how we perceive distance and speed. This fact should hardly come as a surprise because we do know that things are not as we see them. The sun that we see, for instance, is already eight minutes old by the time we see it. This delay is trivial; if we want to know what is going on at the sun now, all we have to do is to wait for eight minutes. We, nonetheless, have tocorrectfor this distortion in our perception due to the finite speed of light before we can trust what we see. What is surprising (and seldom highlighted) is that when it comes to sensing motion, we cannot back-calculate the same way we take out the delay in seeing the sun. If we see a celestial body moving at an improbably high speed, we cannot figure out how fast and in what direction it isreallymoving without making further assumptions. One way of handling this difficulty is to ascribe the distortions in our perception of motion to the fundamental properties of the arena of physicsspace and time. Another course of action is to accept the disconnection between our perception and the underlyingrealityand deal with it in some way. Exploring the second option, we assume an underlying reality that gives rise to our perceived picture. We further model this underlying reality as obeying classical mechanics, and work out our perceived picture through the apparatus of perception. In other words, we do not attribute the manifestations of the finite speed of light to the properties of the underlying reality. Instead, we work out our perceived picture that this model predicts and verify whether the properties we do observe can originate from this perceptual constraint. Space, the objects in it, and their motion are, by and large, the product of optical perception. One tends to take it for granted that perception arises from reality as one perceives it. In this article, we take the position that what we perceive is an incomplete or distorted picture of an underlying reality. Further, we are trying out classical mechanics for the the underlying reality (for which we use terms like absolute, noumenal or physical reality) that does cause our perception to see if it fits with our perceived picture (which we may refer to as sensed or phenomenal reality). Note that we are not implying that the manifestations of perception are mere delusions. They are not; they are indeed part of our sensed reality because reality is an end result of perception. This insight may be behind Goethe’s famous statement, “Optical illusion is optical truth. We applied this line of thinking to a physics problem recently. We looked at the spectral evolution of a GRB and found it to be remarkably similar to that in a sonic boom. Using this fact, we presented a model for GRB as our perception of aluminalboom, with the understanding that it is our perceived picture of reality that obeys Lorentz invariance and our model for the underlying reality (causing the perceived picture) may violate relativistic physics. The striking agreement between the model and the observed features, however, extended beyond GRBs to symmetric radio sources, which can also be regarded as perceptual effects of hypothetical luminal booms. In this article, we look at other implications of the model. We start with the similarities between the light travel time (LTT) effects and the coordinate transformation in Special Relativity (SR). These similarities are hardly surprising because SR is derived partly based on LTT effects. We then propose an interpretation of SR as a formalization of LTT effects and study a few observed cosmological phenomena in the light of this interpretation. #### Similarities between Light Travel Time Effects and SR Special relativity seeks a linear coordinate transformation between coordinate systems in motion with respect to each other. We can trace the origin of linearity to a hidden assumption on the nature of space and time built into SR, as stated by Einstein: “In the first place it is clear that the equations must be linear on account of the properties of homogeneity which we attribute to space and time.Because of this assumption of linearity, the original derivation of the transformation equations ignores the asymmetry between approaching and receding objects. Both approaching and receding objects can be described by two coordinate systems that are always receding from each other. For instance, if a system $K$ is moving with respect to another system $k$ along the positive X axis of $k$, then an object at rest in $K$ at a positive $x$ is receding while another object at a negative $x$ is approaching an observer at the origin of $k$. The coordinate transformation in Einstein’s original paper is derived, in part, a manifestation of the light travel time (LTT) effects and the consequence of imposing the constancy of light speed in all inertial frames. This is most obvious in the first thought experiment, where observers moving with a rod find their clocks not synchronized due to the difference in light travel times along the length of the rod. However, in the current interpretation of SR, the coordinate transformation is considered a basic property of space and time. One difficulty that arises from this interpretation of SR is that the definition of the relative velocity between the two inertial frames becomes ambiguous. If it is the velocity of the moving frame as measured by the observer, then the observed superluminal motion in radio jets starting from the core region becomes a violation of SR. If it is a velocity that we have to deduce by considering LT effects, then we have to employ the extra ad-hoc assumption that superluminality is forbidden. These difficulties suggest that it may be better to disentangle the light travel time effects from the rest of SR. In this section, we will consider space and time as a part of the cognitive model created by the brain, and argue that special relativity applies to the cognitive model. The absolute reality (of which the SR-like space-time is our perception) does not have to obey the restrictions of SR. In particular, objects are not restricted to subluminal speeds, but they may appear to us as though they are restricted to subluminal speeds in our perception of space and time. If we disentangle LTT effects from the rest of SR, we can understand a wide array of phenomena, as we shall see in this article. Unlike SR, considerations based on LTT effects result in intrinsically different set of transformation laws for objects approaching an observer and those receding from him. More generally, the transformation depends on the angle between the velocity of the object and the observer’s line of sight. Since the transformation equations based on LTT effects treat approaching and receding objects asymmetrically, they provide a natural solution to the twin paradox, for instance. #### Conclusions Because space and time are a part of a reality created out of light inputs to our eyes, some of their properties are manifestations of LTT effects, especially on our perception of motion. The absolute, physical reality presumably generating the light inputs does not have to obey the properties we ascribe to our perceived space and time. We showed that LTT effects are qualitatively identical to those of SR, noting that SR only considers frames of reference receding from each other. This similarity is not surprising because the coordinate transformation in SR is derived based partly on LTT effects, and partly on the assumption that light travels at the same speed with respect to all inertial frames. In treating it as a manifestation of LTT, we did not address the primary motivation of SR, which is a covariant formulation of Maxwell’s equations. It may be possible to disentangle the covariance of electrodynamics from the coordinate transformation, although it is not attempted in this article. Unlike SR, LTT effects are asymmetric. This asymmetry provides a resolution to the twin paradox and an interpretation of the assumed causality violations associated with superluminality. Furthermore, the perception of superluminality is modulated by LTT effects, and explains $gamma$ ray bursts and symmetric jets. As we showed in the article, perception of superluminal motion also holds an explanation for cosmological phenomena like the expansion of the universe and cosmic microwave background radiation. LTT effects should be considered as a fundamental constraint in our perception, and consequently in physics, rather than as a convenient explanation for isolated phenomena. Given that our perception is filtered through LTT effects, we have to deconvolute them from our perceived reality in order to understand the nature of the absolute, physical reality. This deconvolution, however, results in multiple solutions. Thus, the absolute, physical reality is beyond our grasp, and any assumed properties of the absolute reality can only be validated through how well the resultant perceived reality agrees with our observations. In this article, we assumed that the underlying reality obeys our intuitively obvious classical mechanics and asked the question how such a reality would be perceived when filtered through light travel time effects. We demonstrated that this particular treatment could explain certain astrophysical and cosmological phenomena that we observe. The coordinate transformation in SR can be viewed as a redefinition of space and time (or, more generally, reality) in order to accommodate the distortions in our perception of motion due to light travel time effects. One may be tempted to argue that SR applies to therealspace and time, not our perception. This line of argument begs the question, what is real? Reality is only a cognitive model created in our brain starting from our sensory inputs, visual inputs being the most significant. Space itself is a part of this cognitive model. The properties of space are a mapping of the constraints of our perception. The choice of accepting our perception as a true image of reality and redefining space and time as described in special relativity indeed amounts to a philosophical choice. The alternative presented in the article is inspired by the view in modern neuroscience that reality is a cognitive model in the brain based on our sensory inputs. Adopting this alternative reduces us to guessing the nature of the absolute reality and comparing its predicted projection to our real perception. It may simplify and elucidate some theories in physics and explain some puzzling phenomena in our universe. However, this option is yet another philosophical stance against the unknowable absolute reality. # Are Radio Sources and Gamma Ray Bursts Luminal Booms? This article was published in the International Journal of Modern Physics D (IJMPD) in 2007. It soon became the Top Accessed Article of the journal by Jan 2008. Although it might seem like a hard core physics article, it is in fact an application of the philosophical insight permeating this blog and my book. This blog version contains the abstract, introduction and conclusions. The full version of the article is available as a PDF file. Journal Reference: IJMP-D Vol. 16, No. 6 (2007) pp. 983–1000. . Abstract The softening of the GRB afterglow bears remarkable similarities to the frequency evolution in a sonic boom. At the front end of the sonic boom cone, the frequency is infinite, much like a Gamma Ray Burst (GRB). Inside the cone, the frequency rapidly decreases to infrasonic ranges and the sound source appears at two places at the same time, mimicking the double-lobed radio sources. Although aluminalboom violates the Lorentz invariance and is therefore forbidden, it is tempting to work out the details and compare them with existing data. This temptation is further enhanced by the observed superluminality in the celestial objects associated with radio sources and some GRBs. In this article, we calculate the temporal and spatial variation of observed frequencies from a hypothetical luminal boom and show remarkable similarity between our calculations and current observations. #### Introduction A sonic boom is created when an object emitting sound passes through the medium faster than the speed of sound in that medium. As the object traverses the medium, the sound it emits creates a conical wavefront, as shown in Figure 1. The sound frequency at this wavefront is infinite because of the Doppler shift. The frequency behind the conical wavefront drops dramatically and soon reaches the infrasonic range. This frequency evolution is remarkably similar to afterglow evolution of a gamma ray burst (GRB). Gamma Ray Bursts are very brief, but intense flashes of $\gamma$ rays in the sky, lasting from a few milliseconds to several minutes, and are currently believed to emanate from cataclysmic stellar collapses. The short flashes (the prompt emissions) are followed by an afterglow of progressively softer energies. Thus, the initial $\gamma$ rays are promptly replaced by X-rays, light and even radio frequency waves. This softening of the spectrum has been known for quite some time, and was first described using a hypernova (fireball) model. In this model, a relativistically expanding fireball produces the $\gamma$ emission, and the spectrum softens as the fireball cools down. The model calculates the energy released in the $\gamma$ region as $10^ {53}$$10^ {54}$ ergs in a few seconds. This energy output is similar to about 1000 times the total energy released by the sun over its entire lifetime. More recently, an inverse decay of the peak energy with varying time constant has been used to empirically fit the observed time evolution of the peak energy using a collapsar model. According to this model, GRBs are produced when the energy of highly relativistic flows in stellar collapses are dissipated, with the resulting radiation jets angled properly with respect to our line of sight. The collapsar model estimates a lower energy output because the energy release is not isotropic, but concentrated along the jets. However, the rate of the collapsar events has to be corrected for the fraction of the solid angle within which the radiation jets can appear as GRBs. GRBs are observed roughly at the rate of once a day. Thus, the expected rate of the cataclysmic events powering the GRBs is of the order of $10^4$$10^6$ per day. Because of this inverse relationship between the rate and the estimated energy output, the total energy released per observed GRB remains the same. If we think of a GRB as an effect similar to the sonic boom in supersonic motion, the assumed cataclysmic energy requirement becomes superfluous. Another feature of our perception of supersonic object is that we hear the sound source at two different location as the same time, as illustrated in Figure 2. This curious effect takes place because the sound waves emitted at two different points in the trajectory of the supersonic object reach the observer at the same instant in time. The end result of this effect is the perception of a symmetrically receding pair of sound sources, which, in the luminal world, is a good description of symmetric radio sources (Double Radio source Associated with Galactic Nucleus or DRAGN). Radio Sources are typically symmetric and seem associated with galactic cores, currently considered manifestations of space-time singularities or neutron stars. Different classes of such objects associated with Active Galactic Nuclei (AGN) were found in the last fifty years. Figure 3 shows the radio galaxy Cygnus A, an example of such a radio source and one of the brightest radio objects. Many of its features are common to most extragalactic radio sources: the symmetric double lobes, an indication of a core, an appearance of jets feeding the lobes and the hotspots. Some researchers have reported more detailed kinematical features, such as the proper motion of the hotspots in the lobes. Symmetric radio sources (galactic or extragalactic) and GRBs may appear to be completely distinct phenomena. However, their cores show a similar time evolution in the peak energy, but with vastly different time constants. The spectra of GRBs rapidly evolve from $\gamma$ region to an optical or even RF afterglow, similar to the spectral evolution of the hotspots of a radio source as they move from the core to the lobes. Other similarities have begun to attract attention in the recent years. This article explores the similarities between a hypotheticalluminalboom and these two astrophysical phenomena, although such a luminal boom is forbidden by the Lorentz invariance. Treating GRB as a manifestation of a hypothetical luminal boom results in a model that unifies these two phenomena and makes detailed predictions of their kinematics. #### Conclusions In this article, we looked at the spatio-temporal evolution of a supersonic object (both in its position and the sound frequency we hear). We showed that it closely resembles GRBs and DRAGNs if we were to extend the calculations to light, although a luminal boom would necessitate superluminal motion and is therefore forbidden. This difficulty notwithstanding, we presented a unified model for Gamma Ray Bursts and jet like radio sources based on bulk superluminal motion. We showed that a single superluminal object flying across our field of vision would appear to us as the symmetric separation of two objects from a fixed core. Using this fact as the model for symmetric jets and GRBs, we explained their kinematic features quantitatively. In particular, we showed that the angle of separation of the hotspots was parabolic in time, and the redshifts of the two hotspots were almost identical to each other. Even the fact that the spectra of the hotspots are in the radio frequency region is explained by assuming hyperluminal motion and the consequent redshift of the black body radiation of a typical star. The time evolution of the black body radiation of a superluminal object is completely consistent with the softening of the spectra observed in GRBs and radio sources. In addition, our model explains why there is significant blue shift at the core regions of radio sources, why radio sources seem to be associated with optical galaxies and why GRBs appear at random points with no advance indication of their impending appearance. Although it does not address the energetics issues (the origin of superluminality), our model presents an intriguing option based on how we would perceive hypothetical superluminal motion. We presented a set of predictions and compared them to existing data from DRAGNs and GRBs. The features such as the blueness of the core, symmetry of the lobes, the transient $\gamma$ and X-Ray bursts, the measured evolution of the spectra along the jet all find natural and simple explanations in this model as perceptual effects. Encouraged by this initial success, we may accept our model based on luminal boom as a working model for these astrophysical phenomena. It has to be emphasized that perceptual effects can masquerade as apparent violations of traditional physics. An example of such an effect is the apparent superluminal motion, which was explained and anticipated within the context of the special theory of relativity even before it was actually observed. Although the observation of superluminal motion was the starting point behind the work presented in this article, it is by no means an indication of the validity of our model. The similarity between a sonic boom and a hypothetical luminal boom in spatio-temporal and spectral evolution is presented here as a curious, albeit probably unsound, foundation for our model. One can, however, argue that the special theory of relativity (SR) does not deal with superluminality and, therefore, superluminal motion and luminal booms are not inconsistent with SR. As evidenced by the opening statements of Einstein’s original paper, the primary motivation for SR is a covariant formulation of Maxwell’s equations, which requires a coordinate transformation derived based partly on light travel time (LTT) effects, and partly on the assumption that light travels at the same speed with respect to all inertial frames. Despite this dependence on LTT, the LTT effects are currently assumed to apply on a space-time that obeys SR. SR is a redefinition of space and time (or, more generally, reality) in order to accommodate its two basic postulates. It may be that there is a deeper structure to space-time, of which SR is only our perception, filtered through the LTT effects. By treating them as an optical illusion to be applied on a space-time that obeys SR, we may be double counting them. We may avoid the double counting by disentangling the covariance of Maxwell’s equations from the coordinate transformations part of SR. Treating the LTT effects separately (without attributing their consequences to the basic nature of space and time), we can accommodate superluminality and obtain elegant explanations of the astrophysical phenomena described in this article. Our unified explanation for GRBs and symmetric radio sources, therefore, has implications as far reaching as our basic understanding of the nature of space and time. Photo by NASA Goddard Photo and Video # Constraints of Perception and Cognition in Relativistic Physics This post is an abridged online version of my article that appears in Galilean Electrodynamics in November, 2008. [Ref: Galilean Electrodynamics, Vol. 19, No. 6, Nov/Dec 2008, pp: 103–117] () Cognitive neuroscience treats space and time as our brain’s representation of our sensory inputs. In this view, our perceptual reality is only a distant and convenient mapping of the physical processes causing the sensory inputs. Sound is a mapping of auditory inputs, and space is a representation of visual inputs. Any limitation in the chain of sensing has a specific manifestation on the cognitive representation that is our reality. One physical limitation of our visual sensing is the finite speed of light, which manifests itself as a basic property of our space-time. In this article, we look at the consequences of the limited speed of our perception, namely the speed of light, and show that they are remarkably similar to the coordinate transformation in special relativity. From this observation, and inspired by the notion that space is merely a cognitive model created out of light signal inputs, we examine the implications of treating special relativity theory as a formalism for describing the perceptual effects due to the finite speed of light. Using this framework, we show that we can unify and explain a wide array of seemingly unrelated astrophysical and cosmological phenomena. Once we identify the manifestations of the limitations in our perception and cognitive representation, we can understand the consequent constraints on our space and time, leading to a new understanding of astrophysics and cosmology. Key words: cognitive neuroscience; reality; special relativity; light travel time effect; gamma rays bursts; cosmic microwave background radiation. #### 1. Introduction Our reality is a mental picture that our brain creates, starting from our sensory inputs [1]. Although this cognitive map is often assumed to be a faithful image of the physical causes behind the sensing process, the causes themselves are entirely different from the perceptual experience of sensing. The difference between the cognitive representation and their physical causes is not immediately obvious when we consider our primary sense of sight. But, we can appreciate the difference by looking at the olfactory and auditory senses because we can use our cognitive model based on sight in order to understand the workings of the ‘lessersenses. Odors, which may appear to be a property of the air we breathe, are in fact our brain’s representation of the chemical signatures that our noses sense. Similarly, sound is not an intrinsic property of a vibrating body, but our brain’s mechanism to represent the pressure waves in the air that our ears sense. Table I shows the chain from the physical causes of the sensory input to the final reality as the brain creates it. Although the physical causes can be identified for the olfactory and auditory chains, they are not easily discerned for visual process. Since sight is the most powerful sense we possess, we are obliged to accept our brain’s representation of visual inputs as the fundamental reality. While our visual reality provides an excellent framework for physical sciences, it is important to realize that the reality itself is a model with potential physical or physiological limitations and distortions. The tight integration between the physiology of perception and its representation in the brain was proven recently in a clever experiment using the tactile funneling illusion [2]. This illusion results in a single tactile sensation at the focal point at the center of a stimulus pattern even though no stimulation is applied at that site. In the experiment, the brain activation region corresponded to the focal point where the sensation was perceived, rather than the points where the stimuli were applied, proving that the brain registered perceptions, not the physical causes of the perceived reality. In other words, for the brain, there is no difference between applying the pattern of the stimuli and applying only one stimulus at the center of the pattern. The brain maps the sensory inputs to regions that correspond to their perception, rather than the regions that physiologically correspond to the sensory stimuli. Sense modality: Physical cause: Sensed signal: Brain’s model: Olfactory Chemicals Chemical reactions Smells Auditory Vibrations Pressure waves Sounds Visual Unknown Light Space, time reality Table I: The brain’s representation of different sensory inputs. Odors are a representation of chemical compositions and concentration our nose senses. Sounds are a mapping of the air pressure waves produced by a vibrating object. In sight, we do not know the physical reality, our representation is space, and possibly time. The neurological localization of different aspects of reality has been established in neuroscience by lesion studies. The perception of motion (and the consequent basis of our sense of time), for instance, is so localized that a tiny lesion can erase it completely. Cases of patients with such specific loss of a part of reality [1] illustrate the fact that our experience of reality, every aspect of it, is indeed a creation of the brain. Space and time are aspects of the cognitive representation in our brain. Space is a perceptual experience much like sound. Comparisons between the auditory and visual modes of sensing can be useful in understanding the limitations of their representations in the brain. One limitation is the input ranges of the sensory organs. Ears are sensitive in the frequency range 20Hz-20kHz, and eyes are limited to the visible spectrum. Another limitation, which may exist in specific individuals, is an inadequate representation of the inputs. Such a limitation can lead to tone-deafness and color-blindness, for instance. The speed of the sense modality also introduces an effect, such as the time lag between seeing an event and hearing the corresponding sound. For visual perception, a consequence of the finite speed of light is called a Light Travel Time (LTT) effect. LLT offers one possible interpretation for the observed superluminal motion in certain celestial objects [3,4]: when an object approaches the observer at a shallow angle, it may appear to move much faster than reality [5] due to LTT. Other consequences of the LTT effects in our perception are remarkably similar to the coordinate transformation of the special relativity theory (SRT). These consequences include an apparent contraction of a receding object along its direction of motion and a time dilation effect. Furthermore, a receding object can never appear to be going faster than the speed of light, even if its real speed is superluminal. While SRT does not explicitly forbid it, superluminality is understood to lead to time travel and the consequent violations of causality. An apparent violation of causality is one of the consequences of LTT, when the superluminal object is approaching the observer. All these LTT effects are remarkably similar to effects predicted by SRT, and are currently taken as ‘confirmationthat space-time obeys SRT. But instead, space-time may have a deeper structure that, when filtered through LTT effects, results in our perception that space-time obeys SRT. Once we accept the neuroscience view of reality as a representation of our sensory inputs, we can understand why the speed of light figures so prominently in our physical theories. The theories of physics are a description of reality. Reality is created out of the readings from our senses, especially our eyes. They work at the speed of light. Thus the sanctity accorded to the speed of light is a feature only of our reality, not the absolute, ultimate reality that our senses are striving to perceive. When it comes to physics that describes phenomena well beyond our sensory ranges, we really have to take into account the role that our perception and cognition play in seeing them. The Universe as we see it is only a cognitive model created out of the photons falling on our retina or on the photo-sensors of the Hubble telescope. Because of the finite speed of the information carrier (namely photons), our perception is distorted in such a way as to give us the impression that space and time obey SRT. They do, but space and time are not the absolute reality. “Space and time are modes by which we think and not conditions in which we live,” as Einstein himself put it. Treating our perceived reality as our brain’s representation of our visual inputs (filtered through the LTT effect), we will see that all the strange effects of the coordinate transformation in SRT can be understood as the manifestations of the finite speed of our senses in our space and time. Furthermore, we will show that this line of thinking leads to natural explanations for two classes of astrophysical phenomena: Gamma Ray Bursts, which are very brief, but intense flashes of $\gamma$ rays, currently believed to emanate from cataclysmic stellar collapses, and Radio Sources, which are typically symmetric and seem associated with galactic cores, currently considered manifestations of space-time singularities or neutron stars. These two astrophysical phenomena appear distinct and unrelated, but they can be unified and explained using LTT effects. This article presents such a unified quantitative model. It will also show that the cognitive limitations to reality due to LTT effects can provide qualitative explanations for such cosmological features as the apparent expansion of the Universe and the Cosmic Microwave Background Radiation (CMBR). Both these phenomena can be understood as related to our perception of superluminal objects. It is the unification of these seemingly distinct phenomena at vastly different length and time scales, along with its conceptual simplicity, that we hold as the indicators of validity of this framework. #### 2. Similarities between LTT Effects & SRT The coordinate transformation derived in Einstein’s original paper [6] is, in part, a manifestation of the LTT effects and the consequence of imposing the constancy of light speed in all inertial frames. This is most obvious in the first thought experiment, where observers moving with a rod find their clocks not synchronized due to the difference in LTT’s along the length of the rod. However, in the current interpretation of SRT, the coordinate transformation is considered a basic property of space and time. One difficulty that arises from this formulation is that the definition of the relative velocity between the two inertial frames becomes ambiguous. If it is the velocity of the moving frame as measured by the observer, then the observed superluminal motion in radio jets starting from the core region becomes a violation of SRT. If it is a velocity that we have to deduce by considering LTT effects, then we have to employ the extra ad-hoc assumption that superluminality is forbidden. These difficulties suggest that it may be better to disentangle the LTT effects from the rest of SRT. Although not attempted in this paper, the primary motivation for SRT, namely the covariance of Maxwell’s equations, may be accomplished even without attributing LTT effects to the properties of space and time. In this Section, we will consider space and time as a part of the cognitive model created by the brain, and illustrate that SRT applies to the cognitive model. The absolute reality (of which the SRT-like space-time is our perception) does not have to obey the restrictions of SRT. In particular, objects are not restricted to subluminal speeds, even though they may appear to us as if they are restricted to subluminal speeds in our perception of space and time. If we disentangle LTT effects from the rest of SRT, we can understand a wide array of phenomena, as shown in this article. SRT seeks a linear coordinate transformation between coordinate systems in motion with respect to each other. We can trace the origin of linearity to a hidden assumption on the nature of space and time built into SRT, as stated by Einstein [6]: “In the first place it is clear that the equations must be linear on account of the properties of homogeneity which we attribute to space and time.Because of this assumption of linearity, the original derivation of the transformation equations ignores the asymmetry between approaching and receding objects and concentrates on receding objects. Both approaching and receding objects can be described by two coordinate systems that are always receding from each other. For instance, if a system K is moving with respect to another system k along the positive X axis of k, then an object at rest in K at a positive x is approaching an observer at the origin of k. Unlike SRT, considerations based on LTT effects result in intrinsically different set of transformation laws for objects approaching an observer and those receding from him. More generally, the transformation depends on the angle between the velocity of the object and the observer’s line of sight. Since the transformation equations based on LTT effects treat approaching and receding objects asymmetrically, they provide a natural solution to the twin paradox, for instance. ##### 2.1 First Order Perceptual Effects For approaching and receding objects, the relativistic effects are second order in speed $\beta$, and speed typically appears as $\sqrt{1-\beta^2}$. The LTT effects, on the other hand, are first order in speed. The first order effects have been studied in the last fifty years in terms of the appearance of a relativistically moving extended body [7-15]. It has also been suggested that the relativistic Doppler effect can be considered the geometric mean [16] of more basic calculations. The current belief is that the first order effects are an optical illusion to be taken out of our perception of reality. Once these effects are taken out or ‘deconvolvedfrom the observations, the ‘realspace and time are assumed to obey SRT. Note that this assumption is impossible to verify because the deconvolution is an ill-posed problemthere are multiple solutions to the absolute reality that all result in the same perceptual picture. Not all the solutions obey SRT. The notion that it is the absolute reality that obeys SRT ushers in a deeper philosophical problem. This notion is tantamount to insisting that space and time are in fact ‘intuitionsbeyond sensory perception rather than a cognitive picture created by our brain out of the sensory inputs it receives. A formal critique of the Kantian intuitions of space and time is beyond the scope of this article. Here, we take the position that it is our observed or perceived reality that obeys SRT and explore where it leads us. In other words, we assume that SRT is nothing but a formalization of the perceptual effects. These effects are not first order in speed when the object is not directly approaching (or receding from) the observer, as we will see later. We will show in this article that a treatment of SRT as a perceptual effect will give us natural solution for astrophysical phenomena like gamma ray bursts and symmetric radio jets. ##### 2.2 Perception of Speed We first look at how the perception of motion is modulated by LTT effects. As remarked earlier, the transformation equations of SRT treat only objects receding from the observer. For this reason, we first consider a receding object, flying away from the observer at a speed $\beta$ of the object depends on the real speed b (as shown in Appendix A.1): $\beta_O ,=, \frac{\beta}{1,+,\beta}$ (1) $\lim_{\beta\to\infty} \beta_O ,=, 1$ (2) Thus, due to LTT effects, an infinite real velocity gets mapped to an apparent velocity $\beta_O=1$. In other words, no object can appear to travel faster than the speed of light, entirely consistent with SRT. Physically, this apparent speed limit amounts to a mapping of $c$ to $\infty$. This mapping is most obvious in its consequences. For instance, it takes an infinite amount of energy to accelerate an object to an apparent speed $\beta_O=1$ because, in reality, we are accelerating it to an infinite speed. This infinite energy requirement can also be viewed as the relativistic mass changing with speed, reaching $\infty$ at $\beta_O=1$. Einstein explained this mapping as: “For velocities greater than that of light our deliberations become meaningless; we shall, however, find in what follows, that the velocity of light in our theory plays the part, physically, of an infinitely great velocity.” Thus, for objects receding from the observer, the effects of LTT are almost identical to the consequences of SRT, in terms of the perception of speed. ##### Time Dilation Figure 1:. Comparison between light travel time (LTT) effects and the predictions of the special theory of relativity (SR). The X-axis is the apparent speed and the Y-axis shows the relative time dilation or length contraction. LTT effects influence the way time at the moving object is perceived. Imagine an object receding from the observer at a constant rate. As it moves away, the successive photons emitted by the object take longer and longer to reach the observer because they are emitted at farther and farther away. This travel time delay gives the observer the illusion that time is flowing slower for the moving object. It can be easily shown (see Appendix A.2) that the time interval observed $\Delta t_O$ is related to the real time interval $\Delta t$ as: $\frac{\Delta t_O}{\Delta t} ,=, \frac{1}{1-\beta_O}$ (3) for an object receding from the observer ($\theta=\pi$). This observed time dilation is plotted in Fig. 1, where it is compared to the time dilation predicted in SR. Note that the time dilation due to LTT has a bigger magnitude than the one predicted in SR. However, the variation is similar, with both time dilations tending to $\infty$ as the observed speed tends to $c$. ##### 2.4 Length Contraction The length of an object in motion also appears different due to LTT effects. It can be shown (see Appendix A.3) that observed length $d_O$ as: $\frac{d_O}{d} ,=, {1-\beta_O}$ (4) for an object receding from the observer with an apparent speed of $\beta_O$. This equation also is plotted in Fig. 1. Note again that the LTT effects are stronger than the ones predicted in SRT. Fig. 1 illustrates that both time dilation and Lorentz contraction can be thought of as LTT effects. While the actual magnitudes of LTT effects are larger than what SRT predicts, their qualitative dependence on speed is almost identical. This similarity is not surprising because the coordinate transformation in SRT is partly based on LTT effects. If LTT effects are to be applied, as an optical illusion, on top of the consequences of SRT as currently believed, then the total observed length contraction and time dilation will be significantly more than the SRT predictions. ##### 2.5 Doppler Shift The rest of the article (the sections up to Conclusions) has been abridged and can be read in the PDF version. () #### 5 Conclusions In this article, we started with an insight from cognitive neuroscience about the nature of reality. Reality is a convenient representation that our brain creates out of our sensory inputs. This representation, though convenient, is an incredibly distant experiential mapping of the actual physical causes that make up the inputs to our senses. Furthermore, limitations in the chain of sensing and perception map to measurable and predictable manifestations to the reality we perceive. One such fundamental constraint to our perceived reality is the speed of light, and the corresponding manifestations, LTT effects. Because space and time are a part of a reality created out of light inputs to our eyes, some of their properties are manifestations of LTT effects, especially on our perception of motion. The absolute, physical reality generating the light inputs does not obey the properties we ascribe to our perceived space and time. We showed that LTT effects are qualitatively identical to those of SRT, noting that SRT only considers frames of reference receding from each other. This similarity is not surprising because the coordinate transformation in SRT is derived based partly on LTT effects, and partly on the assumption that light travels at the same speed with respect to all inertial frames. In treating it as a manifestation of LTT, we did not address the primary motivation of SRT, which is a covariant formulation of Maxwell’s equations, as evidenced by the opening statements of Einstein’s original paper [6]. It may be possible to disentangle the covariance of electrodynamics from the coordinate transformation, although it is not attempted in this article. Unlike SRT, LTT effects are asymmetric. This asymmetry provides a resolution to the twin paradox and an interpretation of the assumed causality violations associated with superluminality. Furthermore, the perception of superluminality is modulated by LTT effects, and explains g ray bursts and symmetric jets. As we showed in the article, perception of superluminal motion also holds an explanation for cosmological phenomena like the expansion of the Universe and cosmic microwave background radiation. LTT effects should be considered as a fundamental constraint in our perception, and consequently in physics, rather than as a convenient explanation for isolated phenomena. Given that our perception is filtered through LTT effects, we have to deconvolute them from our perceived reality in order to understand the nature of the absolute, physical reality. This deconvolution, however, results in multiple solutions. Thus, the absolute, physical reality is beyond our grasp, and any assumed properties of the absolute reality can only be validated through how well the resultant perceived reality agrees with our observations. In this article, we assumed that the absolute reality obeys our intuitively obvious classical mechanics and asked the question how such a reality would be perceived when filtered through LTT effects. We demonstrated that this particular treatment could explain certain astrophysical and cosmological phenomena that we observe. The distinction between the different notions of velocity, including the proper velocity and the Einsteinian velocity, was the subject matter of a recent issue of this journal [33]. The coordinate transformation in SRT should be viewed as a redefinition of space and time (or, more generally, reality) in order to accommodate the distortions in our perception of motion due to LTT effects. The absolute reality behind our perception is not subject to restrictions of SRT. One may be tempted to argue that SRT applies to the ‘realspace and time, not our perception. This line of argument begs the question, what is real? Reality is nothing but a cognitive model created in our brain starting from our sensory inputs, visual inputs being the most significant. Space itself is a part of this cognitive model. The properties of space are a mapping of the constraints of our perception. We have no access to a reality beyond our perception. The choice of accepting our perception as a true image of reality and redefining space and time as described in SRT indeed amounts to a philosophical choice. The alternative presented in the article is prompted by the view in modern neuroscience that reality is a cognitive model in the brain based on our sensory inputs. Adopting this alternative reduces us to guessing the nature of the absolute reality and comparing its predicted projection to our real perception. It may simplify and elucidate some theories in physics and explain some puzzling phenomena in our Universe. However, this option is yet another philosophical stance against the unknowable absolute reality. #### References [1] V.S. Ramachandran, “The Emerging Mind: Reith Lectures on Neuroscience” (BBC, 2003). [2] L.M. Chen, R.M. Friedman, and A. W. Roe, Science 302, 881 (2003). [3] J.A. Biretta, W.B. Sparks, and F. Macchetto, ApJ 520, 621 (1999). [4] A.J. Zensus, ARA&A 35, 607 (1997). [5] M. Rees, Nature 211, 468 (1966). 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Whitney, Galilean Electrodynamics, Special Issues 3, Editor’s Essays, Winter 2005. # The Unreal Universe — Seeing Light in Science and Spirituality We know that our universe is a bit unreal. The stars we see in the night sky, for instance, are not really there. They may have moved or even died by the time we get to see them. This delay is due to the time it takes for light from the distant stars and galaxies to reach us. We know of this delay. The same delay in seeing has a lesser known manifestation in the way we perceive moving objects. It distorts our perception such that something coming towards us would look as though it is coming in faster. Strange as it may sound, this effect has been observed in astrophysical studies. Some of the heavenly bodies do look as though they are moving several times the speed of light, while theirrealspeed is probably a lot lower. Now, this effect raises an interesting questionwhat is therealspeed? If seeing is believing, the speed we see should be the real speed. Then again, we know of the light travel time effect. So we should correct the speed we see before believing it. What then doesseeingmean? When we say we see something, what do we really mean? #### Light in Physics Seeing involves light, obviously. The finite speed of light influences and distorts the way we see things. This fact should hardly come as a surprise because we do know that things are not as we see them. The sun that we see is already eight minutes old by the time we see it. This delay is not a big deal; if we want to know what is going on at the sun now, all we have to do is to wait for eight minutes. We, nonetheless, have tocorrectfor the distortions in our perception due to the finite speed of light before we can trust what we see. What is surprising (and seldom highlighted) is that when it comes to sensing motion, we cannot back-calculate the same way we take out the delay in seeing the sun. If we see a celestial body moving at an improbably high speed, we cannot figure out how fast and in what direction it isreallymoving without making further assumptions. One way of handling this difficulty is to ascribe the distortions in our perception to the fundamental properties of the arena of physicsspace and time. Another course of action is to accept the disconnection between our perception and the underlyingrealityand deal with it in some way. Einstein chose the first route. In his groundbreaking paper over a hundred years ago, he introduced the special theory of relativity, in which he attributed the manifestations of the finite speed of light to the fundamental properties of space and time. One core idea in special relativity (SR) is that the notion of simultaneity needs to be redefined because it takes some time for light from an event at a distant place to reach us, and we become aware of the event. The concept of “Now” doesn’t make much sense, as we saw, when we speak of an event happening in the sun, for instance. Simultaneity is relative. Einstein defined simultaneity using the instants in time we detect the event. Detection, as he defined it, involves a round-trip travel of light similar to Radar detection. We send out light, and look at the reflection. If the reflected light from two events reaches us at the same instant, they are simultaneous. Another way of defining simultaneity is using sensingwe can call two events simultaneous if the light from them reaches us at the same instant. In other words, we can use the light generated by the objects under observation rather than sending light to them and looking at the reflection. This difference may sound like a hair-splitting technicality, but it does make an enormous difference in the predictions we can make. Einstein’s choice results in a mathematical picture that has many desirable properties, thereby making further development elegant. The other possibility has an advantage when it comes to describing objects in motion because it corresponds better with how we measure them. We don’t use Radar to see the stars in motion; we merely sense the light (or other radiation) coming from them. But this choice of using a sensory paradigm, rather than Radar-like detection, to describe the universe results in a slightly uglier mathematical picture. The mathematical difference spawns different philosophical stances, which in turn percolate to the understanding of our physical picture of reality. As an illustration, let us look at an example from astrophysics. Suppose we observe (through a radio telescope, for instance) two objects in the sky, roughly of the same shape and properties. The only thing we know for sure is that the radio waves from two different points in the sky reach the radio telescope at the same instant in time. We can guess that the waves started their journey quite a while ago. For symmetric objects, if we assume (as we routinely do) that the waves started the journey roughly at the same instant in time, we end up with a picture of tworealsymmetric lobes more or less the way see them. But there is different possibility that the waves originated from the same object (which is in motion) at two different instants in time, reaching the telescope at the same instant. This possibility explains some spectral and temporal properties of such symmetric radio sources, which is what I mathematically described in a recent physics article. Now, which of these two pictures should we take as real? Two symmetric objects as we see them or one object moving in such a way as to give us that impression? Does it really matter which one isreal”? Doesrealmean anything in this context? The philosophical stance in implied in special relativity answers this question unequivocally. There is an unambiguous physical reality from which we get the two symmetric radio sources, although it takes a bit of mathematical work to get to it. The mathematics rules out the possibility of a single object moving in such a fashion as to mimic two objects. Essentially, what we see is what is out there. On the other hand, if we define simultaneity using concurrent arrival of light, we will be forced to admit the exact opposite. What we see is pretty far from what is out there. We will confess that we cannot unambiguously decouple the distortions due to the constraints in perception (the finite speed of light being the constraint of interest here) from what we see. There are multiple physical realities that can result in the same perceptual picture. The only philosophical stance that makes sense is the one that disconnects the sensed reality and the causes behind what is being sensed. This disconnect is not uncommon in philosophical schools of thought. Phenomenalism, for instance, holds the view that space and time are not objective realities. They are merely the medium of our perception. All the phenomena that happen in space and time are merely bundles of our perception. In other words, space and time are cognitive constructs arising from perception. Thus, all the physical properties that we ascribe to space and time can only apply to the phenomenal reality (the reality as we sense it). The noumenal reality (which holds the physical causes of our perception), by contrast, remains beyond our cognitive reach. The ramifications of the two different philosophical stances described above are tremendous. Since modern physics seems to embrace a non-phenomenalistic view of space and time, it finds itself at odds with that branch of philosophy. This chasm between philosophy and physics has grown to such a degree that the Nobel prize winning physicist, Steven Weinberg, wondered (in his bookDreams of a Final Theory”) why the contribution from philosophy to physics have been so surprisingly small. It also prompts philosophers to make statements like, “Whether ‘noumenal reality causes phenomenal realityor whether ‘noumenal reality is independent of our sensing itor whether ‘we sense noumenal reality,’ the problem remains that the concept of noumenal reality is a totally redundant concept for the analysis of science. One, almost accidental, difficulty in redefining the effects of the finite speed of light as the properties of space and time is that any effect that we do understand gets instantly relegated to the realm of optical illusions. For instance, the eight-minute delay in seeing the sun, because we readily understand it and disassociate from our perception using simple arithmetic, is considered a mere optical illusion. However, the distortions in our perception of fast moving objects, although originating from the same source are considered a property of space and time because they are more complex. We have to come to terms with the fact that when it comes to seeing the universe, there is no such thing as an optical illusion, which is probably what Goethe pointed out when he said, “Optical illusion is optical truth. The distinction (or lack thereof) between optical illusion and truth is one of the oldest debates in philosophy. After all, it is about the distinction between knowledge and reality. Knowledge is considered our view about something that, in reality, is “actually the case.” In other words, knowledge is a reflection, or a mental image of something external, as shown in the figure below. In this picture, the black arrow represents the process of creating knowledge, which includes perception, cognitive activities, and the exercise of pure reason. This is the picture that physics has come to accept. While acknowledging that our perception may be imperfect, physics assumes that we can get closer and closer to the external reality through increasingly finer experimentation, and, more importantly, through better theorization. The Special and General Theories of Relativity are examples of brilliant applications of this view of reality where simple physical principles are relentlessly pursued using formidable machine of pure reason to their logically inevitable conclusions. But there is another, alternative view of knowledge and reality that has been around for a long time. This is the view that regards perceived reality as an internal cognitive representation of our sensory inputs, as illustrated below. In this view, knowledge and perceived reality are both internal cognitive constructs, although we have come to think of them as separate. What is external is not the reality as we perceive it, but an unknowable entity giving rise to the physical causes behind sensory inputs. In the illustration, the first arrow represents the process of sensing, and the second arrow represents the cognitive and logical reasoning steps. In order to apply this view of reality and knowledge, we have to guess the nature of the absolute reality, unknowable as it is. One possible candidate for the absolute reality is Newtonian mechanics, which gives a reasonable prediction for our perceived reality. To summarize, when we try to handle the distortions due to perception, we have two options, or two possible philosophical stances. One is to accept the distortions as part of our space and time, as SR does. The other option is to assume that there is ahigherreality distinct from our sensed reality, whose properties we can only conjecture. In other words, one option is to live with the distortion, while the other is to propose educated guesses for the higher reality. Neither of these options is particularly attractive. But the guessing path is similar to the view accepted in phenomenalism. It also leads naturally to how reality is viewed in cognitive neuroscience, which studies the biological mechanisms behind cognition. In my view, the two options are not inherently distinct. The philosophical stance of SR can be thought of as coming from a deep understanding that space is merely a phenomenal construct. If the sense modality introduces distortions in the phenomenal picture, we may argue that one sensible way of handling it is to redefine the properties of the phenomenal reality. #### Role of Light in Our Reality From the perspective of cognitive neuroscience, everything we see, sense, feel and think is the result of the neuronal interconnections in our brain and the tiny electrical signals in them. This view must be right. What else is there? All our thoughts and worries, knowledge and beliefs, ego and reality, life and deatheverything is merely neuronal firings in the one and half kilograms of gooey, grey material that we call our brain. There is nothing else. Nothing! In fact, this view of reality in neuroscience is an exact echo of phenomenalism, which considers everything a bundle of perception or mental constructs. Space and time are also cognitive constructs in our brain, like everything else. They are mental pictures our brains concoct out of the sensory inputs that our senses receive. Generated from our sensory perception and fabricated by our cognitive process, the space-time continuum is the arena of physics. Of all our senses, sight is by far the dominant one. The sensory input to sight is light. In a space created by the brain out of the light falling on our retinas (or on the photo sensors of the Hubble telescope), is it a surprise that nothing can travel faster than light? This philosophical stance is the basis of my book, The Unreal Universe, which explores the common threads binding physics and philosophy. Such philosophical musings usually get a bad rap from us physicists. To physicists, philosophy is an entirely different field, another silo of knowledge. We need to change this belief and appreciate the overlap among different knowledge silos. It is in this overlap that we can expect to find breakthroughs in human thought. This philosophical grand-standing may sound presumptuous and the veiled self-admonition of physicists understandably unwelcome; but I am holding a trump card. Based on this philosophical stance, I have come up with a radically new model for two astrophysical phenomena, and published it in an article titled, “Are Radio Sources and Gamma Ray Bursts Luminal Booms?” in the well-known International Journal of Modern Physics D in June 2007. This article, which soon became one of the top accessed articles of the journal by Jan 2008, is a direct application of the view that the finite speed of light distorts the way we perceive motion. Because of these distortions, the way we see things is a far cry from the way they are. We may be tempted to think that we can escape such perceptual constraints by using technological extensions to our senses such as radio telescopes, electron microscopes or spectroscopic speed measurements. After all, these instruments do not haveperceptionper se and should be immune to the human weaknesses we suffer from. But these soulless instruments also measure our universe using information carriers limited to the speed of light. We, therefore, cannot escape the basic constraints of our perception even when we use modern instruments. In other words, the Hubble telescope may see a billion light years farther than our naked eyes, but what it sees is still a billion years older than what our eyes see. Our reality, whether technologically enhanced or built upon direct sensory inputs, is the end result of our perceptual process. To the extent that our long range perception is based on light (and is therefore limited to its speed), we get only a distorted picture of the universe. #### Light in Philosophy and Spirituality The twist to this story of light and reality is that we seem to have known all this for a long time. Classical philosophical schools seem to have thought along lines very similar to Einstein’s thought experiment. Once we appreciate the special place accorded to light in modern science, we have to ask ourselves how different our universe would have been in the absence of light. Of course, light is only a label we attach to a sensory experience. Therefore, to be more accurate, we have to ask a different question: if we did not have any senses that responded to what we call light, would that affect the form of the universe? The immediate answer from any normal (that is, non-philosophical) person is that it is obvious. If everybody is blind, everybody is blind. But the existence of the universe is independent of whether we can see it or not. Is it though? What does it mean to say the universe exists if we cannot sense it? Ahthe age-old conundrum of the falling tree in a deserted forest. Remember, the universe is a cognitive construct or a mental representation of the light input to our eyes. It is notout there,” but in the neurons of our brain, as everything else is. In the absence of light in our eyes, there is no input to be represented, ergo no universe. If we had sensed the universe using modalities that operated at other speeds (echolocation, for instance), it is those speeds that would have figured in the fundamental properties of space and time. This is the inescapable conclusion from phenomenalism. The role of light in creating our reality or universe is at the heart of Western religious thinking. A universe devoid of light is not simply a world where you have switched off the lights. It is indeed a universe devoid of itself, a universe that doesn’t exist. It is in this context that we have to understand the wisdom behind the statement thatthe earth was without form, and voiduntil God caused light to be, by sayingLet there be light. The Quran also says, “Allah is the light of the heavens and the earth,” which is mirrored in one of the ancient Hindu writings: “Lead me from darkness to light, lead me from the unreal to the real.The role of light in taking us from the unreal void (the nothingness) to a reality was indeed understood for a long, long time. Is it possible that the ancient saints and prophets knew things that we are only now beginning to uncover with all our supposed advances in knowledge? I know I may be rushing in where angels fear to tread, for reinterpreting the scriptures is a dangerous game. Such foreign interpretations are seldom welcome in the theological circles. But I seek refuge in the fact that I am looking for concurrence in the metaphysical views of spiritual philosophies, without diminishing their mystical or theological value. The parallels between the noumenal-phenomenal distinction in phenomenalism and the Brahman-Maya distinction in Advaita are hard to ignore. This time-tested wisdom on the nature of reality from the repertoire of spirituality is now reinvented in modern neuroscience, which treats reality as a cognitive representation created by the brain. The brain uses the sensory inputs, memory, consciousness, and even language as ingredients in concocting our sense of reality. This view of reality, however, is something physics is yet to come to terms with. But to the extent that its arena (space and time) is a part of reality, physics is not immune to philosophy. As we push the boundaries of our knowledge further and further, we are beginning to discover hitherto unsuspected and often surprising interconnections between different branches of human efforts. In the final analysis, how can the diverse domains of our knowledge be independent of each other when all our knowledge resides in our brain? Knowledge is a cognitive representation of our experiences. But then, so is reality; it is a cognitive representation of our sensory inputs. It is a fallacy to think that knowledge is our internal representation of an external reality, and therefore distinct from it. Knowledge and reality are both internal cognitive constructs, although we have come to think of them as separate. Recognizing and making use of the interconnections among the different domains of human endeavour may be the catalyst for the next breakthrough in our collective wisdom that we have been waiting for. I recently bought a Sony World Band Radio receiver. It is a beautiful machine with some twenty frequency bands and all kinds of locks and tricks to latch on to distant radio stations. I bought it for my father, who is fond of listening to his radio late into the night. Two days after I bought the radio, my father suffered a severe heart failure. A congestive heart failure (CHF) is not to be confused with a heart attack. The symptoms of a CHF are deceptively similar to an asthma attack, which can be doubly treacherous if the patient already has respiratory troubles because the early care may get directed to the lungs while the troubled heart may be ignored. So I thought I would discuss the symptoms here in the hope that it will help those with aging family members who may otherwise misidentify a potential CHF. Much more information is available on the Internet; try Googlingcongestive heart failure. For asthma patients, a danger sign of a heart failure is persistent breathing difficulty despite inhalation medication. Watch out for breathing trouble that increases when they lie down, and subsides when they sit up. They may have consequent sleeplessness. If they show the symptoms of water retention (swelling in lower limps or neck, unexpected sudden weight gain etc.), and if they have other risk factors (hypertension, irregular heart beat), please do not wait, rush to the hospital. The prognosis for CHF is not good. It is a chronic condition, progressive and terminal. In other words, it is not something we catch like the flu and get better soon. Depending on the stage the patient is, we have to worry about the quality of life, palliative care or even end of life care. Once a heart has started failing, it is difficult to reverse the progression of the onslaught. There are no easy solutions, no silver bullets. What we can concentrate on, really, is the quality of their life. And the grace and dignity with which they leave it. For most of them, it is their last act. Let’s make it a good one. By my father’s bedside now, listening to the Sony, with all these sad thoughts in my head, I remember my first taste of real winter in the fall of 1987 in Syracuse. I was listening to the weatherman of the local radio station (was it WSYR?). While lamenting the temperatures going south, he observed, rather philosophically, “C’mon, we all know there’s only one way the temperatures can go.Yes, we know that there is only one way things can go from here. But we still mourn the passing of a summer full of sunshine and blue skies. The Sony radio plays on, impervious to these doleful musings, with young happy voices dishing out songs and jokes for the benefit of a new generation of yuppie commuters full of gusto and eagerness to conquer a world. Little do they knowit was all conquered many times over during the summers of yester years with the same gusto and passion. The old vanguards step aside willingly and make room for the children of new summers. The new generation has different tastes. They hum to different iTunes on their iPods. This beautiful radio receiver, with most of it seventeen odd short wave bands now silent, is probably the last of its kind. The music and jokes of the next generation have changed. Their hair-do and styles have changed. But the new campaigners charge in with the same dreams of glory as the ones before them. Theirs is the same gusto. Same passion. Perhaps nothing and nobody really passes on. We all leave behind a little bit of ourselves, tiny echoes of our conquests, memories in those dear to us, and miniscule additions to the mythos that will live on. Like teardrops in the rain.	2020-07-12 12:48:43	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 46, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4914049208164215, "perplexity": 943.8056151614682}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1593657138718.61/warc/CC-MAIN-20200712113546-20200712143546-00533.warc.gz"}
https://testbook.com/question-answer/the-average-proportion-non-conforming-of-20-sample--6090e2286f0692d723a4b9ff	# The average proportion non-conforming of 20 samples each of size 100 items is 0.12. The upper control limit for the relevant chart is ______ (round off to 2 decimal places) This question was previously asked in GATE PI 2019 Official Paper View all GATE PI Papers > ## Detailed Solution Concept: P-chart (Proportion or Fraction Defective Chart) • It is used to monitor and control the fraction produced in a process that is defective or non-conforming. • It follows a binomial distribution. • This chart is best suited in cases where inspection is carried out to classify articles as either excepted or rejected. $$Control\;limits = \bar p \pm 3 \times \sqrt {\frac{{\bar p\;\left( {1 - \bar p} \right)}}{n}}$$ Calculation: Given: The average proportion, $$\bar p$$ = 0.12 Size, n = 100 $$Control\;limits = \bar p \pm 3 \times \sqrt {\frac{{\bar p\;\left( {1 - \bar p} \right)}}{n}}$$ Upper Control Limit (UCL) is given by : $$Upper \:control\;limit = 0.12 + 3 \times \sqrt {\frac{{0.12\;\left( {1 - 0.12} \right)}}{100}}$$ $$Upper \:control\;limit = 0.12 + 3 \times \sqrt {\frac{{0.12\;\left( 0.88 \right)}}{100}}$$ ∴ UCL = 0.21	2021-10-17 18:33:22	{"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": 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": 0.697184145450592, "perplexity": 4157.303044316032}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585181.6/warc/CC-MAIN-20211017175237-20211017205237-00665.warc.gz"}
http://thomaslau.xyz/2018/08/03/2018-08-03-many_links_0803/	Versions and SemVer In software, we use version numbers to signal that something has changed. Version numbering schemes go from dead simple (integers that increment with every release, or date stamps) to surprisingly complex (1.0~pre3+dfsg-0.1+b2, 2.1.1+git20160721~8efc468-2, and 1.2.0+LibO5.2.7-1+deb9u4 are a few versions spotted in the wild). But when it comes to software version numbers, the current leader in version numbering schemes is SemVer (or Semantic Versioning). Don’t be fooled, though! Many people claim to know how SemVer works, but have never read the specification. Since this is a critical piece of what we are about to talk about, here is a summary of the spec: Version numbers take the form X.Y.Z, sometimes augmented with additional pre-release and build information: X.Y.Z-AAA#BBB. And each of those fields means something well defined and specific. X is the major number. Changes in this indicate breaking changes to the API (and/or behavior). Y is the minor number. Changes to this number indicate that new features were added, but that no APIs are broken as a result. Z is the patch version. Changes to this indicate that internal changes were made, but that no changes (even compatible changes) were made to the API. These three are the important ones for us. Again, I suggest taking 15 minutes to read the entire spec. Countless projects use a format that looks like SemVer, but many of them ignore the semantics behind the version number. Often, it seems that version numbers are incremented by “gut feel” instead of any consistent semantic: “This feels like a minor version update.” 3. lucene中的PackedInts源码解读-1 http://suichangkele.iteye.com/blog/2427364 4. 【译】追踪同步分片副本 https://www.easyice.cn/archives/243 https://www.easyice.cn/archives/283 5. Lucene 基础理论 http://www.blogjava.net/hoojo/archive/2012/09/06/387140.html https://github.com/kakawait/cas-security-spring-boot-starter 6. https://en.wikipedia.org/wiki/High-availability_cluster Node configurations Active/active — Traffic intended for the failed node is either passed onto an existing node or load balanced across the remaining nodes. This is usually only possible when the nodes use a homogeneous software configuration. Active/passive — Provides a fully redundant instance of each node, which is only brought online when its associated primary node fails.[2] This configuration typically requires the most extra hardware. N+1 — Provides a single extra node that is brought online to take over the role of the node that has failed. In the case of heterogeneous software configuration on each primary node, the extra node must be universally capable of assuming any of the roles of the primary nodes it is responsible for. This normally refers to clusters that have multiple services running simultaneously; in the single service case, this degenerates to active/passive. N+M — In cases where a single cluster is managing many services, having only one dedicated failover node might not offer sufficient redundancy. In such cases, more than one (M) standby servers are included and available. The number of standby servers is a tradeoff between cost and reliability requirements. N-to-1 — Allows the failover standby node to become the active one temporarily, until the original node can be restored or brought back online, at which point the services or instances must be failed-back to it in order to restore high availability. N-to-N — A combination of active/active and N+M clusters, N to N clusters redistribute the services, instances or connections from the failed node among the remaining active nodes, thus eliminating (as with active/active) the need for a ‘standby’ node, but introducing a need for extra capacity on all active nodes. active-active active-passive active-passive也是两个服务器节点, 但是绝大多数时间是active的那个(或者说primary)进行服务, 当primary服务器出问题, 就使用另一个passive服务器作为备用. 7. Why I don’t use JSON ASTs lihaoyi给出的答案，可参考 If you use uJson, you don’t need to choose: 1.Just need case classes? Read/write directly to case classes 2.Need an AST? Read/write directly to the AST 3.Specifically want some other AST that’s not the uJson AST? You can read/write directly to the circe/play-json/etc. ASTs too 4.Don’t need case classes or an AST, and just want to pretty-print JSON? Read/write directly String -> String 5.Don’t need any output at all, and just want to parse+validate your JSON? You can do that too. ASTs are a thing you sometimes want, but they don’t need to be in the critical path if you want some other thing. If you want one, great, if you want something else, a visitor-based library like uJson can give you that something else directly and without fuss. 8. https://www.wired.co.uk/article/human-faeces-poo-as-fertiliser 9. ActivityPub https://activitypub.rocks/ ActivityPub is a decentralized social networking protocol based on the ActivityStreams 2.0 data format. ActivityPub is an official W3C recommended standard published by the W3C Social Web Working Group. It provides a client to server API for creating, updating and deleting content, as well as a federated server to server API for delivering notifications and subscribing to content. 10. https://mp.weixin.qq.com/s?__biz=MzAwODY4OTk2Mg==&mid=2652046740&;idx=1&sn=2b01ecd64c516d59f38253f723bcebda 11. oschina的大约译文 12. Kafka 2.0重磅发布，新特性独家解读 http://www.infoq.com/cn/news/2018/08/kafka2.0-new-features KIP-268：简化 Kafka Streams 升级过程 Kafka Streams 利用 Consumer Rebalance 协议里面的元数据字符串编码诸如任务分配、全局查询、版本升级相关的信息。然而，当编码版本本身改变的时候，就需要进行离线升级。比如之前从 0.10.0 版本向更高级的版本升级的时候，用户就需要将所有的 Streams 程序下线，换上新的 Kafka 版本号，然后在全部重启。 KIP-268 利用 version prob 可以使得旧版本的任务分配者告知其他高版本的成员暂时使用旧版本的 Rebalance 元数据编码，这样就可以让用户依然能够通过 rolling bounce 在线升级 Kafka Streams 的版本。而当所有参与的成员全部升级完毕之后，最后一次 rebalance 会自动切换回新版本的元数据编码。 KIP-279：修补多次 Kafka 分区主本迁移时的日志分歧问题 KIP-283：降低信息格式向下转换时的内存消耗 KIP-223：加入消费者客户端的领先指标 KIP-237：加入更多 Kafka 控制器的健康指标 KIP-290、KIP-227：细粒度前缀通配符访问控制 13. Service Mesh是什么？至今到位的总结 http://blog.brucefeng.info/post/what-is-service-mesh Service Mesh能做什么 Metric和链路追踪 2 为什么有微服务了还要Service Mesh 14. https://www.theatlantic.com/science/archive/2018/08/nasa-culture-optimism-james-webb/566558/?utm_source=feed 15. http://www.sciencemag.org/news/2018/07/beyond-silicon-15-billion-us-program-aims-spur-new-types-computer-chips Silicon computer chips have been on a roll for half a century, getting ever more powerful. But the pace of innovation is slowing. Today the U.S. military’s Defense Advanced Research Projects Agency(DARPA) announced dozens of new grants totaling \$75 million in a program that aims to reinvigorate the chip industry with basic research into new designs and materials, such as carbon nanotubes. Over the next few years, the DARPA program, which supports both academic and industry scientists, will grow to$300 million per year up to a total of \$1.5 billion over 5 years. 16. http://www.discoverdev.io/archive/2018-08-01.html http://www.discoverdev.io/archive/2018-08-02.html http://www.discoverdev.io/archive/2018-08-03.html	2020-08-06 12:24:05	{"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": 0.2947937548160553, "perplexity": 3626.2086122790215}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1596439736962.52/warc/CC-MAIN-20200806121241-20200806151241-00021.warc.gz"}
https://community.wolfram.com/groups/-/m/t/2073305	# Retrieve data from a graph Posted 17 days ago 245 Views \| 2 Replies \| 1 Total Likes \| Friends: I have a question about a (I think) very simple operation which I have not managed to solve.I request, say, the GDP per capita of a country using Shift =:= per capita GDP ColombiaThis returns a lot of information, including a nice beautiful graph with Colombia's GDP per capita from 1960 to 2018. Now, however, I want not only the graph but also all the data points, say in a list. If I select the Graph and right click there is a "get coordinates" option, but apparently it does not allow me to get all the data points.So the question is: how can I get the data from the graph? Thanks in advance Francisco 2 Replies Sort By: Posted 17 days ago Hi Francisco,If you copy the cell as Input it is gdp = Entity["Country", "Colombia"][ EntityProperty["Country", "GDP", {"Date" -> All, "CurrencyUnit" -> "CurrentUSDollar", "PerCapita" -> "PerCapita"}]]; Which evaluates to a TimeSeries. To plot it DateListPlot[gdp] To get the data points gdp["DatePath"]	2020-09-26 05:57:41	{"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": 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": 0.3278422951698303, "perplexity": 1836.129527294854}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1600400234232.50/warc/CC-MAIN-20200926040104-20200926070104-00585.warc.gz"}
https://www.physicsforums.com/threads/how-to-calculate-the-moment-of-inertia-for-complex-3d-shapes.936625/	# How to calculate the moment of inertia for complex 3D shapes #### jm090693 <<<moved from another sub forum, no template>> Hi, I need to calculate the moment of inertia for the component in the attached image so that i can calculate the angular momentum. Is it possible? Overall i am trying to calculate the forces on this lug as it passes around a 3" radius at 2M a second. Thanks, Jack #### Attachments • 106.4 KB Views: 905 Related Introductory Physics Homework News on Phys.org #### anorlunda Mentor Gold Member Where is the axis of rotation? Next time, you must post homework in a homework forum and fill out the template. But even this time, we require that you show some effort before we help. What have you tried so far? What is the axis of rotation? #### haruspex Homework Helper Gold Member 2018 Award The method is to decompose it into simple shapes, find the MoI of each around the chosen axis, and add them up. Sometimes it helps to represent a shape as the difference of two shapes, e.g. for a sphere with a hole treat it as a whole sphere then subtract the MoI of the missing piece. #### jm090693 Hi, I have added some more details here. I did not post it in the homework section as it isn't homework? Thanks, Jack #### haruspex Homework Helper Gold Member 2018 Award Hi, I have added some more details here. I did not post it in the homework section as it isn't homework? Thanks, Jack You won't need it to be all that accurate. Just treat the lug as a rectangular plate, LxW say, (length x width). Let the midpoint of the plate be distance R from the centre of rotation. The moment of inertia would be M(R2+(L2+W2)/12). Not sure how to figure out the stress, though. On the face of it, the angular motion goes from zero to nonzero, and vice versa, in zero time, which means infinite acceleration. So the elasticity of the material is important. Do you really need to calculate this to establish the cause? Is there any other feasible cause? An obvious solution would be to increase the radius of the turn. In principle, the start and end of the turn would be equally stressful. Do the cracks originate at the leading edge or the trailing edge? #### jm090693 You won't need it to be all that accurate. Just treat the lug as a rectangular plate, LxW say, (length x width). Let the midpoint of the plate be distance R from the centre of rotation. The moment of inertia would be M(R2+(L2+W2)/12). Not sure how to figure out the stress, though. On the face of it, the angular motion goes from zero to nonzero, and vice versa, in zero time, which means infinite acceleration. So the elasticity of the material is important. Do you really need to calculate this to establish the cause? Is there any other feasible cause? An obvious solution would be to increase the radius of the turn. In principle, the start and end of the turn would be equally stressful. Do the cracks originate at the leading edge or the trailing edge? Hi, Do you mean treat the leading edge of the lug as a rectangular plate, what about everything behind that front plate or is that excluded? Basically i want a value which i can compare with the material properties to prove that this is why they are failing. I will then increase the surface area of contact between the lug/chain and reduce the centre of gravity closer to the chain attachment before re-calculating. See attached showing some calculations for the acceleration at the tip of the lugs. Absolutely, if the radius of that corner could be increased there would be no problem. Unfortunately due to the application it needs to change direction in a very small window. I believe the stress occurs when the chain changes 90 degrees direction and then once horizontal is in a wear-strip/guide meaning it is fixed rotationally. However the inertia of the lug continues as it is accelerating around the 90 degree bend. The only portion stopping the lug rotating any more past horizontal is the 31mm^2 surface area between the lug/chain. This pressure repeatidly at 450/minute 24/7 is causing a fatigue crack on the lug face. (VIDEO) The yellow lug cracks on the leading edge of the chain attachment & the red lug cracks on the leading edge of the chain attachment. Meaning the above theory must be true as there is no other point on the chain path which has forces in these direcitons. Jack Last edited: #### haruspex Homework Helper Gold Member 2018 Award treat the leading edge of the lug as a rectangular plate No, the whole thing beyond the narrow stalk. The thickness of the plate does not matter. the inertia of the lug continues as it is accelerating around the 90 degree bend Not quite sure what you mean by that. Once it has got going around the bend there is no angular acceleration until the end. The stressful events are starting and finishing the rotation. The video seems to show that the belt rate changes hugely. That would exert similar stresses during the belt accelerations. The yellow lug cracks on the leading edge of the chain attachment & the red lug cracks on the trailing edge I suggest this is because the red lugs act as buffers, sparing the yellow lugs from the deceleration stress. L #### llober No, the whole thing beyond the narrow stalk. The thickness of the plate does not matter. Not quite sure what you mean by that. Once it has got going around the bend there is no angular acceleration until the end. The stressful events are starting and finishing the rotation. The video seems to show that the belt rate changes hugely. That would exert similar stresses during the belt accelerations. I suggest this is because the red lugs act as buffers, sparing the yellow lugs from the deceleration stress. I think the fatigue tension that produces the crack is not due to the lug leaving the turn, but during the turn itself, because at the end of the turn the material is at compression where the crack appears, but is being teared appart during the turn. Anyway, or it's a poltergeist, or is due to the forces produced during the turn... obviously, it's the second reason. You'll calculate the moi ... and what? If you have to turn the piece that way, you'll have to reinforce the piece just where the crack is appearing, because "experimentally" you have determined that the cross section of the piece there is not enough to handle the stress. #### jm090693 The stressful events are starting and finishing the rotation. Yes, precisely. I am concentrating on the stressful event at the end of the rotation. Here the inertia of the lug means it wants to continue rotating or travelling at the higher corner speed. The leading edge of the chain attachment slows the lug down to the linear speed of the horizontal chain. I believe the lug is rotating on the bolt which connects it to the chain attachment at the end of rotation, where the lug want to continue. It's the stress on the lug i want to calculate at this point. The video seems to show that the belt rate changes hugely. That would exert similar stresses during the belt accelerations. My bad i should have explained. This is a high speed camera video, the sections at the start & end are the running speed, the section in the middle has been slowed down so you can see whats going on. I suggest this is because the red lugs act as buffers, sparing the yellow lugs from the deceleration stress. I believe the same theory above of the lugs rotating at the end of rotation is happening on the red lugs too, as the cracks are both on the leading edge of the lug. #### jm090693 I think the fatigue tension that produces the crack is not due to the lug leaving the turn, but during the turn itself, because at the end of the turn the material is at compression where the crack appears, but is being teared appart during the turn. Anyway, or it's a poltergeist, or is due to the forces produced during the turn... obviously, it's the second reason. You'll calculate the moi ... and what? If you have to turn the piece that way, you'll have to reinforce the piece just where the crack is appearing, because "experimentally" you have determined that the cross section of the piece there is not enough to handle the stress. Hi, the chain is driven, if the lug was cracking during the turn it would be on the trailing edge of the lug. At the end of the rotation the bottom of the lug is in tension as the lug is wanting to continue rotating on the chain attachment. I need to prove my theory in calculations so i can confirm my new design with calls prior to paying for new moulds. Thanks, Jack #### haruspex Homework Helper Gold Member 2018 Award I think the fatigue tension that produces the crack is not due to the lug leaving the turn, but during the turn itself No, the stress during the turn is just a bit of tension from centrifugal force. This would be quite modest compared with the very high angular accelerations at start and end of the turn. #### haruspex Homework Helper Gold Member 2018 Award It's the stress on the lug i want to calculate at this point. As I indicated in post #5, to do that you need to find the angular acceleration at transition. This is problematic because it depends on the elasticity of the lug material, and even of the chain. Even without the lugs, the chain itself cannot instantly transition between linear and rotational. If you could inspect the chain shape in great detail you would see some inertia at each end of the turn. The extent of that depends on the chain tension. Maybe we can assume the chain is quite taut and we can ignore this. That leaves us with the lug elasticity. Any data on that? L #### llober If a CAD programd doesn't give you the MoI of this complex 3D shape, then perhaps you can use an experimental method to find an aproximate value. We can use Python to solve numerically the differential equation, first with an estimated value for your MoI, and iterating until the period (T) found experimentally is close enough to the result T solving the differential equation: https://repl.it/@llober/Solid-body-pendulum After, you calculate the MoI at the real axis position, using the parallel axes theorem. Before and after the turn, you have a Kinetic Energy for the moving object, with mass m and a linear speed v ... $$K = \frac{1}{2}m \ v^2$$ In the turn, you have a Rotational Energy ... $$K_r = \frac{1}{2}\ I_s \ ω^2$$ So which is the problem in my opinion? If both energies don't match, then you'll have an inelastic "impact" both when entering the turn, and after when leaving the turn. Greater the mismatch, greater the "hit" to the moving piece. Doing very, very rough numbers, the tension wheel should rotate, for example, at 30 rad/s (now is at about 45 rad/s). If you need to mantain the same linear speed, the radius of the tension wheel should be 1.5 times greater. Take it with a grain (or two) of salt. #### Attachments • 29.1 KB Views: 918 Last edited by a moderator: #### haruspex Homework Helper Gold Member 2018 Award Before and after the turn, you have a Kinetic Energy for the moving object, with mass m and a linear speed v ... $$K = \frac{1}{2}m \ v^2$$ In the turn, you have a Rotational Energy ... $$K_r = \frac{1}{2}\ I_s \ ω^2$$ So which is the problem in my opinion? The energy is not in itself a problem. Forces are the problem. If both energies don't match, Match? As in, same magnitude? What has that to do with anything?! L #### llober The energy is not in itself a problem. Forces are the problem. Match? As in, same magnitude? What has that to do with anything?! You mean that Forces and Energies are not related in anyway? I wonder what part of what I said is not true about the kinetic and rotational energies ... or do you mean that during the turn, the object mantains both energies, so is the sum of the initial Kinetic plus the Rotational? #### Attachments • 9.3 KB Views: 721 Last edited by a moderator: #### haruspex Homework Helper Gold Member 2018 Award I would like to know what part of what I said is not true about the kinetic and rotational energies You wrote that the problem is if the energies do not match. I asked what you meant by that. You mean Forces an Energies don't have anything to do, one concept with the other, and are not related in anyway? Of course they are related, but there is no limit to the energy the object may have without there being any significant stress. L #### llober You wrote that the problem is if the energies do not match. I asked what you meant by that. Of course they are related, but there is no limit to the energy the object may have without there being any significant stress. Ok, let's say that the object enters the turn with a kinetic energy of 1 unit ... this object is not in free movement, because is being pulled by the chain, but when it enters the turn, its energy (kinetic) is 1 unit. While in the turn, its energy, due to its moment of inertia and angular speed forced by the chain, has to be 3 units. Can you tell me where it gets this "extra 2-energy units"? You tell it, of course because there're acting forces. The forces act suddenly, because in a very short time, the energy of the object has to increase from 1 unit to 3 units. At the exit of the turn, the energy of the object has to decrease suddenly from 3 units to 1 unit. (The 3:1 ratio is a rough estimate I've made, based on the drawings of the machine and its elements). Did you understand it? Last edited by a moderator: #### haruspex Homework Helper Gold Member 2018 Award The forces act suddenly, because in a very short time, the energy of the object has to increase from 1 unit to 3 units. Sure, but it's not the increase that matters, it's the rapid change in direction. By your argument, there would be no stress if the energies were of equal magnitude. But it is good to see that you are now acknowledging the importance of the rate of change rather than just the magnitude. Last edited: L #### llober Sure, but it's not the increase that matters, it's the rapid change in direction. By your argument, there would be no stress if the energies were of equal magnitude. But it is good to see that you are now acknowledging the importance of the rate of change rather than just the magnitude. Wow, I didn't negate the importance of the rate of change, because what I said is that the energy of the body has to change from 1 to 3 (for example) at the very start of the turn, and that implicitly means "instantly", which, of course, is not zero time but a short time. And the magnitudes do matter, because the rate is about how those magnitudes change in time ... and we're talking about the, more or less, same "short" time (undetermined), but obviously is not the same to change from 1.0 to 1.01, than from 1.0 to 3.0 in the "same" short time ... I don't get how you can't not understand this. You talk about forces, I talk about energies ... we're talking about the same physics, but we're watching the problem from different points of view, perhaps because I'm an engineer. In my vision, I can give an order of magnitude of the root cause of the problem ... in your vision, it's absolutely obscure, and due to complex forces that act in very short undetermined times ... you can't give any number, not even approximate numbers with that approach. I don't care the forces or its distribution to see what is happening in this case... I know that are there, in a very complex interation, which presumably can only be solved doing a numerical finite element simulation of the case. By the way, yes, I'm saying that if you design properly the system, and the Kinetic energy of the piece when enters the turn is more or less equal to the Rotational energy that will have during the turn (which is a desing choice), then the unique forces that will "smoothly" play are the centripetal forces, that are unavoidable because, of course, you have to turn. At least, I hope that you realize this. Sir, you see the little times and uncertain forces, but don't see the overall picture. Last edited by a moderator: #### haruspex Homework Helper Gold Member 2018 Award if ... the Kinetic energy of the piece when enters the turn is more or less equal to the Rotational energy that will have during the turn, then the unique forces that will "smoothly" play are the centripetal forces No. The forces that execute the transition create large torques. During the turn, the torques have disappeared and only the (much weaker) centripetal force remains. what I said is that the energy of the body has to change from 1 to 3 (for example) at the very start of the turn, Not until post #17. Until then you implied it was all down to the magnitude of the change. Indeed, you originally stated (post #8) that the main stress was during the turn, not at the transition between linear and rotational. L #### llober No. The forces that execute the transition create large torques. During the turn, the torques have disappeared and only the (much weaker) centripetal force remains. Not until post #17. Until then you implied it was all down to the magnitude of the change. Indeed, you originally stated (post #8) that the main stress was during the turn, not at the transition between linear and rotational. Yes, I said in post #8 that the strees was in the turn, and I was wrong because I didn't realize the huge mismatch between the entering kinetic energy and the rotational energy that the piece has to achieve of a sudden, and I assumed only the centripetal forces, that is how the system should work if properly designed. Errare humanum est! "The forces that execute the transition..." ... yes, there's a transition, big and problematic transition. For example, at the exit of the turn, where the energy has to go from 3 (example) to 1 ...that energy is totally lost ... the effect is a "hit", where that extra energy will be dissipated by unelastic deformation forces, vibration, heat, sound waves, etc ... And, by the way, you implied in post #14 that forces and energies are not related! In post #17 I included this drawing... You put words in my mouth that I did not say! According to may view, the centripetal forces (done by the walls) do turn the ball. What you don't seem to realize is that the kinetic energy at the entry of the turn, is the rotational energy during the turn (if the body moves freely). If you substitute the value of ω by v/r, and assume that the ball is a "point mass", you'll see that's the same thing. And for me, given that you have to make a turn and you go to a certain fixed speed, then the best choice is to balance the kinetic entering energy with the rotational energy ... or this is not true? #### Attachments • 9.3 KB Views: 192 Last edited by a moderator: #### haruspex Homework Helper Gold Member 2018 Award you implied in post #14 that forces and energies are not related! I did not. I wrote that the energy is not in itself a problem. Specifically, it is the rate at which the energy is transferred from one mode to another, i.e. the forces. where the energy has to go from 3 (example) to 1 ...that energy is totally lost . No, most of the KE lost by each flange would be transferred back into the belt. In post #17 I included this drawing I assume in this diagram that the ball is frictionless, so is not rotating on its own centre. Or that might as well be the case. That means there is not essentially any rotational KE involved. Yes, you can choose to view it as rotational KE by choosing a an axis off to the side, as you have done, but you can do exactly the same with constant velocity motion. the best choice is to balance the kinetic entering energy with the rotational energy Try this: a rod is moving laterally at a constant speed. One end encounters a rubber stop and bounces elastically. This converts much of the linear KE to rotational, but there is no net change in KE. By your reasoning, there is no stress on the rod, right? L #### llober The definition of Rotational energy is $$E_{rot}=\frac{1}{2}Iω^2$$ where $I$ is the moment of inertia around the axis of rotation considered. https://en.wikipedia.org/wiki/Rotational_energy In post #5, you correctly stated that if we consider the object to be like a plate, then the moment of inertia would be: $$M(R^2+\frac{L^2+W^2}{12})$$ So, the rotational energy, as is officially defined, is: $$E_{rot}=\frac{1}{2}Mω^2(R^2+\frac{L^2+W^2}{12})=\frac{1}{2}MR^2ω^2 + \frac{1}{2}Mω^2\frac{L^2+W^2}{12}$$ As we know that $ω=\frac{v_{cm}}{R}$ $$E_{rot}=\frac{1}{2}Mv_{cm}^2+\frac{1}{2}Mω^2\frac{L^2+W^2}{12}$$ And you can say that this is the Kinetic energy of the center of mass moving around the center of the rotation, plus an additional energy coming from the rotation of the object around its own center of mass. In our case, we consider a plate of about 6" x 2.5". I'll work in SI units from now on, so the plate is 0.15m x 0.06m. The rotational speed for a 2m/s linear speed and 0.044m tension wheel radius is 45.5 rad/s. The radius to the center of mass is 0.127m. The energy per Kg is $$E_{rot}=\frac{1}{2}(0.127)^2(45.5)^2 + \frac{1}{2}(45.5)^2·\frac{(0.15)^2+(0.06)^2}{12}=19 \ m^2s^{-2}$$ Before entering the turn, the linear Kinetic energy per Kg is $$\frac{1}{2}(2)^2 = 2 \ m^2s^{-2}$$ So we end having 9 times more energy just "one instant" after entering the turn, that the energy we had during the linear movement just "one instant" before entering the turn (oops .. I said 3 times in a previous post... sorry). And, to make it worst, that energy is taken away from the object as soon as it leaves the turn, which at 45.5 rad/s happens 0.04 s later. The "minimum stress" to the system, imho, would be to reduce the rotational speed of the drive wheel (if that is feasible), and have the same energy before and during the turn, which doesn't mean that no stress would exist, but means that the own kinetic energy of the entering object would suffice for the turn. But, there's one caveat: if we want to mantain the speed of the drive chain at 2 m/s, if we reduce the rotational speed, we have to increase the radius and so the moment of inertia is affected! At the end, in this particular case, it's not possible to mantain the 2 m/s drive chain speed and balance exactly the initial kinetic energy with the rotational energy, but we can get quite close to the objective For example, with a drive wheel radius of 8.2" (from 1.73" initial) and a rotational speed of 9.6 rad/s (91.7 rpm), we have a rotational energy that is only 2 times greater than the linear kinetic energy for 2 m/s speed (originally, was 9 times greater). #### haruspex Homework Helper Gold Member 2018 Award which doesn't mean that no stress would exist, but means that the own kinetic energy of the entering object would suffice for the turn Your insistence that the magnitude of the change in energy is relevant is not based on any physical law that I am aware of. Consider an inertial frame of reference moving with the linear portion of the belt approaching the turn. In this frame, we go from no KE to the same rotational KE (but a different linear KE). If the energies matched, somehow, in the lab frame they do not in this frame, yet the accelerations, forces and stresses are exactly the same. "How to calculate the moment of inertia for complex 3D shapes" ### Physics Forums Values We Value Quality • Topics based on mainstream science • Proper English grammar and spelling We Value Civility • Positive and compassionate attitudes • Patience while debating We Value Productivity • Disciplined to remain on-topic • Recognition of own weaknesses • Solo and co-op problem solving	2019-05-20 21:36:50	{"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": 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": 0.716375470161438, "perplexity": 734.3948353265935}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1558232256147.15/warc/CC-MAIN-20190520202108-20190520224108-00237.warc.gz"}
http://mathhelpforum.com/discrete-math/115190-fibonacci-number-induction-proof.html	# Math Help - Fibonacci number Induction Proof 1. ## Fibonacci number Induction Proof Prove every third Fibonacci number is even.-- The Fibonacci numbers are the sequence 1,1,2,3,5,8,13,..., given by $f_n$ where $f$(1)=1, and $f$(2)=1, and $f_(n)=f(n-1)+(n-2)($ for all $n$. This is the problem... This is what I have... Base Case: f(3k-2) if k=1 then f(3(1)-2)=f(1)=1 odd # f(3k-1) f(3(1)-1)=f(2)=1 odd # f(3k) f(3(1) =f(3)=2 even # so f(1)+f(2)=f(3) an odd+odd=even Inductive step: Suppose it is true for k+1 f(3(k+1)-2)=f(3k+1) f(3(k+1)-1)=f(3k+2) f(3(k+1)) =f(3K+3) I do not know where to go from here... 2. First let me note that it's very good that you made your induction statement (let's call it $P(k)$) consist of three parts: $P(k)$ is ( $f(3k-2)$ is odd) and ( $f(3k-1)$ is odd) and ( $f(3k)$ is even) The original assertion to prove (let's call it $Q(k)$) is just: $Q(k)$ is ( $f(3k)$ is even) However, this is not enough to prove the induction step, which is the following statement: for all $k \ge 1$, $Q(k)$ implies $Q(k+1)$ Knowing $Q(k)$ just is not enough to be able to prove $Q(k+1)$. Therefore, we strengthen the induction statement from $Q(k)$ to $P(k)$. Now, in proving that $P(k)$ implies $P(k+1)$, we have more to prove (three substatements of $P(k+1)$ instead of just one of $Q(k+1)$), but we also have a stronger assumption $P(k)$. This technique -- strengthening the induction statement in order for the induction step to go through -- is very common in mathematics. End of a long digression. Concerning your proof, why do you say "Suppose it is true for k+1"? Inductive step is usually "suppose it is true for k; we need to prove it for k+1", i.e., one has to prove $P(k)$ implies $P(k+1)$ for all $k\ge 1$. So write carefully what you know, i.e., $P(k)$, and what you need to prove, i.e., $P(k+1)$, and the proof should be pretty straightforward.	2014-04-19 11:15:12	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 30, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9254122972488403, "perplexity": 545.2195473957211}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1397609537097.26/warc/CC-MAIN-20140416005217-00212-ip-10-147-4-33.ec2.internal.warc.gz"}
http://people.cst.cmich.edu/salis1bt/algcomb/2018fall/index.html	# CMU Algebra and Combinatorics Seminar ## Fall 2018 ### Meeting Times Fridays, 11:15am–12:15pm, in Pearce 226. ### Abstracts Speaker: Botong Wang Title: The log-concave conjectures of graphs and matroids Abstract: The chromatic polynomial is an important invariant in graph theory introduced by Birkhoff. A generalization of chromatic polynomial is the characteristic polynomial in matroid theory. It was conjectured by Rota, Heron and Welsh in the 70’s that the coefficients of the characteristic polynomials are log-concave. I will talk about a beautiful proof of this conjecture by Karim Adiprisito, June Huh and Eric Katz. I will also discuss a log-concavity result about the number of independent sets, which is joint work with June Huh and Benjamin Schroter. The key idea to proof the log-concavity properties is to use the Hodge-Riemann relation in algebraic geometry. Speaker: Paramasamy Karuppuchamy Title: Schubert varieties and toric varieties Abstract: Degeneration of Schubert varieties to toric varieties is completed in the paper "Toric degeneration of Schubert varieties" by Philippe Caldero. In an attempt to find a geometric proof of this result we realized that certain Schubert varieties are already toric varieties: A Schubert variety $X_w$ is a toric variety if and only if $w$ is a product of distinct simple reflections. Part of this result can be found (not explicitly mentioned) in Deodhar's article "On some geometric aspects of Bruhat orderings. I. A finer decomposition of Bruhat cell." The author was unaware of this fact while writing his article. In $A_n$ type, Masuda and Lee have different approach to get this result in their recent paper "Generic torus orbit closure in Schubert varieties." In this talk we give an overview of this topic. Speaker: Olivia Dumitrescu Title: Interplay between ribbon graphs and CohFT Abstract: I will review an axiomatic formulation of a 2D TQFT whose formalism is based on the edge-contraction operations on graphs drawn on a Riemann surface (cellular graphs). I will describe a new result, that ribbon graphs provide both cohomological field theory and a visual explanation of Frobenius-Hopf duality, that plays a crucial role in Givental-Teleman's classification theorem of CohFTs. No prerequisite is assumed. This is based on a work in progress with Motohico Mulase. Speaker: Jordan Watts Title: Diffeological Groups Abstract: Lie groups, groups equipped with a smooth manifold structure, are a broadly-studied and import class of groups with a natural connection to Lie algebras. They arise naturally in representation theory, classical mechanics, and other fields. However, there are plenty of groups that do not admit a smooth manifold structure, but we still wish to treat them as though they did… The concept of diffeological group remedies this by equipped these groups, such as the irrational torus and diffeomorphism groups, with a smooth structure that is not necessarily that of a manifold. We can also easily make the natural connection to Lie algebras of these groups. This talk will be designed for graduate students, will discuss Lie groups, diffeological groups, their Lie algebras, and have some helpful examples throughout. Speaker: Dmitry Zakharov Title: Maps to trees and loci in the moduli space of tropical curves Abstract: Tropical geometry is a large and growing field of mathematics that aims to find combinatorial, piecewise-linear analogues of various algebraic and geometric objects. One particularly well-developed correspondence is the one between algebraic curves and metric graphs, which are also called tropical curves. There exist tropical analogues of many constructions and results for algebraic curves, such as meromorphic functions, divisors, linear equivalence, the Riemann–Roch theorem, Jacobians, and moduli spaces. Such objects can be studied using purely combinatorial methods, and results about them can then be used to understand their algebraic analogues. A classical problem in algebraic geometry is the study of loci in the moduli space of algebraic curves consisting of curves admitting linear systems of a particular type. A major difference between tropical and algebraic curves is that the former usually have a much larger collection of principal divisors than the latter. For this reason, the loci of tropical curves admitting specific linear systems have unexpectedly large dimension in moduli. I will talk about an approach to this problem in which, instead of looking at tropical curves with linear systems, we look at tropical curves admitting maps to trees of a particular type. Speaker: Takumi Murayama Title: Frobenius-Seshadri constants and characterizations of projective space Abstract: Mori and Mukai conjectured that projective spaces can be characterized by their intersection theory. While their conjecture has been proved over the complex numbers by Cho, Miyaoka, and Shepherd-Barron, the conjecture is still open in positive characteristic. We will describe how Seshadri constants and a positive-characteristic analogue of these constants, called Frobenius-Seshadri constants, can be used to give a partial answer to Mori and Mukai's conjecture in positive characteristic. Frobenius-Seshadri constants were introduced by Mustata-Schwede and the presenter, and are obtained in a way similar to how Hilbert-Kunz multiplicity is obtained from Hilbert-Samuel multiplicity.	2021-04-22 20:18:58	{"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": 0.49098724126815796, "perplexity": 539.461488204307}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1618039604430.92/warc/CC-MAIN-20210422191215-20210422221215-00043.warc.gz"}
http://www.centelcu.org/Tennessee/beaker-measurement-error.html	At Computer Clinic, we provide excellent in-house or on-site computer services. We are your one-stop friendly neighborhood computer store. Our goal is to help our clients receive immediate computer assistance. Our technicians are eager to help you either in-store, or at your work or home location. We service all major computer brands and when we fix it, it stays fixed. We provide on-site repairs & upgrades, virus/spyware removal, network installation/repair, emergency data recovery, and custom built computers. Call us for fast, reliable, and affordable solutions to the problems that arise with your computer. Address 1760 W Elk Ave Ste 3, Elizabethton, TN 37643 (423) 278-9246 http://www.computerclinictricities.net # beaker measurement error Eidson, Tennessee Let's consider the following table of results. You don't know the accuracy of your measuring device unless you calibrate it, i.e. Note Systematic and random errors refer to problems associated with making measurements. Generated Sun, 02 Oct 2016 01:27:32 GMT by s_hv997 (squid/3.5.20) Belmont, CA: Thomson Brooks/Cole, 2009. accurate(the average is accurate)not preciseprecisenot accurateaccurateandprecise In any measurement, the number of significant figures is critical. Significant figures are a more approximate method of estimating the uncertainty than error propagation. Click here to check your answer to Practice Problem 6 Units \| Errors \| Significant Figures \| Scientific Notation Back to General Chemistry Topic Review Math Skills ReviewSignificant Figures In this case, the main mistake was trying to align one end of the ruler with one mark. How many significant figures does our answer have? 4! a set of measurements that is both precise and accurate? Hint: Change the number to scientific notation. Since Tom must rely on the machine for an absorbance reading and it provides consistently different measurements, this is an example of systematic error. Addition and subtraction: Uncertainty in results depends on the absolute uncertainty of the numbers used in the calculation. The first specifies precision (0.1 mg, usually) and the second specifies a broad target. This is because the liquid leaves the buret at the bottom.The smallest division in this buret is 0.1 mL. There are rigorous statistical tests to determine when a result or datum can be discarded because of wide discrepancy with other data in the set, but they are beyond the scope B. Confidence intervals are calculated with the help of a statistical device called the Student's t. The relative uncertainty in the volume is greater than that of the moles, which depends on the mass measurement, just like we saw in the significant figures analysis. If a person were to approximate the volume of liquid in the following picture to be 43.1 ml, what type of error would their estimate be? Volume measurements made with a 50-mL beaker are accurate to within ±5 mL. This means that the true value of the volume is determined by the experiment to be in the range between 8.95 and 9.01 mL Multiplication and division: Uncertainty in results depends Nevertheless, buret readings estimated to the nearest 0.01 mL will be recorded as raw data in your notebook. You would first weigh the beaker itself. S. Chemistry and Chemical Reactivity. 7th. Appendix A of your textbook contains a thorough description of how to use significant figures in calculations. This is because the spread in the four values indicates that the actual uncertainty in this group of results is greater than that predicted for an individual result, using just the Systematic errors can result in high precision, but poor accuracy, and usually do not average out, even if the observations are repeated many times. Such a calculation is referred to as the percent error of a measurementand is represented by the following formula: $\text{Percent Error} = \dfrac{\text{Experimental Result - Accepted value}}{\text{Accepted Value}} \times 100\%$ Example example: Most people have exactly 10 fingers and 10 toes. You record the sample weight to the 0.1 mg, for example 0.1968 g. A strict following of the significant figure rules resulted in a loss of precision, in this case. Practice Problem 6 Which of the following procedures would lead to systematic errors, and which would produce random errors? (a) Using a 1-quart milk carton to measure 1-liter samples of An equally precise value would be 36.6 mL or 36.4 mL. These errors can be divided into two classes: systematic and random. The left-most significant figure, used to determine the result's significant figures for addition and subtraction, is related to the absolute uncertainty. Substituting the four values above gives Next, we will use Equation 4 to calculate the standard deviation of these four values: Using Equation 5 with N = 4, the standard error You might have read 46 mL; your friend might read the volume as 48 mL. Therefore, our reading error is 0.01 mL. And you might think that the errors arose from only two sources, (1) Instrumental error (How "well calibrated" is the ruler? Since the true value, or bull's eye position, is not generally known, the exact error is also unknowable. Systematic vs. If you had a beaker and some graphite how would you weigh the exact amount of graphite using the weighing of difference procedure? Figure 2: Systematic and random errors. All the answers are correct within the reading error of 1 mL.So, How many significant figures does our volume of 47 1 mL have? For the example of the three weighings, with an average of 6.3302 ± 0.0001 g, the absolute uncertainty is 0.0001 g. compare it against a ruler you knew was accurate. Please try the request again. Add enough solution so that the buret is nearly full, but then simply read the starting value to whatever precision the buret allows and record that value. Therefore, the shots are not precise since they are relatively spread out but they are accurate because they all reached the hole. Random Errors Random errors most often result from limitations in the equipment or techniques used to make a measurement.	2019-01-19 20:27:16	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.5459515452384949, "perplexity": 1114.6101134277785}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1547583681597.51/warc/CC-MAIN-20190119201117-20190119223117-00266.warc.gz"}
https://researchmap.jp/tyo	# Jun'ichi Yokoyama Last updated: Nov 18, 2019 at 16:47 Name Jun'ichi Yokoyama The University of Tokyo Graduate School of Science Research Center for The Early Universe 50212303 ## Awards & Honors Mar 2014 Paper Award, Physical Society of Japan Feb 2013 Inoue Award, Inoue Foundation Mar 1997 JPS Paper Award, Physical Society of Japan Feb 1993 Inoue Research Promotion Award, Inoue Foundation ## Published Papers B. J. Carr, Kazunori Kohri, Yuuiti Sendouda, Jun'ichi Yokoyama Phys. Rev. D 94, 044029 (2016) Apr 2016 The fraction of the Universe going into primordial black holes (PBHs) with initial mass M_* \approx 5 \times 10^{14} g, such that they are evaporating at the present epoch, is strongly constrained by observations of both the extragalactic and Gala... B. J. Carr, Kazunori Kohri, Yuuiti Sendouda, Jun'ichi Yokoyama Phys.Rev.D81:104019,2010 Dec 2009 We update the constraints on the fraction of the Universe going into primordial black holes in the mass range 10^9--10^17 g associated with the effects of their evaporations on big bang nucleosynthesis and the extragalactic photon background. We i... ## Misc Nov 2019 We revisit the gravitational leptogenesis scenario in which the inflaton is coupled to gravity by the Chern-Simons term and the lepton asymmetry is generated through the gravitational anomaly in the lepton number current during inflation. We const... Soichiro Hashiba, Jun'ichi Yokoyama Phys. Lett. B 798, 135024 (2019) May 2019 Incorporating three generations of right-handed Majorana neutrinos to quintessential inflation, we construct a model which simultaneously explains inflation, dark energy, dark matter and baryogenesis. These neutrinos have hierarchical masses \$M_3 ... Soichiro Hashiba, Jun'ichi Yokoyama Phys. Rev. D 99, 043008 (2019) Dec 2018 The purely gravitational dark matter (PGDM) which interacts with the standard model particles only by gravitational interaction has recently been discussed. Due to its feeble interaction, PGDM may be produced mainly by the gravitational particle c... Tomohiro Nakama, Jun'ichi Yokoyama Phys. Rev. D 99, 061303 (2019) Nov 2018 High energy collisions of particles may have created tiny black holes in the early Universe, which might leave stable remnants instead of fully evaporating as a result of Hawking radiation. If the reheating temperature was sufficiently close to th... Soichiro Hashiba, Jun'ichi Yokoyama JCAP 01 (2019) 028 Sep 2018 We calculate the number density and the energy density of a massive scalar particle conformally coupled to gravity produced by gravitational particle creation in the case the kination stage follows inflation. In this model, mode functions are deri... Minxi He, Alexei A. Starobinsky, Jun'ichi Yokoyama Apr 2018 We analyze a two-field inflationary model consisting of the Ricci scalar squared () term and the standard Higgs field non-minimally coupled to gravity in addition to the Einstein term. Detailed analysis of the power spectrum of this model... Tomohiro Nakama, Jun'ichi Yokoyama PTEP 2019 (2019) no.3, 033E02 Mar 2018 It is shown that a positive non-minimal coupling of the Higgs field to gravity can solve the two problems in inflation models in which postinflationary universe is dominated by an energy with stiff equation of state such as a kination, namely, ove... Koichiro Horiguchi, Kiyotomo Ichiki, Jun'ichi Yokoyama May 2017 While the observed nearly scale-invariant initial power spectrum is regarded as one of the favorable evidence of the standard inflationary cosmology, precision observations of the Cosmic Microwave Background (CMB) anisotropies also suggest possibl... Hiroaki W. H. Tahara, Jun'ichi Yokoyama PTEP 2018 (2018) no.1, 013E03 Apr 2017 Gravitational waves from inflation induce polarization patterns in the cosmic microwave background (CMB). It is known that there are only two types of non-Gaussianities of the gravitaional waves in the most general scalar field theories having sec... Yi-Peng Wu, Jun'ichi Yokoyama JCAP05(2018)009 Apr 2017 We investigate loop corrections to the primordial fluctuations in the single-field inflationary paradigm from spectator fields that experience a smooth transition of their vacuum expectation values. We show that when the phase transition involves ... Hossein Bazrafshan Moghaddam, Robert Brandenberger, Jun'ichi Yokoyama Phys. Rev. D 95, 063529 (2017) Dec 2016 We study particle production at the end of inflation in kinetically driven G-inflation model and show that, in spite of the fact that there are no inflaton oscillations and hence no parametric resonance instabilities, the production of matter part... Naritaka Oshita, Jun'ichi Yokoyama Prog. Theor. Exp. Phys. (2016) 051E02 Mar 2016 We revisit the derivation of the Hawking-Moss transition rate. Using the static coordinates, we show that the Euclidean action is entirely determined by the contribution of the entropy of de Sitter space which is proportional to the surface area o... Takahiro Hayashinaka, Tomohiro Fujita, Jun'ichi Yokoyama Mar 2016 We explore Schwinger effect of spin 1/2 charged particles with static electric field in 1+3 dimensional de Sitter spacetime. We analytically calculate the vacuum expectation value of the spinor current which is induced by the produced particles in... Naritaka Oshita, Jun'ichi Yokoyama Phys. Lett. B 785 (2018) 197-200 Jan 2016 We discuss a two-step mechanism to create a new inflationary domain beyond a wormhole throat which is created by a phase transition around an evaporating black hole. The first step is creation of a false vacuum bubble with a thin-wall boundary by ... Tomohiro Fujita, Xian Gao, Jun'ichi Yokoyama Nov 2015 We investigate the cosmological background evolution and perturbations in a general class of spatially covariant theories of gravity, which propagates two tensor modes and one scalar mode. We show that the structure of the theory is preserved unde... Christophe Ringeval, Daisuke Yamauchi, Jun'ichi Yokoyama, Francois R. Bouchet JCAP 1602: 033, 2016 Oct 2015 Cosmic strings formed during inflation are expected to be either diluted over super-Hubble distances, i.e., invisible today, or to have crossed our past light cone very recently. We discuss the latter situation in which a few strings imprint their... Sachiko Kuroyanagi, Takashi Hiramatsu, Jun'ichi Yokoyama JCAP 02 (2016) 023 Sep 2015 We investigate the imprint of reheating on the gravitational wave spectrum produced by self-ordering of multi-component scalar fields after a global phase transition. The equation of state of the Universe during reheating, which usually has differ... Pisin Chen, Teruaki Suyama, Jun'ichi Yokoyama Phys. Rev. D 92, 124016 (2015) Aug 2015 We propose a new scalar-tensor model which induces significant deviation from general relativity inside dense objects like neutron stars, while passing solar-system and terrestrial experiments, extending a model proposed by Damour and Esposito-Far... Taro Kunimitsu, Teruaki Suyama, Yuki Watanabe, Jun'ichi Yokoyama JCAP08(2015)044 Apr 2015 We systematically show that in potential driven generalized G-inflation models, quantum corrections coming from new physics at the strong coupling scale can be avoided, while producing observable tensor modes. The effective action can be approxima... Tsutomu Kobayashi, Masahide Yamaguchi, Jun'ichi Yokoyama Apr 2015 It has been pointed out that the null energy condition can be violated stably in some non-canonical scalar-field theories. This allows us to consider the Galilean Genesis scenario in which the universe starts expanding from Minkowski spacetime and... Kazufumi Takahashi, Jun'ichi Yokoyama Phys. Rev. D 91, 084060 (2015) Mar 2015 f(R) gravity is one of the simplest generalizations of general relativity, which may explain the accelerated cosmic expansion without introducing a cosmological constant. Transformed into the Einstein frame, a new scalar degree of freedom appears ... Takashi Hiramatsu, Yuhei Miyamoto, Jun'ichi Yokoyama Dec 2014 The mechanism of thermal inflation, a relatively short period of accelerated expansion after primordial inflation, is a desirable ingredient for a certain class of particle physics models if they are not to be in contention with the cosmology of t... Nov 2014 We study reheating processes and its cosmological consequences in the Starobinsky model embedded in the old-minimal supergravity. First, we consider minimal coupling between the gravity and matter sectors in the higher curvature theory, and transf... Hayato Motohashi, Alexei A. Starobinsky, Jun'ichi Yokoyama JCAP 09 (2015) 018 Nov 2014 We consider an inflationary scenario where the rate of inflaton roll defined by remains constant. The rate of roll is small for slow-roll inflation, while a generic rate of roll leads to the interesting case of constant-rol... Tomohiro Fujita, Jun'ichi Yokoyama, Shuichiro Yokoyama Nov 2014 We consider the possibility of enhancing the inflationary tensor mode by introducing a spectator scalar field with a small sound speed which induces gravitational waves as a second order effect. We analytically obtain the power spectra of gravitat... Sachiko Kuroyanagi, Kazunori Nakayama, Jun'ichi Yokoyama Prog. Theor. Exp. Phys. (2015) 013E02 Oct 2014 If the tensor-to-scalar ratio of cosmological perturbations takes a large value , which may be inferred by recent BICEP2 result, we can hope to determine thermal history, in particular, the reheating temperature, , after inflat... Kohei Kamada, Yuhei Miyamoto, Daisuke Yamauchi, Jun'ichi Yokoyama Phys. Rev. D 90, 083502 (2014) Jul 2014 The network of cosmic strings generated in a phase transition during inflation enters the scaling regime later than that of usual strings. If it occurs after the recombination, temperature anisotropies of the cosmic microwave background (CMB) at h... Teruaki Suyama, Yi-Peng Wu, Jun'ichi Yokoyama Phys. Rev. D 90, 043514 (2014) Jun 2014 Scalar field with generalized kinetic interactions metamorphoses depending on its field value, ranging from cosmological constant to stiff matter. We show that such a scalar field can give rise to temporal enhancement of the curvature perturbation... Phys. Rev. D 90, 103520 (2014) May 2014 We consider the ghost-free higher order corrections to the Starobinsky model in the old minimal supergravity. In general, higher order corrections cannot be forbidden by symmetries, which likely violate the flatness of the scalaron potential and m... Tomohiro Nakama, Tomohiro Harada, A. G. Polnarev, Jun'ichi Yokoyama Jan 2014 Primordial black holes (PBHs) are an important tool in cosmology to probe the primordial spectrum of small-scale curvature perturbations that reenter the cosmological horizon during radiation domination epoch. We numerically solve the evolution of... Tomohiro Nakama, Tomohiro Harada, A. G. Polnarev, Jun'ichi Yokoyama JCAP01(2014)037 Oct 2013 Primordial black holes (PBHs) are an important tool in cosmology to probe the primordial spectrum of small-scale curvature perturbations that reenter the cosmological horizon during radiation domination epoch. We numerically solve the evolution of... Kohei Kamada, Tsutomu Kobayashi, Taro Kunimitsu, Masahide Yamaguchi, Jun'ichi Yokoyama Phys. Rev. D 88, 123518 (2013) Sep 2013 Higgs G-inflation is a Higgs inflation model with a generalized Galileon term added to the standard model Higgs field, which realizes inflation compatible with observations. Recently, it was claimed that the generalized Galileon term induces insta... Yuhei Miyamoto, Hayato Motohashi, Teruaki Suyama, Jun'ichi Yokoyama Phys. Rev. D 89, 085037 (2014) Aug 2013 We study the dynamics of the oscillating gauged scalar field in a thermal bath. A Langevin type equation of motion of the scalar field, which contains both dissipation and fluctuation terms, is derived by using the real-time finite temperature eff... Yuki Watanabe, Jun'ichi Yokoyama Mar 2013 Reheating after R^2 inflation proceeds through gravitational particle production of conformally noninvariant fields. We argue that the nonvanishing expectation value of flat directions generic in supersymmetric theories break conformal invariance ... Christophe Ringeval, Teruaki Suyama, Jun'ichi Yokoyama Feb 2013 As additional perturbative degrees of freedom, it is known that magnetic fields of inflationary origin can source curvature perturbations on super-Hubble scales. By requiring the magnetic generated curvature to remain smaller than its inflationary... Takeshi Kobayashi, Jun'ichi Yokoyama JCAP02(2013)005 Oct 2012 Cosmic inflation driven by branes wrapping the extra dimensions involves Kaluza-Klein (KK) degrees of freedom in addition to the zero-mode position of the brane which plays the role of the inflaton. As the wrapped brane passes by localized sources... Hayato Motohashi, Teruaki Suyama, Jun'ichi Yokoyama Phys.Rev. D86 (2012) 123514 Oct 2012 We provide a general formalism to calculate the infrared correlators of multiple interacting scalar fields in the de Sitter space by means of the stochastic approach. These scalar fields are treated as test fields and hence our result is applicabl... Taro Kunimitsu, Jun'ichi Yokoyama Phys. Rev. D 86, 083541 (2012) Aug 2012 During inflation in the early universe, the Higgs field continuously acquires long-wave quantum fluctuations. They accumulate to yield a non-vanishing value with an exponentially large correlation length. We study consequences of such Higgs conden... Matthew Lake, Jun'ichi Yokoyama JCAP 09 (2012) 030 [JCAP 08 (2013) E01] Jul 2012 We consider a generalization of the Nielsen-Olesen ansatz, in an abelian-Higgs model with externally coupled charge, which describes strings with twisted magnetic flux lines in the vortex core. The solution does not possess cylindrical symmetry, w... Xian Gao, Tsutomu Kobayashi, Maresuke Shiraishi, Masahide Yamaguchi, Jun'ichi Yokoyama, Shuichiro Yokoyama Jul 2012 We compute the full bispectra, namely both auto- and cross- bispectra, of primordial curvature and tensor perturbations in the most general single-field inflation model whose scalar and gravitational equations of motion are of second order. The fo... Ryo Saito, Masahiro Nakashima, Yu-ichi Takamizu, Jun'ichi Yokoyama Jun 2012 We investigate the possibility that a heavy scalar field, whose mass exceeds the Hubble scale during inflation, could leave non-negligible signatures in the Cosmic Microwave Background (CMB) temperature anisotropy power spectrum through the parame... A. G. Polnarev, Tomohiro Nakama, Jun'ichi Yokoyama JCAP09(2012)027 Apr 2012 For an arbitrary strong, spherically symmetric super-horizon curvature perturbation, we present analytical solutions of the Einstein equations in terms of asymptotic expansion over the ratio of the Hubble radius to the length-scale of the curvatur... Teruaki Suyama, Jun'ichi Yokoyama Apr 2012 There is an observational indication of extragalactic magnetic fields. No known astrophysical process can explain the origin of such large scale magnetic fields, which motivates us to look for their origin in primordial inflation. By solving the l... Kohei Kamada, Yuhei Miyamoto, Jun'ichi Yokoyama Apr 2012 The cosmic string is a useful probe of the early Universe and may give us a clue to physics at high energy scales where any artificial particle accelerators cannot reach. Although one of the most promising tools is the cosmic microwave background,... Hayato Motohashi, Alexei A. Starobinsky, Jun'ichi Yokoyama Phys. Rev. Lett. 110, 121302 (2013) Mar 2012 It is shown that the tension between recent neutrino oscillation experiments, favoring sterile neutrinos with masses of the order of 1 eV, and cosmological data which impose stringent constraints on neutrino masses from the free streaming suppress... Kohei Kamada, Kazunori Nakayama, Jun'ichi Yokoyama Oct 2011 In F-term supergravity inflation models, scalar fields other than the inflaton generically receive a Hubble induced mass, which may restore gauge symmetries during inflation and phase transitions may occur during or after inflation as the Hubble p... Xian Gao, Tsutomu Kobayashi, Masahide Yamaguchi, Jun'ichi Yokoyama Aug 2011 We completely clarify the feature of primordial non-Gaussianities of tensor perturbations in generalized G-inflation, i.e., the most general single-field inflation model with second order field equations. It is shown that the most general cubic ac... Teruaki Suyama, Jun'ichi Yokoyama Jun 2011 If more than one curvaton dominate the Universe at different epochs from each other, curvature perturbations can be temporarily enhanced to a value much larger than the observed one 10^{-5}. The traces of the enhancement may be left as higher orde... Tsutomu Kobayashi, Masahide Yamaguchi, Jun'ichi Yokoyama Prog. Theor. Phys. 126 (2011), 511-529 May 2011 We study generalized Galileons as a framework to develop the most general single-field inflation models ever, Generalized G-inflation, containing yet further generalization of G-inflation, as well as previous examples such as k-inflation, extended... Tsutomu Kobayashi, Masahide Yamaguchi, Jun'ichi Yokoyama Phys.Rev.D83:103524,2011 Mar 2011 We present a comprehensive study of primordial fluctuations generated from G-inflation, in which the inflaton Lagrangian is of the form with . The Lagrangian still gives rise to second-order g... Hayato Motohashi, Alexei A. Starobinsky, Jun'ichi Yokoyama JCAP 1106 (2011) 006 Jan 2011 It is known that scalar-tensor theory of gravity admits regular crossing of the phantom divide line for dark energy, and existing viable models of present dark energy for its particular case -- gravity -- possess one such cross... Hayato Motohashi, Alexei A. Starobinsky, Jun'ichi Yokoyama Int.J.Mod.Phys. D20 (2011) 1347-1355 Jan 2011 We have investigated the evolution of a homogeneous isotropic background of the Universe and inhomogeneous subhorizon matter density perturbations in viable models of present dark energy and cosmic acceleration analytically and numerically.... Takashi Hiramatsu, Masahiro Kawasaki, Toyokazu Sekiguchi, Masahide Yamaguchi, Jun'ichi Yokoyama Phys.Rev.D83:123531, 2011 Dec 2010 Cosmological evolution of axionic string network is analyzed in terms of field-theoretic simulations in a box of 512^3 grids, which are the largest ever, using a new and more efficient identification scheme of global strings. The scaling parameter... Kohei Kamada, Tsutomu Kobayashi, Masahide Yamaguchi, Jun'ichi Yokoyama Phys.Rev.D83:083515,2011 Dec 2010 A new class of inflation models within the context of G-inflation is proposed, in which the standard model Higgs boson can act as an inflaton thanks to Galileon-like non-linear derivative interaction. The generated primordial density perturbation ... Yu-ichi Takamizu, Jun'ichi Yokoyama Phys.Rev.D83:043504,2011 Nov 2010 We clarify the behavior of curvature perturbations in a nonlinear theory in case the inflaton temporarily stops during inflation. We focus on the evolution of curvature perturbation on superhorizon scales by adopting the spatial Tsutomu Kobayashi, Masahide Yamaguchi, Jun'ichi Yokoyama Phys.Rev.Lett.105:231302,2010 Aug 2010 We propose a new class of inflation model, G-inflation, which has a Galileon-like nonlinear derivative interaction of the form in the Lagrangian with the resultant equations of motion being of second order. It is ... Hayato Motohashi, Alexei A. Starobinsky, Jun'ichi Yokoyama Prog. Theor. Phys. 124 (2010), 541-546 May 2010 The effect of massive neutrinos on matter power spectrum is discussed in the context of gravity. It is shown that the anomalous growth of density fluctuations on small scales due to the scalaron force can be compensated by free streaming of... Larissa Lorenz, Jerome Martin, Jun'ichi Yokoyama Phys.Rev.D82:023515,2010 Apr 2010 Stochastic effects during inflation can be addressed by averaging the quantum inflaton field over Hubble-patch sized domains. The averaged field then obeys a Langevin-type equation into which short-scale fluctuations enter as a noise term. We solv... Hayato Motohashi, Alexei A. Starobinsky, Jun'ichi Yokoyama Prog. Theor. Phys. 123 (2010), 887-902 Feb 2010 Evolution of a background space-time metric and sub-horizon matter density perturbations in the Universe is numerically analyzed in viable models of present dark energy and cosmic acceleration. It is found that viable models generically exh... Hayato Motohashi, Alexei A. Starobinsky, Jun'ichi Yokoyama Proceedings of the 19nth Workshop on General Relativity and Gravitation in Japan, 239-242 (2010) http://www2.rikkyo.ac.jp/web/jgrg19/index.html Feb 2010 We present numerical calculation of the evolution of a background space-time metric and sub-horizon matter density perturbations in viable gravity models of present dark energy and cosmic acceleration. We found that viable models genericall... Ryo Saito, Jun'ichi Yokoyama Prog.Theor.Phys.123:867-886,2010 Dec 2009 We investigate features of Gravitational Waves (GWs) induced by primordial density fluctuations with a large amplitude peak associated with formation of Primordial Black Holes (PBHs). It is shown that the spectrum of induced GW is insensitive to t... Kiyotomo Ichiki, Ryo Nagata, Jun'ichi Yokoyama Phys.Rev.D81:083010,2010 Nov 2009 As a test of the standard inflationary cosmology, which generically predicts nearly scale-invariant spectrum of primordial curvature fluctuations, we perform Markov-Chain Monte-Carlo analysis to search for possible modulations in the power spectru... Masahiro Nakashima, Kazuhide Ichikawa, Ryo Nagata, Jun'ichi Yokoyama JCAP 1001:030,2010 Oct 2009 We investigate constraints on the time variation of the fine structure constant between the recombination epoch and the present epoch, \Delta\alpha/\alpha \equiv (\alpha_{rec} - \alpha_{now})/\alpha_{now}, from cosmic microwave background (CMB) ta... Kazunori Nakayama, Jun'ichi Yokoyama JCAP 1001:010,2010 Oct 2009 We study observational implications of the stochastic gravitational wave background and a non-Gaussian feature of scalar perturbations on the curvaton mechanism of the generation of density/curvature fluctuations, and show that they can determine ... Hayato Motohashi, Alexei A. Starobinsky, Jun'ichi Yokoyama Int.J.Mod.Phys.D18:1731-1740,2009 May 2009 For a class of viable cosmological models in gravity which deviation from the Einstein gravity decreases as a inverse power law of the Ricci scalar for large , an analytic solution for density perturbations in the matter component du... Martin Lemoine, Jerome Martin, Jun'ichi Yokoyama Phys.Rev.D80:123514,2009 Apr 2009 We set constraints on moduli cosmology from the production of dark matter -- decay, assuming the modulus remains light during inflation. We find that the moduli problem be... Martin Lemoine, Jerome Martin, Jun'ichi Yokoyama Europhys.Lett.89:29001,2010 Mar 2009 Moduli fields generically produce strong dark matter -- radiation and baryon -- radiation isocurvature perturbations through their decay if they remain light during inflation. We show that existing upper bounds on the magnitude of such fluctuation... Ryo Nagata, Jun'ichi Yokoyama Phys.Rev. D79 (2009) 043010 Dec 2008 The primordial curvature fluctuation spectrum is reconstructed by the maximum likelihood reconstruction method using the five-year Wilkinson Microwave Anisotropy Probe data of the cosmic microwave background temperature anisotropy. We apply the co... Ryo Saito, Jun'ichi Yokoyama Phys.Rev.Lett.102:161101,2009 Dec 2008 Formation of significant number of primordial black holes (PBHs) is realized if and only if primordial density fluctuations have a large amplitude, which means that tensor perturbations generated from these scalar perturbations as a second order e... Masahiro Nakashima, Ryo Nagata, Jun'ichi Yokoyama Prog.Theor.Phys.120:1207-1215,2008 Oct 2008 The constraints on the time variation of the fine structure constant at recombination epoch relative to its present value, , are obtained from the an... Ryo Nagata, Jun'ichi Yokoyama Phys.Rev.D78:123002,2008 Sep 2008 The primordial curvature fluctuation spectrum is reconstructed by the cosmic inversion method using the five-year WMAP data of the cosmic microwave background temperature anisotropy. We apply the covariance matrix analysis and decompose the recons... Ryo Saito, Jun'ichi Yokoyama, Ryo Nagata JCAP 0806:024,2008 Apr 2008 We show a text-book potential for single-field inflation, namely, the Coleman-Weinberg model can induce double inflation and formation of primordial black holes (PBHs), because fluctuations that leave the horizon near the end of first inflation ar... Kazunori Nakayama, Shun Saito, Yudai Suwa, Jun'ichi Yokoyama JCAP 0806:020,2008 Apr 2008 Thermal history of the universe after big-bang nucleosynthesis (BBN) is well understood both theoretically and observationally, and recent cosmological observations also begin to reveal the inflationary dynamics. However, the epoch between inflati... Nemanja Kaloper, Lorenzo Sorbo, Jun'ichi Yokoyama Phys.Rev.D78:043527,2008 Mar 2008 We revisit inflation in induced gravity. Our focus is on models where the low scale Planck mass is completely determined by the breaking of the scaling symmetry in the field theory sector. The Higgs-like field which breaks the symmetry with a GUT-... Phys.Rev.D78:043502,2008 Mar 2008 We investigate the Affleck-Dine mechanism when multiple flat directions have large values simultaneously. We consider in detail the case when both and flat directions are operative with a non-renormalizable superpotential. In case ... Kazunori Nakayama, Shun Saito, Yudai Suwa, Jun'ichi Yokoyama Phys.Rev.D77:124001,2008 Feb 2008 It is shown that space-based gravitational wave detectors such as DECIGO and/or Big Bang Observer (BBO) will provide us with invaluable information on the cosmic thermal history after inflation and they will be able to determine the reheat tempera... Jerome Martin, Jun'ichi Yokoyama JCAP 0801:025,2008 Nov 2007 We consider the generation of large-scale magnetic fields in slow-roll inflation. The inflaton field is described in a supergravity framework where the conformal invariance of the electromagnetic field is generically and naturally broken. For each... Prog.Theor.Phys.122:969-986,2010 Jun 2009 We study the fine-tuning problem on the initial condition of inflation in the minimal supersymmetric standard model (MSSM), associated with the narrowness of the slow-roll region. We consider two cases before the onset of the MSSM inflation, namel... Toshihiro Kawaguchi, Masahiro Kawasaki, Tsutomu Takayama, Masahide Yamaguchi, Jun'ichi Yokoyama Mon.Not.Roy.Astron.Soc.388:1426-1432,2008 Nov 2007 Formation of primordial black holes (PBHs) on astrophysical mass scales is a natural consequence of inflationary cosmology if the primordial perturbation spectrum has a large and negative running of the spectral index as observationally inferred t... Takeshi Chiba, Yoshiaki Himemoto, Masahide Yamaguchi, Jun'ichi Yokoyama Phys.Rev.D76:043516,2007 May 2007 We calculate the signal-to-noise ratio (SNR) of the stochastic gravitational-wave background in an extreme case that its spectrum has a sharp falloff with its amplitude close to the detection threshold. Such a spectral feature is a characteristic ... Kiyotomo Ichiki, Masahide Yamaguchi, Jun'Ichi Yokoyama Phys.Rev.D75:084017,2007 Nov 2006 Effects of neutrino free streaming is evaluated on the primordial spectrum of gravitational radiation taking both neutrino chemical potential and masses into account. The former or the lepton asymmetry induces two competitive effects, namely, to i... Takeshi Chiba, Tatsuo Kobayashi, Masahide Yamaguchi, Jun'ichi Yokoyama Phys.Rev.D75:043516,2007 Oct 2006 Recent astrophysical observations indicate that the proton-electron mass ratio and the fine structure constant have gone through nontrivial time evolution. We discuss their time variation in the context of a dilaton runaway scenario with gauge cou... Bo Feng, Jun-Qing Xia, Jun'ichi Yokoyama JCAP 0705:020,2007 Aug 2006 We probe the scale dependence of the primordial spectrum in the light of the three-year WMAP (WMAP3) alone and WMAP3 in combination with the other cosmological observations such as galaxy clustering and Type Ia Supernova (SNIa). We pay particular ... Masahiro Kawasaki, Tsutomu Takayama, Masahide Yamaguchi, Jun'ichi Yokoyama Phys.Rev.D74:043525,2006 May 2006 We numerically investigate density perturbations generated in the smooth hybrid new inflation model, a kind of double inflation model that is designed to reproduce the running spectral index suggested by the WMAP results. We confirm that this mode... Masahide Yamaguchi, Jun'ichi Yokoyama Phys.Rev. D74 (2006) 043523 Dec 2005 Any one-field inflation is actually realized in a multifield configuration because the inflaton must have couplings with other fields to reheat the universe and is coupled to all other fields at least gravitationally. In all single inflaton models... Phys.Rev.D73:043507,2006 Dec 2005 We discuss a D-term inflation scenario where a right-handed sneutrino can be an inflaton field leading to a viable inflation and leptogenesis, with a minimal form of Kähler potential. The decay of an inflaton sneutrino can non-thermally create l... Osamu Seto, Jun'ichi Yokoyama Phys.Rev. D73 (2006) 023508 Aug 2005 The influence of higher-order terms in the Kähler potential of the supergravity D-term inflation model on the density perturbation is studied. We show that these terms can make the inflaton potential flatter, which lowers the energy scale of i... Noriyuki Kogo, Misao Sasaki, Jun'ichi Yokoyama Prog.Theor.Phys.114:555-572,2005 Apr 2005 We investigate the question of how tightly we can constrain the cosmological parameters by using the `cosmic inversion'' method in which we directly reconstruct the power spectrum of primordial curvature perturbations, , from the temperatur... Kazuharu Bamba, J. Yokoyama Feb 2005 Generation of large-scale magnetic fields is studied in dilaton electromagnetism in noncommutative inflationary cosmology, taking into account the effects of the spacetime uncertainty principle motivated by string theory. We show that it is possib... Tomo Takahashi, Masahide Yamaguchi, Jun'ichi Yokoyama, Shuichiro Yokoyama Phys.Lett.B678:15-19,2009 May 2009 We investigate density fluctuations in a scenario with gravitino dark matter in the framework of modulated reheating, which is known to generate large non-Gaussianity. We show that gravitino dark matter is disfavored if primordial fluctuations hav... Kazunori Kohri, Masahiro Yamaguchi, Jun'ichi Yokoyama Phys.Rev.D72:083510,2005 Feb 2005 We propose a new scenario of non-thermal production of neutralino cold dark matter, in which the overproduction problem of lightest supersymmetric particles (LSPs) in the standard thermal history is naturally solved. The mechanism requires a heavy... Kazuharu Bamba, J. Yokoyama Phys.Rev. D70 (2004) 083508 Sep 2004 The generation of large-scale magnetic fields is studied in dilaton electromagnetism in noncommutative inflationary cosmology taking into account the effects of the spacetime uncertainty principle motivated by string theory. We show that it is pos... Jun'ichi Yokoyama Phys.Rev.Lett.96:171301,2006 Jan 2006 It is shown that the coherent field oscillation of moduli fields with weak or TeV scale masses can dissipate its energy efficiently if they have a derivative coupling to standard bosonic fields in a thermal state. This mechanism may provide a new ... Noriyuki Kogo, Misao Sasaki, Jun'ichi Yokoyama Phys.Rev.D70:103001,2004 Sep 2004 We develop a new method to reconstruct the power spectrum of primordial curvature perturbations, , by using both the temperature and polarization spectra of the cosmic microwave background (CMB). We test this method using several mock primor... Kazunori Kohri, Masahiro Yamaguchi, Jun'ichi Yokoyama Phys.Rev. D70 (2004) 043522 Mar 2004 We study the cosmological consequences of generic scalar fields like moduli which decay only through gravitationally suppressed interactions. We consider a new production mechanism of gravitinos from moduli decay, which might be more effective tha... Masahide Yamaguchi, Jun'ichi Yokoyama Phys.Rev.D70:023513,2004 Feb 2004 It is shown that in a smooth hybrid inflation model in supergravity adiabatic fluctuations with a running spectral index with on a large scale and on a smaller scale can be naturally generated, as favored by the first-year data o... Kazuharu Bamba, J. Yokoyama Phys.Rev.D69:043507,2004 Oct 2003 The generation of large-scale magnetic fields is studied in dilaton electromagnetism in inflationary cosmology, taking into account the dilaton's evolution throughout inflation and reheating until it is stabilized with possible entropy production.... Jun'ichi Yokoyama Phys.Lett. B635 (2006) 66-71 Oct 2005 We calculate the dissipation rate of a coherently oscillating scalar field in a thermal environment using nonequilibrium quantum field theory and apply it to the reheating stage after cosmic inflation. It is shown that the rate is nonvanishing eve... Jun'ichi Yokoyama Phys.Rev. D70 (2004) 103511 Jun 2004 Relaxation process of a coherent scalar field oscillation in the thermal bath is investigated using nonequilibrium quantum field theory. The Langevin-type equation of motion is obtained which has a memory term and both additive and multiplicative ... Noriyuki Kogo, Makoto Matsumiya, Misao Sasaki, Jun'ichi Yokoyama Astrophys.J.607:32-39,2004 Sep 2003 We reconstruct the primordial spectrum of the curvature perturbation, , from the observational data of the Wilkinson Microwave Anisotropy Probe (WMAP) by the cosmic inversion method developed recently. In contrast to conventional parameter-f...	2019-12-11 11:00:01	{"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": 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": 0.6356694102287292, "perplexity": 5930.396308427461}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540530857.12/warc/CC-MAIN-20191211103140-20191211131140-00270.warc.gz"}
http://mathschallenge.net/full/consecutive_composites	## Consecutive Composites #### Problem Although there are infinitely many primes it is a most remarkable fact that there can always be found a sequence of $n$ consecutive composite numbers. For example, there are thirteen consecutive composite numbers between the primes 113 and 127. Prove that there exists a sequence of $n$ consecutive composite numbers for any finite value $n$. #### Solution Given that $n$ is a positive integer it is clear that $n! = n \times (n - 1) \times (n - 2) \times ... \times 3 \times 2 \times 1$, is divisible by all the integers $1, 2, 3, ... , n$. So it follows that $n! + 2$ is divisible by 2, $n! + 3$ is divisible by 3, ... , $n! + n$ is divisible by $n$. Therefore $(n+1)! + 2, (n+1)! + 3, ... , (n+1)! + n, (n+1)! + (n+1)$ is a sequence of $n$ consecutive composite numbers. Note that although this proves the existence of a sequence of $n$ consecutive composites it does not find the first such sequence. For example, using this method $n$ = 5, 6! = 720, so 722, 723, 724, 725, and 726 are all composite, but the first sequence of five consecutive composites is 24, 25, 26, 27, and 28. Also be aware that as $n$ gets larger, the size of the list of consecutive composite numbers increases. However, as $n$ always remains finite, the size of the set remains finite and so this proof does not demonstrate an end to primes. Problem ID: 325 (26 Jun 2007) Difficulty: 3 Star Only Show Problem	2015-03-30 01:02:34	{"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": 0.8292650580406189, "perplexity": 190.27962907973262}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1427131298871.15/warc/CC-MAIN-20150323172138-00214-ip-10-168-14-71.ec2.internal.warc.gz"}
http://alieco-spa.com/hus3f1c/a7c3c5-best-linear-unbiased-estimator-characteristics	The expectation $\mx X\BETA$ is trivially estimable $\mx y_f$ is said to be unbiasedly predictable. Minimizing $$J$$ with respect to $$\textbf{a}$$ is equivalent to setting the first derivative of $$J$$ w.r.t $$\textbf{a}$$ to zero. Example: Suppose X 1;X 2; ;X n is an i.i.d. \mx B(\mx X : \SIGMA \mx X^{\bot}) = (\mx X : \mx{0}) , $$J = \textbf{a}^T \textbf{C} \textbf{a} + \lambda(\textbf{a}^T \textbf{s} -1) \;\;\;\;\;\;\;\;\;\; (11)$$. \mx{V}_{21} & \mx V_{22} $\C(\mx A),$ Any given sample mean may underestimate or overestimate μ, but there is no systematic tendency for sample means to either under or overestimate μ. www.springer.com It is unbiased 3. see, e.g., for $\mx y_f$ if and only if there exists a matrix $\mx L$ such that $\mx A^{+},$ under $\{ \mx y, \, \mx X\BETA, \, \mx I_n \}$ the $\OLSE$ of In practice, knowledge of PDF of the underlying process is actually unknown. \end{equation}. \end{equation} Marshall and Olkin (1979, p. 462)], i.e., that the difference B - A is a symmetric nonnegative definite matrix. Isotalo and Puntanen (2006, p. 1015). \mx y \\ \cov( \mx{G} \mx y) \leq_{ {\rm L}} \cov( \mx{L} \mx y) \quad 5.2, Th. and $\M_{2} = \{ \mx y, \, \mx X\BETA, \, \mx V_2 \}$, Heidelberg: The term σ ^ 1 in the numerator is the best linear unbiased estimator of σ under the assumption of normality while the term σ ^ 2 in the denominator is the usual sample standard deviation S. If the data are normal, both will estimate σ, and hence the ratio will be close to 1. there exists a matrix $\mx A$ such that $\mx{K}' = \mx{A}\mx{X}$, i.e., \end{pmatrix} = Such a property is known as the Gauss-Markov theorem, which is discussed later in multiple linear regression model. $\mx B \mx y$ is the $\BLUE$ for $\mx X\BETA$ if and only if So they are termed as the Best Linear Unbiased Estimators (BLUE). is the best linear unbiased predictor ($\BLUP$) for $\mx y_f$ Even if the PDF is known, finding an MVUE is not guaranteed. and it can be expressed as $\BETAH = (\mx X' \mx X) ^{-}\mx X' \mx y,$ between the Best Linear Unbiased Estimate (BLUE) 2 Motivation for BLUE Except for Linear Model case, the optimal MVU estimator might: 1. not even exist 2. be difficult or impossible to find ⇒ Resort to a sub-optimal estimate BLUE is one such sub-optimal estimate Idea for BLUE: 1. \def\BLUE}{\small\mathrm{BLUE}} Thus, the entire estimation problem boils down to finding the vector of constants – $$\textbf{a}$$. \end{equation} [\OLSE vs. \BLUE] Consider the general linear model \M =\{\mx y,\,\mx X\BETA,\,\mx V\}. where \mx X \in \rz^{n \times p} and \mx Z \in \rz^{n \times q} are \mx y = \mx X \BETA + \EPS, We may not be sure how much performance we have lost – Since we will not able to find the MVUE estimator for bench marking (due to non-availability of underlying PDF of the process). some statements which involve the random vector \mx y, these random effects with \end{pmatrix}. The nonnegative in \M, and \EPS_f is an m \times 1 random Find the best one (i.e. \mx V & \mx X \\ \mx X' & \mx 0 this is what we would like to find ). the column space, x[n] = s[n] \theta + w[n] \;\;\;\;\;\;\;\;\;\; (5), Here , $$w[n]$$ is zero mean process noise , whose PDF can take any form (Uniform, Gaussian, Colored etc., ). It is also worth noting that the matrix \mx G satisfying Here $$\textbf{a}$$ is a vector of constants whose value we seek to find in order to meet the design specifications. Then the estimator \mx{Gy} is the \BLUE for \mx X\BETA if and only if there exists a matrix \mx{L} \in \rz^{p \times n} so that \mx G is a solution to In our for any linear unbiased estimator \BETA^{} of \BETA; here \mx{H} = \mx P_{\mx X} and \mx{M} = \mx I_n - \mx H. \begin{pmatrix} to denote the orthogonal projector (with respect to the standard This assumption addresses the … Springer Science+Business Media, LLC. Just the first two moments (mean and variance) of the PDF is sufficient for finding the BLUE. Discount not applicable for individual purchase of ebooks. \C(\mx A^{\bot}) = \NS(\mx A') = \C(\mx A)^{\bot}. of attention in the literature, \mx{V}^+ and \mx{H} and \mx{M} = \mx I_n - \mx H may be interchanged.). Consider the linear models In terms of Pandora's Box (Theorem 2), \mx A \mx y = \BLUP(\GAMMA) as \end{align} Street West, Montréal (Québec), Canada H3A 2K6. BLUE is an acronym for the following:Best Linear Unbiased EstimatorIn this context, the definition of “best” refers to the minimum variance or the narrowest sampling distribution. by \end{pmatrix}. Then the linear estimator \mx{Ay} Unbiasedness is discussed in more detail in the lecture entitled Point estimation. where for $\mx G$ if and only if $\C(\mx X : \mx V) = \rz^n.$ For the equality The expectation and the covariance matrix are where \mx 0 \\ We present below six characterizations for the $\OLSE$ and \text{for all } \mx{L} \colon $\mx {W}= \mx V + \mx X\mx U\mx X'$ and $\mx U$ is any arbitrary conformable "det" denotes estimators; those which have are called estimable parametric functions, $\mx y$ belongs to the subspace $\C(\mx X : \mx V)$ It can be used to derive the Kalman filter, the method of Kriging used for ore reserve estimation, credibility theory used to work out insurance premiums, and Hoadley's quality measurement plan used to estimate a quality index. where "$\leq_\text{L}$" refers to the Löwner partial ordering. \mx{V}_{21} & \mx V_{22} Linearity constraint was already given above. for $\mx X\BETA$ is defined to be \E(\mx{Ay}) = \mx{AX}\BETA = \mx K' \BETA \E(\GAMMA) = \mx 0_q , \quad \mx{L} $\mx y_f = \mx X_f\BETA +\EPS_f ,$ BLUE. $\def\NS{ {\mathscr N}}\def\OLSE{ {\small\mathrm{OLSE}}}$ then = \{ \mx y,\, \mx X\BETA + \mx Z\GAMMA, \, \mx D,\,\mx R \}.$\begin{equation} the best linear unbiased estimator, \mx G_2 = \mx{H} - \mx{HVM}(\mx{MVM})^{-}\mx{M} + \mx F_{2}[\mx{I}_n - \mx L for all$\BETA\in\rz^{p}.$In general, it is a method of estimating random effects.$\mx A',$In econometrics, Ordinary Least Squares (OLS) method is widely used to estimate the parameters of a linear regression model. Mitra, Sujit Kumar and Moore, Betty Jeanne (1973). if and only if$\mx{A}$satisfies the equation Consider the general linear model$ \M =\{\mx y,\,\mx X\BETA,\,\mx V\}$. = \mx A(\mx A'\mx A)^{-}\mx A'$ and For the equality Here A 0$is an unknown constant.$\mx A^{-},$\quad \text{or in short } \end{pmatrix} The distinction arises because it is conventional to talk about estimating fixe… So your problem boils down to $$\text{ Minimize }\qquad\sum_{i=1}^n c_i^2 \\\quad\qquad\text{ subject to }\qquad\sum_{i=1}^n c_i=\sqrt{\frac{\pi}{2}}$$ Can you proceed now, …$\mx X\BETA$is trivially the$\BLUE$; this result is often called \begin{equation} Tiga asumsi dasar yang tidak boleh dilanggar oleh regresi linear berganda yaitu : 1. The Gauss-Markov theorem states that under the five assumptions above, the OLS estimator b is best linear unbiased.$\def\E{E}$Consider the mixed model known matrices,$\BETA \in \rz^{p}$is a vector of unknown fixed \end{equation}$ \C(\mx K ) \subset \C(\mx X')$. effects,$\GAMMA$is an unobservable vector ($q$elements) of \mx A(\mx X : \SIGMA \mx X^{\bot}) = (\mx 0 : \mx{D}\mx{Z}' \mx X^{\bot}). following proposition and related discussion, see, e.g., As is well known, a statistic FY is said to be the best linear unbiased estimator (BLUE) of Xif E (FY) = Xand D (FY) -L D (GY) for every GY such that E (GY) = X Here A -L B means that A is below B with respect to the Lner partial ordering [cf. \begin{equation} statements need hold only for those values of$\mx y$that belong and the null space, \mx A' \\ since Anderson (1948), \begin{equation} Moreover,$\def\rz{ {\mathbf{R}}} \def\SIGMA{\Sigma} \def\var{ {\rm var}}Then the following statements are equivalent: Notice that obviously \begin{align} The OLS estimator is an efficient estimator. which would provide an unbiased and in some sense "best" estimator Reprinted with permission from Lovric, Miodrag (2011), Least squares theory using an estimated dispersion matrix and its application to measurement of signals. Encyclopedia of Statistical Science. The best linear unbiased estimator (BLUE) of the vector {\displaystyle \beta } of parameters {\displaystyle \beta _ {j}} is one with the smallest mean squared error for every vector {\displaystyle \lambda } of linear combination parameters. It is also efficient amongst all linear estimators, as well as all estimators that uses some function of the x.\NS(\mx A)$Gauss--Markov estimation with an incorrect dispersion matrix. \mx Z \mx D \\ Watson (1967), 1. \begin{pmatrix} Linear Models – Least Squares Estimator (LSE), Multipath channel models: scattering function, Minimum Variance Unbiased Estimator (MVUE), Minimum Variance Unbiased Estimators (MVUE), Likelihood Function and Maximum Likelihood Estimation (MLE), Score, Fisher Information and Estimator Sensitivity, Introduction to Cramer Rao Lower Bound (CRLB), Cramer Rao Lower Bound for Scalar Parameter Estimation, Applying Cramer Rao Lower Bound (CRLB) to find a Minimum Variance Unbiased Estimator (MVUE), Cramer Rao Lower Bound for Phase Estimation, Normalized CRLB - an alternate form of CRLB and its relation to estimator sensitivity, Cramer Rao Lower Bound (CRLB) for Vector Parameter Estimation, The Mean Square Error – Why do we use it for estimation problems, How to estimate unknown parameters using Ordinary Least Squares (OLS), Essential Preliminary Matrix Algebra for Signal Processing.$ \mx{G}\mx X = \mx{X}.$Contact Us.$\def\rank{ {\rm rank}} \def\tr{ { \rm trace}}$Restrict estimate to be unbiased 3. BLUP was derived by Charles Roy Henderson in 1950 but the term "best linear unbiased predictor" (or "prediction") seems not to have been used until 1962. \mx{A}(\mx{X} : \mx{V} \mx X^{\bot}) = (\mx X_f : \mx{V}_{21} \mx X^{\bot} ). Puntanen, Simo; Styan, George P. H. and Werner, Hans Joachim (2000). The equation (1) has a unique solution However, not all parametric functions have linear unbiased Consider now two linear models observations,$\BETA$is the same vector of unknown parameters as Untuk menghasilkan keputusan yang BLUE maka harus dipenuhi diantaranya tiga asumsi dasar.$\mx A \mx y$is the$\BLUP$for$\GAMMA$if and only if$\C(\mx A)^{\bot},$in the following form, see is the Best Linear Unbiased Estimator (BLUE) if εsatisﬁes (1) and (2). \begin{pmatrix} Rao (1967), Linear regression models have several applications in real life. We have discussed Minimum Variance Unbiased Estimator (MVUE) in one of the previous articles. matrix such that$\C(\mx W) = \C(\mx X : \mx V).\def\BLUP}{\small\mathrm{BLUP}}$the Moore--Penrose inverse, [Pandora's Box] Anderson, T. W. (1948). If$\mx V$is positive definite, Then$\OLSE(\mx{X}\BETA) = \BLUE(\mx{X}\BETA)$if and only if any one of the following six equivalent conditions holds.$\EE(\EPS ) = \mx 0,and \end{pmatrix} \begin{align} \frac{\partial J}{\partial \textbf{a}} &= 2\textbf{C}\textbf{a} + \lambda \textbf{s}=0 \\ & \Rightarrow \boxed {\textbf{a}=-\frac{\lambda}{2}\textbf{C}^{-1}\textbf{s}} \end{align} \;\;\;\;\;\;\;\;\;\; (12), $$\textbf{a}^T \textbf{s} = -\frac{\lambda}{2}\textbf{s}^{T}\textbf{C}^{-1} \textbf{s}=1 \Rightarrow \boxed {-\frac{\lambda}{2}=\frac{1}{\textbf{s}^{T}\textbf{C}^{-1} \textbf{s}}} \;\;\;\;\;\;\;\;\;\; (13)$$, Finally, from $$(12)$$ and $$(13)$$, the co-effs of the BLUE estimator (vector of constants that weights the data samples) is given by, $$\boxed{a = \frac{\textbf{C}^{-1}\textbf{s}}{\textbf{s}^{T}\textbf{C}^{-1} \textbf{s}}} \;\;\;\;\;\;\;\;\;\; (14)$$, The BLUE estimate and the variance of the estimates are as follows, $$\boxed{ \hat{\theta}_{BLUE} =\textbf{a}^{T} \textbf{x} = \frac{\textbf{C}^{-1}\textbf{s} \textbf{x}}{\textbf{s}^{T}\textbf{C}^{-1} \textbf{s}}} \;\;\;\;\;\;\;\;\;\; (15)$$, $$\boxed {var(\hat{\theta})= \frac{1}{\textbf{s}^{T}\textbf{C}^{-1} \textbf{s}} } \;\;\;\;\;\;\;\;\;\; (16)$$. The random- and fixed-effects estimators (RE and FE, respectively) are two competing methods that address these problems. \mx 0\BLUE(\mx X\BETA) = \mx X(\mx X' \mx V^{-1} \mx X)^{-} \mx X' \mx V^{-1} \mx y.\mx M$. where$\SIGMA= \mx Z\mx D\mx Z' + \mx R$. \E(\EPS) = \mx 0_n \,, \quad As discussed above, in order to find a BLUE estimator for a given set of data, two constraints – linearity & unbiased estimates – must be satisfied and the variance of the estimate should be minimum. 2. Active 1 year, 11 ...$ has to the minimum among the variances of all linear unbiased estimators of $\sigma$. Division Headquarters 315 N Racine Avenue, Suite 501 Chicago, IL 60607 +1 866-331-2435 Email: [email protected], Department of Mathematics and Statistics, $\mx{K}' \BETA$ is estimable $\def\C{ {\mathscr C}}$ \var(\betat_i) \le \var(\beta^{}_i) \,, \quad i = 1,\dotsc,p , Haslett and Puntanen (2010b, 2010c). (One covariance matrix is said to be larger than another if their difference is positive semi-definite.) On the theory of testing serial correlation. \mx L $\mx X' \mx X \BETAH = \mx X' \mx y$; hence $\EPS$ is an unobservable vector of random errors This article considers sevaral estimators, including the Best Linear Unbiased Estimator (BLUE), for an SQD. For some further references from those years we may mention \mx y_f Projectors, generalized inverses and the BLUE's. $\sigma^2=1.$. In terms of Pandora's Box (Theorem 2), $\mx{Ay}$ is the $\BLUP$ Consider the model If $\mx X$ has full column rank, then $\BETA$ is estimable \begin{equation} we will use the symbols The mean of the above equation is given by, $$E(x[n]) = E(s[n] \theta) = s[n] \theta \;\;\;\;\; \;\;\;\;(6)$$, $$E[\hat{\theta}] =\sum_{n=0}^{N} a_n E \left( x[n] \right) = \theta \sum_{n=0}^{N} a_n s[n] = \theta \textbf{a}^T \textbf{s} = \theta \;\;\;\;\;\;\;\; (7)$$, $$\theta \textbf{a}^T \textbf{s} = \theta \;\;\;\;\;\;\; (8)$$, The above equality can be satisfied only if, $$\textbf{a}^T \textbf{s} =1 \;\;\;\;\;\;\; (9)$$. \tr [\cov(\BETAT)] \le \tr [\cov(\BETA^{})] , \qquad = A study of the influence of the `natural restrictions' on estimation problems in the singular Gauss--Markov model. under two partitioned models, see \M_{\mathrm{mix}} \mx V & \mx{V}_{12} \\ Theorem 4. Combining both the constraints $$(1)$$ and $$(2)$$ or $$(3)$$, $$E[\hat{\theta}] =\sum_{n=0}^{N} a_n E \left( x[n] \right) = \textbf{a}^T \textbf{x} = \theta \;\;\;\;\;\;\;\; (4)$$. with probability $1$; this is the consistency condition process we derive the hyetograph associated with any given flood discharge Q, using best linear unbiased estimation (BLUE) theory. \end{equation} Unbiasedness. \begin{pmatrix} The Gauss-Markov theorem famously states that OLS is BLUE. Note that even if θˆ is an unbiased estimator of θ, g(θˆ) will generally not be an unbiased estimator of g(θ) unless g is linear or aﬃne. 30% discount is given when all the three ebooks are checked out in a single purchase (offer valid for a limited period). This page was last edited on 29 March 2016, at 20:18. \begin{pmatrix} Consider a data model, as shown below, where the observed samples are in linear form with respect to the parameter to be estimated. \mx X' \mx{V}_{12} \\ Click here for more information. Rao (1971, Th. \begin{equation} He is a masters in communication engineering and has 12 years of technical expertise in channel modeling and has worked in various technologies ranging from read channel, OFDM, MIMO, 3GPP PHY layer, Data Science & Machine learning. $\def\GAMMA{\gamma}$ In this article we consider the general linear model see, e.g., Two matrix-based proofs that the linear estimator, Rao, C. Radhakrishna (1967). It can further be shown that the ordinary least squares estimators b0 and b1 possess the minimum variance in the class of linear and unbiased estimators. Effect of adding regressors on the equality of the BLUEs under two linear models. \tag{1} Find the linear estimator that is unbiased and has minimum variance This leads to Best Linear Unbiased Estimator (BLUE) To find a BLUE estimator, full knowledge of PDF is not needed. Email: [email protected], https://encyclopediaofmath.org/index.php?title=Best_linear_unbiased_estimation_in_linear_models&oldid=38515. Theorem 2. \begin{equation} A linear predictor $\mx{Ay}$ is said to be unbiased for $\mx y_f$ if The conditional mean should be zero.A4. Haslett, Stephen J. and Puntanen, Simo (2010a). \mx{G}(\mx{X} : \mx{V}\mx{X}^{\bot} ) = (\mx{X} : \mx{0}). holds for all $\mx B$ such that $\mx{By}$ is an unbiased linear A mixed linear model can be presented as \cov\begin{pmatrix} \end{pmatrix} the following ways: A widely used method for prediction of complex traits in animal and plant breeding is $\mx A$ and $\mx B$ as submatrices. Just repeated here for convenience. $\mx{BX} = \mx{I}_p. This is a typical Lagrangian Multiplier problem, which can be considered as minimizing the following equation with respect to $$\textbf{a}$$ (Remember !!!$\cov(\GAMMA,\EPS) = \mx 0_{q \times p}$and \det[\cov(\BETAT)] \le \det[\cov(\BETA^{})], \iff inner product) onto covariance matrix Let$\mx K' \BETA$be a given vector of parametric functions specified error vector associated with new observations. definite (possibly singular) matrix$\mx V $is known. of the linear model, \mx G_1 = \mx{X}(\mx{X}'\mx{W}^{-}\mx{X})^{-}\mx{X}'\mx{W}^{-} That is, the OLS estimator has smaller variance than any other linear unbiased estimator.$\var$refers to the variance and \begin{pmatrix}$\def\EE{E}$4.4 Feedback 4. 3.3, Th.$\BETA$and Discount can only be availed during checkout.$\M_f$, where The new observations are assumed to follow where$\mx F_{1}$and$\mx F_{2}$are arbitrary Consider the general linear model$ \M =\{\mx y,\,\mx X\BETA,\,\mx V\}$. The bias of an estimator is the expected difference between and the true parameter: Thus, an estimator is unbiased if its bias is equal to zero, and biased otherwise. An unbiased linear estimator$\mx{Gy} for all $\mx{B}$ such that (1) with expectation Christensen (2002, p. 283), $\C(\mx A).$ \C(\mx V_2\mx X^{\bot}) = \C(\mx V_1 \mx X^\bot). $\OLSE(\mx K' \BETA) = \mx K' \BETAH,$ 5.5), that Zyskind (1967) $\mx{X}_f\BETA$ is a given estimable parametric function. \end{pmatrix}. denote an $m\times 1$ unobservable random vector containing Marshall and Olkin (1979, p. 462)], i.e., that the difference B - A is a symmetric nonnegative definite matrix. Relative e ciency: If ^ 1 and ^ 2 are both unbiased estimators of a parameter we say that ^ 1 is relatively more e cient if var(^ 1)	2021-10-25 17:54:24	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "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": 0.9878585338592529, "perplexity": 1869.1911191073934}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1634323587719.64/warc/CC-MAIN-20211025154225-20211025184225-00638.warc.gz"}
https://in.mathworks.com/help/stats/classreg.learning.partition.regressionpartitionedlinear-class.html	# RegressionPartitionedLinear Package: classreg.learning.partition Superclasses: `RegressionPartitionedModel` Cross-validated linear regression model for high-dimensional data ## Description `RegressionPartitionedLinear` is a set of linear regression models trained on cross-validated folds. To obtain a cross-validated, linear regression model, use `fitrlinear` and specify one of the cross-validation options. You can estimate the predictive quality of the model, or how well the linear regression model generalizes, using one or more of these “kfold” methods: `kfoldPredict` and `kfoldLoss`. Every “kfold” method uses models trained on in-fold observations to predict the response for out-of-fold observations. For example, suppose that you cross-validate using five folds. In this case, the software randomly assigns each observation into five roughly equally sized groups. The training fold contains four of the groups (that is, roughly 4/5 of the data) and the test fold contains the other group (that is, roughly 1/5 of the data). In this case, cross-validation proceeds as follows: 1. The software trains the first model (stored in `CVMdl.Trained{1}`) using the observations in the last four groups and reserves the observations in the first group for validation. 2. The software trains the second model (stored in `CVMdl.Trained{2}`) using the observations in the first group and last three groups. The software reserves the observations in the second group for validation. 3. The software proceeds in a similar fashion for the third through fifth models. If you validate by calling `kfoldPredict`, it computes predictions for the observations in group 1 using the first model, group 2 for the second model, and so on. In short, the software estimates a response for every observation using the model trained without that observation. Note Unlike other cross-validated, regression models, `RegressionPartitionedLinear` model objects do not store the predictor data set. ## Construction `CVMdl = fitrlinear(X,Y,Name,Value)` creates a cross-validated, linear regression model when `Name` is either `'CrossVal'`, `'CVPartition'`, `'Holdout'`, or `'KFold'`. For more details, see `fitrlinear`. ## Properties expand all Cross-Validation Properties Cross-validated model name, specified as a character vector. For example, `'Linear'` specifies a cross-validated linear model for binary classification or regression. Data Types: `char` Number of cross-validated folds, specified as a positive integer. Data Types: `double` Cross-validation parameter values, e.g., the name-value pair argument values used to cross-validate the linear model, specified as an object. `ModelParameters` does not contain estimated parameters. Access properties of `ModelParameters` using dot notation. Number of observations in the training data, specified as a positive numeric scalar. Data Types: `double` Data partition indicating how the software splits the data into cross-validation folds, specified as a `cvpartition` model. Linear regression models trained on cross-validation folds, specified as a cell array of `RegressionLinear` models. `Trained` has k cells, where k is the number of folds. Data Types: `cell` Observation weights used to cross-validate the model, specified as a numeric vector. `W` has `NumObservations` elements. The software normalizes the weights used for training so that `sum(W,'omitnan')` is `1`. Data Types: `single` \| `double` Observed responses used to cross-validate the model, specified as a numeric vector containing `NumObservations` elements. Each row of `Y` represents the observed response of the corresponding observation in the predictor data. Data Types: `single` \| `double` Other Regression Properties Categorical predictor indices, specified as a vector of positive integers. `CategoricalPredictors` contains index values indicating that the corresponding predictors are categorical. The index values are between 1 and `p`, where `p` is the number of predictors used to train the model. If none of the predictors are categorical, then this property is empty (`[]`). Data Types: `single` \| `double` Predictor names in order of their appearance in the predictor data, specified as a cell array of character vectors. The length of `PredictorNames` is equal to the number of variables in the training data `X` or `Tbl` used as predictor variables. Data Types: `cell` Response variable name, specified as a character vector. Data Types: `char` Response transformation function, specified as `'none'` or a function handle. `ResponseTransform` describes how the software transforms raw response values. For a MATLAB® function or a function that you define, enter its function handle. For example, you can enter ```Mdl.ResponseTransform = @function```, where `function` accepts a numeric vector of the original responses and returns a numeric vector of the same size containing the transformed responses. Data Types: `char` \| `function_handle` ## Methods kfoldLoss Regression loss for observations not used in training kfoldPredict Predict responses for observations not used for training ## Copy Semantics Value. To learn how value classes affect copy operations, see Copying Objects. ## Examples collapse all Simulate 10000 observations from this model `$y={x}_{100}+2{x}_{200}+e.$` • $X=\left\{{x}_{1},...,{x}_{1000}\right\}$ is a 10000-by-1000 sparse matrix with 10% nonzero standard normal elements. • e is random normal error with mean 0 and standard deviation 0.3. ```rng(1) % For reproducibility n = 1e4; d = 1e3; nz = 0.1; X = sprandn(n,d,nz); Y = X(:,100) + 2X(:,200) + 0.3randn(n,1);``` Cross-validate a linear regression model. To increase execution speed, transpose the predictor data and specify that the observations are in columns. ```X = X'; CVMdl = fitrlinear(X,Y,'CrossVal','on','ObservationsIn','columns');``` `CVMdl` is a `RegressionPartitionedLinear` cross-validated model. Because `fitrlinear` implements 10-fold cross-validation by default, `CVMdl.Trained` contains a cell vector of ten `RegressionLinear` models. Each cell contains a linear regression model trained on nine folds, and then tested on the remaining fold. Predict responses for out-of-fold observations and estimate the generalization error by passing `CVMdl` to `kfoldPredict` and `kfoldLoss`, respectively. ```oofYHat = kfoldPredict(CVMdl); ge = kfoldLoss(CVMdl)``` ```ge = 0.1748 ``` The estimated, generalization, mean squared error is 0.1748. To determine a good lasso-penalty strength for a linear regression model that uses least squares, implement 5-fold cross-validation. Simulate 10000 observations from this model `$y={x}_{100}+2{x}_{200}+e.$` • $X=\left\{{x}_{1},...,{x}_{1000}\right\}$ is a 10000-by-1000 sparse matrix with 10% nonzero standard normal elements. • e is random normal error with mean 0 and standard deviation 0.3. ```rng(1) % For reproducibility n = 1e4; d = 1e3; nz = 0.1; X = sprandn(n,d,nz); Y = X(:,100) + 2X(:,200) + 0.3randn(n,1);``` Create a set of 15 logarithmically-spaced regularization strengths from $1{0}^{-5}$ through $1{0}^{-1}$. `Lambda = logspace(-5,-1,15);` Cross-validate the models. To increase execution speed, transpose the predictor data and specify that the observations are in columns. Optimize the objective function using SpaRSA. ```X = X'; CVMdl = fitrlinear(X,Y,'ObservationsIn','columns','KFold',5,'Lambda',Lambda,... 'Learner','leastsquares','Solver','sparsa','Regularization','lasso'); numCLModels = numel(CVMdl.Trained)``` ```numCLModels = 5 ``` `CVMdl` is a `RegressionPartitionedLinear` model. Because `fitrlinear` implements 5-fold cross-validation, `CVMdl` contains 5 `RegressionLinear` models that the software trains on each fold. Display the first trained linear regression model. `Mdl1 = CVMdl.Trained{1}` ```Mdl1 = RegressionLinear ResponseName: 'Y' ResponseTransform: 'none' Beta: [1000x15 double] Bias: [-0.0049 -0.0049 -0.0049 -0.0049 -0.0049 -0.0048 ... ] Lambda: [1.0000e-05 1.9307e-05 3.7276e-05 7.1969e-05 ... ] Learner: 'leastsquares' Properties, Methods ``` `Mdl1` is a `RegressionLinear` model object. `fitrlinear` constructed `Mdl1` by training on the first four folds. Because `Lambda` is a sequence of regularization strengths, you can think of `Mdl1` as 15 models, one for each regularization strength in `Lambda`. Estimate the cross-validated MSE. `mse = kfoldLoss(CVMdl);` Higher values of `Lambda` lead to predictor variable sparsity, which is a good quality of a regression model. For each regularization strength, train a linear regression model using the entire data set and the same options as when you cross-validated the models. Determine the number of nonzero coefficients per model. ```Mdl = fitrlinear(X,Y,'ObservationsIn','columns','Lambda',Lambda,... 'Learner','leastsquares','Solver','sparsa','Regularization','lasso'); numNZCoeff = sum(Mdl.Beta~=0);``` In the same figure, plot the cross-validated MSE and frequency of nonzero coefficients for each regularization strength. Plot all variables on the log scale. ```figure [h,hL1,hL2] = plotyy(log10(Lambda),log10(mse),... log10(Lambda),log10(numNZCoeff)); hL1.Marker = 'o'; hL2.Marker = 'o'; ylabel(h(1),'log_{10} MSE') ylabel(h(2),'log_{10} nonzero-coefficient frequency') xlabel('log_{10} Lambda') hold off``` Choose the index of the regularization strength that balances predictor variable sparsity and low MSE (for example, `Lambda(10)`). `idxFinal = 10;` Extract the model with corresponding to the minimal MSE. `MdlFinal = selectModels(Mdl,idxFinal)` ```MdlFinal = RegressionLinear ResponseName: 'Y' ResponseTransform: 'none' Beta: [1000x1 double] Bias: -0.0050 Lambda: 0.0037 Learner: 'leastsquares' Properties, Methods ``` `idxNZCoeff = find(MdlFinal.Beta~=0)` ```idxNZCoeff = 2×1 100 200 ``` `EstCoeff = Mdl.Beta(idxNZCoeff)` ```EstCoeff = 2×1 1.0051 1.9965 ``` `MdlFinal` is a `RegressionLinear` model with one regularization strength. The nonzero coefficients `EstCoeff` are close to the coefficients that simulated the data. ## Version History Introduced in R2016a	2022-07-05 14:00:18	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 6, "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": 0.8188030123710632, "perplexity": 1935.4248392890277}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1656104576719.83/warc/CC-MAIN-20220705113756-20220705143756-00134.warc.gz"}
https://l3ouu4n9.github.io/post/hackthebox/2020-05-13-bank/	From nmap, there are ssh, dns and http service. This is the web page. Take a look at the DNS zone transfer data for bank.htb. There are multiple domains. We can add nameserver 10.10.10.29 in /etc/resolv.conf to access those domains. In bank.htb, there is a login page. I use gobuster to see if there are other pages. In the directory balance-transfer, there are lots of transfer records. Most of their size are 58X, but I found one with size 257. It contains the credential. We can login with the credential. From the source code, there is a comment saying .htb file can be executed as .php file. In support.php, we can upload files. So I upload a php reverse shell and named it reverse.htb. Access the web page, we get the shell as user www-data. When searching SUID bit binaries, I notice that there is /var/htb/bin/emergency, which is odd. When I execute it, I become root, and that is the box.	2022-01-18 05:22:38	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.18489226698875427, "perplexity": 4110.856261330497}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1642320300722.91/warc/CC-MAIN-20220118032342-20220118062342-00265.warc.gz"}
https://plainmath.net/90587/the-question-is-f-x-x-x-1-g-x-1-x	# The question is: f(x)=x/x−1, g(x)=1/x, h(x)=x2−1. Find f o g o h and state its domain. The question is: $f\left(x\right)=\frac{x}{x-1}$ $g\left(x\right)=\frac{1}{x}$ $h\left(x\right)={x}^{2}-1$ Find $f\circ g\circ h$ and state its domain. The answer the textbook states is that the domain is all real values of $x$, except $±1$ and $±\sqrt{2}$. However surely the domain excludes $0$ as well, since $g\left(0\right)$ is undefined. 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 Maleah Lester You're not inputting $x$ into $g$, though. You're inputting $h\left(x\right)$. So yes, $g\left(0\right)$ is undefined, which means that whatever values of $x$ makes $h\left(x\right)=0$ is not part of the domain. That's why they exclude $±1$. metal1fc $h\left(x\right)={x}^{2}-1$ has a value for all $R$ $g\left(h\left(x\right)\right)=\frac{1}{{x}^{2}-1}$ which does not have a value at $x=±\sqrt{1}$ $f\left(g\left(h\left(x\right)\right)\right)=\frac{\frac{1}{{x}^{2}-1}}{\frac{1}{{x}^{2}-1}-1}=\frac{1}{1-{x}^{2}+1}=\frac{1}{2-{x}^{2}}$ which does not have a value at $x=±\sqrt{2}$ Hence the total domain is $x\in R-\left\{±\sqrt{1},±\sqrt{2}\right\}$	2022-09-30 20:05:22	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 81, "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": 0.8545804619789124, "perplexity": 472.73610346861716}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1664030335504.22/warc/CC-MAIN-20220930181143-20220930211143-00023.warc.gz"}
https://recipes.tidymodels.org/reference/step_discretize.html	step_discretize creates a specification of a recipe step that will convert numeric data into a factor with bins having approximately the same number of data points (based on a training set). Usage step_discretize( recipe, ..., role = NA, trained = FALSE, num_breaks = 4, min_unique = 10, objects = NULL, options = list(), skip = FALSE, id = rand_id("discretize") ) Arguments recipe A recipe object. The step will be added to the sequence of operations for this recipe. ... One or more selector functions to choose variables for this step. See selections() for more details. role Not used by this step since no new variables are created. trained A logical to indicate if the quantities for preprocessing have been estimated. num_breaks An integer defining how many cuts to make of the data. min_unique An integer defining a sample size line of dignity for the binning. If (the number of unique values)/(cuts+1) is less than min_unique, no discretization takes place. objects The discretize() objects are stored here once the recipe has be trained by prep(). options A list of options to discretize(). A default is set for the argument x. Note that using the options prefix and labels when more than one variable is being transformed might be problematic as all variables inherit those values. skip A logical. Should the step be skipped when the recipe is baked by bake()? While all operations are baked when prep() is run, some operations may not be able to be conducted on new data (e.g. processing the outcome variable(s)). Care should be taken when using skip = TRUE as it may affect the computations for subsequent operations. id A character string that is unique to this step to identify it. Value An updated version of recipe with the new step added to the sequence of any existing operations. Tidying When you tidy() this step, a tibble with columns terms (the selectors or variables selected) and value (the breaks) is returned. Other discretization steps: step_cut() Examples library(modeldata) data(biomass) biomass_tr <- biomass[biomass$dataset == "Training",] biomass_te <- biomass[biomass$dataset == "Testing",] rec <- recipe(HHV ~ carbon + hydrogen + oxygen + nitrogen + sulfur, data = biomass_tr) %>% step_discretize(carbon, hydrogen) rec <- prep(rec, biomass_tr) binned_te <- bake(rec, biomass_te) table(binned_te\$carbon) #> #> bin_missing bin1 bin2 bin3 bin4 #> 0 22 17 25 16 tidy(rec, 1) #> # A tibble: 10 × 3 #> terms value id #> <chr> <dbl> <chr> #> 1 carbon -Inf discretize_hhrhR #> 2 carbon 44.7 discretize_hhrhR #> 3 carbon 47.1 discretize_hhrhR #> 4 carbon 49.7 discretize_hhrhR #> 5 carbon Inf discretize_hhrhR #> 6 hydrogen -Inf discretize_hhrhR #> 7 hydrogen 5.20 discretize_hhrhR #> 8 hydrogen 5.78 discretize_hhrhR #> 9 hydrogen 6.05 discretize_hhrhR #> 10 hydrogen Inf discretize_hhrhR	2022-05-16 08:34:16	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.28874221444129944, "perplexity": 7306.717378759309}, "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/1652662510097.3/warc/CC-MAIN-20220516073101-20220516103101-00302.warc.gz"}
https://www.gamedev.net/forums/topic/56386-help-array-question/	• ### Announcements #### Archived This topic is now archived and is closed to further replies. # Help! Array Question... ## Recommended Posts webmunkey 122 Hey, I was just wondering how I could "delete" an entry in an array and have the entries after it move their index number down a place to take the place of the deleted entry. Example: int array[3] = {1, 2, 3}; array[0] + 1 = x In this example x would equal 2... but how could I change my array so that it would look like this: delete_array entry 1 // now array looks like this array[3] = {2, 3} array[0] + 1 = x; Now x would equal 3... but how can I do this? Please help! It is extremely important for me to implement this feature so I can finish my game! Thanks... -Jesse ##### Share on other sites cyberg 122 maybe you could use CArray from MFC if you use this else what you have to do is a loop like that for(i=delete_entry;i{ array=array[i+1]; } and I dont think that you can do something like array[0] + 1 = x; it will give you an error... cyberg ##### Share on other sites cyberg 122 grrrrrrrr html tag!!!! what I have to do to post my code correctly? ##### Share on other sites Guest Anonymous Poster this shifts everything down 1. you can play around with this theme to create a general array element deletor. int t[] = {1,2,3}; int s = sizeof(t)-sizeof(int); memcpy( t, &t[1], s-1 ); Sandman 2210 ##### Share on other sites Beer Hunter 712 webmunkey - The Anonymous Poster knows his stuff. memcpy is a good choice. cyberg - To get the < and > characters on this board, you have to type < and > . It can seem annoying, but such is life when html is enabled. Sandman - Webmunkey seems to only be learning the language. Give ''em a chance to understand the basics before doing anything more complicated. I know that linked lists aren''t really that hard, but they can seem complicated to a beginner. ##### Share on other sites BeerNutts 4400 Actually, You can''t use memcpy in this situation. If you read in K&R, you''ll see you should use memmove when memory overlaps each other. You''ll see location &t[1] is a part of t, so you should use memmove. We had a problem here at work where someone was using memcpy and the source was part of the destination. ##### Share on other sites jacksonh 122 I would highly reccomend Linked Lists, if you don''t know how to use them find a tutorial on them or get ahold of me. ##### Share on other sites webmunkey 122 Thanks everyone! I'm not familiar with linked lists yet, those will come later, but I did use the Anonymous Poster's algoritm and it worked great after I changed it to what BeerNutts said. Thanks again, -Jesse Edited by - webmunkey on August 3, 2001 12:08:20 PM ##### Share on other sites merlin9x9 174 Why does everyone always seem to forget the STL? Use vector or list. ##### Share on other sites Guest Anonymous Poster Am I stupid or is what webmunkey describing similar to what my book calls a Q(queue). You have an array and then there are a number of rules describing how data may enter and leave and what may and may not be done. I thought it was a common programming exercise and that it also mimicks the real stack in memory, assemblers, correct me if I''m wrong. Of course, a linked list will work, but since you started out with an array you may want to keep an index number in each node. int array[X] = {1,2,3}; for(i=0;i<(X-1);i++){ array = array[i+1]; } array[i]=0; /* final result array[X] = {2,3,0} / ##### Share on other sites Beer Hunter 712 BeerNutts - memcpy will screw you over if the start of the destination lies inside the source block. But for sliding data down an array, the start of the destination lies outside that block, so you''re safe to use it for that purpose. Try it out for yourself and see. ##### Share on other sites BeerNutts 4400 Beer Hunter (BTW, nice name) I'm not sure of the implementation of memcpy vs memmove in the standard libraries, but considering memmove is described as: "same as memcpy, except it works when objects overlap." would lead me to always use memove if two objects overlapped, regardless of destination or source start. I guess I'm just conservative, but, for embedded systems, better safe than sorry. Nutts Edit: I actually looked at the old code that caused the problem, and it is exactly like the above problem. I believe it was intermittent, but memcpy was the culprit. We changed this piece of code: memcpy( &gTaskList[i], &gTaskList[i+1], sizeof( tTaskEntry ) gulNumTasks - 1 ); to this memmove( &gTaskList[i], &gTaskList[i+1], sizeof( tTaskEntry ) * gulNumTasks - 1 );	2017-08-21 06:52:10	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.1789788156747818, "perplexity": 4108.568617723404}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886107720.63/warc/CC-MAIN-20170821060924-20170821080924-00143.warc.gz"}
https://www.biostars.org/p/265141/	smallRNA low percentage of mapping, N at the beninning of the reads and kmers 1 0 Entering edit mode 5.2 years ago noeD ▴ 130 Hello! I am working with smallRNA data. I have analyzed the fastq with fastqc, and I saw that there were illumina small RNA 3' adapter, in fact my sequence length distribution were centered on 51. Therefore I have used cutadapt in order to remove that adapter and my sequence length distribution changed: https://drive.google.com/file/d/0B4m6-7p8GFwIa3B5VGRmT3FjUDA/view?usp=sharing After that I aligned my reads against reference genome (hg38) with botwie, using default parameters, in order to see how it performed. I obtained a very very low percentage of mapped read (0.30%). I have checked again my fastq file with fastqc and I saw that there were several kmers at the end of the reads. Is it normal? I have upload all images from fastqc at this link: https://drive.google.com/open?id=0B4m6-7p8GFwIQmppNjNQVm5BRVU Are there other adapter that I should trim? At the beginning of the reads I saw that in some case there were N, should I trim them? I reported here and extract of my fastq: @HISEQ2500:231:C9L77ACXX:1:2316:21153:100286 1:N:0:NTAGCT AAGCCGCCAGTTGAAGAACTGT + <7<B00<<0<BFBFFIIIIIII @HISEQ2500:231:C9L77ACXX:1:2316:21183:100346 1:N:0:NTAGCT CTCCAGGCCGAGGAC + <B<<<0<<BB<0<BB @HISEQ2500:231:C9L77ACXX:1:1101:1376:1894 1:N:0:CTAGCT NAGCTTATCAGACTGATGTTGA + #00BBFFFFFFFFFFFIIIIBF @HISEQ2500:231:C9L77ACXX:1:1101:1314:1913 1:N:0:CTAGCT NGCTACATCTGGCTACTGGGTCT + #0<FFFFFFFFFFIIIIIIIIII As you can see, in the first read there isn't a N at the beginning of the read, but it is presented in the index of the reads. In the last read exactly the opposite is happening: N at the beginning of the read, but not in the index of the reads. How should fix that issue? Best smallRNA RNA-Seq alignment • 1.7k views 2 Entering edit mode small RNA data analysis requires pre-processing of the data in specific ways (based on the kit used etc). You may want to try a dedicated pipeline (e.g. miRquant or miRdeep2 ) for this purpose. 2 Entering edit mode 5.2 years ago You should trim leading/trailing Ns; they never help alignment, and are particularly bad with Bowtie as it allows very few mismatches. You can do that by quality-trimming to a q-score of 2. On the other hand, if the exact starting position of the read is important, just discard the reads containing Ns. As far as adapter contamination goes... if you successfully trimmed using Illumina's Small RNA adapters as a reference, I don't see a point in trying other adapter sequences as well. You can also remove reads with Ns in the barcodes, or barcodes that do not exactly match the expected barcodes, during the demultiplexing process. I recommend this if you are multiplexing, to prevent crosstalk between libraries. It's also possible to remove them after the fact - BBDuk has the flags "barcodefilter" and "barcodes" for that purpose. If crosstalk is not a problem for the experiment, there's no reason to remove them. As for the low alignment rate, it's hard to say what might cause that (could be that the library is mostly not human, for example). I'd suggest trying other aligners (bowtie2, bwa-aln, BBMap) to see if they improve things, and you might try BLASTing some of the longest unaligned reads to nt/RefSeq to see what they hit, though that's much more useful with longer reads.	2022-09-25 05:16:08	{"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": 0.2947472929954529, "perplexity": 2568.4402285113038}, "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/1664030334514.38/warc/CC-MAIN-20220925035541-20220925065541-00058.warc.gz"}
https://math.meta.stackexchange.com/questions/26663/how-can-i-quit-this-site	# How can I quit this site? [duplicate] This question already has an answer here: I have had enough of this site so I want to quit by blocking/deleting this account... can someone help me? I guess that I should provide a few legitimate reasons why I want to do this. 1. There has been a lot of instances where I have posted an interesting question (some of which are not answered till date) but it's not upvoted. 2. When I was totally new as a user I found myself helpless by the comments and downvotes of relatively senior guys and also in a few cases when they haven’t been able to answer my question they have typed a number of times: so what does your question mean or so what is your question, although it is perfectly clear to understand. 3. Also there have been instances in the previous case where they have gone to the extent to gang up and close my question. 4. I am very new to MathJax codes but although I typed the question so that there was no problem in reading and understanding the mathematical idea behind it, someone (I don’t want to name) edited it and commented (and I quote) “ Where on earth do you get your ideas of how to write MathJax code?”. I find this particular person among a few others to be extremely keen to edit questions where those changes are not at all necessary and comment in such a manner. I had to mention his name and literally beg his pardon : “I am a new-learner on MathJax codes, so please pardon my mistakes.” 5. When I edited some question for betterment someone commented : “I could answer your edited question but then perhaps you would change it again and we would go on forever.” , is this the real attitude of this site – since you have edited your question, I know the answer of your question but I won’t let you know!!! 6. Although it is vivid from the previous statements but I’ll still mention it in a different point : I have found some people (OBVIOUSLY NOT EVERYBODY) very aggressive and rude, at the end of the day I am here to learn Mathematics not to indulge in any sort of quarreling. So if anyone knows how to block / delete my account and also delete all my activities till date, then please let me know. ## marked as duplicate by Namaste, Joel Reyes Noche, JonMark Perry, Surb, kingW3Jul 24 '17 at 8:44 • Contact us and use the "I need to delete my user profile" option. But anything you have contributed to Mathematics Stack Exchange has been licensed to Stack Exchange in perpetuity (see Terms of Service). – user642796 Jul 23 '17 at 9:29 • See also: Delete My Account. – Martin Sleziak Jul 23 '17 at 9:39 • Looking at your 18 not deleted questions (I can't access deleted posts yet) people were either helpful or neutral except the 4th point you mentioned (the last question). Also notice that you can flag comments for being rude and if you think your question was closed unjustly you can post it here on meta (there is a post called something like reopen, undeletion requests). Anyway if you still wish to leave this site farewell and best regards. – kingW3 Jul 23 '17 at 10:49 • Lmao with the MathJax, I do think most people are understanding of such errors. – Simply Beautiful Art Jul 23 '17 at 11:45 • Sometimes when I ask seemingly interesting questions that get about 5 views within the first hour (-_-) I try asking/advertising a bit in the main chat room, people there sure know lots of things. And you don't need upvotes on your questions for it to be interesting. – Simply Beautiful Art Jul 23 '17 at 11:48 • "Also there have been instances in the previous case where they have gone to the extent to gang up and close my question." - I don't think people gang up to close your questions, rather, questions that are in the process of being closed will often be sent to the review queue, where people then decide if the question is worthy of being closed or not. Generally, I recommend referring here when deciding if your questions are up to site standards. – Simply Beautiful Art Jul 23 '17 at 11:53 • For the 5th point, just be sure your answers don't change the question to the extent that it invalidates previous answers. Note that your questions aren't meant just for you, as other people will eventually see them as well, perhaps due to having the same question as you did. IMHO, editing questions so that some answers become useless really destroys the point of the site, which would be to create a network of questions and corresponding answers that can be referred to and learned from, not just personal learning, but for everyone. – Simply Beautiful Art Jul 23 '17 at 12:00 • I think the ganging up comment is very apt - it is precisely the behaviour I have observed by existing users to new users. What is worse is that the existing users who do this down-voting or voting to CLOSE sometimes have no experience or knowledge in the subject area, and may be closing off perfectly reasonable questions, in circumstances where they likely know less about the subject area than the OP. That is ultimately a reflection of the breadth of the mathSE site, and the demands made on all users to try and help out. – wolfies Jul 23 '17 at 18:27 • Try to think it one more time, I would not like that you to leave this site. Courage and good luck. – user243301 Jul 23 '17 at 20:20 • @wolfies I think I've recently encountered just this rant of yours very very recently. – Namaste Jul 23 '17 at 21:30 • Phil Jones: If you are concerned about an issue or two, that's what meta is for. If you came here to make public your declaration of deletion-hood, this isn't to place to say that. If you want to leave the site, you've now been given instructions on how to do so, and "reasons" for doing so are not required. You've got the guidelines to delete your account. No need to make your exit so dramatic. – Namaste Jul 23 '17 at 21:32 • Phil Jones: I also understand the when one feels attacked, it is certainly understandable to retreat. The user you speak of is not highly regarded; and I'm sorry if I sounded a bit "cold" to you, in my comment above. Please sit on it... wait... before seeking to have your account deleted. It seems that you are doing that, and I'm very happy you haven't acted on deleting your account. – Namaste Jul 24 '17 at 20:58 • .....To be honest, there have been a handful of times where I felt I needed to leave the site, and three of those times I used "contact us" in order to have my account deleted. But I never was certain enough to write "delete me" in my profile. In the end, I knew I really didn't want to leave the site, but was rather reacting to the poor conduct of others. Hold your head high, and don't let the user that attacked you have to power to decide your departure. – Namaste Jul 24 '17 at 21:01 To delete you account, go to the relevant help-center page and follow the instructions. In your case this amounts to follow the "contact us" link (as explained in a comment). Check whether you receive an email response and follow the instructions there. Then there may be a small wait, but eventually you account will be gone. (It may also happen more directly as your account is relatively new.) However, as explained on that link your content will not be deleted. It will be made anonymous though. Please do not try to delete your content manually before leaving; this would be undone and thus it would just waste your time and that of the person that has to undo it. On the general situation, many users here are rather direct and cut it short. Reasons for this include, personal style, culture, and last but not least efficiency, when they do the same thing over and over again. While this may seems rude, the intent is most of the time not to be rude. Incidentally, the user you would not name has themselves repeatedly expressed concern about poor treatment of (new) users. I am sure their intent was to be helpful, and the sentence you quote expressed an honest surprise and curiosity how you could get the idea to write in this form. To look at it in a positive way, this feedback (even if not expressed in the most polite form) can help you to improve your skills. • On the general situation, many users here are rather direct and cut it short. Reasons for this include, personal style, culture, and last but not least efficiency, when they do the same thing over and over again ...........In effect, what you are saying is that mathSE has grown too large and too cumbersome, attracts too many posts, is under-resourced to deal with them, and allows people who are skilled in area A (but not B) to moderate content on B, leading to unsatisfactory outcomes. – wolfies Jul 23 '17 at 18:32 • I would not say that it is too large in an absolute sense (SO is about an order of magnitude larger and still functions). But, yes, the site has grown quite large, has a rapid turnover, and this does influence the style of the site (towards the direct and short). I find the last part astonishing given the current situation. A main concern for them were issues with their use of MathJax, or rather the lack of it. The editor(s) seemed competent in making their corrections; OP did not like the way they approached it or the fact that corrections were made at all. – quid Jul 23 '17 at 19:13 • Furthermore, from what I can see, I do not feel that OP had that poor a reception on the site. But maybe I looked at the wrong places. Part of the problem might be that some users have unreasonable expectations on what level of "service" one can expect on a free site mostly run by volunteers. Sure, the site should be welcoming, but I feel it is not untowardly to expect that users try to make a good faith effort to get familiar with and then to comply with the customs of the site. Finally,it's not always necessary to understand the math in detail to be able to see that a question is lacking. – quid Jul 23 '17 at 19:32 • All worthy sentiments which I completely agree with. But then, at the same time, there is a complaint from a person so unhappy with their treatment that they wish to cancel their entire account (such is their level of discontent), the same downvoting of the complaint ('see no evil, hear no evil'), and an apparent unwillingness to seek ways to accommodate those concerns. This person is not the first to raise these concerns - nor will they be the last. – wolfies Jul 24 '17 at 4:14 • You did not reply to my question if you bothered to check who the user is they complain about specifically. It so happens it is the user that in recent times was most vocal in complaining on meta about poor treatment of (new) users. This goes to show that in practice things are quite a bit more complicated. Finally, one should not forget that not every complaint is automatically reasonable. I invested some time to look through the posts of OP to see how they were treated. Did you? If yes, could you point to specific problems? – quid Jul 24 '17 at 8:49 • So, @wolfies If I were to post on meta with a complaint about users like you who see things from a very immature perspecive, and the many, many users who participate here because users like you do their homework for them, and demonstrate just upset I am by my treatment from users like you, and in addition, indeed, I am so upset that I may quit the site.... Does that, therefore, motivate you to speak on my behalf, and acting like you know what I need? Perhaps "I'm so unhappy with my treatment that I wish to cancel my account". Do we therefore conclude my complaint is valid? – Namaste Jul 24 '17 at 22:10	2019-04-19 20:39:28	{"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": 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": 0.37425947189331055, "perplexity": 770.527435391117}, "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-2019-18/segments/1555578528058.3/warc/CC-MAIN-20190419201105-20190419223105-00348.warc.gz"}
https://www.physicsforums.com/threads/why-do-morphisms-in-category-of-rings-respect-identity.612987/	# Why do morphisms in category of rings respect identity 1. Jun 10, 2012 ### algebrat Hi, I'm looking for intuition and/or logic as to why we would want or need morphisms according to axioms in category theory, to imply that in the category of rings, they must preserve the identity (unless codomain is "0"). Thank you very much. 2. Jun 11, 2012 ### algebrat Okay, I think I thought it was more complicated then it was. I'm sure I was interested for some specific reason, but I guess morphism in concrete categories are just structure preserving functions (I do not have an in depth source to check yet). So the identity part is as Artin says, there aren't necessarily a lot of inverses, so we need to explicitly mention that 1 maps to 1. f(x)=f(x)f(1)→f(x)(1-f(x))=0. For instance we are not in an integral domain so we should require that f(1)=1. Okay, now that I write that, I don't get it. Are all morphisms by definition, in categroy of rings with identity, necessarily taking identity to identity, or is this a stronger restriction that Artin has requested, as opposed to definition of morphism. 3. Jun 11, 2012 ### Sankaku The way I understand it, if the category itself demands that all its members have multiplicative identity, then the morphisms will have to respect that. Otherwise, the image of your morphism may not be in the category any longer. If you are in the category of rings without 1 (Pseudo-rings, Rngs), then you aren't restricted this way. I am still working on categories myself, so others may be able to give you a deeper explanation. 4. Jun 11, 2012 ### alexfloo I'm still not 100% certain what your question is, so first I'll give my interpretation of the question. I believe you are asking why the axioms of general ring homomorphism (f(x)f(y)=f(xy) and f(x)+f(y)=f(x+y)) imply the additional axiom of ring homomorphism for unital rings (f(1)=1), in the case that both rings have a unit. If this is your question, then the simple answer is that they don't. A general ring homomorphism R→S actually does not need to be a unital ring homomorphism just because R and S are unital. For instance, the zero function between any two rings is a (pseudo-)ring homomorphism, but it is never a unital ring homomorphism unless the codomain is the zero ring (since we would need 0=1). Put in other words, the category of rings is a subcategory of the category of pseudo-rings. However, it is not a full subcategory. 5. Jun 11, 2012 ### alexfloo Another interpretation of your question is "Why do we include this additional axiom f(1)=1 explicitly?" If that's what you meant, then the answer is essentially the same. It's because it's not implied by the other ones. When we're talking about a concrete category of algebraic objects, a full subcategory basically corresponds to the case that no new axioms of homomorphism are necessary. Since the "extra structure" (the unit) is not automatically preserved, we need two things: 1. On the algebra side, we add an extra rule to our homomorphism. 2. On the categorical side, we cut out the morphisms that didn't satisfy that rule, making our subcategory no longer full. 6. Jun 11, 2012 ### algebrat Thank you!! 7. Jun 12, 2012 ### Sankaku Yes, very nicely said. 8. Jun 12, 2012 ### algebrat To be sure, I was thanking you equally Sankaku!!	2018-08-15 01:32:59	{"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": 0.9228287935256958, "perplexity": 627.482720027356}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221209755.32/warc/CC-MAIN-20180815004637-20180815024637-00444.warc.gz"}
https://developer.here.com/documentation/data-client-library/dev_guide/client/data-integrity.html	# Check Data Integrity The HERE platform offers standard ways to ensure data integrity on partition level. The layer configuration for all layer types supports two optional fields which are Digest (checksum) and CRC. ### Note Digest and CRC are two different things. Digest is used for security to prevent human tampering. CRC is used for safety to prevent bit flips by computer hardware or network transportation. You can use both fields at the same time. ## Digest When retrieving a partition, your application can check if the partition data was not tampered by a human. To do that, your application should calculate the checksum of the data and compare it with the checksum (Digest) contained within the partition. If these two checksum values match, the retrieved data is consistent with the partition that was uploaded. Data Client Library has the algorithms for Digest calculation built in, so your application should use these. Note that, by default, a partition does not contain a checksum. You need to explicitly choose the checksum algorithm in the layer configuration. If the checksum setting in the layer configuration is not "Undefined" then Data Client Library automatically calculates checksum using the selected algorithm prior to upload of partition. When choosing a Digest algorithm, consider the following: • SHA-256 is recommended for applications where strong data security is required. • MD5 and SHA-1 is acceptable when the purpose of applying a hash is to verify data integrity during transit. ## CRC When retrieving a partition, your application can check if one or more bits of the partition data were flipped by computer hardware or network transportation. To do that, your application should calculate the CRC of the data and compare it with the CRC contained within the partition. If these two CRC values match, the retrieved data is consistent with the partition that was uploaded. Data Client Library has the algorithms for CRC calculation built in, so your application should use these. Note that, by default, a partition does not contain a CRC. You need to explicitly choose the CRC algorithm in the layer configuration. If the CRC setting in the layer configuration is not "None" then the Data Client Library automatically calculates CRC using the selected algorithm prior to upload of partition. Currently only one CRC algorithm is supported: • CRC-32C (Castagnoli), see e.g. CRC algorithms. Please note that this CRC is padded with zeros to a fixed length of 8 characters. Please also note that CRC is stored as a string, e.g. if your calculated CRC is the uint32 value of 0x1234af then the CRC which is actually stored in the partition metadata is the string "001234af". ## Further Information For more information about the possible Digest and CRC algorithms, see the Update Catalog of the Data API Reference. Here, you will find configuration information about other layer types as well. ## Examples One can compute a checksum value of a blob and correlate it to a new partition as follows: Scala Java val bufferedBlob: BufferedBlob = ByteArrayData(Array.emptyByteArray) val writeEngine = DataEngine().writeEngine(hrn) val blobChecksum: Future[Option[String]] = writeEngine.blobChecksum(layerName, bufferedBlob) val singlePartition = Source .fromFuture(blobChecksum) .map(checksum => NewPartition("12345", layerName, bufferedBlob, checksum = checksum)) writeEngine.publish(singlePartition) BufferedBlob bufferedBlob = new ByteArrayData(new byte[] {}); WriteEngine writeEngine = DataEngine.get(myActorSystem).writeEngine(hrn); CompletionStage<Optional<String>> blobChecksum = writeEngine.blobChecksum(layerName, bufferedBlob); Source<PendingPartition, NotUsed> singlePartition = Source.fromCompletionStage(blobChecksum) .map( checksum -> new NewPartition.Builder() .withPartition("12345") .withLayer(layerName) .withData(bufferedBlob) .withChecksum(checksum) .build()); writeEngine.publish(singlePartition); A partition can be retrieved and its checksum checked with the following code: Scala Java val dataEngine = DataEngine() val writeEngine = dataEngine.writeEngine(hrn) val blobChecksum: Future[Option[String]] = readEngine .getDataAsBytes(partition) .flatMap(bytes => writeEngine.blobChecksum(layerName, ByteArrayData(bytes))) blobChecksum.map(checksum => checksum == partition.checksum) DataEngine dataEngine = DataEngine.get(myActorSystem); WriteEngine writeEngine = dataEngine.writeEngine(hrn); blobChecksum.toCompletableFuture().get().equals(partition.getChecksum());	2021-09-25 19:02:00	{"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": 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": 0.5042918920516968, "perplexity": 4203.528784054186}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057733.53/warc/CC-MAIN-20210925172649-20210925202649-00696.warc.gz"}
https://www.askiitians.com/forums/IIT-JEE-Entrance-Exam/ques-a-point-mass-starts-moving-in-a-straight-lin_99956.htm	#### Thank you for registering. One of our academic counsellors will contact you within 1 working day. Click to Chat 1800-5470-145 +91 7353221155 CART 0 • 0 MY CART (5) Use Coupon: CART20 and get 20% off on all online Study Material ITEM DETAILS MRP DISCOUNT FINAL PRICE Total Price: Rs. There are no items in this cart. Continue Shopping # ques) A point mass starts moving in a straight line with constant acceleration. After time to the acc changes its sign, remaining the same in magnitude. Dtermine the time t from the beginning of motion in which the point mass returns to its initial problems. Arun Kumar IIT Delhi 7 years ago initial position you mean. Hi Keerthi, $\\ x_1=1/2a*t_0^2 \\v=at_0 \\v=0\,at\,t=t_0 \\x_2=v^2/2a=1/2(at_0^2) \\now\,(1/2)at^2=at_0^2 \\=>t=t_0\sqrt2\\ T_{total}=2t_0+\sqrt2t_0$ Thanks & Regards Arun Kumar IIT Delhi	2021-10-22 04:58:13	{"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": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.34933164715766907, "perplexity": 10792.678112584439}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585450.39/warc/CC-MAIN-20211022021705-20211022051705-00259.warc.gz"}
https://collegephysicsanswers.com/openstax-solutions/cows-milk-produced-near-nuclear-reactors-can-be-tested-little-100-pci-131textrmi	Question Cow’s milk produced near nuclear reactors can be tested for as little as 1.00 pCi of $^{131}\textrm{I}$ per liter, to check for possible reactor leakage. What mass of $^{131}\textrm{I}$ has this activity? $8.12\times 10^{-18}\textrm{ g}$ Solution Video	2020-04-04 03:22:15	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5045949220657349, "perplexity": 3949.468545011929}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1585370519111.47/warc/CC-MAIN-20200404011558-20200404041558-00410.warc.gz"}
http://math.stackexchange.com/questions/308651/isomorphism-between-source-of-kernels-of-parallel-arrow-of-a-pullback-square	# Isomorphism between (source of) kernels of parallel arrow of a pullback square, by adjunction Let $\mathcal A$ be an abelian category, $\alpha_1 \colon A_1 \to B$, $\alpha_2 \colon A_2 \to B$ two morphisms and $A_1 \leftarrow A_1\times_B A_2 \to A_2$ their pullback (with morphisms, say, $p_1, p_2$). I can easily show, by hands, that the unique morphism $\varphi\colon\text{Ker}\, p_2 \to \text{Ker}\,\alpha_1$ induced by the universal property of $\ker\alpha_1 \colon\text{Ker}\,\alpha_1 \to A_1$ is an isomorphism. I have been told that, actually, this is because "limits preserve limits" (or "limits commute with limits"). I know that the limit functor $\lim\colon\mathcal{A^D}\to A$ is right-adjoint to the costant functor $\Delta\colon \mathcal{A}\to \mathcal{A^D}$, but I cannot see how to prove the above result using it. Can you provide any reference, also in order to become familiar with this "limits twists"? - The phrase "limits commute with limits" means the following: Let $C$ be a complete category and $I,J$ small categories. If $F : I \times J \to C$ is a functor, then there are canonical isomorphisms $\lim_i \lim_j F(i,j) \cong \lim_{i,j} F(i,j) \cong \lim_j \lim_i F(i,j)$. A proof can be found in the book by Mac Lane, but basically it is an easy exercise playing around with universal properties. Nothing really happens. As a special case of this, one gets the canonical isomorphism $(X \times_S S') \times_{S'} T \cong X \times_S T$ in an arbitrary category where these fiber products make sense (see also MO/80797). If $T$ is a zero object, this means that the kernel of $X \times_S S' \to S'$ is the same as the kernel of $X \to S$. Actually, isn't $(X \times_S S') \times_{S'} T \cong X \times_S T$ the pullback pasting lemma? This isomorphism has a somewhat different flavour than, say, $\ker (f \times g) \cong \ker (f) \times \ker (g)$. – Zhen Lin Feb 20 '13 at 8:43	2015-04-26 19:37:14	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9758242964744568, "perplexity": 228.05610229937085}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246655962.81/warc/CC-MAIN-20150417045735-00240-ip-10-235-10-82.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/158487/function-that-magnifies-small-changes-and-compresses-large-changes	# Function that magnifies small changes and compresses large changes I need a function (for a heatmap algorithm) that takes a percentage difference between two values, and returns a number between 0 and 1. The output will be used in coloring parts of the screen. The output will be passed to an alpha channel. The catch is, most of the time the percentage value I want to plot will be small and it will be hard to discern the difference between the colors. That's why I want to magnify small values and compress the large ones. The output should be something like this: f(0) = 0 f(1) = 1 f(0.001) = 0.1 f(0.002) = 0.2 f(0.7) = 0.75 f(0.8) = 0.78 ... Do you see the pattern? I've tried log, but I need the function to be defined at f(0). I've also tried x^a where a < 1, but in order to get the magnification I need I need to use a << 1 and that makes all the output so small it can't be discerned or the resolution is so small that the graphics assume it's the same color. - Well, the $\log$ function and the power law are the usual suspects for this sort of thing. So if you want something different that works better for your data, it would help if you told us a little more about just how big or small these numbers that you're dealing with can be. – Rahul Jun 15 '12 at 1:21 Another nice function to experiment with is $arctan(x)$. For example, if the region to be magnified is centred around $h$, try $arctan((x-h)*k)/\pi+1/2$ for different values of $k$. – Théophile Jun 15 '12 at 1:52 Both answers suggested so far are based on $f(x)=g(ax)/g(a)$ for some nondecreasing function $g$ such that $g(0)=0$, with $a\gt0$. Rick uses $g(t)=t/(1+t)$ and E.O. uses $g(t)=\log(1+t)$. – Did Jun 15 '12 at 7:14 I second Rahul's request for more information. For example: do you want the magnification ratio to grow without bound, as the numbers approach zero, or do you want to set a maximum, so that e.g. $f(0.002)$ is roughly twice as large as $f(0.001)$? Same at the other end: do you also want to set a lower bound to the compression ratio? Is it ok, if everything above, say $0.8$ is mapped to the same color, so $f(x)>255/256$ for all $x>0.8$ (assuming 24bit true color or 256 level greyscale)? Or what would you use there in place of $0.8$? – Jyrki Lahtonen Jun 15 '12 at 7:19 For example the average $$f(x)=\frac{x^a+x}2$$ for some $a$ between $0$ and $1$ will magnify the lower end, but at the same time make sure that the compression ratio is $2:1$ at worst. However, this is non-linear for small inputs, so you won't have $$f(2\delta)\approx 2 f(\delta)$$ for small values of $\delta$. With this function the magnification ratio grows without bound as $\delta\to0$. – Jyrki Lahtonen Jun 15 '12 at 7:22 Any function that goes through $(0,0)$ and $(1,1)$ passing above the diagonal satisfies the basic shape you are looking for. If you also want the function to be symmetric, perhaps the simplest choice is a hyperbolic segment such as $f(x)={{(n+1) x}\over{n x+1}}$ where $n>0$ plotted here for $n=10,5,$ and $1$: As a result, by choosing $n$ you can select the amount compression/magnification. - +1: This is a simple way of controlling the maximum magnification ($1:(n+1)$) as well as the minimum compression ($(n+1):1$). Let's wait for the OP to comment, whether this meets the requirements. – Jyrki Lahtonen Jun 15 '12 at 7:32 Picking up on your idea of using $\log$, since you want $f(0)=0$ and $f(1)=1$ the function must be of the form $$f(x)=\frac{1}{\log\left(\frac{b-1}{b}\right)}\log\left(\frac{b-x}{b}\right), b\in(-\infty,0)$$ Depending on how much emphasis you want to put on the smaller numbers you can change the value of $b$. If you want a lot of emphasis on the smaller numbers you want $b$ to be close to 0. Here is an animation to illustrate what I mean: $$b\in[-5,-0.001]$$ If you want a lot of emphasis on the smaller numbers, this is an example when $b=-0.005$: Note that in the example you give, you want $f(0.8)=0.78$ and $f(1)=1$. That would mean that you would need to emphasize numbers when to $x$ is close to $1$. This would possibly defeat the purpose of your function. For a function to do what you wish $f(x)\ge x, \forall x\in[0,1]$	2014-07-11 14:52:57	{"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": 0.7698399424552917, "perplexity": 249.84676387795272}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1404776427615.55/warc/CC-MAIN-20140707234027-00013-ip-10-180-212-248.ec2.internal.warc.gz"}
http://tex.stackexchange.com/questions/98052/umlauts-in-math-mode/98061	# Umlauts in math mode I'm trying to use German umlauts in math mode but get the known error: LaTeX Warning: Command \" invalid in math mode on input line ##. So I could think of two possible solutions: 1. Use \text instead of \mathrm. But this wouldn't be the right solution because this text will be changed according to the surrounding text and the text I try to write ("Empfänger", German word for "receiver") should appear as a superscript to a field variable. 2. Replace the letter ä with "a. But this leads to LaTeX not recognizing the ligature of fä. Since I'm using the lmodern package, the letter ä is still shown, so the resulting PDF is what I would expect. Are there some elegant and correct solutions to this merely cosmetic problem? My code for this would be: \documentclass{scrartcl} % Kodierung \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{lmodern} % Sprache (neue deutsche Rechtschreibung) \usepackage[ngerman]{babel} % Mathematik \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \begin{document} % reference Empfänger % creates warning but is correct \begin{align} \mathrm{Empfänger} \end{align} % creates no warning but has no ligature \begin{align} \mathrm{Empf"anger} \end{align} \end{document} - Welcome to TeX.SE! Instead of \mathrm{Empf"anger}, try writing \text{Empf"anger}. The \text command, incidentally, is provided by the amsmath package, which you're already loading. – Mico Feb 13 '13 at 14:48 You can use the text formatting commands (in your case, you need \textrm) in math mode. I don't post it as an answer because I don't know if it is best practice. – T. Verron Feb 13 '13 at 14:49 Side note: "a expands to \ddot a in math mode. – Qrrbrbirlbel Feb 13 '13 at 14:50 I don't exactly why you don't want to use \text but probably you are looking for \text{\normalfont Empfänger}. – Ulrike Fischer Feb 13 '13 at 14:50 @UlrikeFischer : As I understand it, the text is supposed to be part of a notation. You don't want a notation to be different depending on whether you're in a theorem or a proof. – T. Verron Feb 13 '13 at 14:54 As you load the ams packages, you can use \textnormal: \documentclass{scrartcl} % Kodierung \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{lmodern} % Sprache (neue deutsche Rechtschreibung) \usepackage[ngerman]{babel} % Mathematik \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \begin{document} Test \begin{align}$$\textnormal{Empf"anger}$$\end{align} test \itshape Test $$\textnormal{Empf"anger}$$ test \end{document} As the example shows, this is immune to the surounding text changes. - You don't need ams to use it: \textnormal is defined in the latex kernel. – Ulrike Fischer Feb 13 '13 at 15:12 @UlrikeFischer True, but with ams it will resize appropriately in sub- and suprescripts too. – Andrew Swann Feb 13 '13 at 15:37 Instead of using \mathrm you should use \text to write text in a math environment. That should not have any problems with umlauts. - If the text around the equation is italics (e.g. a theorem), \text will print the text as italics too. – T. Verron Feb 13 '13 at 14:49 Maybe not the most elegant solution, but definitely package-independent is to define a new command: \newcommand{\aumlaut}{\text{\textit{\"a}}} If you have inputenc with UTF-8 for instance, you could of course have ä straightaway. - This only takes care of cases where the surrounding text is in italics. See for example \textbf{$\aumlaut$}. – T. Verron Feb 13 '13 at 15:18 That's right. You will have to define another command for each case. – Count Zero Feb 13 '13 at 15:39	2015-03-06 17:47:43	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 3, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.9359580874443054, "perplexity": 3091.7122844523287}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1424936469305.48/warc/CC-MAIN-20150226074109-00007-ip-10-28-5-156.ec2.internal.warc.gz"}
http://tzsm.hsvmev.de/velocity-pressure-formula-metric.html	# Velocity Pressure Formula Metric Don't forget that the four velocity is a unit vector, so you can (using the metric) to get an equation relating the components to each other, hence get another equation with the density $\rho. Note that in each equation a new term involving a pressure-strain rate has appeared as the first term on the right-hand side. You’re going to have now the pressure at the exit. Can you help me translate, program, or host these calculators? [Hide this request] Check out our spreadsheet version of this calculator: Download Spreadsheet Open Google Sheets version View All Spreadsheets. Dynamic pressure and velocity head Engineering ToolBox - Resources, Tools and Basic Information for Engineering and Design of Technical Applications! - the most efficient way to navigate the Engineering ToolBox!. Drilling Speeds and Feeds. ): Round Duct Equivalent Diameter (in. Includes 53 different calculations. V = Flow Velocity D = Pipe Diameter H L = Head Loss g = Acceleration Of. The Manning equation can be used because the stormwater flows under gravity. U I = velocity in expanded section of inlet distributor, m/s (ft/sec) U L = velocity in external piping, m/s (ft/sec) K = Dimensional constant = 0. Speed of Sound at nasa. Equation Manning’s equation is used to relate the average channel (conduit) velocity to energy loss, Sf = hf/L. Online Ductulator Free Duct Size Calculator. Thus pressure. The pressure drop over a length of pipe or hose can be calculated using the following formula, which for ease of calculation uses metric units. ) General case To change to US Customary units multiply by = 1 (metric) or 1. Re: Gas Velocity in a pipe I don't know if this will help you guys but here I go, as much of you guys I had and still have a lot of troubles with flow of fluids xD, I was designing a transport system and I'd some troubles calculating the velocty of gas 'cause I was working with a high pressure gas flow (actual flow/area doesn't work 'cause at those conditions the gas isn't ideal), after a long. Linear momentum, p, is defined as the product of mass and velocity: p = mv. Thus, we select these locations in such a way as to be able to specify the maximum amount of information possible at. 72 psi plus the frictional dP of 7. These calculators use these formulas to determine the water flow rate and time to fill a container with a garden hose. Pressure In (P1) Considering the direction of the fluid, we define P1 as the pressure (gauge or absolut) existing in the pipeline. Speed, velocity: meter per second: m/s kilometer per hour: km/h: 1 km/h = 0. 2 meters per second. Velocity is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of bodies. This force, which is due to the continuous, random motion of molecules, is known as fluid pressure. Velocity pressure only accounts for a small percentage of the total pressure, and any deviation from the Hazen-Williams formula can be correct for with all of the other safety factors in the NFPA 13 standard. Pressure and force are related, and so you can calculate one if you know the other by using the physics equation, P = F/A. Determine the maximum discharge over a broad-crested weir 60 meters long having 0. 00 Corrected Volumetric Flowrate cfm 40. = Velocity of water in the pipe = Frictional resistance per unit area at unit velocity Consider sections (1-1) and (2-2) of the pipe Let, = Intensity of pressure at section (1-1) = Intensity of pressure at section (2-2) A little consideration will show that p 1 and p 2 would have been equal, if there would have been no frictional resistance. √= Square root of the number to the right. In this article we learn how to perform pump calculations in both imperial and metric units to assess pumping performance following the change of flow rate, pump speed, head pressure and power. Flow rates are usually measured by the volume of water that passes each minute. % add to clean pressure drop. Mass (M) x Volume (V) is not the formula for anything. Into that, you enter your individual weight and height measurements and your age. 72 psi plus the frictional dP of 7. To convert and calculate metric and imperial unitsfor distance, length, weight, capacity, temperature, velocity. VP - Velocity Pressure: expressed in inches water gauge. NOTICE: The modern lift equation and the lift equation used by the Wright brothers in 1900 are slightly different. 2) The pressure at the bottom of a cylinder that contains a gas is P = 735. Approximate pressure for each schedule number may be computed from the formula. Formulas and Calculations for Drilling, Production, and Workover, All the Formulas You Need to Solve Drilling and Production Problems, Fourth Edition provides a convenient reference for oil field workers who do not use formulas and calculations on a regular basis, aiming to help reduce the volume of materials they must carry to the rig floor or. p 2 calculate pressure in position two z 2 calculate height in position two v 2 calculate velocity in position two Available only if "use flow rate q in the calculation" is selected. P S = Static Pressure. This is the resulting pressure generated by the specified force and area and is calculated by dividing the force by the area. 0 m/s and a pressure of 200000 Pa. 48979591 speed of sound (at std. Hg) Q = A x V Q = ft3/minute (CFM) A = duct area (ft2) V = Velocity, ft/minute (fpm). The Bernoulli's Velocity calculator uses Bernoulli's equation to compute velocity (V 1) based on the following parameters. Advanced Fluid Systems has taken great care to verify that the conversions and calculations on this page are correct. 2 kg/m 3) (35 m/s) 2 (10 m 2) = 7350 N = 7. In a free plane sound wave the pressure P and particle velocity U are in phase and proportional to each other. Bernoulli's equation relates a moving fluid. According to Bernoulli's equation, the pressure in a fluid will tend to decrease if its velocity increases. This is a quantity that is conserved when there are no external forces acting. By multiplying air velocity by the cross section area of a duct, you can determine the air volume flowing past a point in the duct per unit of time. 1 inches of water (25 Pa) now allows the use of standard fan curves. The equation illustrates that momentum is directly proportional to an object's mass and directly proportional to the object's velocity. 2001-01-01. C, the constant, lets you know that the sum of a fluid's static pressure and dynamic pressure, multiplied by the fluid's velocity squared, is constant at all points along the flow. equals a Flow Velocity of 3,468 Ft/Min. Origin of Erosional Velocity Formula in Pipes Rev. 25; 181 ft/min; Determine annular pressure losses, psi: P = (1. Take coefficient of discharge as 0. We therefore need to understand how we quantify perturbations to the stress-energy tensor. 0 m/s and a pressure of 200000 Pa. ) Density of (Gas). The origin of the x-y axes is chosen to be at point 2, hence y 2 = 0 and y 1 = H. (Eq 5)$\frac{L}{D}=0. It uses a differential pressure principle for the measurement using the known or measured static pressure and total pressure differences know as the dynamic pressure. , Clovis, California, USA ABSTRACT: The generalized procedure for the calculation of the pressure drop along tunnels is by using Atkinson’s equation. The left side of the equation contains what we call the driving head. increases Total Pressure, such as undersizing outlets or increasing the air velocity, also increases sound level. Basic Hydraulic Principles Chapter 1 R = A / Pw = 4. The pressure at this point is the leak-off pressure and is used to estimate formation fracture pressure. 0 Full Specs Visit Site External Download Site. Ductulator Calculator Air duct also termed as ductulator is a passage that is used to provide fresh air into rooms, buildings etc. These heads include heads due to a pump (if present), elevation, pressure, and velocity. This ultra calculator is special by allowing you to choose among a great variety of units. For example, if you drive a car for a distance of 70 miles in one hour, your average velocity equals 70 mph. ca Page 2 Proof: In a circular pipe shear causes the pressure loss. air flow rate remains the same with a change in air density. The dynamic pressure in water with - temperature 20 o C - density 1000 kg/m 3 and velocity 5 m/s - can be calculated as p d = 1/2 (1000 kg/m 3 ) (5 m/s) 2 = 12500 Pa. This is PSIG (gauge pressure) plus 14. hand-held calculator. Coefficient A. Velocity Head Loss Coefficient Flow Model. , if the flow is steady), the fluid leaves through the hole in the pipe at a velocity of v = √2_g_h, where g is acceleration due to gravity, 9. PSIA = Absolute pressure. Recommended fluid velocities through pipes and hoses in hydraulic systems are as follows: Service suction/intake return pressure/discharge Velocity (ft/sec) 2-4 4 13 7 - 18. Retooling: In the 1980s, much of the American automobile industry switched from the English system of measurement to the metric system of measurement. Darcy-Weisbach •Major losses (pipe. The equation above can be used to calculate water flow rates for pipe sizes and lengths with different pipe materials and pressure driving forces, using the Hazen Williams equation as demonstrated in the table above. Fluid Mechanics is the study of the application of forces upon fluids. , in 100F increments. The left side of the equation contains what we call the driving head. This taper opening acts as a wedge together with the cutting ring which will, with pressure from the rotation of the nut, be forced to cut into the tube and form a visible collar. VP = Velocity pressure, in. Also, to find density of the gas the ideal gas equation was used, relating pressure and temperature. Water Velocity Calculator. 2017-11-01. ) General case To change to US Customary units multiply by = 1 (metric) or 1. K value #2. The Bernoulli equation can be adapted to a streamline from the surface (1) to the orifice (2): p 1 / γ + v 1 2 / (2 g) + h 1. In the following equation (Bernoulli's equation) each of the terms is a head term: elevation head h, pressure head p and v elocity head v2 /2g. Example-1: A galvanized duct with an equvilant length of 79 feet is required by design have a pressure drop of 0. It is used in the design of water pipe systems [1] such as fire sprinkler systems , [2] water supply networks , and irrigation systems. Thus if the terms are positive and the exit velocity is greater than the entry velocity (that is, v 2 > v 1), the exit pressure must be lower than the entry pressure (that is, P 2 < P 1). Email this Article. Darcy-Weisbach •Major losses (pipe. For Bernoulli applications, please see our Bernoulli Calculator with Applications. P/U = Z where Z = radiation impedance of air which is 406 Rayl metric. Imperial Units SI Units. The common unit and term for this velocity is surface feet per minute, abbreviated sfm. & +=#∗,∗ℎ Angular velocity of circular motion, where T is the period of the motion. c 2004 International Press Vol. Result will be displayed. Welcome! This page is useful to meteorologists who need help remembering units, constants, and abbreviations. * Available in Metric and Imperial Units * *** Available in English, Français, Español, Italiano, Deutsch. Full line catalog of Industrial controls including pressure gages, manometers, pressure-switches, transmitters, flowmeters, flow-switches, pitot tubes, anemometers. Prepare numerical calculation problems for IWCF Well Control examination by practicing these problems. A wind load refers to the intensity of the force that wind applies to a structure. Size the line based on water usage (flow requirements), the pressure drop you will be able to tolerate, and the cost of the pipe. 6T c m/s (where T c is the temperature in degrees Celsius and m/s is velocity in meters per second) provides a good approximation. Air velocity (distance traveled per unit of time) is usually expressed in Linear Feet per Minute (LFM). Fluid velocity in the annulus is determined by using: where: D1 = hole or casing I. Impact Force from Falling Object Even though the application of conservation of energy to a falling object allows us to predict its impact velocity and kinetic energy, we cannot predict its impact force without knowing how far it travels after impact. If the only pressure for the pipe is that of a pitometer, that value will be the velocity pressure of the fluid. HVAC Duct Friction Equations Page 1 of 6. 4327 x 10—7) x MW x Lx V2 Dh — Dp where P = annular pressure losses, psi L = length, ft Dh = hole or casing ID, in. It applies in all types of duct and holds account particularly on the operating conditions and specific Characteristics on the ductworks, such as:. Other units are used in SI:tonne (t), electronvolt (eV), and the atomic mass unit (u). This Tech Briefprovides basic examples of these formulas and conversion factors. Every tool manufacturer has a recommended table of sfm values for their tools. The pressure drop over a length of pipe or hose can be calculated using the following formula, which for ease of calculation uses metric units. Sound of Speed in Air — the formula v sound in air = 331. 5)0791 x (1. 33 bar/100 stks). The magnitude of pressure wave in a metallic piping system is much higher due to the higher modulus of elasticity of these materials. 1 oxygen 159 20. 8 kilonewtons per cubic meter (9. Pressure drop in conductors is an important consideration for the designer especially in systems where long pipe or hose runs are necessary. If a zero (0) is put in any blank, the equation will solve for this variable. These formulas are common practice rules of thumb and provide theoretical values to which actual values will likely differ. Concentration Increasing conc. Includes 53 different calculations. Welcome to CALCULATOR EDGE, an online FREE Engineering Calculators for Engineers and Students : worldwide, Our website features more than few hundred calculators for solving complex equations and. The PISO (Pressure Implicit with split Operator) algorithm was employed for the pressure-velocity coupling. 92) T act = measured dry bulb temperature of the actual airstream, °F P act = absolute pressure of the actual airstream, in. Download Pressure Unit Converter our powerful software utility that helps you make easy conversion between more than 2,100 various units of measure in more than 70 categories. Below are equations commonly used for pipe friction head loss calculation. Result will be displayed. Notice that the free air delivery of fans is the airflow delivered with no resistance. All you need to know is the fluid’s speed and height at those two points. If you have previous experience with the University of Wyoming Spur Gear Design Calculator you might consider going directly to the input routine. Units of Measurement/Pressure. So our first step is to find out the start pressure. Until now, to calculate the Volume flow rate or conveyed volume of a gear pump, technicians had to look up the formulas, pick up a calculator, enter the calculation according to. 9 SQRT( 1/ density of the ﬂuid), density in lbs per cubic feet. The velocity pressure expressed in Ft of the fluid flowing = V^2/2g (Equation 1) Where V = Ft/sec g = 32. The pressure exerted by a column of liquid of height h and density ρ is given by the hydrostatic pressure equation p = ρgh, where g is the gravitational acceleration. In extreme cases this may lead to cavitation when the local pressure is less than the vapour pressure of a liquid. Frequently these terms are used interchangeably and it may entirely depend on a particular project or company standards to identify all the pressure relief devices either as 'safety valves' or as 'relief valves' or sometimes even as 'safety relief valves'. MOD = PCD / N. Several density conversions. The Manning equation is useful together with the equation: Q = VA, to determine the storm sewer diameter and slope that will give full pipe flow rate greater than the design flow, and full pipe flow velocity greater than the required design minimum, typically 3 ft/sec. Equations displayed for easy reference. Velocity in pipe (u) Linear Velocity of the fluid in pipe. This could be the pressure in a street water main, a pump, or the pressure at the start of a branch take off. ) to Velocity (FPM) CALCULATIONS The Pitot tube is the primary standard used to calibrate all other air-ﬂow measuring devices. Full line catalog of Industrial controls including pressure gages, manometers, pressure-switches, transmitters, flowmeters, flow-switches, pitot tubes, anemometers. 49 is a units conversion factor. If the Kill Mud Weight or Leak Off values are to be used in subsequent calculations, use the rounded value in the future calculation. Pipe diameter:. The unit assigned per currently set under Unit System – ft/min for Imperial and m/s for Metric. Discover a universal assistant for all of your unit conversion needs - download the free demo version right away!. The Hazen-Williams equation is an empirical relationship which relates the flow of water in a pipe with the physical properties of the pipe and the pressure drop caused by friction. Bernoulli’s equation states mathematically that if a fluid is flowing through a tube and the tube diameter decreases, then the velocity of the fluid increases, the pressure decreases, and the mass flow (and therefore volumetric flow) remains constant so long as the air density is constant. In order to use this constant one has to meet certain qualifacations, otherwise the final results could be bogus! Example: For a duct with a 28-by-20-in. I know that velocity pressure can be calculated from dynamic pressure according to the potential energy of pressure (from Bernoulli's equation): $$P = \frac{1}{2} \rho \overline{V}^2$$ where P = dynamic pressure in Pascals, rho = density in kg/m^3, and V = velocity in m/s. At the nozzle the pressure decreases to atmospheric pressure (101300 Pa), there is no change in height. Thus if the terms are positive and the exit velocity is greater than the entry velocity (that is, v 2 > v 1), the exit pressure must be lower than the entry pressure (that is, P 2 < P 1). What is the Dynamic Pressure at that speed/altitude combination? According to Glenn Research Center’s AtmosModeler Simulator , that speed is 1,806 km/hr (501. ): Round Duct Equivalent Diameter (in. of one substance favors reaction that produces less of that substance Common Acids Acid Formula Strength Hydrochloric (muriatic) acid HCl strong Nitric acid HNO 3 strong Sulfuric acid H 2 SO 4 strong Acetic acid CH 3 COOH weak Citric acid C. Pressure Drop Online-Calculator for small mobiles. This calculator also computes the Reynolds number, Friction Factor, and pressure drop through the pipe and will account for either laminar or turbulent flow. 075 = lb/ft^3 density of air and in. List of Pressure category units. Mass (M) x Volume (V) is not the formula for anything. 5 mps Pressure lines to 200 bar is 4. It is for manual use and does no calculations. Hg) Q = A x V Q = ft3/minute (CFM) A = duct area (ft2) V = Velocity, ft/minute (fpm). the Rouse formula in which the actual depth H is replaced by H'=1. Like speed, its units are length and time, but direction is also involved in the equation. Radiologists often use Doppler ultrasound to measure flow velocity change in a blood vessel with a stenosis- but I would. SI Metric Units The velocity of gas flowing through an orifice becomes choked (and is also referred to as sonic velocity) when the ratio of the absolute upstream pressure to the absolute downstream pressure is equal to or greater than [ ( k + 1 ) / 2 ] k / ( k - 1 ), where k is the specific. VELOCITY AND PRESSURE DROP IN PIPES. ΔP = The Velocity Pressure measured by the pressure sensor. If the maximum discharge head of a pump is say 10 meter, the fluid will be able to get to 10 m of height above the discharge and the flow will be zero. Pressure is defined as the force applied which is perpendicular to the surface of the object per area over which the force is distributed. % add to clean pressure drop. This difference, called the dynamic pressure, can be used to calculate the fluid velocity at the point of measurement. Every tool manufacturer has a recommended table of sfm values for their tools. Whilst every effort has been made in building this velocity & speed converter, we are not to be held liable for any special, incidental, indirect or consequential damages or monetary losses of any kind arising out of or in connection with the use of the converter tools and information derived from the web site. Note that (a) the windward pressure varies along the height of the building, while the leeward pressure is assumed to be constant; and that (b) the leeward pressure is negative, i. When calculating Pressure Reduction Schedule, ROUND DOWN to two decimal places (for example, round down 1. Formulas Velocity of oil flow in pipe: V = GPM x 0. When the fluid’s static pressure reaches liquid vapor pressure, vapor cavities (bubbles) form and grow until they violently implode by the recovered pressure downstream to the valve seat. Linear Velocity (V): The calculator returns velocity in meters per second. Pressure Loss Through Perforated Plate (Air) In many applications of perforated plate, the estimated energy loss or pressure loss through perforated plates is one of the design considerations. Piston / Bore Diameter Rod Diameter. Dynamic pressure is the kinetic energy per unit volume of a fluid particle. 2 meters per second. Example: 10 gpm flow across a pressure reducing valve with a 300 psi drop = 1. Formula to find pressure gradient for a given mud density. Plastic pipe,. When we start any hydraulic calculation for a water based fire protection systems such as a fire sprinklers, water mist systems the k-factor formula is the first which we will need to use and as it is so fundamental all fire protection engineers must have a good understanding of how it works. wc D = Density, lbs/ft3, 0. Dynamic pressure and velocity head Engineering ToolBox - Resources, Tools and Basic Information for Engineering and Design of Technical Applications! - the most efficient way to navigate the Engineering ToolBox!. " Sounds impressive, but in traditional English terms a pascal is only 0. 320833 x GPM (Flow Through Conduit). Metric Association (USMA) has all sorts of useful information and history on the metric system. c 2004 International Press Vol. The conversion chart for pipe Note the different weights or piping are grouped under various schedule numbers. NOTICE: The modern lift equation and the lift equation used by the Wright brothers in 1900 are slightly different. P is brake mean effective pressure, in PSI L is piston stroke, in feet A is the area of one piston, in square. • Top eq'n shown in units: ft3/sec = ft2 x ft. While fluid power formulas are useful tools for specifying system components and capabilities; other factors such as mechanical efficiencies, fluid dynamics and material limitations must also be considered. In testing for temperature degradation, the damper is heated in the closed position for 30 minutes and then cycled to see that it operates as intended. As shown in the figure, the jet narrows over a short distance beyond the orifice that is comparable with the jet diameter to form what is generally known as a vena contracta --that is. 60 km/h to the north). As usual, here at www. Aside from mph, velocity can likewise be represented in terms of other units. V = 4005 x √0. F w = 1/2 ρ v 2 A = 1/2 (1. Choose Category. HOME > PLAIN SLIDING LINEAR BEARINGS > PRESSURE VELOCITY DRY SLIDING (PLAIN) BUSHES - PRESSURE VELOCITY: PV (Pressure Velocity) PV is the unit of measure determined by two factors; Pressure (PSI) and Velocity (SFM) Formula: P = W/(dxb) W = Static load on a single bearing. It is defined as the flow rate in cubic meters per hour [m3/h] of water at a temperature of 16º celsius with a pressure drop across the valve of 1 bar. Acceleration is a measure of how fast velocity is changing, so we can think of it as the change in velocity over time. PUMP SYSTEM FORMULAS (IMPERIAL UNITS) (change to metric units) This page contains many applets to help you do calculations and to SAVE YOU TIME. Formulas and Calculations 6 Convert pressure, psi, into mud weight, ppg using meters as the unit of measure Annular velocity (AV), ft/min Formula 1. The average of that number is your velocity. Pipe Water Velocity and Minimum Pipe Diameter. (The fact that Jira provides this metric is a good reason to get rid of Jira. In addition to rapid valve closure or opening, sudden air release and pump start up or shut down can create water hammer. Functional integral for non-Lagrangian systems. 3208 ÷ A A = GPM x 0. wc Calculating Air Velocity with density correction (Actual) V = 1096. Seems straightforward enough, but when I plug in my pressure readings, I'm getting too high of a result for Q (I get a result I'd expect for a fan, and not the dribbling of air I'm actually getting from the pipe). Inches per minute to Inches per second Conversion Example. The velocity of a rocket can be estimated with the formula below: Δv = ve * ln(m0 / mf) where. CONVERSION CHART Velocity Pressure (IN. ΔP = The Velocity Pressure measured by the pressure sensor. I know that velocity pressure can be calculated from dynamic pressure according to the potential energy of pressure (from Bernoulli's equation): $$P = \frac{1}{2} \rho \overline{V}^2$$ where P = dynamic pressure in Pascals, rho = density in kg/m^3, and V = velocity in m/s. 075) (Equation 2) Where 62. By comparing the pressure drop given by the Ergun equation to the pressure drop for minimum fluidization, you can calculate the superficial velocity necessary for fluidization. • Top eq'n shown in units: ft3/sec = ft2 x ft. It is defined as the flow rate in cubic meters per hour [m3/h] of water at a temperature of 16º celsius with a pressure drop across the valve of 1 bar. Section 2: Design 6. cc Formula: a x a x. Re: Gas Velocity in a pipe I don't know if this will help you guys but here I go, as much of you guys I had and still have a lot of troubles with flow of fluids xD, I was designing a transport system and I'd some troubles calculating the velocty of gas 'cause I was working with a high pressure gas flow (actual flow/area doesn't work 'cause at those conditions the gas isn't ideal), after a long. The average velocity at which the water flows through the pores is the ratio of volume flow rate to the average area of voids A v on a cross section normal to the macroscopic direction of flow. • Top eq'n shown in units: ft3/sec = ft2 x ft. An Excel Template to Calculate Water Flow Rates for Pipe Sizes and lengths. 387 x 6 = 4942. Formulas for gear calculation – external gears Contents: Relationship between the involute elements Determination of base tooth thickness from a known thickness and vice-versa. Manometer's are used to measure the difference between dynamic and static pressures and may be configured as a u-tube, a single tube, or inverted depending on the application. wc D = Density, lbs/ft3, 0. Recall that the pressure, velocity, and elevation at each of these locations appear in the Engineering Bernoulli Equation. Origin of Erosional Velocity Formula in Pipes Rev. Note that (a) the windward pressure varies along the height of the building, while the leeward pressure is assumed to be constant; and that (b) the leeward pressure is negative, i. Pascal (Pa) is the metric SI unit of pressure and the standard pressure unit in the MKS metric system, equal to one newton per square meter or one "kilogram per meter per second per second. It applies in all types of duct and holds account particularly on the operating conditions and specific Characteristics on the ductworks, such as:. Mass (M) x Volume (V) is not the formula for anything. More appro-priately, this is expressed as 9. 00 Corrected Velocity Pressure "wg 42. Using a BMI calculator such as the one on this page, you enter your weight and height measurements. com, we have a calculator that will do all the work for you. generates 8 foot-pounds. Where v is flow velocity, rho is density, and P is pressure. Although you can use a simple formula to calculate wind loads from wind speed, building designers, engineers and constructors must incorporate many additional calculations to ensure their structures won't blow over in a high wind. Total Pressure : The total work a fan must do to move a specified volume of air against the static pressure plus the velocity pressure is defined as total pressure of the system and is expressed in millimeters or mm wg. Manometer's are used to measure the difference between dynamic and static pressures and may be configured as a u-tube, a single tube, or inverted depending on the application. VELOCITY AND PRESSURE DROP IN PIPES. The Marczak equation has the advantage that all coefficients are expressed to 6 decimal places rather than the 8 decimal places of Bilaniuk and Wong. This means Friedmann's equation remains valid at all events in u, where the metric takes the form given in Equation. 33 bar/100 stks). BAR= A metric unit of pressure = 105n/sq. Simply enter the three known measures to calculate the fourth. PUBLISHED BY THE NATIONAL ENVIRONMENTAL SERVICES CENTER. The Joukowski impulse equation is used to calculate the resulting pressure when the liquid velocity that drops to zero upon contacting a closed valve. hand-held calculator. The viscosity of a fluid is is one of the most important aspects for longer life of the hydraulic system. This Tech Briefprovides basic examples of these formulas and conversion factors. 077–132, March 2004 007 COSMOLOGY, BLACK HOLES AND SHOCK WAVES BEYOND THEHUBBLELENGTH∗ JOE. essentially you divide the equation for velocity by time in a mathematically exact way. The flow rate depends on the area of the pipe or channel that the liquid is moving through, and the velocity of the liquid. 3215 ) / (12 x. Pressure vs. Calculator Use. The Flow Velocity is then determined with the following equation: V = 4005 x √ΔP V = Flow Velocity in feet per minute. Thus, we select these locations in such a way as to be able to specify the maximum amount of information possible at. The wave velocity, v, is the velocity at which the shape of the wave propagates in space. K value #2. BAR= A metric unit of pressure = 105n/sq. 2: Composition of Dry Air Gas Partial Pressure [Torr] Percent [by volume] nitrogen 593 78. When the dynamic pressure is divided by the product of fluid density and acceleration due to gravity, g, the result is called velocity head, which is used in head equations like the one used for pressure head and hydraulic head. Metric theory of relativity (MTR) is the theory that describes the transformation of physical quantities in the laws of motion, including the laws of mechanics, electrodynamics and the covariant theory of gravitation, based on the spatial-temporal relations in arbitrary reference frames. By substituting the dimensions of force, area, length and velocity in (3. to its suction side and move it to the discharge side. (Eq 5) \$\frac{L}{D}=0. useful formulas & fan laws basic fan laws • effect of fan speed 1. This taper opening acts as a wedge together with the cutting ring which will, with pressure from the rotation of the nut, be forced to cut into the tube and form a visible collar. The above formula & step by step calculation may useful for users to understand how the values are being used in the formula to find the head loss, however, when it comes to online for quick calculations, this Darcy Weisbach head loss calculator helps the user to perform & verify such calculations as quick as possible. For the purpose of this exercise use schedule 40 steel pipe. 3208 ÷ V GPM = A x V ÷ 0. 92 inches Hg. ) Gallons U. Temperature (T) Temperature existing inside the pipe. Ductulator calculats duct dimensions using equal friction method or velocity method. Tables 1 and 2 summarize measurements for DBF and SPS flyash, respectively. Starting from an idea of S. Velocity pressure only accounts for a small percentage of the total pressure, and any deviation from the Hazen-Williams formula can be correct for with all of the other safety factors in the NFPA 13 standard. This conversion table package allows conversion from one set of units to another. When the size of a pipe and the quantity of water it will discharge in a given time are known, the mean velocity of efflux can be found by the formula: v=q. Water Hammer - Maximum Surge Pressure for Water: Maximum Surge Pressure for Water: Pressure Wave Velocity: Fluid Velocity Change: Acceleration of Gravity: where, P s = Maximum Surge Pressure for Water, a = Pressure Wave Velocity, ΔV = Fluid Velocity Change, g = Acceleration of Gravity. Calculate Air Velocity. 67, 1992", and a mole fraction of carbon dioxide of 0. The other input values, like velocity and reference length, are not necessary unless you want to perform the rest of the calculations available. To find the pressure find the mass of air hitting unit area of the wall per second and multiply by the change of velocity. 3208 ÷ A V is oil velocity in feet per second; GPM is flow in gallons per minute; A is inside area of pipe in square inches Charles' Law for behaviour of gases P1V1 = P2V2 or T1P2 = T2P1 T1, P1 and V1 are initial temperature, pressure, and volume, and T2, P2, and V2 are final conditions. Length = 6500 ft Hole size = 8. = An arbitrary number to account for hose size See Table. com, we have a calculator that will do all the work for you. Fluid Power Formula ) OR D 11 2 2 1 1 Miscellaneous Fluid Power Formula Fluid Pressure= Force (Pounds) Unit Area (in2) Velocity Through Pipes in Feet/Second (FPS)= 0. CALCULATION OF CYLINDER FORCES - METRIC BASED PRODUCTS General Formula The cylinder output forces are derived from the following formula: Prior to selecting the cylinder bore size, properly size the piston rod for tension (pull) or compression (push) loading. ) Back to: Units. Recall that the pressure, velocity, and elevation at each of these locations appear in the Engineering Bernoulli Equation. 50 m, what is the density of the gas? Answer: The density can be found by rearranging the pressure formula:. Inserting a propeller into the water flow will cause friction, resulting in pressure or head losses in the pipeline. Horsepower for driving a pump For every 1 HP of drive, the equivalent of 1 GPM @ 1500 PSI can be produced. The result is: a = Vmax^2 * 2Xmp x / (2*Xmp + x)^3 a is the acceleration, measured in inches per second per second x is how far the bullet base moves from rest until it exits the muzzle, measured in inches Vmax is the velocity the bullet would reach if the. H 1 is the total hydraulic energy (rig pump) required to counteract all friction energy (loss) starting at the Kelly hose (surface line) and Kelly, down the drillstring, through the bit nozzles. edu is a platform for academics to share research papers. Magnehelic® Differential Pressure Gage and Pitot Tube Measuring Air Velocity. of one substance favors reaction that produces less of that substance Common Acids Acid Formula Strength Hydrochloric (muriatic) acid HCl strong Nitric acid HNO 3 strong Sulfuric acid H 2 SO 4 strong Acetic acid CH 3 COOH weak Citric acid C. Dynamic pressure is density times the fluid velocity squared times one-half, assuming no friction and steady fluid flow throughout. / /	2019-11-23 01:06:32	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.5741694569587708, "perplexity": 1324.7204417469964}, "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-2019-47/segments/1573496672313.95/warc/CC-MAIN-20191123005913-20191123034913-00202.warc.gz"}
http://www.realmagick.com/euler-lagrange-equation-proof/	realmagick.com The shrine of knowledge. # Euler Lagrange Equation Proof A selection of articles related to euler lagrange equation proof. Beth Moore Beth Moore, founder and pastor of internationally known Living Proof Ministries, was born Wanda Elizabeth Green on June 16, 1957 in an army base at Green Bay, Wisconsin. Beth is the fourth of five children and was raised in Arkadelphia. >> Modern Science >> Real Interviews The Book of the Law, Liber 220, Part III III;1. Abrahadabra; the reward of Ra Hoor Khut. III;2. There is division hither homeward; there is a word not known. Spelling is defunct; all is not aught. Beware! Hold! Raise the spell of Ra-Hoor-Khuit! III;3. Now let it be first understood that I... Thelema >> The Book of the Law Sensible Ideas: The Knowledge Behind the Perception Do objects exist independently of consciousness? If so, what is the nature of these objects? These questions shall be investigated through the conflicting philosophies of John Locke and George Berkeley. Preliminary Quotations from Locke and Berkeley: Primary... Religion & Philosophy >> Philosophy Magick and Science Many people wonder how magick works. To the non-believer, they say that there is no proof that spells work. And those abilities such as telekinesis are illusionary or product of imagination. They ask us to use scientific means to explain how exactly these... Mystic Sciences >> Magick >> Modern Science Every Thought is a Prayer Psychic power. We use it every day. We cause things to happen; we shape events with our mind. We can do this, because every thought is a prayer. Thoughts are a powerful form of energy. They are very real and very potent. Look for their energy not just in deep... >> Psychic Abilities Euler Lagrange Equation Proof is described in multiple online sources, as addition to our editors' articles, see section below for printable documents, Euler Lagrange Equation Proof books and related discussion. ## Suggested Pdf Resources ON THE METRICS AND EULER-LAGRANGE EQUATIONS OF ∂t g−1(t)=−Dg−1(t)v(t),g(0)=g0,g(1)=g1. ∫ 1. 0. The Euler-Lagrange equation The Euler-Lagrange equation a b t ya yb. Figure 2.1: Possible paths joining the two fixed points (a, ya) and (b, yb). Lagrange's Equation Mar 30, 2006 Lagrange equations of motion. Theorem 3.1. 1 The Euler Lagrange Equations 1 The Euler Lagrange Equations. Many interesting models can be created from classical mechanics problems in which the simple motions of objects are studied. M2A1: A reduced form of the Euler-Lagrange equation ## Suggested Web Resources EulerLagrange equation - Wikipedia, the free encyclopedia The derivation of the one-dimensional EulerLagrange equation is one of the classic proofs in mathematics. Simple Derivation of Euler-Lagrange Equations Jun 21, 2008 Simple Derivation of Euler-Lagrange Equations Classical Physics discussion. PlanetMath: derivation of Euler-Lagrange differential equation [parent] derivation of Euler-Lagrange differential equation (elementary), ( Derivation) . $t_0 \in (a,b)$ . In other words, q satisfies the Euler-Lagrange equation. ON THE METRICS AND EULER-LAGRANGE EQUATIONS OF ∂t g−1(t)=−Dg−1(t)v(t),g(0)=g0,g(1)=g1. ∫ 1. 0. Euler-Lagrange Differential Equation -- from Wolfram MathWorld Great care has been taken to prepare the information on this page. Elements of the content come from factual and lexical knowledge databases, realmagick.com library and third-party sources. We appreciate your suggestions and comments on further improvements of the site.	2016-02-07 15:22:29	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5682836174964905, "perplexity": 4387.199171890119}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701149548.13/warc/CC-MAIN-20160205193909-00246-ip-10-236-182-209.ec2.internal.warc.gz"}
http://www.transum.org/Software/MathsMenu/Starter.asp?ID_Starter=77	# Key Eleven ## An Advanced Mathematics Lesson Starter Of The Day Choose 4 keys on your calculator key pad that are positioned in the four corners of a rectangle. Use these keys to type in a 4-digit number going round your rectangle either clockwise or anticlockwise. Eg. 3146 Check that your 4-digit number is a multiple of eleven. Can you prove that all 4-digit numbers formed this way are multiples of 11? Is this strange result also true for numbers formed from keys on the corners of a parallelogram? Note 1: The rectangles referred to above have horizontal or vertical sides. Note 2: A divisibility test for eleven is to consider the alternating digit sum of the number. If this is divisible by eleven then the original number is also divisible by eleven. For example, if a four digit number has digits a, b, c and d then the alternating digit sum is a - b + c - d. Share Topics: Starter How did you use this starter? Can you suggest how teachers could present or develop this resource? Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for Maths teachers anywhere in the world. If you don't have the time to provide feedback we'd really appreciate it if you could give this page a score! We are constantly improving and adding to these starters so it would be really helpful to know which ones are most useful. Simply click on a button below: Excellent, I would like to see more like this Good, achieved the results I required Satisfactory Didn't really capture the interest of the students Not for me! I wouldn't use this type of activity. This starter has scored a mean of 2.0 out of 5 based on 1 votes. Previous Day \| This starter is for \| Next Day Let the rectangle have a width of $$w$$ key spacings and a height of $$h$$ key spacings. (The example in the diagram above has $$w = 2$$ and $$h = 1$$ ) Let the number be formed by going around the rectangle in a clockwise direction starting from the top left (unlike the example in the diagram above). Let the first digit of the 4-digit number be $$x$$. The alternating digit sum is $$x - (x+w) + (x+w+h) - (x+h)$$ $$= x - x - w + x + w + h - x - h$$ $$= 0$$ (which is a multiple of 11 so the 4-digit number must also be a multiple of 11) Also any cyclic permutation of the 4-digit number will be divisible by 11 (accounting for the example in the diagram above) This should also be applicable to parallelograms How did you use this starter? Can you suggest how teachers could present or develop this resource? Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for Maths teachers anywhere in the world. Click here to enter your comments. Your access to the majority of the Transum resources continues to be free but you can help support the continued growth of the website by doing your Amazon shopping using the links on this page. Below is an Amazon search box and some items chosen and recommended by Transum Mathematics to get you started. ## Texas Instruments Nspire Calculator This handheld device and companion software are designed to generate opportunities for classroom exploration and to promote greater understanding of core concepts in the mathematics and science classroom. TI-Nspire technology has been developed through sound classroom research which shows that "linked multiple representation are crucial in development of conceptual understanding and it is feasible only through use of a technology such as TI-Nspire, which provides simultaneous, dynamically linked representations of graphs, equations, data, and verbal explanations, such that a change in one representation is immediately reflected in the others. For the young people in your life it is a great investment. Bought as a gift for a special occasion but useful for many years to come as the young person turns into an A-level candidate then works their way through university. more... The analytics show that more and more people are accessing Transum Mathematics via an iPad as it is so portable and responsive. The iPad has so many other uses in addition to solving Transum's puzzles and challenges and it would make an excellent gift for anyone. The redesigned Retina display is as stunning to look at as it is to touch. It all comes with iOS, the world's most advanced mobile operating system. iPad Pro. Everything you want modern computing to be. more... Click the images above to see all the details of these items and to buy them online. For Students: For All:	2019-02-19 09:07:17	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.2209651619195938, "perplexity": 863.7575389402909}, "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-2019-09/segments/1550247489729.11/warc/CC-MAIN-20190219081639-20190219103639-00230.warc.gz"}
http://academic.hep.com.cn/fcse/EN/Y2017/V11/I3	Frontiers of Chemical Science and Engineering ISSN 2095-0179 ISSN 2095-0187(Online) CN 11-5981/TQ 2015 Impact Factor: 1.043 Cover Story 2017, Volume 11 Issue 3 The authors of Special Topic on environment and sustainable development come from China, USA, Canada, Italia, the Netherlands and Belgium, thereby forming a union of chemical engineering from China, Europe, and North America. #### Current Issue , Volume 11 Issue 3 Preface Select Special Topic on environment and sustainable development Front. Chem. Sci. Eng.. 2017, 11 (3): 291-292. https://doi.org/10.1007/s11705-017-1667-6 Abstract HTML PDF (46KB) Select Combining innovative science and policy to improve air quality in cities with refining and chemicals manufacturing: The case study of Houston, Texas, USA David T. Allen Front. Chem. Sci. Eng.. 2017, 11 (3): 293-304. https://doi.org/10.1007/s11705-017-1660-0 Abstract HTML PDF (967KB) In Houston, a combination of urban emissions from a city of 4 million people, coupled with emissions from extensive petroleum refining and chemical manufacturing, leads to conditions for photochemistry that are unique in the United States, and historically, the city had experienced some of the highest ozone concentrations recorded in the United States. Large air quality field studies (the Texas Air Quality Studies or TexAQS I and II) were conducted to determine root causes of the high ozone concentrations. Hundreds of air quality investigators, from around the world, deployed instruments on aircraft, on ships, and at fixed ground sites to make extensive air quality measurements; detailed photochemical modeling was used to interpret and assess the implications of the measurements. The Texas Air Quality Studies revealed that both continuous and episodic emissions of light alkenes, which came to be called highly reactive volatile organic compounds, played a critical role in the formation of ozone and other photochemical oxidants in the region. Understanding and quantifying the role of these emissions in regional air quality required innovations in characterizing emissions and in photochemical modeling. Reducing emissions required innovative policy approaches. These coupled scientific and policy innovations are described, and the result, substantially cleaner air for Houston, is documented. The lessons learned from the Houston air quality experience are relevant to cities with similar population and industrial profiles around the world. Select Key findings of the 2016 symposium on the frontiers of chemical science and engineering: Environment and sustainable development Zhongming Lu, Duo Li, John C. Crittenden Front. Chem. Sci. Eng.. 2017, 11 (3): 305-307. https://doi.org/10.1007/s11705-017-1666-7 Abstract HTML PDF (227KB) RESEARCH ARTICLE Select Electrochemistry during efficient copper recovery from complex electronic waste using ammonia based solutions Zhi Sun, Hongbin Cao, Prakash Venkatesan, Wei Jin, Yanping Xiao, Jilt Sietsma, Yongxiang Yang Front. Chem. Sci. Eng.. 2017, 11 (3): 308-316. https://doi.org/10.1007/s11705-016-1587-x Abstract HTML PDF (331KB) Leaching selectivity during metal recovery from complex electronic waste using a hydrochemical process is always one of the generic issues. It was recently improved by using ammonia-based leaching process, specifically for electronic waste enriched with copper. This research proposes electrodeposition as the subsequent approach to effectively recover copper from the solutions after selective leaching of the electronic waste and focuses on recognising the electrochemical features of copper recovery. The electrochemical reactions were investigated by considering the effects of copper concentration, scan rate and ammonium salts. The diffusion coefficient, charge transfer coefficient and heterogeneous reaction constant of the electrodeposition process were evaluated in accordance with different solution conditions. The results have shown that electrochemical recovery of copper from ammonia-based solution under the conditions of selective electronic waste treatment is charge transfer controlled and provide bases to correlate the kinetic parameters with further optimisation of the selective recovery of metals from electronic waste. Select The influence of curing conditions on the mechanical properties and leaching of inorganic polymers made of fayalitic slag Remus I. Iacobescu, Valérie Cappuyns, Tinne Geens, Lubica Kriskova, Silviana Onisei, Peter T. Jones, Yiannis Pontikes Front. Chem. Sci. Eng.. 2017, 11 (3): 317-327. https://doi.org/10.1007/s11705-017-1622-6 Abstract HTML PDF (417KB) This study reports on the impact of the curing conditions on the mechanical properties and leaching of inorganic polymer (IP) mortars made from a water quenched fayalitic slag. Three similar IP mortars were produced by mixing together slag, aggregate and activating solution, and cured in three different environments for 28 d: a) at 20 °C and relative humidity (RH) ~ 50% (T20RH50), b) at 20 °C and RH≥90% (T20RH90) and c) at 60 °C and RH ~ 20% (T60RH20). Compressive strength (EN 196-1) varied between 19 MPa (T20RH50) and 31 MPa (T20RH90). This was found to be attributed to the cracks formed upon curing. Geochemical modelling and two leaching tests were performed, the EA NEN 7375 tank test, and the BS EN 12457-1 single batch test. Results show that Cu, Ni, Pb, Zn and As leaching occurred even at high pH, which varied between 10 and 11 in the tank test’s leachates and between 12 and 12.5 in the single batch’s leachates. Leaching values obtained were below the requirements for non-shaped materials of Flemish legislation for As, Cu and Ni in the single batch test. Select The preparation and performance of lignin-based activated carbon fiber adsorbents for treating gaseous streams Min Song, Wei Zhang, Yongsheng Chen, Jinming Luo, John C. Crittenden Front. Chem. Sci. Eng.. 2017, 11 (3): 328-337. https://doi.org/10.1007/s11705-017-1646-y Abstract HTML PDF (343KB) Two types of lignin-based carbon fibers were prepared by electrospinning method. The first was activated with Fe3O4 (LCF-Fe), and the second was not activated with Fe3O4 (LCF). Gas phase adsorption isotherms for toluene on LCF-Fe and LCF were studied. The gas phase adsorption isotherm for 0% RH showed LCF-Fe have about 439 mg/g adsorption capacity which was close to that of commercially available activated carbon (500 mg/g). The Dubinin-Radushkevich equation described the isotherm data very well. Competitive adsorption isotherms between water vapor and toluene were measured for their RH from 0 to 80%. The effect of humidity on toluene gas-phase adsorption was predicted by using the Okazaki et al. model. In addition, a constant pattern homogeneous surface diffusion model (CPHSDM) was used to predict the toluene breakthrough curve of continuous flow-packed columns containing LCF-Fe, and its capacity was 412 mg/g. Our study, which included material characterization, adsorption isotherms, kinetics, the impact of humidity and fixed bed performance modeling, demonstrated the suitability of lignin-based carbon fiber for volatile organic compound removal from gas streams. Select Removal of dyes from wastewater by growing fungal pellets in a semi-continuous mode Tao Lu, Qilei Zhang, Shanjing Yao Front. Chem. Sci. Eng.. 2017, 11 (3): 338-345. https://doi.org/10.1007/s11705-017-1644-0 Abstract HTML PDF (350KB) To increase the efficiency of dye removal from wastewater using mycelial pellets, a bubble column reactor with a simple structure was designed and efficiently used to remove dyes from solution containing dyes. The mycelial pellets were prepared by marine fungus Aspergillus niger ZJUBE-1. Eight dyes were tested as dye targets for the adsorption capacity of mycelial pellets and good removal results were obtained. Eriochrome black T was selected as a model dye for characterizing the adsorption processes in detail. The measurement results of Zeta potential and FT-IR analysis indicate that the electrostatic attraction may play a key role in the biosorption process. The bubble column reactor was utilized to study the batch dye-removal efficiency of mycelial pellets. A re-culture process between every two batches, which was under non-sterile condition, successfully enhanced the utilization of mycelium biomass. The dye removal rate is 96.4% after 12 h in the first batch and then decreases slowly in the following batches. This semi-continuous mode, which consists of commutative processes of dye-removal and re-culture, has some outstanding advantages, such as low power consumption, easy operation, high dye removal rate, and efficient biomass utilization. Select A comparative study on polypropylene separators coated with different inorganic materials for lithium-ion batteries Linghui Yu, Jiansong Miao, Yi Jin, Jerry Y.S. Lin Front. Chem. Sci. Eng.. 2017, 11 (3): 346-352. https://doi.org/10.1007/s11705-017-1648-9 Abstract HTML PDF (328KB) Coating commercial porous polyolefin separators with inorganic materials can improve the thermal stability of the polyolefin separators and hence improve the safety of lithium-ion batteries. Several different inorganic materials have been studied for the coating. However, there lacks a study on how different inorganic materials affect the properties of separators, in terms of thermal stability and cell performance. Herein, we present such a study on coating a commercial polypropylene separator with four inorganic materials, i.e., Al2O3, SiO2, ZrO2 and zeolite. All inorganic coatings have improved thermal stability of the separators although with differences. The coating layers add 28%–45% of electrical resistance compared with the pure polypropylene separator, but all the cells prepared with the coated polypropylene separators have the same electrical chemical performance as the uncoated separator in terms of rate capability and capacities at different temperatures. Select Characterization of landfilled stainless steel slags in view of metal recovery Xuan Wang, Daneel Geysen, Tom Van Gerven, Peter T. Jones, Bart Blanpain, Muxing Guo Front. Chem. Sci. Eng.. 2017, 11 (3): 353-362. https://doi.org/10.1007/s11705-017-1656-9 Abstract HTML PDF (740KB) The slag samples taken from landfill, which originated from different metallurgical processes, have been characterized in this study. The slags were categorized as electric arc furnace (EAF) slag, argon oxygen decarburization/metal refining process slag and vacuum oxygen decarburization slag based on chromium content and basicity. EAF slags have higher potential in metal recovery than the other two slags due to its higher iron and chromium contents. The size of the iron-chromium-nickel alloy particles varies from a few µm up to several cm. The recoveries of large metal particles and metal-spinel aggregates have potential to make the metal recovery from landfilled slags economically viable. Select Synthesis and in vivo nematocidal evaluation of novel 3-(trifluoromethyl)-1H-pyrazole-4-carboxamide derivatives Wen Zhao, Jiahua Xing, Tianming Xu, Weili Peng, Xinghai Liu Front. Chem. Sci. Eng.. 2017, 11 (3): 363-368. https://doi.org/10.1007/s11705-016-1595-x Abstract HTML PDF (137KB) Pyrazole carboxamide derivatives represent an important class of fungicides in agrochemicals. To find more novel structural pyrazole carboxamides, a novel series of 3-(trifluoromethyl)-1H-pyrazole-4-carboxamide compounds were prepared from ethyl 4,4,4-trifluoroacetoacetate and triethyl orthoformate as starting materials. All the products were characterized by Fourier transform infrared spectroscopy, 1H nuclear magnetic resonance (NMR), 13C NMR, 19F NMR and mass spectrography. The bioassay results showed these fluorine-containing pyrazole carboxamides have a weak fungicidal activity but some of them exhibit a good nematocidal activity against M. incognita. Select Alkali-thermal gasification and hydrogen generation potential of biomass Alexander B. Koven, Shitang S. Tong, Ramin R. Farnood, Charles Q. Jia Front. Chem. Sci. Eng.. 2017, 11 (3): 369-378. https://doi.org/10.1007/s11705-017-1662-y Abstract HTML PDF (267KB) Generating hydrogen gas from biomass is one approach to lowering dependencies on fossil fuels for energy and chemical feedstock, as well as reducing greenhouse gas emissions. Using both equilibrium simulations and batch experiments with NaOH as a model alkaline, this study established the technical feasibility of converting various biomasses (e.g., glucose, cellulose, xylan and lignin) into H2-rich gas via catalyst-free, alkali-thermal gasification at moderate temperatures (as low as 300 °C). This process could produce more H2 with less carbon-containing gases in the product than other comparable methods. It was shown that alkali-thermal gasification follows$Cx HyOz+ 2xNaOH+(x−z)H2 O=$ $(2x+y/2−z )H2+x Na2 CO 3$, with carbonate being the solid product which is different from the one suggested in the literature. Moreover, the concept of hydrogen generation potential (H2-GP)—the maximum amount of H2 that a biomass can yield, was introduced. For a given biomass CxHyOz, the H2-GP would be moles of H2. It was demonstrated experimentally that the H2-GP was achievable by adjusting the amounts of H2O and NaOH, temperature and pressure. Select Design, synthesis, biological activity and density function theory study of pyrazole derivatives containing 1,3,4-thiadiazole moiety Xiaoming Ding, Zhiwen Zhai, Luping Lv, Zhaohui Sun, Xinghai Liu Front. Chem. Sci. Eng.. 2017, 11 (3): 379-386. https://doi.org/10.1007/s11705-017-1634-2 Abstract HTML PDF (223KB) A variety of pyrazole derivatives containing 1,3,4-thiadiazole moiety were synthesized under microwave irradiation, and their structures were confirmed by 1H NMR and HRMS. They were evaluated for herbicidal and antifungal activities, and the results indicated that two compounds with a phenyl group (6a) and 4-tert-butylphenyl group (6n) possess good herbicidal activity for dicotyledon Brassica campestris and Raphanus sativus with the inhibition of 90% for root and 80%–90% for stalk at 100 ppm respectively. The structure-activity relationship of compounds 6a and 6n was also studied by density function theory method. Select Merits and limitations of TiO2-based photocatalytic pretreatment of soils impacted by crude oil for expediting bioremediation Yu Yang, Hassan Javed, Danning Zhang, Deyi Li, Roopa Kamath, Kevin McVey, Kanwartej Sra, Pedro J.J. Alvarez Front. Chem. Sci. Eng.. 2017, 11 (3): 387-394. https://doi.org/10.1007/s11705-017-1657-8 Abstract HTML PDF (293KB) Heavy hydrocarbons (HHCs) in soils impacted by crude oil spills are generally recalcitrant to biodegradation due to their low bioavailability and complex chemical structure. In this study, soils were pretreated with varying concentrations of ultraviolet radiation A (UVA) or ultraviolet radiation C (UVC) activated titanium dioxide (TiO2) (1%–5%) under varying moisture conditions (0%–300% water holding capacity (WHC)) to enhance biodegradation of HCCs and shorten remediation timeframes. We demonstrate that pretreatment of impacted soils with UVC-activated TiO2 in soil slurries could enhance bioremediation of HHCs. ?Total petroleum hydrocarbon (TPH) removal after 24 h exposure to UVC (254 nm and 4.8 mW/cm2) was (19.1±1.6)% in slurries with 300% WHC and 5 wt-% TiO2. TPH removal was non-selective in the C15-C36 range and increased with moisture content and TiO2 concentration. In a 10-d bioremediation test, TPH removal in treated soil increased to (26.0±0.9)%, compared to (15.4±0.8)% for controls without photocatalytic pre-treatment. Enhanced biodegradation was also confirmed by respirometry. This suggests that addition of UVC-activated TiO2 to soil slurries can transform recalcitrant hydrocarbons into more bioavailable and biodegradable byproducts and increase the rate of subsequent biodegradation. However, similar results were not observed for soils pretreated with UVA activated TiO2. This suggests that activation of TiO2 by sunlight and direct addition of TiO2 to unsaturated soils within landfarming setting may not be a feasible approach. Nevertheless, less than 1% of UVA (7.5 mW/cm2) or UVC (1.4 mW/cm2) penetrated beyond 0.3 cm soil depth, indicating that limited light penetration through soil would hinder the ability of TiO2 to enhance soil bioremediation under land farming conditions. Select Multi-functional 3D N-doped TiO2 microspheres used as scattering layers for dye-sensitized solar cells Zijian Cui, Kaiyue Zhang, Guangyu Xing, Yaqing Feng, Shuxian Meng Front. Chem. Sci. Eng.. 2017, 11 (3): 395-404. https://doi.org/10.1007/s11705-017-1643-1 Abstract HTML PDF (506KB) Three-dimensional TiO2 microspheres doped with N were synthesized by a simple single-step solvothermal method and the sample treated for 15 h (hereafter called TMF) was then used as scattering layers in the photoanodes of dye-sensitized solar cells (DSSCs). The TMF was characterized using scanning electron microscopy, high resolution transmission electron microscopy, Brunauer-Emmett-Teller measurements, X-ray diffraction, and X-ray photoelectron spectroscopy. The TMF had a high surface area of 93.2 m2·g−1 which was beneficial for more dye-loading. Five photoanode films with different internal structures were fabricated by printing different numbers of TMF scattering layers on fluorine-doped tin oxide glass. UV-vis diffuse reflection spectra, incident photon-to-current efficiencies, photocurrent-voltage curves and electrochemical impedance spectroscopy were used to investigate the optical and electrochemical properties of these photoanodes in DSSCs. The presence of nitrogen in the TMF changed the TMF microstructure, which led to a higher open circuit voltage and a longer electron lifetime. In addition, the presence of the nitrogen significantly improved the light utilization and photocurrent. The highest photoelectric conversion efficiency achieved was 8.08%, which is much higher than that derived from typical P25 nanoparticles (6.52%). Select Predictive calculations of gas solubility and permeability in glassy polymeric membranes: An overview Matteo Minelli, Maria Grazia De Angelis, Giulio C. Sarti Front. Chem. Sci. Eng.. 2017, 11 (3): 405-413. https://doi.org/10.1007/s11705-017-1615-5 Abstract HTML PDF (271KB) The possibility to evaluate in a predictive way the relevant transport properties of low molecular weight species, both gases and vapors, in glassy polymeric membranes is inspected in detail, with particular attention to the methods recently developed based on solid thermodynamic basis. The solubility of pure and mixed gases, diffusivity and permeability of single gases in polymer glasses are examined, considering in particular poly(2,6-dimethyl-1,4-phenylene oxide) as a relevant test case. The procedure clearly indicates what are the relevant physical properties of the polymer matrix and of the penetrants required by the calculations, which can be obtained experimentally through independent measurements. For gas and vapor solubility, the comparison with direct experimental data for mixed gases points out also the ability to account for the significant variations that solubility-selectivity experiences upon variations of pressure and/or feed composition. For gas and vapor permeability, the comparison with direct experimental data shows the possibility to account for the various different trends observed experimentally as penetrant pressure is increased, including the so-called plasticization behavior. The procedure followed for permeability calculations leads also to clear correlations between permeability and physical properties of both polymer and penetrant, based on which pure predictive calculations are reliably made. Select A knowledge reasoning Fuzzy-Bayesian network for root cause analysis of abnormal aluminum electrolysis cell condition Weichao Yue, Xiaofang Chen, Weihua Gui, Yongfang Xie, Hongliang Zhang Front. Chem. Sci. Eng.. 2017, 11 (3): 414-428. https://doi.org/10.1007/s11705-017-1663-x Abstract HTML PDF (516KB) Root cause analysis (RCA) of abnormal aluminum electrolysis cell condition has long been a challenging industrial issue due to its inherent complexity in analyzing based on multi-source knowledge. In addition, accurate RCA of abnormal aluminum electrolysis cell condition is the precondition of improving current efficiency. RCA of abnormal condition is a complex work of multi-source knowledge fusion, which is difficult to ensure the RCA accuracy of abnormal cell condition because of dwindling and frequent flow of experienced technicians. In view of this, a method based on Fuzzy-Bayesian network to construct multi-source knowledge solidification reasoning model is proposed. The method can effectively fuse and solidify the knowledge, which is used to analyze the cause of abnormal condition by technicians providing a clear and intuitive framework to this complex task, and also achieve the result of root cause automatically. The proposed method was verified under 20 sets of abnormal cell conditions, and implements root cause analysis by finding the abnormal state of root node, which has a maximum posterior probability by Bayesian diagnosis reasoning. The accuracy of the test results is up to 95%, which shows that the knowledge reasoning feasibility for RCA of aluminum electrolysis cell. Select Performance monitoring of non-gaussian chemical processes with modes-switching using globality-locality preserving projection Xin Peng, Yang Tang, Wenli Du, Feng Qian Front. Chem. Sci. Eng.. 2017, 11 (3): 429-439. https://doi.org/10.1007/s11705-017-1675-6 Abstract HTML PDF (531KB) In this paper, we propose a novel performance monitoring and fault detection method, which is based on modified structure analysis and globality and locality preserving (MSAGL) projection, for non-Gaussian processes with multiple operation conditions. By using locality preserving projection to analyze the embedding geometrical manifold and extracting the non-Gaussian features by independent component analysis, MSAGL preserves both the global and local structures of the data simultaneously. Furthermore, the tradeoff parameter of MSAGL is tuned adaptively in order to find the projection direction optimal for revealing the hidden structural information. The validity and effectiveness of this approach are illustrated by applying the proposed technique to the Tennessee Eastman process simulation under multiple operation conditions. The results demonstrate the advantages of the proposed method over conventional eigendecomposition-based monitoring methods. Select Molecular dynamics study of water diffusion in an amphiphilic block copolymer with large difference in the blocks’ glass transition temperatures Yang Zhou, Phillip Choi Front. Chem. Sci. Eng.. 2017, 11 (3): 440-447. https://doi.org/10.1007/s11705-017-1626-2 Abstract HTML PDF (367KB) Isothermal-isobaric molecular dynamics simulation was used to study the diffusion mechanism of water in polyurethane-block-poly(N-isopropyl acrylamide) (PU-block-PNIPAm) with a hydrophobic PU/hydrophilic PNIPAm mass ratio of 1.4 to 1 at 298 K and 450 K. Here, the experimental glass transition temperature (Tg) of PU is 243 K while that of PNIPAm is 383 K. Different amounts of water up to 15 wt-% were added to PU-block-PNIPAm. We were able to reproduce the specific volumes and glass transition temperatures (250 K and 390 K) of PU-block-PNIPAm. The computed self-diffusion coefficient of water increased exponentially with increasing water concentration at both temperatures (i.e., following the free volume model of Fujita). It suggested that water diffusion in PU-block-PNIPAm depends only on its fractional free volume despite the free volume inhomogeneity. It is noted that at 298 K, PU is rubbery while PNIPAm is glassy. Regardless of temperature, radial distribution functions showed that water formed clusters with sizes in the range of 0.2–0.4 nm in PU-block-PNIPAm. At low water concentrations, more clusters were found in the PU domain but at high water concentrations, more in the PNIPAm domain. It is believed that water molecules diffuse as clusters rather than as individual molecules. REVIEW ARTICLE Select Recent advances in SERS detection of perchlorate Jumin Hao, Xiaoguang Meng Front. Chem. Sci. Eng.. 2017, 11 (3): 448-464. https://doi.org/10.1007/s11705-017-1611-9 Abstract HTML PDF (680KB) Perchlorate has recently emerged as a widespread environmental contaminant of healthy concern. Development of novel detection methods for perchlorate with the potential for field use has been an urgent need. The investigation has shown that surface-enhanced Raman scattering (SERS) technique has great potential to become a practical analysis tool for the rapid screening and routine monitoring of perchlorate in the field, particularly when coupled with portable/handheld Raman spectrometers. In this review article, we summarize progress made in SERS analysis of perchlorate in water and other media with an emphasis on the development of SERS substrates for perchlorate detection. The potential of this technique for fast screening and field testing of perchlorate-contaminated environmental samples is discussed. The challenges and possible solutions are also addressed, aiming to provide a better understanding on the development directions in the research field. Select Towards Cr(VI)-free anodization of aluminum alloys for aerospace adhesive bonding applications: A review Shoshan T. Abrahami, John M. M. de Kok, Herman Terryn, Johannes M. C. Mol Front. Chem. Sci. Eng.. 2017, 11 (3): 465-482. https://doi.org/10.1007/s11705-017-1641-3 Abstract HTML PDF (778KB) For more than six decades, chromic acid anodizing (CAA) has been the central process in the surface pre-treatment of aluminum for adhesively bonded aircraft structures. Unfortunately, this electrolyte contains hexavalent chromium (Cr(VI)), a compound known for its toxicity and carcinogenic properties. To comply with the new strict international regulations, the Cr(VI)-era will soon have to come to an end. Anodizing aluminum in acid electrolytes produces a self-ordered porous oxide layer. Although different acids can be used to create this type of structure, the excellent adhesion and corrosion resistance that is currently achieved by the complete Cr(VI)-based process is not easily matched. This paper provides a critical overview and appraisal of proposed alternatives to CAA, including combinations of multiple anodizing steps, pre- and post anodizing treatments. The work is presented in terms of the modifications to the oxide properties, such as morphological features (e.g., pore size, barrier layer thickness) and surface chemistry, in order to evaluate the link between fundamental principles of adhesion and bond performance. Select Materials sustainability for environment: Red-mud treatment Brajendra Mishra, Sumedh Gostu Front. Chem. Sci. Eng.. 2017, 11 (3): 483-496. https://doi.org/10.1007/s11705-017-1653-z Abstract HTML PDF (409KB) Bayer’s process revolutionized the extraction of aluminum from the bauxite ores. However, the hydrothermal extraction of alumina is associated with the generation of a byproduct, red-mud consisting of undissolved solids composed of iron oxides, sodium alumino silicates, titania, silica and rare earth elements. The accumulation of red-mud (or bauxite residue) in the world is 30 billion metric tons produced at a rate of 125 million tons per annum (2013). Utilization of red-mud for constructional purposes, wastewater treatment, metallurgical products, and pigments are listed. Metallurgical processing efforts of red-mud to generate various value added products such as pig iron, direct reduced iron slag wool, magnetite, titania, iron carbides are presented in the article. Select Standard method design considerations for semi-quantification of total naphthenic acids in oil sands process affected water by mass spectrometry: A review Kevin A. Kovalchik, Matthew S. MacLennan, Kerry M. Peru, John V. Headley, David D.Y. Chen Front. Chem. Sci. Eng.. 2017, 11 (3): 497-507. https://doi.org/10.1007/s11705-017-1652-0 Abstract HTML PDF (393KB) Naphthenic acids are a complex class of thousands of naturally occurring aliphatic and alicyclic carboxylic acids found in oil sands bitumen and in the wastewater generated from bitumen processing. Dozens of analytical methods have been developed for the semi-quantification of total naphthenic acids in water samples. However, different methods can give different results, prompting investigation into the comparability of the many methods. A review of important methodological features for analyzing total naphthenic acids is presented and informs the design of future standard methods for the semi-quantification of total naphthenic acids using mass spectrometry. The design considerations presented are a synthesis of discussions from an Environment and Climate Change Canada (ECCC) led taskforce of 10 laboratory experts from government, industry and academia during April 2016 and subsequent discussions between University of British Columbia and ECCC representatives. Matters considered are: extraction method, solvent, pH, and temperature; analysis instrumentation and resolution; choice of calibration standards; use of surrogate and internal standards; and use of online or offline separation prior to analysis. The design considerations are amenable to both time-of-flight and Orbitrap mass spectrometers. 22 articles	2020-06-05 04:30:18	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 3, "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": 0.5252733826637268, "perplexity": 10433.907198632485}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590348492427.71/warc/CC-MAIN-20200605014501-20200605044501-00385.warc.gz"}
http://cs.stackexchange.com/questions/4618/what-piece-am-i-missing-to-turn-this-idea-into-a-programming-language	# What piece am I missing to turn this idea into a programming language? I've been doing some reading (I'll name drop along the way) and have selected a few scattered ideas that I think could be cobbled together into a nifty esoteric programming language. But I'm having some difficulty assembling the parts. Kleene's Theorem states: Any Regular Set can be recognized by some Finite-State Machine (Minsky 4.3). Minsky's Theorem 3.5: Every Finite-State machine is equivalent to, and can be "simulated by", some neural net. "There is a natural way to represent any forest as a binary tree." (Knuth, v1, 333). And according to Bentley (Programming Pearls, p.126) a binary tree can be encoded as a flat array. So I'm imagining an array of bit-fields (say 4 bits so it can easily be worked with in hexadecimal). Each field indicates a type of automaton, and the positions of the array encode (via an intermediary binary tree representation) a forest which approximates (? missing piece ?) the power of a graph. I'm somewhat bewildered by the possibilities of automaton sets to try, and of course the fun Universal Automata require three inputs (I worked up an algorithm inspired by Bentley to encode a ternary tree implicitly in a flat array, but it feels like the wrong direction). So I'd appreciate any side-bar guidance on that. Current best idea: the normal set: and or xor not nand nor, with remaining bits used for threshold weights on the inputs. So the big piece I'm missing is a formalism for applying one of these nibble-strings to a datum. Any ideas or related research I should look into? Edit: My theoretical support suggests that the type of computations will probably be limited to RL acceptors (and maybe generators, but I haven't thought that through). So, I tried to find an example to flesh this out. The C int isdigit(int c) function performs a logical computation on (in effect) a bit-string. Assuming ASCII, where the valid digits are 0x30 0x31 0x32 0x33 0x34 0x35 0x36 0x37 0x38 0x39, so bit 7 must be off, bit 6 must be off, bit 5 must be on, and bit 4 must be on: these giving us the 0x30 prefix; then bit 3 must be off (0-7) or if bit 3 is on, bit 2 must be off and bit 1 must be off (suppressing A-F), and don't care about bit 0 (allowing 8 and 9). If you represent the input c as a bit-array (c[0]..c[7]), this becomes ~c[7] & (~c[6] & (c[5] & (c[4] & (~c[3] \| (~c[2] & ~c[1]))))) Arranging the operators into a tree (colon (:) represents a wire since pipe (\|) is logical or), c[7] 6 5 4 3 2 1 0 ~ ~ : : ~ ~ ~ : & : : : & & : \| & : & My thought based on this is to insert "input lead" tokens into the tree which receive the values of the input bit assigned in a left-to-right manner. And I also need a ground or sink to explicitly ignore certain inputs (like c[0] above). This leads me to make NOT (~) a binary operator which negates the left input and simply absorbs right input. And in the course of trying this, I also realized the necessity for a ZERO token to build masks (and to provide dummy input for NOTs). So the new set is: &(and) \|(or) ^(xor) ~(not x, sink y) 0(zero) I(input) So the tree becomes (flipping up for down) ^ & & & \| I 0 & I ~ & & I I 0 ~ ~ ~ ~ I 0 I 0 I 0 I 0 = = = = = = = = 7 6 5 4 3 2 1 0 Which encodes into the array (skipping the "forest<=>tree" part, "_" represents a blank) _ ^ & & & \| I 0 & I ~ & _ _ _ _ & I _ _ I 0 ~ ~ _ _ _ _ _ _ _ _ ~ ~ _ _ _ _ _ _ _ _ _ _ I 0 I 0 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ I 0 I 0 The tree->array encoding always put the root in array(1) so with zero-indexed array, there's a convenient blank at the beginning that could be used for linkage, I think. With only 6 operators, I suppose it could be encoded in octal. By packing a forest of trees, we could represent a chain of acceptors each applied on the next input depending on the result of the previous. - I think the problem is that you are focussing on low-level operations and how to compress them into a small number of bits. This is not what a programming language is about. A programming language offers abstractions to help write programs. Think about what those are. – Dave Clarke Sep 20 '12 at 17:47 And syntax. You are missing syntax. – Dave Clarke Sep 20 '12 at 18:00 (must admit am slightly mystified why this question has significant votes.) maybe start with requirements! you seem to be devising a variant of circuits and gates. circuits use DAGs (directed acyclic graphs) but you seem more interested in trees, which in boolean logic are modelled with what are known as formulas. so you might want to study the classes of circuits that are already known. usually new constructions will be equivalent or fall "close" to existing classes. sometimes new classes are invented for various purposes but usually only by what you might call pro cs theorists – vzn Sep 22 '12 at 18:19 Iuser (or @Dave), please clean up the question. It is really difficult to understand what is being asked here. Please use an informative title. – Kaveh Oct 9 '12 at 3:30 @Kaveh. Regrettably, I think you're right. I will not be offended if this Q is closed as not constructive. – luser droog Oct 9 '12 at 4:23 This sounds more like the makings of a computational model rather than a programming language, as such, perhaps in the same way that the quantum computation can form the basis of a programming language such as the quantum lambda calculus. • What kinds of computation are you trying to perform? • How can these computations be composed? • How can the results of these computations be stored? How can they be used in subsequent computations? • How can I represent this syntactically? • Does my chosen syntax have a clear denotational meaning? Or clear operational meaning? • Do my syntactic constructs compose nicely? Are they sufficiently orthogonal and free from arbitrary constraints? • Can I modularise computations in my language? • What forms of abstraction does my language offer? There are a number of places you could use as a starting point: • The imperative language WHILE --- a simple language with variables, assignment, sequencing, if statements, and while statements. Maybe later add procedures. • The lambda calculus. • Combinatorial logic. Programs consist of combinators which are plugged together. No variables. • Something like Prolog or Datalog. Or you could be completely wild and ignore these and see how far you get (which could result in something interesting). - +1 Lots to work with, thank you. I'm gonna keep this open to try to solicit more, but you're definitely the front-runner. :) – luser droog Sep 20 '12 at 15:53 Ahh, the ever persistent question of how to design a language. For years I never had a good solid answer to that until I fond Cognitive dimensions of notations. The author list out various dimensions, think orthogonal properties, which should be considered when designing a language. I wish I had found this decades ago. Enjoy. - That looks very useful. Thankyou. – luser droog Sep 22 '12 at 4:05 It seems similar to the Kantian categories. – luser droog Jan 21 '13 at 6:26 From the point of view of filling-out the machine model, the three criteria of Turing Completeness (Böhm-Jacopini theorem) appear useful. • Sequence • Selection • Iteration or Recursion It's clearly #3 that's missing at present, the fanciful "linkage" mentioned above. Edit: This doesn't really help dig me out of the hole with this question, but ... I was reading an old book on digital computer fundamentals and found this very nice representation of all possible functions of two binary inputs: x y F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15 F16 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 C = 0 1 2 3 4 5 6 7 8 9 A B C D E F (function code) So take the function F n, store n as C = n - 1, and the Universal Binary Function is: C x,y = ( C >> ( 2 * x + y ) & 1 ). And it fits in one hex digit! Edit: There's another piece missing from the above: representing a neural-net as a forest. Combining the above Universal Binary Function table with the isdigit(c) example from the question gives [0] used for "size" 8 [1] tree root 8 8 = X & Y (AND) [2] [3] 7 = ~(X & Y) (NAND) 8 7 1 = ~(X \| Y) (NOR) [4] [5] [6] [7] 1 8 7 C C = X (ignore Y) [8] [9] [10] [11] [12] [13] [14] [15] c7 c6 c5 c4 c3 c2 c1 c0 Where each function is applied to its two children, and the 8 input values propagate upward toward the root, collapsing into the yielded value. So, the (1 bit) isdigit(8bit c) function encodes into this array (with the argument row omitted). 8:8 8 7 1 8 7 C [. . . . . . . .] For building data-types, I'm inspired by this brief description of the types in Konrad Zuse's Plankalkül, Binary n-bit numbers in the Plankalkül were represented by type S1.n. Another special type was used for floating-binary numbers, namely, SΔ1 = (S1.3, S1.7, S1.22). The first three-bit component here was for signs and special markers‐‐ indicating, for example, whether the number was real or imaginary or zero; the second was for a seven-bit exponent in two's complement notation; and the final 22 bits represented the 23-bit fraction part of a normalized number, with the redundant leading-'1' bit suppressed. -- Knuth, "The Early Development of Programming Languages" in Selected Papers on Computer Languages, p. 10. - Although the assumption is reasonable, I don't see the OP mention the wish to build a Turing-complete language. – Raphael Oct 7 '12 at 12:59 @Raphael: This answer is by the OP. – Dave Clarke Oct 7 '12 at 14:41 @DaveClarke Haha, my bad. The earlier comment should then read: not every programming language has to be Turing complete, so #3 can be more subtle. – Raphael Oct 7 '12 at 15:40 I posed a Kolmogorov-complexity challenge on codegolf.SE to generate the Universal Binary Function table, and as expected, APL and J are well-suited to this area. – luser droog Jul 26 '13 at 7:03	2015-03-30 17:28:59	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.5458419919013977, "perplexity": 1293.37846530632}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1427131299515.96/warc/CC-MAIN-20150323172139-00194-ip-10-168-14-71.ec2.internal.warc.gz"}
https://www.statistics-lab.com/%E7%BB%9F%E8%AE%A1%E4%BB%A3%E5%86%99%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84%E4%BD%9C%E4%B8%9A%E4%BB%A3%E5%86%99data-structure%E4%BB%A3%E8%80%83linked-lists/	statistics-lab™ 为您的留学生涯保驾护航在代写数据结构data structure方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数据结构data structure方面经验极为丰富，各种代写数据结构data structure相关的作业也就用不着说。 • Statistical Inference 统计推断 • Statistical Computing 统计计算 • (Generalized) Linear Models 广义线性模型 • Statistical Machine Learning 统计机器学习 • Longitudinal Data Analysis 纵向数据分析 • Foundations of Data Science 数据科学基础 By the time we traverse the complete list (for creating the hash table), we can find the list length. I ct us say the list length is $M$. To find $n^{\text {th }}$ from the end of linked list, we can convent this to $(M-n+1)^{\text {th }}$ from the beginuing. Sance we already know the length of the list, it is just a matter of retuming $(M-n+1)^{\text {th }}$ key value from the hash table. ‘ Iine Complexity: I inve lor creating the hash table, $T(m)=()(m)$. Space Cionaplexity: Since we need to create a hash table of size $m$, O( $m)$. Problem-4 Can we use Problem-3 approach for solving Problem-2 without creating the hash table? Solution Yeg. If we observe the Problenh3 solution, what we are actually doing is finding the size of the linked list. That means we are using the hash table to find the size of the linked list. We can find the length of the linked list just by starting at the head node and traversing the list. So, we can find the length of the list without creating the hash table. After finding the length, compute $M-n+1$ and with one more scan we can get the $(M-n+1)^{\text {th }}$ node from the beginning. This solution needs two scans: one for finding the length of the list and the other for finding $(M-n+1)^{t h}$ node from the hegianing- Time Complexity: Time for finding the length + Time for finding the $(M-n+1)^{\text {th }}$ node from the beginning. Therefore, $T(n=O(n)+$ $O(n) \approx O(n)$. Space Complexity: $O(1)$. Hence, no need to create the hash table. Problem5 Can we solve Problen-2 in one scan? Solution: Yes. Efficient Approach: Use two pointers $p N$ thNode and $p$ Temp. Initially, both point to head node of the list. $p N$ thNode starts moving only alter $p$ Temp has made $n$ mones. From there both mowe forward until $p$ Temp reaches the end of the list. As a result, $p N$ thNode points to $n^{\text {th }}$ node from the end of the linked list. Notet At any poant of time both move one node at a time. ## 统计代写\|数据结构作业代写data structure代考\|Efficient Approach Solution: Yes. Fincient Approed QMemoryleas Appronch): The space conplexity can be reduced to O(1) by considering two pointers at differeat speed – a slow pointer and a fast pointer. The slow poanter moves one step at a time while the fast pointer mowes two steps at a tine. This problem was solved by Floyd. The solution is named the Floyd cycle finding algorithm. It uses two pointers moving at different speeds to walk the linked list. If there is no cycle in the list, the fast poanter will eventually reach the ead and we can return false in this case. Now consider a cyclic list and imagiae the slow and fast pointers are two rumers racing around a circle track. Once they enter the loop they are expected to neet, which denotes that there is a loop. This works becanse the only way a faster movang pointer would point to the sanae location as a slower moving pointer is if sonachow the entire list or a part of it is circular. Think of a tortoise and a hare rumaing on a track. The faster numang hare will catch up with the tortoise if they are ruming in a loop. As an exanple, consider the followanng exmple and trace out the Floyd algonthm. From the diagrans below we can see that after the final step they are meeting at sone point in the loop which nay not be the starting point of the loop. Note: slowPtr (tortoise) moves one pointer at a tine and fastPtr (hare) mones two pointers at a tine. ## 统计代写\|数据结构作业代写data structure代考\|Algorithm Create two stacks one for the first list and one for the second list. Traverse the first list and push all the node addresses onto the first stack. Traverse the second list and push all the node addresses onto the second stack. Now both stacks contain the node address of the corresponding lists. Now compare the top node address of both stacks. If they are the same, take the top elements from both the slacks and keep them in some temporary variable (since both node addresses are node, it is enough if we use one temporary varable). Continue this process until the top node addresses of the stacks are not the same. This point is the one where the lists merge into a single list. Return the value of the tenporary variable. ## 数据结构代写 ‘ 线复杂性：我想创建哈希表，吨(米)=()(米). Space Cionaplexity：因为我们需要创建一个大小为米，这（米). ## 广义线性模型代考 statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。 ## MATLAB代写 MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。	2023-04-01 23:19:13	{"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": 0.3222501873970032, "perplexity": 1944.8326016069354}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1679296950363.89/warc/CC-MAIN-20230401221921-20230402011921-00528.warc.gz"}
https://gamedev.stackexchange.com/questions/20791/how-do-i-change-my-gravity-and-jumping-logic-to-put-the-character-at-a-specific?noredirect=1	# How do I change my gravity and jumping logic to put the character at a specific height at a specific point in time? I'm making a 2D platformer on Android using libGDX. One thing I've had a problem with recently is gravity and jumping. I found a few tutorials on the internet and was able to come up with this... public static final int JUMP_HEIGHT = 64; public static final float JUMP_GRAVITY= -6.0f; public static final float MAX_GRAVITY = 20.0f; public static final float GRAVITY= 19.0f; private void updateGravity(float delta) { if(Gdx.input.isTouched() && !jumping) { jumping = true; gravity = Config.JUMP_GRAVITY; } if(jumping) { // change frame to jump sprite.setRegion(walkFrames[0]); sprite.translate(0, (-1gravity)); // runner has landed? if(sprite.getY() <= idleY) { jumping = false; sprite.setPosition(Config.RUNNER_X, idleY); } // slowdown / fall if (gravity < Config.MAX_GRAVITY) gravity += Config.GRAVITYdelta; } } The code works and my character will jump and fall smoothly. Although the time it takes for him to jump and comedown is independent from delta the height he jumps is not. If the device lags while jumping he will jump only a portion of how high he should. For the life of me I cannot get it to make sure it jumps a specific height (JUMP_HEIGHT) and in a specific time independent of the delta. What is your gravity variable supposed to represent? Gravity (in physics) is acceleration, which is distance / time squared. You're translating your unit by some arbitrary number (gravity), and not taking into account time. So you're treating your acceleration value as if it were a distance value. A naive solution is simply to multiply your gravity by delta (so imagine your gravity is in distance/time), so it's more-or-less framerate independent, but there will be integration errors. Also if your framerate is variable, your end jump height might be different. Usually what I do in these situations is that if your translation would put you beyond your jump height, just clamp that frame's jump height to the max jump height. This would mean that your guy jumping would move at a constant speed (other than the final frame if you do clamping) instead of accelerating up or down. Now if you want your jumping to be more an acceleration rather than a velocity, you want to do some math like this article describes: http://gafferongames.com/game-physics/integration-basics/ It would involve you adding some velocity/acceleration variables to your character and doing a bit more math, but it's pretty simple. In that scenerio you would always apply gravity to the player, and your jump would be a separate acceleration impulse upwards. • +1 if only for physical unit consistency; if you don’t have it, you have a problem. – sam hocevar Dec 9 '11 at 0:44 • Thanks a bunch. I'm somewhat new to gamedev but I think I'm on the right track thanks to your help. – Lienau Dec 9 '11 at 1:41 Have a read of this too: http://www.niksula.hut.fi/~hkankaan/Homepages/gravity.html • A link alone is not an answer. This is better suited as a comment. – bummzack Dec 9 '11 at 10:06 • I'd argue that it is an answer because the linked page explains exactly what needs to be done to rectify the problem, and any further text I may add would have just repeated info from that. – Maximus Minimus Dec 9 '11 at 10:20	2020-10-24 15:49:47	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.42991799116134644, "perplexity": 1434.7158556930508}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1603107883636.39/warc/CC-MAIN-20201024135444-20201024165444-00122.warc.gz"}
https://www.physicsforums.com/threads/complex-analysis-integrating-rational-functions.311729/	# Complex Analysis: Integrating rational functions ## Homework Statement Hi all. My question has to do with integrating rational functions over the unit circle. My example is taken from here (page 2-3): http://www.maths.mq.edu.au/~wchen/lnicafolder/ica11.pdf We wish to integrate the following $$\int_0^{2\pi } {\frac{{d\theta }}{{a + \cos \theta }}}.$$ According to the .pdf, we integrate along a unit circle, and we define $z = e^{i\theta}$. When rewriting the integrating using z, we find that the integrand has to poles: One inside the unit circle and one outside. When we use the residue theorem, we only use the pole inside the unit circle. My question: How are we even allowed just to say: "We choose only to integrate along a unit circle, and thus we only look at poles inside this circle"? If I claim that we should integrate along a circle big enough to include all poles, then who can say argument against my claim? I hope you understand me. Best regards, Niles. Related Calculus and Beyond Homework Help News on Phys.org Landau The reasoning is not "we must integrate along a unit circle, therefore we substitute z=e^(it)", but "let's substitute z=e^(it), then - since t ranges from 0 to 2*pi - we are integrating along a unit circle". If you want to substitue z=Re^(it), so that you integrate along a non-unit circle, you could try that, but the question is whether that will work. The substitution z=e^(it) is easy to work with. If you want to substitue z=Re^(it), so that you integrate along a non-unit circle, you could try that, but the question is whether that will work. The substitution z=e^(it) is easy to work with. I have not yet seen a proof of that substituting z = exp(it) works as well, only that it apparently makes things easy. Landau	2020-02-26 04:37:14	{"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": 0.9643374085426331, "perplexity": 524.7896186914228}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1581875146186.62/warc/CC-MAIN-20200226023658-20200226053658-00293.warc.gz"}
http://www.paulaquinon.com/2fnsz3v/fe9116-oxidation-number-of-cu-in-cu2o	oxidation number of cu in cu2o How old was queen elizabeth 2 when she became queen? Copper(II) oxide (cupric oxide, CuO), a black powder. The effe Hope this helps~ Here, we reveal key steps for the promotion of this reaction by water when tuning the selectivity of a well-defined CeO 2 /Cu 2 O/Cu(111) catalyst from carbon monoxide and carbon dioxide to methanol under a reaction environment with methane, oxygen, and water. The remarkably enhanced photocatalytic water oxidation activity over such assembled [email protected] 2 O/WO 3 composite photocatalyst was observed. The oxidation number of metallic copper is zero. In its compounds, the most common oxidation number of Cu is +2.Less common is +1. Chemistry. Copper has a +1 or +2 charge. Highly selective oxidation of methane to methanol has long been challenging in catalysis. When it's with another element, ( as in copper(II) chloride), the oxidation number changes. In this study, [email protected] 2 O/WO 3 composite photocatalyst was constructed via coupling Cu 2 O onto the edged (200) and (020) facets of square-like WO 3 nanoplates and followed by photodeposition of Pt onto Cu 2 O. The Cu2O glucose electrode was prepared by in situ electrical oxidation in an alkaline solution, in which Cu2O nanoparticles were deposited on the electrode surface to form a thin film, followed by the growth of Cu(OH)2 nanorods or nanotubes. References; Contributors ; By the end of gen chemistry, calculating oxidation states of different metals should be pretty familiar. 1 0. Answer: Oxidation state of P is +5. In almost all cases, oxygen atoms have oxidation numbers of -2. 400 mV) together with an oxidation current that is up to six times higher, compared to the Cu/Cu 2 O non‐porous film with the same loading. Lol, what I mean is, they give you copper(II) chloride. Cu is oxidized. The thickness of the starting copper film works as a control parameter to determine the thickness (d) of the resulting Cu 2 O film. Find an answer to your question Oxidation number of cu in cucl2 AbhishekDixit67351 is waiting for your help. Explanation: Rules for Oxidation Numbers : The oxidation number of a free element is always zero. The as‐synthesized Cu/Cu 2 O foams showed superior electrocatalytic activity towards the glycerol oxidation reaction, as demonstrated by the significant negative shift in the onset potential (ca. Catalyst Characterization The Fe3O4 nanoparticles prepared with trisodium citrate and sodium acetate presented diameters of about 100 nm by a solvothermal method. The oxidation number of copper depends on its state. Number of times cited according to CrossRef: 3. Cuprous oxide is any copper oxide in which the metal is in the +1 oxidation state. Promising low-temperature activity of an oxygen-activated zeolite, Cu-ZSM-5, has recently been reported in this selective oxidation and the active site in this reaction correlates with an absorption feature at 22,700 cm−1. The oxidation number of copper decreases from $$+2$$ to $$0$$. The Cu(NO3)2 is an ionic compound with overall oxidation number “0”. Here, CO oxidation reaction on Cu 2 O(1 1 1) surface without and with biaxial strain was investigated with density functional theory. The Cu/TiO2 catalyst exhibited remarkably high activity and an overall CO oxidation could be achieved at <100 °C. Verify the change in oxidation number.....2Cu2O(s)+Cu2S(s)---6Cu (s)+SO2(s)? The oxidation behavior of copper has therefore received considerable interest for a very long time 1-3.At temperatures above 600 °C, it is believed that the oxidation is controlled by the lattice diffusion of copper ions through a Cu 2 O layer 4-6. 2Cu2O(s)+O2(g) 4CuO(s) The Change In Enthalpy Upon Reaction Of 3.93 G Cu2O(s) Is −4.01 KJ . The oxidation was carried out at approximately 125 μm oxygen pressure while the reduction was done at the same pressure using H 2 gas. The oxidatuon no of oxygen is (-2) always, when two oxygens are not present , in presence of two oxygen atoms the oxidation no of oxygen is (-1). Oxidation number (also called oxidation state) is a measure of the degree of oxidation of an atom in a substance (see: Rules for assigning oxidation numbers). PHYSICAL REVIEW B 84, 125308 (2011) Atomistic simulations of copper oxidation and Cu/Cu 2O interfaces using charge-optimized many-body potentials Bryce Devine,1 Tzu-Ray Shan( 1 ), Yu-Ting Cheng( ),1 Alan J. H. McGaughey,1, 2Minyoung Lee, Simon R. Phillpot, 1and Susan B. Sinnott ,* 1Department of Materials Science and Engineering, University of Florida, Gainesville, FL 32611-6400, USA The bottom line: the electrons in Cu2O and CuO are shared. As a mineral, it is known as tenorite.It is a product of copper mining and the precursor to many other copper-containing products and chemical compounds. The (II) means it's a +2 charge. Therefore we know that the sum of all the oxidation numbers of Cu, N, and O is equal to 0. Y2O3 nano-particles with typical size of 30 nm were successfully formed into the Cu intra-grains and around grain boundaries, and the density of the Y2O3 nano-particles increased by the Cu2O addition. We find that nearly monodisperse copper oxide nanoparticles prepared via the thermal decomposition of a Cu(I) precursor exhibit exceptional activity toward CO oxidation in CO/O2/N2 mixtures. ; When oxygen is part of a peroxide, its oxidation number is -1. . The oxidation of copper is an important issue both for academic research and industrial applications. KW - MA-HIP. Here, we adopt a two-step method involving the reduction of Cu2+ at alkal The resulting Cu@Cu2O nanomaterial was successfully applied in the oxidation of catechol with high catalytic activity. The oxidation number of Cu in CuO is II, i.e. 1 Introduction. Assign an oxidation number of -2 to oxygen (with exceptions). Cu-based materials have been regarded as suitable oxygen carrier (OC) candidates in chemical looping combustion as a result of their high reactivity. H +1 N +5 O -2 3 + Cu +1 2 O -2 → Cu +2 ( N +5 O -2 3 ) 2 + N +2 O -2 + H +1 2 O -2 2. There are a few exceptions to this rule: When oxygen is in its elemental state (O 2), its oxidation number is 0, as is the case for all elemental atoms. If the question is Cu2, then it is equivalent to Cu and the oxidation number for any element is zero. 50 (2011) 12294–12298) that Cu 2 O octahedra are more active than Cu 2 O cubes in CO oxidation, we found that Cu 2 O cubes exposing {100} surface showed significantly higher activity than Cu 2 O octahedra exposing {111} surface in the reaction. The oxidation number goes from 0 in Cu to +2 in CuSO4. Well, the oxidation number of an element is always zero. ... Cu/CuO.1,2) Plascencia et al. I'm trying to write the formula for copper {II}Chloride and don't know which oxidation number i should use for copper. A number of studies on the oxidation of copper at a wide range of temperature have been conducted using various oxidizing atmospheres including pure oxygen and water vapor. Thermal oxidation (Cu → Cu 2 O). Cu(II)O (Copper(I) oxide (cuprous oxide, Cu2O), a red powder. Cu2O addition promoted the oxidation of Y from the Cu6Y compound, forming Y2O3 nano-particles. The oxidation of the Cu thin films was achieved by thermal treatment in a thermal furnace. A black solid, it is one of the two stable oxides of copper, the other being Cu 2 O or copper(I) oxide (cuprous oxide). As the average size of the cubic Cu 2 O nanocrystals decreases from 1029 nm to 34 nm, the dominant active sites contributing to the catalytic activity switch from face sites to edge sites. Xiuping Su, Wei Chen, Yanna Han, Duanchao Wang, Juming Yao, In-situ synthesis of Cu2O on cotton fibers with antibacterial properties and reusable photocatalytic degradation of dyes, Applied Surface Science, 10.1016/j.apsusc.2020.147945, (147945), (2020). Consider the reaction below between elemental iron and copper sulfate: $\ce{Fe} + \ce{CuSO_4} \rightarrow \ce{FeSO_4} + \ce{Cu}$ In the course of the reaction, the oxidation number of $$\ce{Fe}$$ increases from zero to $$+2$$. Oxidation number is a "mathematical construct" which is NOT the ionic charge, but a number that is useful in understanding redox reactions and for predicting the formulas of compounds. Cuprous oxide (Cu2O) is an important p-type semiconductor and widely used in the electrocatalytic field. Strain engineering has been a powerful strategy to manipulate the catalytic reaction activity. Cu 2 O nanocrystals undergo in situ surface oxidation forming CuO thin films during CO oxidation. Driven by the depletion of crude oil, the direct oxidation of methane to methanol has been of considerable interest. The present study reports on the in situ oxidation of copper to Cu 2 O and subsequent reduction to metallic copper in an environmental scanning electron microscope. Results and Discussion 2.1. The oxidation number of a monatomic ion equals the charge of the ion. In this case, it's +2 because the problem said so. Copper can also have oxidation numbers of +3 and +4.. ON = +2: Examples are CuCl₂, CuO, and CuSO₄.See, for example KW - Cu alloy. KW - CuY compound. Question: The Oxidation Of Copper(I) Oxide, Cu2O(s) , To Copper(II) Oxide, CuO(s) , Is An Exothermic Process. S is reduced. The pressure of the furnace was about 1 atm while temperature was maintained at 150 °C. The barriers heights of rate-determining steps (rds) in Mars-van-Krevelen, Langmuir-Hinshelwood and Eley-Rideal mechanism are strain sensitive. A series of Me/TiO2 samples (Me = V, Cr, Mn, Fe, Co, Ni, Cu and Zn) with designed Me : TiO2 ratio of 1 : 10 were prepared by a photodeposition method and studied for the oxidation of CO. The oxidation number goes from +6 in H2SO4 to +4 in SO2. Changes in Oxidation Number in Redox Reactions. Ed. Abstract. 1. If the question is Cu2+, then the oxidation number is +2. The oxidation number of Hydrogen (H) is +1, but it is -1 in when combined with less electronegative elements. Copper(II) oxide or cupric oxide is the inorganic compound with the formula CuO. studied copper oxidation in the temperature range of 300–1000°C. However, the limited number of active sites and its poor conductivity make it difficult to further improve its catalytic activity. The bonds are polar, since oxygen is more electronegative, but the electrons are shared, none the less. 1 Answer Activity and an overall CO oxidation could be achieved at < 100 °C all the numbers. Since oxygen is more electronegative, but the electrons in Cu2O and CuO are shared the... Oxidation of catechol with high catalytic activity is, they give you copper ( )., ( as in copper ( II ) oxide ( Cu2O ) is +1 ( copper II... Methane to methanol has been of considerable interest important p-type semiconductor and widely in. Copper decreases from \ ( 0\ ) in Cu to +2 in CuSO4 is equivalent Cu! Nanocrystals undergo in situ surface oxidation forming CuO thin films was achieved by treatment. 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High catalytic activity 100 nm by a solvothermal method O nanocrystals undergo situ...	2021-04-21 08:34:44	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.6162058115005493, "perplexity": 5466.690964377475}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039526421.82/warc/CC-MAIN-20210421065303-20210421095303-00397.warc.gz"}
http://hermes.roua.org/hermes-pub/arxiv/05/04/034/article.xhtml	## Discrete L'Hospital's rule ### A.G. Ramm Mathematics Department, Kansas State University, Manhattan, KS 66506-2602, USA [email protected] $\text{}$ $\text{}$ 1 Introduction The aim of this paper is to formulate discrete analog of L'Hospital's rule and describe some of its applications. While the usual L'Hospital's rule is taught to all undergraduate students studying calculus, its discrete analog apparently was not in the literature. Since the L'Hospital's rule proved to be very useful in many applications, one may think that its discrete analog will also be useful. We start by stating the usual, well-known L'Hospital's rule, so that the reader could see the similarities in the formulation of this rule and its discrete analog. Then we formulate this discrete analog. In Section 2 we prove the discrete analog and illustrate it by some examples. After this note has been written, the author learned that in [1, p.67, there is a result of O.Stolz, which is the same as Theorem 2 below. Our proof is slightly different from the proof in [1. The application, given in our Example 2 is of interest in the theory of the dynamical systems method ([2). A version of the usual L'Hospital's rule is the following theorem. Theorem 1. Assume that: 1) the functions $F$ and $G$ are continuously differentiable on the interval $I=\left(a,a+h\right)$ , where $h>0$ and $a$ are real numbers, $f:={F}^{\prime }$ , $g:={G}^{\prime }$ , and $\begin{array}{c}limF=limG=\infty ,lim:={lim}_{x\to a,x>a},\end{array}$ (1.1) 2) $\begin{array}{c}lim\frac{f}{g}=L,g\left(x\right)\ne 0\forall x\in I.\end{array}$ (1.2) Then there exists the limit: $\begin{array}{c}lim\frac{F}{G}=L.\end{array}$ (1.3) The proof of Theorem 1 can be found in any calculus text and does not need a reference. Let us now formulate the discrete analog of the above theorem. Theorem 2. Let ${f}_{j}>0$ and ${g}_{j}>0$ be sequences of numbers, ${F}_{n}:={\sum }_{j=1}^{n}{f}_{j}$ , ${G}_{n}:={\sum }_{j=1}^{n}{g}_{j}$ . Assume that: $\begin{array}{c}{lim}_{n\to \infty }{F}_{n}={lim}_{n\to \infty }{G}_{n}=\infty ,\end{array}$ (1.4) and $\begin{array}{c}{lim}_{n\to \infty }\frac{{f}_{n}}{{g}_{n}}=L.\end{array}$ (1.5) Then $\begin{array}{c}{lim}_{n\to \infty }\frac{{F}_{n}}{{G}_{n}}=L.\end{array}$ (1.6) The similarity of Theorems 1 and 2 is obvious. Remark: One can write equation (1.5) as $\begin{array}{c}{lim}_{n\to \infty }\frac{{F}_{n}-{F}_{n-1}}{{G}_{n}-{G}_{n-1}}=L.\end{array}$ (1.7) Thus, the role of the derivative of $F$ is played by the difference ${F}_{n}-{F}_{n-1}$ . 2 Proofs Fix an arbitrary small $\varepsilon >0$ . Using assumption (1.5), find $M:=M\left(\varepsilon \right)$ , such that $\begin{array}{c}L-\varepsilon <\frac{{f}_{j}}{{g}_{j}}M.\end{array}$ (2.1) Denote ${F}_{nM}:={\sum }_{M}^{n}{f}_{j}$ , and define ${G}_{nM}$ similarly. Using assumption (1.4), find $N:=N\left(\varepsilon \right)$ , such that $\begin{array}{c}\frac{{F}_{M}}{{F}_{nM}}<\varepsilon ,\frac{{G}_{M}}{{G}_{nM}}<\varepsilon ,\forall n>N.\end{array}$ (2.2) Now one gets: $\begin{array}{c}\frac{{F}_{n}}{{G}_{n}}=\frac{{F}_{M}+{F}_{nM}}{{G}_{M}+{G}_{nM}}=\frac{{F}_{nM}}{{G}_{nM}}\frac{1+\frac{{F}_{M}}{{F}_{nM}}}{1+\frac{{G}_{M}}{{G}_{nM}}},\end{array}$ (2.3) and $\begin{array}{c}1+{\varepsilon }_{1}:=\frac{1-\varepsilon }{1+\varepsilon }<\frac{1+\frac{{F}_{M}}{{F}_{nM}}}{1+\frac{{G}_{M}}{{G}_{nM}}}<\frac{1+\varepsilon }{1-\varepsilon }:=1+{\varepsilon }_{2},\end{array}$ (2.4) where ${\varepsilon }_{1}=O\left(\varepsilon \right)$ and ${\varepsilon }_{2}=O\left(\varepsilon \right)$ , as $\varepsilon \to 0$ . Using assumption (1.5), one gets $\begin{array}{c}L-\varepsilon \le {min}_{j\ge M}\frac{{f}_{j}}{{g}_{j}}\le \frac{{F}_{nM}}{{G}_{nM}}\le {max}_{j\ge M}\frac{{f}_{j}}{{g}_{j}}\le L+\varepsilon .\end{array}$ (2.5) Since $\varepsilon >0$ is arbitrarily small, equation (1.6) follows from relations (2.2)-(2.5). Theorem 2 is proved. $\square$ Consider examples of applications of Theorem 2. Example 1. By Theorem 2, one has ${lim}_{n\to \infty }\frac{{\sum }_{j=1}^{n}\frac{{j}^{m}}{1+{j}^{m+1}}}{{\sum }_{j=1}^{n}\frac{{j}^{p}}{1+{j}^{p+1}}}={lim}_{n\to \infty }\frac{{n}^{m}/\left(1+{n}^{m+1}\right)}{{n}^{p}/\left(1+{n}^{p+1}\right)}=1.$ Example 2. In applications the following differential inequality is used (see, e.g., [2): $\begin{array}{c}{g}_{n+1}\le \left(1-{a}_{n}\right){g}_{n}+{b}_{n},n=1,2,3,.....\end{array}$ (2.6) Assume that $\begin{array}{c}0<{a}_{n}<1,{lim}_{n\to \infty }\frac{{b}_{n-1}}{{a}_{n}}=0,{\sum }_{n=1}^{\infty }{a}_{n}=\infty .\end{array}$ (2.7) Note that assumptions (2.7) imply ${lim}_{n\to \infty }{b}_{n}=0$ . Using assumptions (2.7) one can apply Theorem 2 and conclude that $\begin{array}{c}{lim}_{n\to \infty }{\sum }_{k=1}^{n-1}{{b}_{k}}^{n}{\prod }_{j=k+1}\left(1-{a}_{j}\right)=0.\end{array}$ (2.8) This result implies, that ${lim}_{n\to \infty }{g}_{n}=0$ under the assumptions (2.7), where ${g}_{n}$ is a sequence solving inequality (2.6). Let us discuss in detail the application of Theorem 2 in this example. From (2.6) by induction one gets: $\begin{array}{c}{g}_{n+1}\le {b}_{n}+{\sum }_{k=1}^{n-1}{{b}_{k}}^{n}{\prod }_{j=k+1}\left(1-{a}_{j}\right)+{{g}_{1}}^{n}{\prod }_{j=1}\left(1-{a}_{j}\right).\end{array}$ (2.9) Assumption (2.7) implies that $\begin{array}{c}{lim}_{n\to \infty }{b}_{n}=0\text{and}{lim}_{n\to \infty }{{g}_{1}}^{n}{\prod }_{j=1}\left(1-{a}_{j}\right)=0.\end{array}$ (2.10) Let us write the term ${J}_{n}:={\sum }_{k=1}^{n-1}{b}_{k}{\prod }_{j=k+1}^{n}\left(1-{a}_{j}\right)$ in the form: ${J}_{n}=\frac{{\sum }_{k=1}^{n-1}{b}_{k}{\prod }_{j=1}^{k}\left(1-{a}_{j}{\right)}^{-1}}{{\prod }_{j=1}^{n}\left(1-{a}_{j}{\right)}^{-1}}$ . We want to apply Theorem 2 in order to prove that $\begin{array}{c}{lim}_{n\to \infty }{J}_{n}=0.\end{array}$ (2.11) The denominator in ${J}_{n}$ tends to infinity. If the numerator in ${J}_{n}$ is bounded, then (2.11) follows. If this numerator tends to infinity, then one has assumption (1.4) satisfied. To check assumption (1.5) with $L=0$ , one calculates the limit: ${lim}_{n\to \infty }\frac{{b}_{n-1}{\prod }_{j=1}^{n-1}\left(1-{a}_{j}{\right)}^{-1}}{{\prod }_{j=1}^{n}\left(1-{a}_{j}{\right)}^{-1}\left[1-\left(1-{a}_{n}\right)\right]}=li{m}_{n\to \infty }\frac{{b}_{n-1}\left(1-{a}_{n}\right)}{{a}_{n}}=0.$ At the last step asssumption (2.7) was used. So, Theorem 2 yields the desired conclusion (2.11). The discussion of Example 2 is completed. $\square$ References 1. Fikhtengolts, G., Course of differential and integral calculus, vol.1, Fizmatgiz, Moscow, 1962. 2. Ramm, A. G. , Dynamical systems method for ill-posed equations with monotone operators, Comm. in Nonlinear Sci. and Numer. Simulation, 10, N2, (2005). $\text{}$ 2000 Math subject classification: 26A24, 26D15 $\text{}$ Key words: L'Hospital rule, inequalities	2021-01-23 07:00:23	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 56, "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": 0.9087297320365906, "perplexity": 1093.6844834655406}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703536556.58/warc/CC-MAIN-20210123063713-20210123093713-00465.warc.gz"}
http://34.106.105.83/wiki/Lebesgue-Stieljes_Measures	# Lebesgue-Stieljes Measures Given ${\displaystyle F:R\rightarrow R}$ nondecreasing and right contiuous, define an outer measure by ${\displaystyle \mu _{F}^{}(A)=\inf \left\{\sum _{i}\mu _{F}^{}(\left(a,b\right])\ :\ A\subset \bigcup _{i}\left(a,b\right]\right\}}$ where ${\displaystyle \mu _{F}^{}(\left(a,b\right])=F(b)-F(a)}$ and the infimum taken over all coverings of A by countably many semiopen intervals. By Carathéodory's Theorem, we know that ${\displaystyle \mu _{F}:=\left.\mu _{F}^{}\right\|_{M_{\mu _{F}^{*}}}}$ is a measure. This measure is sometimes called the Lebesgue–Stieltjes measure associated with F.[1]	2022-07-03 14:26:40	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 4, "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": 0.938316285610199, "perplexity": 348.9660304481988}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1656104244535.68/warc/CC-MAIN-20220703134535-20220703164535-00660.warc.gz"}
https://ai.stackexchange.com/questions/21388/what-are-the-conditions-of-convergence-of-temporal-difference-learning	# What are the conditions of convergence of temporal-difference learning? In reinforcement learning, temporal difference seem to update the value function in each new iteration of experience absorbed from the environment. What would be the conditions for temporal-difference learning to converge in the end? How is it guaranteed to converge? Any intuitive understanding of those conditions that lead to the convergence? There are different TD algorithms, e.g. Q-learning and SARSA, whose convergence properties have been studied separately (in many cases). In some convergence proofs, e.g. in the paper Convergence of Q-learning: A Simple Proof (by Francisco S. Melo), the required conditions for Q-learning to converge (in probability) are the Robbins-Monro conditions 1. $$\sum_{t} \alpha_t(s, a) = \infty$$ 2. $$\sum_{t} \alpha_t^2(s, a) < \infty,$$ where $$\alpha_t(s, a)$$ is the learning rate at time step $$t$$ (that can depend on the state $$s$$ and action $$a$$), and that each state is visited infinitely often. (The Robbins-Monro conditions (1 and 2) are due to Herbert Robbins and Sutton Monro, who started the field of stochastic approximation in the 1950s, with the paper A Stochastic Approximation Method. The fields of RL and stochastic approximation are related. See this answer for more details.) However, note again that the specific required conditions for TD methods to converge may vary depending on the proof and the specific TD algorithm. For example, the Robbins-Monro conditions are not assumed in Learning to Predict by the Methods of Temporal Differences by Richard S. Sutton (because this is not a proof of convergence in probability but in expectation). Moreover, note that the proofs mentioned above are only applicable to the tabular versions of Q-learning. If you use function approximation, Q-learning (and other TD algorithms) may not converge. Nevertheless, there are cases when Q-learning combined with function approximation converges. See An Analysis of Reinforcement Learning with Function Approximation by Francisco S. Melo et al. and SBEED: Convergent Reinforcement Learning with Nonlinear Function Approximation by Bo Dai et al. • Hi, thanks for the reply! Is there any requirement on the environment? What if the environment is totally random, and for each state, action, there is not even a fixed value in essence? – MJeremy May 26 at 2:55 • @MJeremy I don't know what you mean exactly by "totally random". There are some restrictions on the environment in certain proofs. For example, in the paper Convergence of Q-learning: A Simple Proof, F. Melo e.g. assumes that the reward function is deterministic. So, the assumptions probably vary from one proof to the other. It's probably a better idea that you read the specific papers that provide the proofs, if you want to know more details. – nbro Jun 2 at 15:29	2020-10-25 02:39:33	{"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": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 6, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8769786357879639, "perplexity": 607.3759361155152}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1603107885126.36/warc/CC-MAIN-20201025012538-20201025042538-00594.warc.gz"}
http://mathhelpforum.com/advanced-applied-math/109057-probabilistic-model.html	1. ## Probabilistic Model How would one create a probabilistic model for the following scenario: the probability that the temperature will change in the next 5 minute interval is calculated. 23% of the time the temperature goes up, 30% of the time the temperature goes down and 47% of the time the temperature stays the same. 2. Originally Posted by morganfor How would one create a probabilistic model for the following scenario: the probability that the temperature will change in the next 5 minute interval is calculated. 23% of the time the temperature goes up, 30% of the time the temperature goes down and 47% of the time the temperature stays the same. $t_{n+1}=t_n+r$ where r is a RV with the required distribution (as you don't say naything about the sizes of the changes all we can say is that it is +/ve with prob 0.23, -ve with probability 0.30, and 0 with probability 0.47). CB	2016-10-21 15:02:07	{"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": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7641448974609375, "perplexity": 262.9506614278709}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988718284.75/warc/CC-MAIN-20161020183838-00372-ip-10-171-6-4.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/polar-coordinates-and-unit-vectors.965419/	# I Polar coordinates and unit vectors 1. Feb 1, 2019 ### fog37 Hello, I get that both polar unit vectors, $\hat{r}$ and $\hat{\theta}$, are unit vectors whose directions varies from point to point in the plane. In polar coordinates, the location of an arbitrary point $P$ on the plane is solely given in terms of one of the unit vector, the vector $\hat{r}$. Fo example, $P=3 \hat{r}$. But how do we know where the point is? We only know it is 3 units away from the origin but don't know in which direction. Don't we need a component for the angular unit vector $\hat{\theta}$ as well? Shouldn't the position of point $$P=r \hat{r} + \theta \hat{\theta}$$ where the components are $(r, \theta)$? I understand that the two unit vectors are orthogonal to each other and their direction depends on which point we are considering in the plane... thanks! 2. Feb 1, 2019 ### Staff: Mentor No, that's not true. The radial unit vector gives only the distance from the origin -- the point could be anywhere on a circle of that radius. We don't, unless we also know the angle. 3. Feb 1, 2019 ### PeroK You do need two coordinates to define the point. But, the radial unit vector at that point is a function of $\theta$. In other words, $\theta$ implicitly determines the direction of $\hat{r}$. 4. Feb 1, 2019 ### fog37 Hi Perok, do you have a simple example to share? By just reading $P= 3r \hat{r}$, I don't see how I can identify the specific point on the circumference of radius $r=3$. I see how the two polar unit vectors go in pair and if we know where one points we automatically know the other. But how do we have enough information from just stating $P=r \hat{r}$ without the component for $\hat{\theta}$? 5. Feb 1, 2019 ### PeroK Let's take acceleration in uniform circular motion about the origin. We have, at each point on the circle: $\vec{a} = -a\hat{r}$ However, $\hat{r}$ depends on the coordinates. So, if you want to refer to a specific point, you also need to specify $\theta$. The acceleration vector in this case has zero component in the $\hat{\theta}$ direction. 6. Feb 1, 2019 ### PeroK Here's another way to look at it. First when we have: $\vec{r} = x \hat{x} + y \hat{y}$ Then $x, y$ are variables, depending on position, and the Cartesian unit vectors are constant. You must specify both variables to identify a specific vector. And, when we have: $\vec{r} = r \hat{r}$ Then both $r$ and $\hat{r}$ are variables. Again you must specify both variables to identify a specific vector. In this case however, it is the coordinate $\theta$ that species $\hat{r}$. 7. Feb 1, 2019 ### fog37 I see, thanks. That makes sense. Writing the point as $0 \hat{x} +2 \hat{y}$ involves the Cartesian coordinates $(0,2)$. In polar, the two coordinates are $(2, \pi/2)$ but simply writing $2 \hat{r}$ would not be sufficient since the dependence of $\hat{r}$ on $\theta$ is implicit and not specified... 8. Feb 1, 2019 ### fog37 So would the correct notation be $$r \hat{r}(\theta) = 2 \hat{r}(\theta = \pi/2)$$ to identify the point $P$ described above? I have never seen it written like this.... 9. Feb 13, 2019 at 4:09 PM ### Cryo Yes it would. The difficulties you are having simply go to show that in general "position vectors" can be difficult to define unless you are using suitable coordinate system. In polar coordinates, for example, you can define a vector that connects origin to some other point. But you cannot define a vector that connects the points $\theta=0$ and $\theta=\pi/2$ on a unit circle. In other coordinate systems, e.g. parabolic or bipolar, even the vector from origin to a point will no longer be definable. :-) 10. Feb 13, 2019 at 6:10 PM ### DaveE I don't think "position" (a point in space) and "vectors" (a direction and magnitude) have any inherent relationship. You can use positions in space to describe a vector, like the 2-D vector from (0,0) to (1,1), but this vector isn't "at" (0,0). In this example it is the same vector as the vector described as connecting (0,1) to (1,2). The direction and length are the same, so the vectors are the same. 11. Feb 13, 2019 at 7:13 PM ### Cryo That was the point I was trying to make :-). This logic works well on abstract level, but try to apply it to space addressed by, say, polar coordinates and spanned by polar basis vectors (which are position dependent). In fact, I would say that vectors are always "at" somewhere when it comes to space. More specifically, the only "safe" way I know how to define a vector is in the tangent space of a manifold (here Eucledian space). This definition specifically links vectors vectors to a point where the tangent space is defined	2019-02-17 22:54:42	{"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": 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": 0.9277282953262329, "perplexity": 317.0947034959383}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1550247482788.21/warc/CC-MAIN-20190217213235-20190217235235-00299.warc.gz"}
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Firstly check if the stack height of	2023-01-27 16:48:27	{"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": 0.2403835952281952, "perplexity": 7589.529173166811}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764495001.99/warc/CC-MAIN-20230127164242-20230127194242-00007.warc.gz"}
https://zbmath.org/?q=an:1011.20019	## On $$S$$-quasinormally embedded subgroups of finite groups.(English)Zbl 1011.20019 This paper considers the consequences of a finite group $$G$$ having special families of subgroups that are $$S$$-quasinormally embedded in $$G$$. Sample result: Theorem 3.1. Let $$G$$ be a finite group and let $$p$$ be the smallest prime dividing $$\|G\|$$. Then the following are equivalent: (a) $$G$$ is $$p$$-nilpotent; (b) The maximal subgroups of the Sylow $$p$$-subgroups of $$G$$ are $$S$$-quasinormally embedded in $$G$$. ### MSC: 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure 20D30 Series and lattices of subgroups 20D40 Products of subgroups of abstract finite groups Full Text: ### References: [1] Asaad, M., On maximal subgroups of Sylow subgroups of finite groups, Comm. in algebra, 26, 3647-3652, (1998) · Zbl 0915.20008 [2] Ballester-Bolinches, A.; Pedraza-Aguilera, M.C., Sufficient conditions for supersolvability of finite groups, J. pure appl. algebra, 127, 113-118, (1998) · Zbl 0928.20020 [3] Deskins, W.E., On quasinormal subgroups of finite groups, Math. Z., 82, 125-132, (1963) · Zbl 0114.02004 [4] Doerk, K.; Hawkes, T., Finite solvable groups, (1992), Walter de Gruyter Berlin [5] Gorenstein, D., Finite groups, (1968), Harper & Row New York · Zbl 0185.05701 [6] Huppert, B., Endliche gruppen I, (1979), Springer Berlin · Zbl 0412.20002 [7] Kegel, O.H., Sylow-gruppen und subnormalteiler endlicher gruppen, Math. Z., 78, 205-221, (1962) · Zbl 0102.26802 [8] Srinivasan, S., Two sufficient conditions for supersolvability of finite groups, Israel J. math, 35, 210-214, (1980) · Zbl 0437.20012 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.	2023-02-02 20:54:28	{"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": 0.7063543796539307, "perplexity": 2688.688206003833}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1674764500041.18/warc/CC-MAIN-20230202200542-20230202230542-00566.warc.gz"}
http://mathhelpforum.com/differential-geometry/78653-normed-space-print.html	# Normed space • March 14th 2009, 10:01 AM math8 Normed space Let X be a normed space, F be a closed linear subspace of X, Let z be in X, z is not in F. Let S={x+az:a is in the field Phi}=Span of F and z We show S is closed. I would define a function f : S--> Phi by f(x+az)=a and show that \|f\|< or equal to 1/d where d= distance(z,F) hence f is in S* (dual of S). So we let {x_n +a_n z} be a sequence converging to w (x_n is in F,a_n is in Phi). We show w is in S. We note {f(x_n +a_n z)} is a cauchy sequence. But I am not sure how to proceed. • March 14th 2009, 12:58 PM Opalg Quote: Originally Posted by math8 Let X be a normed space, F be a closed linear subspace of X, Let z be in X, z is not in F. Let S={x+az:a is in the field Phi}=Span of F and z We show S is closed. I would define a function f : S--> Phi by f(x+az)=a and show that \|f\|< or equal to 1/d where d= distance(z,F) hence f is in S* (dual of S). So we let {x_n +a_n z} be a sequence converging to w (x_n is in F,a_n is in Phi). We show w is in S. We note {f(x_n +a_n z)} is a cauchy sequence. But I am not sure how to proceed. That's a very good start. In fact, $(f(x_n +a_n z))$ is a Cauchy sequence of scalars, and since the scalar field (presumably $\mathbb{R}$ or $\mathbb{C}$) is complete, that sequence must converge to a scalar a. In other words, $a_n\to a$. Then $x_n = w-a_nz\to w-az = y$ say, as n→∞. But F is closed, and therefore y∈F. Thus $w = y + az \in S$. • March 14th 2009, 02:05 PM math8 Thanks, this makes a lot of sense. But how do I show that \|f\|< or equal to 1/d where d= distance(z,F) I know it involves some corollary to the Hahn-Banach theorem: " If Y is a linear subspace of the normed space X, x is in X and d=dist(x,Y)>0, then there exists x* in X* such that x(x)=1 and \|x\|=1/d and x\|Y=0 (x restricted to Y is 0)". I think maybe the reason we want this to be true is that in that case, f would be bounded hence continuous, thus f is in the dual S. But how does this help? • March 14th 2009, 02:25 PM Opalg Quote: Originally Posted by math8 Thanks, this makes a lot of sense. But how do I show that \|f\|< or equal to 1/d where d= distance(z,F) I know it involves some corollary to the Hahn-Banach theorem: " If Y is a linear subspace of the normed space X, x is in X and d=dist(x,Y)>0, then there exists x in X* such that x(x)=1 and \|x\|=1/d and x\|Y=0 (x restricted to Y is 0)". I think maybe the reason we want this to be true is that in that case, f would be bounded hence continuous, thus f is in the dual S. But how does this help? Yes, that is exactly what you need. In fact, if you apply the H–B theorem as you have stated it, taking Y to be the subspace F and x to be the element z, then the functional x satisfies x\|F=0 and x(z)=1. So for each element v in F, x(v+az) = a. In other words, x is the functional f that you want (and you need it to be continuous because you want it to take a Cauchy sequence to a Cauchy sequence). • March 14th 2009, 03:52 PM math8 Thanks a lot that is very helpful :D	2016-05-31 18:50:40	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.93147873878479, "perplexity": 1305.9484529668587}, "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-2016-22/segments/1464052868853.18/warc/CC-MAIN-20160524012108-00133-ip-10-185-217-139.ec2.internal.warc.gz"}
http://compgroups.net/comp.databases.oracle.server/-correct-term-for-a-1-1-relation/1518111	f #### "Correct" term for a 1:1 relationship between a "database" and an "instance" where > 1 such things are on the same physical server? What is the "correct" term for a 1:1 relationship between a "database" and an "instance" where there are at least two such "things" on the same physical server? Nearly all the Oracle docs and books define a database something like the following: DATABASE - a collection of datafiles and oracle config files; useless without an "instance" to access the database. INSTANCE - a collection of background procs and memory structures, used to access a "database." Where I work, people typically call each "thing" mentioned in my Subject: line an "instance" or a "database instance." What is the "correct" term for having more than one of these "things" on the same, unpartitioned physical server? This may be splitting hairs; but when people don't or can't come to terms, confusion ensues--and that's typically a Bad Thing. I noticed, or rather a colleague of mine did, that even Tom Kyte mixes terms. In the book Expert Oracle Database Architecture (2005 edition) on page 50, in the bullet-pointed definitional paragraph for Instance, he throws the word "database instance" into the mix. A "no no" according to my colleague (don't shoot the messenger, please); because the vernacular seems to be "instance" or "database instance" for the each "thing." I like to be precise with terms, but then there are business stakeholders who use technically imprecise terms; even DBAs seem to make this mistake. Would it be less confusing to all concerned simply refer to an "Oracle SID" to indicate a 1:1 pairing of an "instance" and a "database" where there are 2 or more of these "things" on a single, unpartitioned server? Help! I want to "get it right" without being pedantic and trying to educate every last stakeholder in a large organization. Maybe education is the "right" thing to do, but when even DBAs use the term "instance" incorrectly... well, that might not end well. :-) Dana 0 7/22/2009 12:24:48 PM comp.databases.oracle.server 22978 articles. 1 followers. 12 Replies 1252 Views Similar Articles [PageSpeed] 44 On Jul 22, 8:24=A0am, dana <[email protected]> wrote: > What is the "correct" term for a 1:1 relationship between a "database" > and an "instance" where there are at least two such "things" on the > same physical server? > > Nearly all the Oracle docs and books define a database something like > the following: > > DATABASE - a collection of datafiles and oracle config files; useless > without an "instance" to access the database. > > INSTANCE - a collection of background procs and memory structures, > used to access a "database." > > Where I work, people typically call each "thing" mentioned in my > Subject: line an "instance" or a "database instance." What is the > "correct" term for having more than one of these "things" on the same, > unpartitioned physical server? > > This may be splitting hairs; but when people don't or can't come to > terms, confusion ensues--and that's typically a Bad Thing. > > I noticed, or rather a colleague of mine did, that even Tom Kyte mixes > terms. In the book Expert Oracle Database Architecture (2005 edition) > on page 50, in the bullet-pointed definitional paragraph for Instance, > he throws the word "database instance" into the mix. A "no no" > according to my colleague (don't shoot the messenger, please); because > the vernacular seems to be "instance" or "database instance" for the > each "thing." > > I like to be precise with terms, but then there are business > stakeholders who use technically imprecise terms; even DBAs seem to > make this mistake. > > Would it be less confusing to all concerned simply refer to an "Oracle > SID" to indicate a 1:1 pairing of an "instance" and a "database" where > there are 2 or more of these "things" on a single, unpartitioned > server? > > Help! I want to "get it right" without being pedantic and trying to > educate every last stakeholder in a large organization. Maybe > education is the "right" thing to do, but when even DBAs use the term > "instance" incorrectly... well, that might not end well. :-) > > Dana In most practical uses you can use the term database to refer to both the physical database and the instance. The term instance is short for database instance which is just a way of specifying the instance for a specific database when you need to make a distinction From 11g Glossary database: Collection of data that is treated as a unit. The purpose of a database is to store and retrieve related information. Each Oracle database instance accesses only one database. instance: A system global area (SGA) and the Oracle Database background processes constitute an Oracle database instance. Every time a database is started, a system global area is allocated and Oracle Database background processes are started. The SGA is deallocated when the instance shuts down. Notice the use of the term Oracle database instance appears in both definitions. HTH -- Mark D Powell -- 0 Mark.Powell (1630) 7/22/2009 1:17:58 PM Thanks Mark. That was helpful. You wrote: > In most practical uses you can use the term database to refer to both the physical database and the instance. .... > database: Collection of data that is treated as a unit. The purpose of > a database is to store and retrieve related information. Each Oracle > database instance accesses only one database. So when people loosely use the term "instance" or "database instance" they're not completely off base, are they? If each instance can ever only access one, and only one, database, then it shouldn't be "considered harmful" to speak this way? Or is the definition implying that a particular instance can only access one and only one database at any one time? I think this was what Tom Kyte was saying in the rest of that book section. I need to re-read it, and will. Here's another question for you: 1) Would there ever be any practical reason for two instances (procs + SGA) to access the same database (collection of data treated as a unit) on the same, unpartitioned physical server? 2) Tangentially related, while I'm at it--is the chief and most obvious benefit of RAC High Availability? Are claims that RAC can be used for scalability mostly hype? Last I checked, there was controversy over this one--guess I need to re-read the "Why You Don't Need RAC" article; never used RAC and I'm not sure I can even articulate precisely what it does let alone how it does it. How does RAC relate to the concepts of "instance" and "database"? > instance: A system global area (SGA) and the Oracle Database > background processes constitute an Oracle database instance. Every > time a database is started, a system global area is allocated and > Oracle Database background processes are started. The SGA is > deallocated when the instance shuts down. > Notice the use of the term Oracle database instance appears in both definitions. Dana Dana 0 7/22/2009 1:37:31 PM On Jul 22, 9:17=A0am, Mark D Powell <[email protected]> wrote: > On Jul 22, 8:24=A0am, dana <[email protected]> wrote: > > > > > > > What is the "correct" term for a 1:1 relationship between a "database" > > and an "instance" where there are at least two such "things" on the > > same physical server? > > > Nearly all the Oracle docs and books define a database something like > > the following: > > > DATABASE - a collection of datafiles and oracle config files; useless > > without an "instance" to access the database. > > > INSTANCE - a collection of background procs and memory structures, > > used to access a "database." > > > Where I work, people typically call each "thing" mentioned in my > > Subject: line an "instance" or a "database instance." What is the > > "correct" term for having more than one of these "things" on the same, > > unpartitioned physical server? > > > This may be splitting hairs; but when people don't or can't come to > > terms, confusion ensues--and that's typically a Bad Thing. > > > I noticed, or rather a colleague of mine did, that even Tom Kyte mixes > > terms. In the book Expert Oracle Database Architecture (2005 edition) > > on page 50, in the bullet-pointed definitional paragraph for Instance, > > he throws the word "database instance" into the mix. A "no no" > > according to my colleague (don't shoot the messenger, please); because > > the vernacular seems to be "instance" or "database instance" for the > > each "thing." > > > I like to be precise with terms, but then there are business > > stakeholders who use technically imprecise terms; even DBAs seem to > > make this mistake. > > > Would it be less confusing to all concerned simply refer to an "Oracle > > SID" to indicate a 1:1 pairing of an "instance" and a "database" where > > there are 2 or more of these "things" on a single, unpartitioned > > server? > > > Help! I want to "get it right" without being pedantic and trying to > > educate every last stakeholder in a large organization. Maybe > > education is the "right" thing to do, but when even DBAs use the term > > "instance" incorrectly... well, that might not end well. :-) > > > Dana > > In most practical uses you can use the term database to refer to both > the physical database and the instance. =A0The term instance is short > for database instance which is just a way of specifying the instance > for a specific database when you need to make a distinction > > From 11g Glossary > > database: Collection of data that is treated as a unit. The purpose of > a database is to store and retrieve related information. Each Oracle > database instance accesses only one database. > > instance: A system global area (SGA) and the Oracle Database > background processes constitute an Oracle database instance. Every > time a database is started, a system global area is allocated and > Oracle Database background processes are started. The SGA is > deallocated when the instance shuts down. > > Notice the use of the term Oracle database instance appears in both > definitions. > > HTH -- Mark D Powell --- Hide quoted text - > > - Show quoted text - PS - if you have multiple Oracle database instances running on one server then there is no specific term for this environment other than using, "multiple Oracle database instances" or "multiple Oracle databases" to refer to this setup. You may also have multiple Oracle software installations and multiple Oracle Homes. HTH -- Mark D Powell -- 0 Mark.Powell (1630) 7/22/2009 1:41:18 PM On Jul 22, 8:37=A0am, dana <[email protected]> wrote: > Here's another question for you: > > 1) Would there ever be any practical reason for two instances (procs + > SGA) to access the same database (collection of data treated as a > unit) on the same, unpartitioned physical server? > > > Dana > > Dana That, I believe, is the basic definition of RAC -- two or more instances accessing a single database. David Fitzjarrell 0 oratune (523) 7/22/2009 2:29:17 PM On Jul 22, 6:37=A0am, dana <[email protected]> wrote: > Thanks Mark. That was helpful. You wrote: > > > In most practical uses you can use the term database to refer to both t= he physical database and the instance. > > ... > > > database: Collection of data that is treated as a unit. The purpose of > > a database is to store and retrieve related information. Each Oracle > > database instance accesses only one database. > > So when people loosely use the term "instance" or "database instance" > they're not completely off base, are they? If each instance can ever > only access one, and only one, database, then it shouldn't be > "considered harmful" to speak this way? Or is the definition implying > that a particular instance can only access one and only one database > at any one time? I think this was what Tom Kyte was saying in the rest > of that book section. I need to re-read it, and will. > > Here's another question for you: > > 1) Would there ever be any practical reason for two instances (procs + > SGA) to access the same database (collection of data treated as a > unit) on the same, unpartitioned physical server? > > 2) Tangentially related, while I'm at it--is the chief and most > obvious benefit of RAC High Availability? Are claims that RAC can be > used for scalability mostly hype? Last I checked, there was > controversy over this one--guess I need to re-read the "Why You Don't > Need RAC" article; never used RAC and I'm not sure I can even > articulate precisely what it does let alone how it does it. How does > RAC relate to the concepts of "instance" and "database"? > > > instance: A system global area (SGA) and the Oracle Database > > background processes constitute an Oracle database instance. Every > > time a database is started, a system global area is allocated and > > Oracle Database background processes are started. The SGA is > > deallocated when the instance shuts down. > > Notice the use of the term Oracle database instance appears in both def= initions. > > Dana > There are also other Oracle things called instances that are not what we think of as database instances. For example, an ASM instance is a bunch of memory structures for controlling storage. OC4J has instances. The Grid API has instances. I don't even know what Beehive is, but it has instances. So all those books are wrong, outside a narrow database-centric view. Since we are all running this open-source stuff along with the database now, we, most especially DBA's, need precision in communication. Of course, if everyone says something, the language changes, regardless of whether it is stupid or counterproductive. The change itself can be interesting. jg -- @home.com is bogus. The I-Pod of soda machines will dumbfound the person in front of you. http://www3.signonsandiego.com/stories/2009/jul/22/futuristic-100-flavor-co= ke-dispenser-tested/?uniontrib 0 joel-garry (4553) 7/22/2009 5:28:28 PM On Jul 22, 8:24=A0am, dana <[email protected]> wrote: > What is the "correct" term for a 1:1 relationship between a "database" > and an "instance" where there are at least two such "things" on the > same physical server? > > Nearly all the Oracle docs and books define a database something like > the following: > > DATABASE - a collection of datafiles and oracle config files; useless > without an "instance" to access the database. > > INSTANCE - a collection of background procs and memory structures, > used to access a "database." > > Where I work, people typically call each "thing" mentioned in my > Subject: line an "instance" or a "database instance." What is the > "correct" term for having more than one of these "things" on the same, > unpartitioned physical server? > > This may be splitting hairs; but when people don't or can't come to > terms, confusion ensues--and that's typically a Bad Thing. > > I noticed, or rather a colleague of mine did, that even Tom Kyte mixes > terms. In the book Expert Oracle Database Architecture (2005 edition) > on page 50, in the bullet-pointed definitional paragraph for Instance, > he throws the word "database instance" into the mix. A "no no" > according to my colleague (don't shoot the messenger, please); because > the vernacular seems to be "instance" or "database instance" for the > each "thing." > > I like to be precise with terms, but then there are business > stakeholders who use technically imprecise terms; even DBAs seem to > make this mistake. > > Would it be less confusing to all concerned simply refer to an "Oracle > SID" to indicate a 1:1 pairing of an "instance" and a "database" where > there are 2 or more of these "things" on a single, unpartitioned > server? > > Help! I want to "get it right" without being pedantic and trying to > educate every last stakeholder in a large organization. Maybe > education is the "right" thing to do, but when even DBAs use the term > "instance" incorrectly... well, that might not end well. :-) > > Dana You cannot talk about something as complicated as Oracle and get things right technically without being verbose and wordy and extremely specific. Most Oracle DBA ( well many anyhow ) understand the differences between instances and databases but tend to get sloppy at times when talking to management, developers, IT operators, etc. Might be best to count things on a server by server bases ( any VM's on a server should count as a server ) and define how many instances each server supports in these categories: single instance oracle databases and multiple instance oracle databases. The multiple instance databases are typically RAC although there is still some OPS ( Oracle Parallel Server ) hanging around still. 0 johnbhurley (2707) 7/22/2009 11:26:25 PM Thanks Mark, Joel, and John. I hadn't thought about non-database instances. > Of course, if everyone says something, the language changes, regardless of whether it is stupid or counterproductive. > The change itself can be interesting. True. I think the Oxford English Dictionary staff changes a word's definition when a particular threshold of the new definition's occurence in print and other media has been met. Dana 0 7/23/2009 5:26:12 PM On Jul 22, 10:29=A0am, ddf <[email protected]> wrote: > On Jul 22, 8:37=A0am, dana <[email protected]> wrote: > > > Here's another question for you: > > > 1) Would there ever be any practical reason for two instances (procs + > > SGA) to access the same database (collection of data treated as a > > unit) on the same, unpartitioned physical server? > > > Dana > > > Dana > > That, I believe, is the basic definition of RAC -- two or more > instances accessing a single database. > > David Fitzjarrell But with RAC you would expect each instance to be running concurrently on different servers. While some Oracle 'experts' have managed to create a RAC setup with multiple instances on a single server the set up is non-standard, unsupported, and for demonstartion purposes only. On a UNIX platform it used to be fairly easy to change the instance idenifier, SID, that was used to identify a running Oracle instance used to access a database. That is you could shut the instance down, make a few quick changes, and start a differently named instance then access the same database you were just working with from a different instance name. Only one instance can access a non-RAC database at a time. There was little practical application for this functionality. HTH -- Mark D Powell -- 0 Mark.Powell (1630) 7/24/2009 7:58:49 PM On Jul 24, 12:58=A0pm, Mark D Powell <[email protected]> wrote: > On Jul 22, 10:29=A0am, ddf <[email protected]> wrote: > > > > > On Jul 22, 8:37=A0am, dana <[email protected]> wrote: > > > > Here's another question for you: > > > > 1) Would there ever be any practical reason for two instances (procs = + > > > SGA) to access the same database (collection of data treated as a > > > unit) on the same, unpartitioned physical server? > > > > Dana > > > > Dana > > > That, I believe, is the basic definition of RAC -- two or more > > instances accessing a single database. > > > David Fitzjarrell > > But with RAC you would expect each instance to be running concurrently > on different servers. =A0While some Oracle 'experts' have managed to > create a RAC setup with multiple instances on a single server the set > up is non-standard, unsupported, and for demonstartion purposes only. > > On a UNIX platform it used to be fairly easy to change the instance > idenifier, SID, that was used to identify a running Oracle instance > used to access a database. =A0That is you could shut the instance down, > make a few quick changes, and start a differently named instance then > access the same database you were just working with from a different > instance name. =A0Only one instance can access a non-RAC database at a > time. =A0There was little practical application for this functionality. > > HTH -- Mark D Powell -- Wasn't there something about oltp tuning during the day, then batch/ report at night? Something is poking at the back of my brain, maybe it wasn't oracle. jg -- @home.com is bogus http://www.reuters.com/article/rbssTechMediaTelecomNews/idUSN21425536200907= 22 0 joel-garry (4553) 7/24/2009 11:11:04 PM joel garry schreef: > On Jul 24, 12:58 pm, Mark D Powell <[email protected]> wrote: >> On Jul 22, 10:29 am, ddf <[email protected]> wrote: >> >> >> >>> On Jul 22, 8:37 am, dana <[email protected]> wrote: >>>> Here's another question for you: >>>> 1) Would there ever be any practical reason for two instances (procs + >>>> SGA) to access the same database (collection of data treated as a >>>> unit) on the same, unpartitioned physical server? >>>> Dana >>>> Dana >>> That, I believe, is the basic definition of RAC -- two or more >>> instances accessing a single database. >>> David Fitzjarrell >> But with RAC you would expect each instance to be running concurrently >> on different servers. While some Oracle 'experts' have managed to >> create a RAC setup with multiple instances on a single server the set >> up is non-standard, unsupported, and for demonstartion purposes only. >> >> On a UNIX platform it used to be fairly easy to change the instance >> idenifier, SID, that was used to identify a running Oracle instance >> used to access a database. That is you could shut the instance down, >> make a few quick changes, and start a differently named instance then >> access the same database you were just working with from a different >> instance name. Only one instance can access a non-RAC database at a >> time. There was little practical application for this functionality. >> >> HTH -- Mark D Powell -- > > Wasn't there something about oltp tuning during the day, then batch/ > report at night? Something is poking at the back of my brain, maybe > it wasn't oracle. > > jg > -- > @home.com is bogus > http://www.reuters.com/article/rbssTechMediaTelecomNews/idUSN2142553620090722 Taken to the letter, a database on (backup) tape or other media is still a database, but not an instance. Shakespeare (What's in a name?) 0 whatsin (688) 7/25/2009 11:04:56 AM Thanks Mark, Joel, and Will. :-) 0 7/27/2009 10:47:16 AM On Jul 24, 7:11=A0pm, joel garry <[email protected]> wrote: > On Jul 24, 12:58=A0pm, Mark D Powell <[email protected]> wrote: > > > > > > > On Jul 22, 10:29=A0am, ddf <[email protected]> wrote: > > > > On Jul 22, 8:37=A0am, dana <[email protected]> wrote: > > > > > Here's another question for you: > > > > > 1) Would there ever be any practical reason for two instances (proc= s + > > > > SGA) to access the same database (collection of data treated as a > > > > unit) on the same, unpartitioned physical server? > > > > > Dana > > > > > Dana > > > > That, I believe, is the basic definition of RAC -- two or more > > > instances accessing a single database. > > > > David Fitzjarrell > > > But with RAC you would expect each instance to be running concurrently > > on different servers. =A0While some Oracle 'experts' have managed to > > create a RAC setup with multiple instances on a single server the set > > up is non-standard, unsupported, and for demonstartion purposes only. > > > On a UNIX platform it used to be fairly easy to change the instance > > idenifier, SID, that was used to identify a running Oracle instance > > used to access a database. =A0That is you could shut the instance down, > > make a few quick changes, and start a differently named instance then > > access the same database you were just working with from a different > > instance name. =A0Only one instance can access a non-RAC database at a > > time. =A0There was little practical application for this functionality. > > > HTH -- Mark D Powell -- > > Wasn't there something about oltp tuning during the day, then batch/ > report at night? =A0Something is poking at the back of my brain, maybe > it wasn't oracle. > > jg > -- > @home.com is bogushttp://www.reuters.com/article/rbssTechMediaTelecomNews= /idUSN21425536...- Hide quoted text - > > - Show quoted text - Yes. You would use a separate pfile then just referred to an init.ora so you would want to use a separate instance identifier, SID, at the OS level so that Oracle would select the right init.ora automatically. The alternate was to use the same SID but specify the init.ora via the startup command. -- Mark D Powell -- 0 Mark.Powell (1630) 7/28/2009 1:55:51 PM Similar Artilces: Hi , Hope you are doing great. Please let me take this opportunity to introduce myself, Iam Karthik working with BhanInfoi Inc, a NY based company. We have consultants on our bench on various technologies, my request is to add me to your distribution list and kindly do send me the requirements. i have the below list available 1. Mainframe 2. Java 3.. Financial Analyst 4. Data Architect If there is any vendor ship agreement which has to be signed then I would like to take an opportunity to represent my company and expect your cooperation... ... 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"In" "Out" and "Trash" I just bought a new computer and I re-installed Eudora Light on my new computer. But when I open Eudora, the "In", "Out" and "Trash" links are not on the left side of the screen the way they were on my old computer. How can I get these links back on the left side of the screen? Thank you. On 25 Mar 2007 09:49:22 -0700, "abx" <[email protected]> wrote: >I just bought a new computer and I re-installed Eudora Light on my new >computer. But when I open Eudora, the "In", "Out" and "Trash" links >are ... Does it need a ";" at the very after of "if" and "for" write code like: int main(void) { int a=10; if(a<20) {} } Compiler ok on dev-cpp . don't we have to add a ";" after if statement? marsarden said: > write code like: > > int main(void) > { > int a=10; > if(a<20) > {} > } > > Compiler ok on dev-cpp . don't we have to add a ";" after if > statement? The syntax for 'if' is: if(expression) statement There is no semicolon after the ) but before the statement. The statement is either a normal statement (which can be empty), ending in a semicolon:- if(expr) ... (mapcar 'quote '("1" "2" "3" "4")) (mapcar 'quote '("1" "2" "3" "4")) returns ((quote "4") (quote "4") (quote "4") (quote "4")) Interesting and (for me) unexpected. Because (mapcar 'read '("1" "2" "3" "4")) returns (1 2 3 4) and (mapcar 'princ '("1" "2" "3" "4")) gives 1234("1" "2" "3" "4") Why isn't (mapcar 'quote '("1" "2" "3" "4")) returning ((quote "1") (quote "2") (quote "3") (quote "4")) Tom Haws www.hawsedc.com Probably has to do with the fact that 'arg and (quote arg) are equivalent, and LISP gets confused by the construct 'quote, which is about the same as (quote (quote arg)). But I don't pretend to know all of the mechanics of the error; the results are somewhat different in R14, BTW: Command: (mapcar 'quote '("1" "2" "3" "4")) ((<Subr: #22e3e40> "4") (<Subr: #22e3e40> "4") (<Subr: #22e3e40> "4") (<Subr: #22e3e40> "4")) ___ "Tom Haws" <[email protected]> wrote in message news:[email protected]... > (mapcar 'quote '("1" "2" "3"... elementvise division "1./a" versus "1 ./ a" Hi, I wonder why there is a difference in the way scilab interprets "1./a" and "1 ./ a". Is this intentionally or is it a bug? -->a=[2 3 4] a = ! 2. 3. 4. ! -->1./a ans = ! 0.0689655 ! ! 0.1034483 ! ! 0.1379310 ! -->1 ./ a ans = ! 0.5 0.3333333 0.25 ! --> -Torbj�rn. Torbj�rn Pettersen wrote: > Hi, > I wonder why there is a difference in the way scilab interprets "1./a" > and "1 ./ a". Is this intentionally or is it a bug? > > -->a=[2 3 4] > a = > > ! 2. 3. 4. ! > > -->1./a > ans = > > ! 0.0689655 ! > ! 0.1034483 ! > ! 0.1379310 ! > > -->1 ./ a > ans = > > ! 0.5 0.3333333 0.25 ! > It's feature plus an unlucky situation. 1./a is intepreted as (1.0)/a which by definition computes the pseudo-inverse of a . Contrary to that 1 ./ a is the (./) operator. It's expanded into ones(a) ./ a which gives elementwise devision. So it's just that 1. is a floating point constant. There is no problem with say a./b ! Helmut Jarausch Lehrstuhl fuer Numerische Mathematik RWTH - Aachen University D 52056 Aachen, Germany Hm yes you are right... but I'm not sure I like that feature - It took to long to figure out where the error was :-) -Torbj�rn. Helmut Jarausch wrote: > Torbj�rn Pettersen wr... Changing "Figure 1" to "Scheme 1" Hi I'm trying to complete a 1-page abstract and am using floatflt to wrap the figures. I have tried to find the answers to the following, but time is of the essence at the moment. My question is regarding one of the captions: I want "Scheme 1" to replace "Figure 2" in the 2nd image I have included. Advice on this is much appreciated and also how to remove the colon after "Figure X:"/"Scheme X:". This is what is written: \usepackage[rflt]{floatflt} \setlength\figgutter{25pt} \newcommand{\incffig}[1]{\includegraphics[width=\floatfltwidth]{#1}} \begin{document} some text \begin{floatingfigure}[l]{0.55\linewidth} \incffig{thefigure} \def\fnum@figure{thecaption} \end{floatingfigure} some more text Hi, I used the caption2 with luck to change the figure captions in an earlier project, but caption2 seems to be obsolete and superseeded by the caption package. In any case, maybe one of them may do the trick for you. Best, Jon Nat wrote: > Hi > > I'm trying to complete a 1-page abstract and am using floatflt to wrap > the figures. I have tried to find the answers to the following, but > time is of the essence at the moment. My question is regarding one of > the captions: I want "Scheme 1" to replace "Figure 2" in the 2nd image > I have included. Advice on this is much appreciated and also how to > remove the colon after "Figure X:"/"Scheme X:". > > This... why have both "." and "->" ? I used to remember why c++ needed both ? Could somebody help me here ? For example class A{ f(); }; A* aa; You could do either "aa->f()" or "(aa).f()". So why does C++ need both operators. Raj "raj" <[email protected]> schreef in bericht news:[email protected]... > I used to remember why c++ needed both ? > Could somebody help me here ? > > For example > > class A{ > f(); > }; > > A aa; > > You could do either "aa->f()" or "(aa).f()". So why does C++... "bb = bb + 1" vs "bb += 1" Hi ! The following code does not compile (as expected): class SimpleJava { public static void main(String args[]) { System.out.println("SimpleJava ...."); byte bb = 0; bb = bb + 1; System.out.println("bb = " + bb); } } While the following one is compiling: class SimpleJava { public static void main(String args[]) { System.out.println("SimpleJava ...."); byte bb = 0; bb += 1; System.out.println("bb = " + bb); } } The += operator is specific to each primitive, so it "know" how to work with integers. The ope... How to alias "n1" to "n 1" and "n2" to "n 2" and so on (with style) I have the following bash function: function ..() { counter=0 case "$1" in [0-9][0-9] ) while [$counter -lt $1 ] do cd .. counter=$(($counter+1)) done ;; ) echo "staying where I am, give me a number (<99) next time :)" ;; esac } The allows me to move up, say 4 levels, using ".. 4" Now, typing the space annoys me so I define the following aliases: alias ..='.. 1' alias ..2='.. 2' alias ..3='.. 3' alias ..4='.. 4' alias ..5='.. 5' Not very elegant I con... "order by" And "limit = 1" Folks, I want to SELECT the single highest value of x. What does "Order by" really order, the full set of table rows before the single selection is returned, or only the returned row? (which wd clearly be useless in this case.) Or is there better SQL? Thanks. AS ashore wrote: > Folks, I want to SELECT the single highest value of x. What does > "Order by" really order, the full set of table rows before the single > selection is returned, or only the returned row? (which wd clearly be > useless in this case.) > > Or is there better SQL? Thanks. > ... fnmatch("\\\\", "\", 0) == 1 ??? FYI fnmatch("\\\\", "\", 0) in DJGPP returns 1(no match) -- the escaping functionality seems to be broken. The same function call returns 0 (match) in Linux (Red Hat 9 kernel 2.4.20) -- what it should be. Alex Alexei A. Frounze wrote: > FYI fnmatch("\\\\", "\", 0) in DJGPP returns 1(no match) -- the obvious correction (missed one slash): fnmatch("\\\\", "\\", 0) > escaping functionality seems to be broken. > The same function call returns 0 (match) in Linux (Red Hat 9 kernel > 2.4.20) -- what it should be. Alex ... Urgent JAVA Requirement in """"""NEW YORK""""""""" Hello Partners, How are you ? Please find the requirement below. Location : NY Duration : 8 mnths Rate :Open Job description: Java/J2EE Web Service Developer =B7 (4+ years of application development experience in Java/J2EE and Web service technologies. =B7 Experience with spring & Hibernate. =B7 Experience with J2EE Application Server (preferably Web logic). =B7 Preferable Aqua logic DSP Experience =B7 Preferable Sonic ESB Composite Service experience Experience w... "use constant X=>(1,2);" or "use constant X=>[1,2];"? What's better? use constant X=>(1,2); or use constant X=>[1,2]; On Sat, 8 Dec 2007 09:44:59 -0800 (PST), Victor Porton <[email protected]> wrote: >What's better? > > use constant X=>(1,2); > >or > > use constant X=>[1,2]; What's better? my @x = (1,2); or my$x = [1,2]; Michele -- {$_=pack'B8'x25,unpack'A8'x32,$a^=sub{pop^pop}->(map substr (($a\|\|=join'',map--$\|x\$_,(unpack'w',unpack'u','G^<R<Y]YB=' ..'KYU;EVH[.FHF2W+#"\Z*5TI/ER<Z`S(G.DZZ9OX0Z')=~/./g)x... Web resources about - "Correct" term for a 1:1 relationship between a "database" and an "instance" where > 1 such things are on the same physical server? - comp.databases.oracle.server Resources last updated: 3/12/2016 6:51:27 AM	2020-01-20 20:49:49	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.4088900685310364, "perplexity": 7280.647145352653}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250599789.45/warc/CC-MAIN-20200120195035-20200120224035-00385.warc.gz"}
https://askthetask.com/981/estimates-websites-previous-websites-previous-websites-previous	0 like 0 dislike Nate estimates that a search for websites about music produces 500,000 websites. If this is a 12% increase over the previous year, explain how you would find the number of websites for the previous year. Then find the number of websites the previous year. $$500000 = a(1.12)^1\\\frac{500000}{1.12} = 446,428.571$$	2023-02-03 06:10:02	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8242374658584595, "perplexity": 4876.413695713194}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1674764500044.16/warc/CC-MAIN-20230203055519-20230203085519-00588.warc.gz"}
https://otrecoding.github.io/OTrecod/reference/power_set.html	A function that gives the power set $$P(S)$$ of any non empty set S. power_set(n, ordinal = FALSE) ## Arguments n an integer. The cardinal of the set ordinal a boolean. If TRUE the power set is only composed of subsets of consecutive elements, FALSE (by default) otherwise. ## Value A list of $$2^n -1$$ subsets (The empty set is excluded) ## References Devlin, Keith J (1979). Fundamentals of contemporary set theory. Universitext. Springer-Verlag ## Author Gregory Guernec [email protected] ## Examples # Powerset of set of 4 elements set1 <- power_set(4) # Powerset of set of 4 elements by only keeping # subsets of consecutive elements set2 <- power_set(4, ordinal = TRUE)	2023-03-26 19:33:46	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 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": 0.7301227450370789, "perplexity": 5649.87327594976}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296946445.46/warc/CC-MAIN-20230326173112-20230326203112-00382.warc.gz"}
http://mathhelpforum.com/calculus/27627-hard-integral.html	# Math Help - Hard Integral 1. ## Hard Integral I am trying to work out the following integral: $ \int_{0}^{\infty}xe^{ax -bx^2} dx $ I have tried making several substitutions e.g. $u = ax -bx^2$ but have got stuck each time. Any help greatly appreciated Cheers 2. ## Completing the square You'll want to complete the square in the exponent: use $u=x-\frac{a}{2b}$. Then $x=u+\frac{a}{2b}$, and the integral becomes: $\int_{0}^{\infty}xe^{ax-bx^2}\,dx=\int_{-\frac{a}{2b}}^{\infty}\left(u+\frac{a}{2b}\right)e ^{\frac{a^2}{4b}-bu^2}\,du$ $\,\,=e^{\frac{a^2}{4b}}\int_{-\frac{a}{2b}}^{\infty}\left(u+\frac{a}{2b}\right)e ^{-bu^2}\,du$ Does that help? --Kevin C. 3. Originally Posted by peterpan I am trying to work out the following integral: $ \int_{0}^{\infty}xe^{ax -bx^2} dx $ I have tried making several substitutions e.g. $u = ax -bx^2$ but have got stuck each time. Any help greatly appreciated Cheers What are the chances .....?? Virtually the exact same question was asked here a while back. I love it when you can recycle .... I wonder if your question is NOT what you've posted, but some other question like ....... finding the moments of a distribution? Hmmmm ...... You're trying to get those moments by first finding the moment generating function ......? Big mistake. Unless you totally want to work with erf(x) ...... (I'm betting you don't). So how to get those pesky moments ...... You've gotta go back to first principles, sport. And it's your lucky day because the hard work's been done for you already ....	2016-02-10 12:04:10	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4721963107585907, "perplexity": 963.6407062308677}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701159155.63/warc/CC-MAIN-20160205193919-00033-ip-10-236-182-209.ec2.internal.warc.gz"}
https://www.preprints.org/manuscript/201801.0072/v1	Preprint Article Version 1 This version is not peer-reviewed A Molecular Dynamics Investigation of the Effect of Pressure and Orientation on the Cu Consumption in Cu-Cu3Sn Interface under Isothermal Ageing and Its Dissipative Mechanisms during Traction Version 1 : Received: 7 January 2018 / Approved: 9 January 2018 / Online: 9 January 2018 (03:27:22 CET) How to cite: Liang, L.; Zhang, J.; Xu, Y.; Zhang, Y.; Wang, W.; Yang, J. A Molecular Dynamics Investigation of the Effect of Pressure and Orientation on the Cu Consumption in Cu-Cu3Sn Interface under Isothermal Ageing and Its Dissipative Mechanisms during Traction. Preprints 2018, 2018010072 (doi: 10.20944/preprints201801.0072.v1). Liang, L.; Zhang, J.; Xu, Y.; Zhang, Y.; Wang, W.; Yang, J. A Molecular Dynamics Investigation of the Effect of Pressure and Orientation on the Cu Consumption in Cu-Cu3Sn Interface under Isothermal Ageing and Its Dissipative Mechanisms during Traction. Preprints 2018, 2018010072 (doi: 10.20944/preprints201801.0072.v1). Abstract In this paper, the nanoscale dissipative mechanisms of a Cu pad in a Ball Grid Array (BGA) packaging structure during isothermal ageing and uniaxial tension were investigated by the molecular dynamics (MD) method and experiments. From the result of the isothermal ageing test, a nonuniform consumption of Cu and large amount of Kirkendall voids were observed at the interface of Cu and Cu3Sn. To study the effect of pressure and orientation on this phenomenon, MD simulations were conducted on four types of Cu-Cu3Sn interface structures with different orientations of Cu. By comparing the diffusion coefficients of atoms in those cases, it was found that the tensile stress would inhibit the diffusion of atoms, whereas compressive stress would accelerate it, and this would be more significant under a larger magnitude of stress and temperature. Note that, in the model with the (101) surface Cu at the interface, both Cu and Cu3Sn have a higher diffusion coefficient compared with the model with (001) surface Cu. Thus, the orientation of Cu will also contribute to the uniform consumption of the pad. Uniaxial tension simulation combined with DXA and CSP analyses on those models also shows the model with (001) surface Cu has a greater mechanical reliability in our simulations and related experiments. Subject Areas interface structure; molecular dynamics; diffusion coefficient; uniaxial tension; orientation	2020-03-30 23:20:55	{"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": 0.8159403204917908, "perplexity": 3788.584423851615}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370497309.31/warc/CC-MAIN-20200330212722-20200331002722-00025.warc.gz"}
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1016	## On-line path planning with optimal C-space discretization • This paper is based on a path planning approach we reported earlier for industrial robot arms with 6 degrees of freedom in an on-line given 3D environment. It has on-line capabilities by searching in an implicit and descrete configuration space and detecting collisions in the Cartesian workspace by distance computation based on the given CAD model. Here, we present different methods for specifying the C-space discretization. Besides the usual uniform and heuristic discretization, we investigate two versions of an optimal discretization for an user-predefined Cartesian resolution. The different methods are experimentally evaluated. Additionally, we provide a set of 3- dimensional benchmark problems for a fair comparison of path planner. For each benchmark, the run-times of our planner are between only 3 and 100 seconds on a Pentium PC with 133 MHz. ### Additional Services Author: Dominik Henrich, Christian Wurll, Heinz Wörn urn:nbn:de:hbz:386-kluedo-9703 Article English 1998 1998 Technische Universität Kaiserslautern 2000/04/04 AG-RESY ; PARO ; SKALP ; discretization; industrial robots ; on-line algorithms ; path planning ; search algorithms Fachbereich Informatik 0 Informatik, Informationswissenschaft, allgemeine Werke / 00 Informatik, Wissen, Systeme / 004 Datenverarbeitung; Informatik AG RESY Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011 $Rev: 13581$	2016-10-27 17:03:39	{"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": 0.1895497888326645, "perplexity": 7115.6683724080485}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988721355.10/warc/CC-MAIN-20161020183841-00109-ip-10-171-6-4.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/169671/operatornameannm-otimes-a-n-operatornameannm-operatornameannn?answertab=votes	# $\operatorname{Ann}(M\otimes_A N)=\operatorname{Ann}M+\operatorname{Ann}N$? In the course of working on an exercise in Atiyah-MacDonald (exercise 3 on p. 31), I've come to the belief that, for $A$ an arbitrary commutative ring and $M,N$ arbitrary $A$-modules, $$\operatorname{Ann}(M\otimes_A N)=\operatorname{Ann}M+\operatorname{Ann}N$$ where $\operatorname{Ann}M$ is the annihilator of $M$, etc. While I think it is trivial that $\operatorname{Ann}(M\otimes_A N)\supset\operatorname{Ann}M+\operatorname{Ann}N$, I am having trouble establishing the reverse inclusion and would appreciate your help in doing so, if it is true, or if it is false I would appreciate a counterexample. Here is what I have thought about so far: • On an intuitive level, because $M\otimes_A N$ is "the biggest/freest" bilinear image of $M\times N$, it should not be annihilated except by the smallest ideal of $A$ that annihilates either of $M$ and $N$, and this is $\operatorname{Ann}M+\operatorname{Ann}N$. I think this is why I believe the claim to be true. • My attempts to create an actual proof have focused on the universal properties of the relevant objects: (a) there is an $A$-bilinear map $M\times N\rightarrow M\otimes_A N$ such that any $A$-bilinear map from $M\times N$ factors through this one; and (b) any homomorphism from $A$ that factors through both $A/\operatorname{Ann}M$ and $A/\operatorname{Ann}N$ also factors through $A/(\operatorname{Ann}M+\operatorname{Ann}N)$. This is the forest I've gotten lost in. The homomorphisms from $A$ I can think of are into the endomorphism rings of $M,N,M\otimes_A N$. I haven't seen how to put them together with bilinear maps to show I'm aiming for. Again, either an indication of how you'd complete the proof (preferably in terms of the universal properties, so I can see how to do what I've been trying to do) or a counterexample would be much appreciated. - This is an excellent question, and your intuition is not too far off. Note that the counterexamples given in the other answers involve a non-finitely generated module ($\mathbb Q$ thought of as a $\mathbb Z$-module), and constructions like annihilators behave much better for finitely generated modules than for non-finitely generated ones (as a general rule). E.g. Suppose that $M = R/I$ and $N = R/J$ are cyclic. Then $M\otimes N = R/(I+J),$ and so your conjectured formula is true in that case. As you observe, always $\operatorname{Ann} M + \operatorname{Ann} N \subset \operatorname{Ann} M\otimes N$. Regarding the converse, what is true for general finitely generated $R$-modules $M$ and $N$ is that $\operatorname{Ann}M\otimes N$ is contained in the radical of $\operatorname{Ann} M + \operatorname{Ann} N$. So your conjecture becomes correct if you restrict to f.g. modules, and take radicals of both sides. [To see this, use the fact that for a finitely generated module, the radical of $\operatorname{Ann} M$ is equal to the intersection of those prime ideals $\mathfrak p$ for which $\kappa(\mathfrak p) \otimes_R M \neq 0$, where $\kappa(\mathfrak p)$ denotes the fraction field of $R/\mathfrak p$.] I don't think that we can do much better than this though, in the finitely generated but non-cyclic case. To see the kind of things that can happen, choose three ideals $I_1,I_2,J,$ and let $M = R/I_1 \oplus R/I_2$ and $N = R/J$. Then $\operatorname{Ann} M = I_1\cap I_2$, $\operatorname{Ann} N = J$, and $M\otimes N = R/I_1+J \oplus R/I_2 + J$, so that $\operatorname{Ann} M\otimes N = (I_1 + J) \cap (I_2 +J).$ Now we always have $(I_1\cap I_2) + J \subset (I_1 + J) \cap (I_2 + J) \subset \operatorname{rad}\bigl((I_1 \cap I_2) + J\bigr),$ but the first inclusion is typically not an equality (because the lattice of ideals in $R$ is always modular, but typically not distributive). So if you take a counterexample to the distributive property for some lattice of ideals, feeding it into the above example will give a counterexample to the precise form of your conjecture (i.e. equality on the nose, rather than up to taking radicals) involving only finitely generated modules. - The containment $\mathrm{Ann}_A(M)+\mathrm{Ann}_A(N)\subseteq\mathrm{Ann}_A(M\otimes_AN)$ can be proper. For an example, take $A=\mathbf{Z}$, $M=\mathbf{Z}/2\mathbf{Z}$, and $N=\mathbf{Q}$. Then $\mathrm{Ann}_A(M)=2\mathbf{Z}$, and $\mathrm{Ann}_A(N)=0$, but $M\otimes_AN=\mathbf{Z}/2\mathbf{Z}\otimes_{\mathbf{Z}}\mathbf{Q}=0$ because the left tensor factor is killed by $2$ but $\mathbf{Q}$ is divisible. So $\mathrm{Ann}_A(M\otimes_AN)=\mathbf{Z}$. - Consider the $\mathbb{Z}$-modules $M:=\mathbb{Z}/m\mathbb{Z}$ and $N:=\mathbb{Q}$. Then if I am not terribly mistaking: $\text{ann}M=m\mathbb{Z}$, $\text{ann}N=0$, $M\otimes_\mathbb{Z} N=0$. //Edit: I was too slow :) -	2013-05-24 22:43:31	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9149941205978394, "perplexity": 126.9104820337446}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1368705195219/warc/CC-MAIN-20130516115315-00068-ip-10-60-113-184.ec2.internal.warc.gz"}
http://aas.org/archives/BAAS/v30n4/aas193/913.htm	AAS Meeting #193 - Austin, Texas, January 1999 Session 46. Photometric Observations of Variable Stars Display, Thursday, January 7, 1999, 9:20am-6:30pm, Exhibits Hall 1 ## [46.02] The Rise and Fall of Mu Velorum T.R. Ayres, R.A. Osten, A. Brown (CASA) \noindent Mu~Velorum (HD~93497) is a close visual pair consisting of a G5~III primary and a fainter companion, currently 2\arcsec\ apart, with an orbital period of about 140~yr. The distance to the system is 36~pc. Mu~Vel~A is a 3\,M\odot giant, in the Hertzsprung gap beyond the rapid braking zone'' just redward of G0~III. Mu~Vel recently was the source of a giant EUV flare, caught during a 12-day pointing in March~1998 by the Deep Survey telescope of the Extreme Ultraviolet Explorer.\/ The outburst rose in less than half a day, and decayed with an e-folding time of several days. The peak flux, ~0.3~cnts~s-1 in the DS band 80--180~Å, was approximately twice the quiescent level recorded during the previous ten days of observation. The size and long duration of the event are very unusual for a mid-G giant; in fact, more typical of the extremes\/ seen among the hyperactive short-period RS~Canum Venaticorum binaries. Although the secondary star is classified as a G2~V, published visual magnitude differences, and the enhanced 1900~Å\ continuum (\mu~Vel was detected by IUE, but AB were not resolved), suggest that it falls earlier on the MS (perhaps F5~V) and indeed itself might be a pair. If \mu~Vel~B is a short-period double, then arguably it could be the source of the giant flare. Unfortunately, little is known about the secondary owing to the small separation of the visual components of \mu~Vel, and their large difference in brightness. We discuss the flare event, EUV spectra obtained in quiescence and during the flare decay, and the nature of the puzzling secondary.\\[2mm] \noindent This work was supported by NASA grant NAG5-3226.	2014-11-28 21:42:50	{"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": 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": 0.7399401068687439, "perplexity": 6733.1924635422165}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931011030.89/warc/CC-MAIN-20141125155651-00216-ip-10-235-23-156.ec2.internal.warc.gz"}
https://www.metabunk.org/threads/bugs-and-suggestions-for-metabunk-org.1959/page-12#post-186573	# Bugs and Suggestions for Metabunk.org #### tinkertailor ##### Senior Member. I'm a bit reluctant to add more ratings, as there's quite a few already. Maybe if there was more call for it. I'd be interested. #### cloudspotter ##### Senior Member. I'm getting that problem where a few read threads are still in bold looking like they're unread. I've noticed it only seems to happen with threads where the last poster is some bloke called Mick West. The Ghost Balloon thread is one example #### deirdre ##### Senior Member. I'm getting that problem where a few read threads are still in bold looking like they're unread. I've noticed it only seems to happen with threads where the last poster is some bloke called Mick West. The Ghost Balloon thread is one example i'm ok in IE. and Chrome. Windows 7. is it still happening @cloudspotter ? #### cloudspotter ##### Senior Member. i'm ok in IE. and Chrome. Windows 7. is it still happening @cloudspotter ? Just viewed this thread in Chrome and IE and it still shows up like this #### cloudspotter ##### Senior Member. Just viewed this thread in Chrome and IE and it still shows up like this And the forum shows like this #### Trailspotter ##### Senior Member. I'm getting that problem where a few read threads are still in bold looking like they're unread. I've noticed it only seems to happen with threads where the last poster is some bloke called Mick West. The Ghost Balloon thread is one example I have the same bug on my laptop and iPhone, both running Safari. I reported it more than year ago, somewhere at page 8 of this thread. @Mick West said it is Xenoforo bug. #### cloudspotter ##### Senior Member. I have the same bug on my laptop and iPhone, both running Safari. I reported it more than year ago, somewhere at page 8 of this thread. @Mick West said it is Xenoforo bug. Yeah it has been ongoing off and on for a while but this is the first I've noticed any pattern to it #### Mick West Staff member I don't know why it happens. However you can flush it out by doing "Mark Forum Read" from within the forum that's showing the problem. Then you can do that forum, or all of them. (Which I just did for you, so it will be clear now) #### skephu ##### Senior Member. I get "server error" on any forum page when I'm logged in on an Android tablet. If I'm not logged in or use a desktop, there is no error. #### Mick West Staff member I get "server error" on any forum page when I'm logged in on an Android tablet. If I'm not logged in or use a desktop, there is no error. Is this a new thing? Have you used the tablet before and had it work? #### skephu ##### Senior Member. Is this a new thing? Have you used the tablet before and had it work? Yes, it's new. But now as I think about it, it may be related to Chrome beta which I recently installed? #### Mick West Staff member Yes, it's new. But now as I think about it, it may be related to Chrome beta which I recently installed? Entirely possible. Seems to be something to do with an add-on I have that logs user-agents, which would be affected by a new browser, or possibly a bug in the browser. I'll scan the logs. #### Mick West Staff member I updated the add-on, try it now #### Steve Funk ##### Senior Member. Since I moved and changed to Comcast, my internet reception in general has been worse, with a lot of short duration crashes or pauses. Metabunk in particular seems harder than other sites to navigate in, and more likely to freeze up. Is there anything that I can do other than change service? #### deirdre ##### Senior Member. Since I moved and changed to Comcast, my internet reception in general has been worse, with a lot of short duration crashes or pauses. Metabunk in particular seems harder than other sites to navigate in, and more likely to freeze up. Is there anything that I can do other than change service? I have horrible speeds and usually MB is fine even when im running Netflix and playing a flash heavy game. When I do bog up I run my cleaners and reboot. If still slow (server side) I turn off all background programs/extentions I don't really need running and don't open any other tabs besides MB etc. Back in the day with dial up, you can turn off your graphics (prevent graphics from downloading) but this doesn't sound like something you want to live with. add: ps my mom has Comcast the biggest package and my stuff works way worse at her house. she had the techs come in and everything. she even got the boosters which did help a bit. but still I'm (my own laptop ) laggier at her house than here. we can't figure out why. http://www.wikihow.com/Make-Your-Wireless-Internet-Connection-Faster-(Comcast) Last edited: #### jaydeehess ##### Senior Member. I am paying for a 15 Mbps connection via Shaw cable in Canada, which shares many infrastructure set ups that Comcast uses. I have no issues with MB. Speedtest indicates that right now i am gett 15.99 Mbps download. #### Jedo ##### Member Done! [TEX] c=\sqrt{a^2+b^2} [/TEX] LaTeX: $e=mc^2$ The tex/latex feature seems to be working only for very simple expressions, but even then randomly only. see LaTeX: $x$ LaTeX: $x=$ LaTeX: $x=x$ LaTeX: $x_0=x$ $$x$$ $$x=$$ $$x=ysa$$ $$x_0=x$$ $$E=mc^2$$ can't see what I do wrong. #### qed ##### Senior Member The LaTeX facility appears to be broken. #### derwoodii ##### Senior Member. um i have been using this link http://news.metabunk.org/ for news as it did open pages of multiple sources but as of late it seems not to work to well,, some pages dont load up is it nqr or can it be tweaked #### Mick West Staff member um i have been using this link http://news.metabunk.org/ for news as it did open pages of multiple sources but as of late it seems not to work to well,, some pages dont load up is it nqr or can it be tweaked Over the last year most news sites have disabled embedding in frames, so you need to add an extension to your browser to disable the x-frame headers. I use this one: Fixed! #### derwoodii ##### Senior Member. ah not with my skills,,, can you links the patch you made or do best to show how i add it to address i tried a few guess attempts but no joy #### Mick West Staff member ah not with my skills,,, can you links the patch you made or do best to show how i add it to address i tried a few guess attempts but no joy What browser do you use? All I did was add that addon to Chrome. I don't really know about other browsers, but there should be something similar. #### derwoodii ##### Senior Member. What browser do you use? All I did was add that addon to Chrome. I don't really know about other browsers, but there should be something similar. Browser? um i dont know i did try to add on the string and the page said was Compatible with your device... gimmi a mo Ill get back do my best to learn the how to do later as off to work now #### derwoodii ##### Senior Member. ah ha had to press the add to chrome button top right Doh i was cutting & pasting script to address line all fixed much thanks #### Trailspotter ##### Senior Member. I stay logged in to Metabunk all time on my laptop and automatically log in back when I restart the computer or browser. opens a new tab in the same Safari window, in which I have to log in again to reply. Why is this? PS If I open other Metabunk pages in a new tab, I am automatically logged in. Last edited: #### Mendel ##### Senior Member. I stay logged in to Metabunk all time on my laptop and automatically log in back when I restart the computer or browser. opens a new tab in the same Safari window, in which I have to log in again to reply. Why is this? #### Mick West Staff member Hmm, the site used to be set to "force www." I shall investigate. #### Trailspotter ##### Senior Member. Thanks. Manually adding 'www' to URL got me logged in. #### Mick West Staff member The www/login bug should be fixed now. All URLs will have www automatically prepended to them #### Mendel ##### Senior Member. Agent K's post https://www.metabunk.org/threads/coronavirus-statistics-cases-mortality-vs-flu.11154/post-238879 did not make it onto my "latest activity" list. I did get an alert that indicates he had quoted me before he edited that post. I've seen posts before that "latest activity" did not seem to track, but I always assumed this was because of moderation releasing those late; but Agent K is a regular poster, so I doubt that's the cause. The post was made 2:18am, last edited 2:30am in my time zone. Last edited: #### deirdre ##### Senior Member. I've seen posts before that "latest activity" did not seem to track, but I always assumed this was because of moderation releasing those late; but Agent K is a regular poster, so I doubt that's the cause. There is a setting members can chose ..that disables seeing them in the feed. It's not fixable at this time.. I already 'complained'. That's why i liked his post about the navy ufo video released... was afraid noone would even know he posted it. #### Rory ##### Senior Member. Suggestion: the face that's used for 'funny': seems a bit much for me - seems too much like 'laughing at' rather than 'with'. #### deirdre ##### Senior Member. too american minstrel. i think it would offend some people. i think anything will continue to be used to mock people. (i didnt think you wre laughing at my fear of my cousins ) this one seems pretty neutral, but when xenoforo updates, wouldnt they all go back to the default ones? #### Rory ##### Senior Member. (i didnt think you were laughing at my fear of my cousins) Indeed. I was laughing at the joke. #### Rory ##### Senior Member. Suggestion: Election 2020 is top of the forums list but it's kind of passed now. Maybe move down or add as a sub-forum of General Discussion/Current Events/Conspiracy Theories? #### deirdre ##### Senior Member. might as well just start a political forum, since midterms are coming up!!! #### FatPhil ##### Senior Member. Bug: Image scaling in replies The images in my reply are not scaled to fit the panel, but in the post I'm responding to they are. I think the two should have the same behaviour (and find scaled preferable). Mendel's HTML and my HTML for the <table> that contains the images are identical, so it doesn't seem to be a problem with what's generated, so I guess it's a CSS selector issue where we differ. I can sniff more...	2022-09-26 05:57:10	{"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": 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": 0.1756671518087387, "perplexity": 4718.006492447941}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030334802.16/warc/CC-MAIN-20220926051040-20220926081040-00359.warc.gz"}
https://www.csdn.net/tags/MtjaMgysMDg2MjItYmxvZwO0O0OO0O0O.html	• Given a positive integer, output its complement number. The complement strategy is to flip the bits of its binary representation. Note: The given integer is guaranteed to fit within the range of a 32-... Given a positive integer, output its complement number. The complement strategy is to flip the bits of its binary representation. Note: The given integer is guaranteed to fit within the range of a 32-bit signed integer. You could assume no leading zero bit in the integer’s binary representation. Example 1: Input: 5 Output: 2 Explanation: The binary representation of 5 is 101 (no leading zero bits), and its complement is 010. So you need to output 2. Example 2: Input: 1 Output: 0 Explanation: The binary representation of 1 is 1 (no leading zero bits), and its complement is 0. So you need to output 0. 因为不是简单的补码，所以我们必须从高位开始判断，0跳过，1开始，start便是这个flag。然后我们翻转的方法就是对应位异或一个1即可 /** * @param {number} num * @return {number} / var findComplement = function(num) { let start = false; for (let i = 31; i >= 0; --i) { if (num & (1 << i)) start = true; if (start) num ^= (1 << i); } return num; }; 展开全文 • 2’s complement 与 1’s complement 2’s complement 是按位取反加一 1’s complement 是按位取反（不加一） 2’s complement 与 1’s complement 2’s complement 是按位取反加一 1’s complement 是按位取反（不加一）展开全文 • To change the binary complement to be signed decimal data. In RTL, 对于负数取反加1 In C or MATLAB, 对于负数，直接减最大值。 Ex, 14 bits signed data changes to decimal data. To change the binary complement to be signed decimal data. In RTL, 对于负数取反加1 In C or MATLAB, 对于负数，直接减最大值。 Ex, 14 bits signed data changes to decimal data. 展开全文 • 1’s Complement和2’s Complement的区别： 1’s Complement表示按位取反； 2’s Complement表示按位取反后再加1 1’s Complement和2’s Complement的区别： 1’s Complement表示按位取反； 2’s Complement表示按位取反后再加1 展开全文 • 在网上查了一下关于1’s Complement和2’s Complement的区别，大部分答案都回答的不是很明确，谷歌了一下，如上图所示： 1’s Complement表示按位取反； 2’s Complement表示按位取反后再加1（其实就是取补码的操作... • complement 方法功能：返回一个函数，即给定函数 fn 的逻辑补集。有点抽象，这里补集的意思可以理解成相反，如，偶数的相反是奇数，正数的相反是负数。实现 const complement = fn => (...args) => !fn(...... • Multi-image Deblurring Using Complement • Given a positive integer, output its complement number. The complement strategy is to flip the bits of its binary representation. Note: The given integer is guaranteed to fit within the range of a 3... • 476. Number Complement Given a positive integer, output its complement number. The complement strategy is to flip the bits of its binary representation. Note: The given integer is guara • 我的专栏Halcon算子会持续更新，重点翻译和讲解所有Halcon算子，整个专栏和Halcon实例分析向...使用算子Complement来计算输入区域的补集，如果'clip_region'的系统标志为'false'(参见我的博客专栏Halcon算子set_sys... • Given a positive integer, output its complement number. The complement strategy is to flip the bits of its binary representation. Note: The given integer is guaranteed to fit within the range of a 3... • C Function Complement <stdlib.h> - qsort qsort函数C语言编译器函数库自带的快速排序函数。 void qsort(voidbase,size_t num,size_t width,int(__cdeclcompare)(const void,const void*)); blog for ... • 1’s complement of a binary number is another binary number obtained by toggling all bits in it, i.e., transforming the 0 bit to 1 and the 1 bit to 0. Let numbers be stored using 4 bits 1’s ... • 原题链接在这里：...Given a positive integer, output its complement number. The complement strategy is to flip the bits of its binary representation. Note: ... ...	2021-09-20 11:29:48	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.255894273519516, "perplexity": 4371.501151380986}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057036.89/warc/CC-MAIN-20210920101029-20210920131029-00375.warc.gz"}
https://eprint.iacr.org/2017/925	Resettably-Sound Resettable Zero Knowledge in Constant Rounds Wutichai Chongchitmate, Rafail Ostrovsky, and Ivan Visconti Abstract In FOCS 2001 Barak et al. conjectured the existence of zero-knowledge arguments that remain secure against resetting provers and resetting verifiers. The conjecture was proven true by Deng et al. in FOCS 2009 under various complexity assumptions and requiring a polynomial number of rounds. Later on in FOCS 2013 Chung et al. improved the assumptions requiring one-way functions only but still with a polynomial number of rounds. In this work we show a constant-round resettably-sound resettable zero-knowledge argument system, therefore improving the round complexity from polynomial to constant. We obtain this result through the following steps. 1. We show an explicit transform from any $\ell$-round concurrent zero-knowledge argument system into an $O(\ell)$-round resettable zero-knowledge argument system. The transform is based on techniques proposed by Barak et al. in FOCS 2001 and by Deng et al. in FOCS 2009. Then, we make use of a recent breakthrough presented by Chung et al. in CRYPTO 2015 that solved the longstanding open question of constructing a constant-round concurrent zero-knowledge argument system from plausible polynomial-time hardness assumptions. Starting with their construction $\Gamma$ we obtain a constant-round resettable zero-knowledge argument system $\Lambda$. 2. We then show that by carefully embedding $\Lambda$ inside $\Gamma$ (i.e., essentially by playing a modification of the construction of Chung et al. against the construction of Chung et al.) we obtain the first constant-round resettably-sound concurrent zero-knowledge argument system $\Delta$. 3. Finally, we apply a transformation due to Deng et al. to $\Delta$ obtaining a resettably-sound resettable zero-knowledge argument system $\Pi$, the main result of this work. While our round-preserving transform for resettable zero knowledge requires one-way functions only, both $\Lambda, \Delta$ and $\Pi$ extend the work of Chung et al. and as such they rely on the same assumptions (i.e., families of collision-resistant hash functions, one-way permutations and indistinguishability obfuscation for P/poly, with slightly super-polynomial security). Available format(s) Category Cryptographic protocols Publication info Keywords zero knowledgeresettable ZKresettable soundnessconstant-Roundindistinguishability obfuscation Contact author(s) wutichai @ cs ucla edu History Short URL https://ia.cr/2017/925 CC BY BibTeX @misc{cryptoeprint:2017/925, author = {Wutichai Chongchitmate and Rafail Ostrovsky and Ivan Visconti}, title = {Resettably-Sound Resettable Zero Knowledge in Constant Rounds}, howpublished = {Cryptology ePrint Archive, Paper 2017/925}, year = {2017}, note = {\url{https://eprint.iacr.org/2017/925}}, url = {https://eprint.iacr.org/2017/925} } Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content.	2022-10-05 00:14:56	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7858738303184509, "perplexity": 2672.819674059496}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337529.69/warc/CC-MAIN-20221004215917-20221005005917-00077.warc.gz"}
https://crypto.stackexchange.com/questions/61181/a-query-on-learning-with-errorslwe-problem	# A query on Learning with errors(LWE) problem In generating an LWE sample, we do $s\xleftarrow{$}\mathbb{Z}_q^{n}, A \xleftarrow{$}\mathbb{Z}_q^{n \times m}~$and $e\xleftarrow{$}\mathbb{{\chi}^{m}}$Then we compute$b^T$=$s^TA$+$e^T$and the sample$(A,b)\approx$truly random sample. Now suppose we have a fixed matrix (not random and public)$A \in\mathbb{Z}_q^{n \times m}$. We choose$R\xleftarrow{$}\mathbb{Z}_q^{n \times m}$ and compute $A' = A + R$ and the generate the LWE sample as $(A',b')$. Will the LWE assumption still hold? If it doesn't hold, then is there a way to mask matrix $A$? • When you say Now suppose we have a fixed matrix (not random and public) , what are the entries in the matrix if it is not random? Or do you mean a random but publicly known matrix? – Ella Rose Jul 31 '18 at 1:35 • A deterministic algorithm is run to generate the entries of $A$ and then it's made public. – chelsea Jul 31 '18 at 3:15 • What does $\chi^m$ mean? – kodlu Jul 31 '18 at 3:24 • The error vector is drawn from a distribution (Gaussian) – chelsea Jul 31 '18 at 3:28 • if R is uniformly random , then so is A+R. – user27950 Jul 31 '18 at 5:32 It is easy to reduce this problem to LWE, since adding any element to a uniformly random value gives a uniformly random distribution. For example, here, $(A',b')$ is distributed the same as if $A'$ was drawn uniformly randomly, and $R$ set as $A'-A$. You're now in the setting of the LWE assumption and can replace $b'$ by a uniformly random value without anybody (PPT) noticing it. With more details: $$R\gets\mathbb{Z}_q^{n\times m}, A' := A + R, b'^t := s^t A'+ e^t, e\gets \chi^m,$$ is distributed identically to: $$A'\gets\mathbb{Z}_q^{n\times m}, R := A' - A, b'^t := s^t A' + e^t, e\gets \chi^m,$$ which is computationally indistinguishable from: $$A'\gets\mathbb{Z}_q^{n\times m}, R := A' - A, b' \gets \mathbb{Z}_q^{m}.$$ You can then switch back to normal generation of $A'$: the previous distribution is identical to: $$R\gets\mathbb{Z}_q^{n\times m}, A' := A + R, b' \gets \mathbb{Z}_q^{m}.$$ • I am not sure that $(R, b')$ is a valid LWE sample because here $b'$ doesn't have the format $s^T R + e$, since there is that additional term $s^T A$... – Hilder Vítor Lima Pereira Aug 1 '18 at 8:12 • $(A',b')$ is the LWE sample, $R$ is built from public values $A'$ and $A$. – Florian Bourse Aug 1 '18 at 8:23 • Yes, $(A', b')$ is a LWE sample. But what I understood from "You're now in the setting of the LWE assumption" is that $(R, b')$ is a LWE sample. So I think I didn't get it. Thus, please, could you explain what you mean by that and by "$R$ set as $A' - A$"? – Hilder Vítor Lima Pereira Aug 1 '18 at 8:43 • I edited the answer to make it more formal – Florian Bourse Aug 1 '18 at 8:56	2019-11-22 08:34:58	{"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": 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": 0.8496329188346863, "perplexity": 902.7009397562558}, "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-2019-47/segments/1573496671245.92/warc/CC-MAIN-20191122065327-20191122093327-00348.warc.gz"}
https://atarnotes.com/forum/index.php?topic=164549.msg948808	August 19, 2017, 05:13:52 pm 2 Members and 5 Guests are viewing this topic. #### beau77bro • Forum Obsessive • Posts: 249 • Respect: +4 ##### Re: 3U Maths Question Thread « Reply #2025 on: May 19, 2017, 10:12:38 pm » 0 Hi! Very rusty with locus and param, need someone to help with these 1) Find the gradients of the common tangents of the parabola y= x(x-4) and the circle (x-2)^2 + y^2 = 4 (Im not quite sure, do i diff both then simultaneous or?) ive been trying to help my friend do this question all day. would these be the tangents we're finding? i think the question makes sense-ish but is this what common tangents means or is it when it's a tangent when they intesect? either way geogebra (even tho roughly done seems to think that the gradients are the answers) assuming the other side is identical. but how do u actually derive the answer? i've tried equatting the gradients, trying to do sub in y=mx +b and then making the discriminant = 0 and trying to do simultaneous but it goes no where? i dont know if im adding any thing relevant but maybe the diagram will help legorgo, or someone else work it out? #### RuiAce • ATAR Notes HSC Lecturer • Moderator • ATAR Notes Legend • Posts: 4847 • I don't write essays. I write proofs. • Respect: +381 ##### Re: 3U Maths Question Thread « Reply #2026 on: May 20, 2017, 12:27:22 am » 0 ive been trying to help my friend do this question all day. would these be the tangents we're finding? i think the question makes sense-ish but is this what common tangents means or is it when it's a tangent when they intesect? either way geogebra (even tho roughly done seems to think that the gradients are the answers) assuming the other side is identical. but how do u actually derive the answer? i've tried equatting the gradients, trying to do sub in y=mx +b and then making the discriminant = 0 and trying to do simultaneous but it goes no where? i dont know if im adding any thing relevant but maybe the diagram will help legorgo, or someone else work it out? Those are just the tangents to each curve, at the points of intersection. A common tangent is when the tangents to both curves (at their points of intersection) are the same line. The curves are essentially tangential to each other when this appears. Interested in Mathematics Tutoring? Message for details! ATAR: 98.60 Currently studying: Bachelor of Science (Advanced Mathematics)/Bachelor of Science (Computer Science) @ UNSW Formerly studying: Bachelor of Actuarial Studies/Bachelor of Science (Advanced Mathematics) #### jamonwindeyer • ATAR Notes HSC Lecturer • Great Wonder of ATAR Notes • Posts: 6382 • Electrical Engineer by day, AN Enthusiast by Night • Respect: +627 ##### Re: 3U Maths Question Thread « Reply #2027 on: May 20, 2017, 12:43:58 am » 0 Those are just the tangents to each curve, at the points of intersection. A common tangent is when the tangents to both curves (at their points of intersection) are the same line. The curves are essentially tangential to each other when this appears. Nah, I think that diagram is showing lines that happen to be tangents of both the circle and parabola, but not at the same point. It's not what I'd define as a common tangent either but it makes the question answerable at least! As to finding it - I'm honestly not sure! On first thought, you need a point on the parabola $x_1,y_1$, and a point on the parabola $x_2,y_2$, such that the gradient of the interval joining the two points is equal to the gradient of the curve at both points. To me that's two equations with four variables. You could then use the equations of the curves (for example, $y_1=x_1(x_1-4)$) to get rid of two of the variables, leaving two equations in two variables - Simulteneously solve (I anticipate that algebra to be horrendous) #### RuiAce • ATAR Notes HSC Lecturer • Moderator • ATAR Notes Legend • Posts: 4847 • I don't write essays. I write proofs. • Respect: +381 ##### Re: 3U Maths Question Thread « Reply #2028 on: May 20, 2017, 12:46:01 am » 0 Nah, I think that diagram is showing lines that happen to be tangents of both the circle and parabola, but not at the same point. It's not what I'd define as a common tangent either but it makes the question answerable at least! As to finding it - I'm honestly not sure! On first thought, you need a point on the parabola $x_1,y_1$, and a point on the parabola $x_2,y_2$, such that the gradient of the interval joining the two points is equal to the gradient of the curve at both points. To me that's two equations with four variables. You could then use the equations of the curves (for example, $y_1=x_1(x_1-4)$) to get rid of two of the variables, leaving two equations in two variables - Simulteneously solve (I anticipate that algebra to be horrendous) I tried to reverse engineer it to see if that's what they meant, but I couldn't quite get it to work. Maybe I had to do it from first principles. $\text{I gave a hint that two of the solutions of}\\ (x-2)^2 + [x(x-4)]^2=4\\\text{ must}\textit{ by inspection be }x=0, x=4\\ \text{which makes the equation much easier to solve (just do polynomial division or something)}$ $\text{But it's still a lot more work to do}\\ \text{Plus I really prefer just using implicit differentiation rather than dealing with square roots}$ « Last Edit: May 20, 2017, 12:49:24 am by RuiAce » Interested in Mathematics Tutoring? Message for details! ATAR: 98.60 Currently studying: Bachelor of Science (Advanced Mathematics)/Bachelor of Science (Computer Science) @ UNSW Formerly studying: Bachelor of Actuarial Studies/Bachelor of Science (Advanced Mathematics) #### RuiAce • ATAR Notes HSC Lecturer • Moderator • ATAR Notes Legend • Posts: 4847 • I don't write essays. I write proofs. • Respect: +381 ##### Re: 3U Maths Question Thread « Reply #2029 on: May 20, 2017, 12:52:09 am » 0 \text{Actually, here's a clever way of solving the resultant quartic}\\ \begin{align}(x-2)^2 + [x(x-4)]^2 &=4\\ (x^2-4x+4) + [x(x-4)]^2&=4\\ x(x-4) + x(x-4)x(x-4)&=0\\ x(x-4)(1+x(x-4))&=0\\ x(x-4)(1-4x+x^2)&=0\end{align} « Last Edit: May 20, 2017, 09:58:39 am by RuiAce » Interested in Mathematics Tutoring? Message for details! ATAR: 98.60 Currently studying: Bachelor of Science (Advanced Mathematics)/Bachelor of Science (Computer Science) @ UNSW Formerly studying: Bachelor of Actuarial Studies/Bachelor of Science (Advanced Mathematics) #### beau77bro • Forum Obsessive • Posts: 249 • Respect: +4 ##### Re: 3U Maths Question Thread « Reply #2030 on: May 20, 2017, 09:54:38 am » 0 \text{Actually, here's a clever way of solving the resultant quartic}\\ \begin{align}(x-2)^2 + [x(x-4)]^2 &=4\\ (x^2-4x+4) + [x(x-4)]^2&=4\\ x(x-4) - x(x-4)x(x-4)&=0\\ x(x-4)(1-x(x-4))&=0\\ x(x-4)(1+4x-x^2)&=0\end{align} sorry to ask this but where does the negative come from in the third line? sorry to ask this but where does the negative come from in the third lin #### RuiAce • ATAR Notes HSC Lecturer • Moderator • ATAR Notes Legend • Posts: 4847 • I don't write essays. I write proofs. • Respect: +381 ##### Re: 3U Maths Question Thread « Reply #2031 on: May 20, 2017, 09:58:47 am » 0 Fixed Interested in Mathematics Tutoring? Message for details! ATAR: 98.60 Currently studying: Bachelor of Science (Advanced Mathematics)/Bachelor of Science (Computer Science) @ UNSW Formerly studying: Bachelor of Actuarial Studies/Bachelor of Science (Advanced Mathematics) #### beau77bro • Forum Obsessive • Posts: 249 • Respect: +4 ##### Re: 3U Maths Question Thread « Reply #2032 on: May 20, 2017, 02:23:59 pm » +1 I've gotten to here but Idk where to go from there #### beau77bro • Forum Obsessive • Posts: 249 • Respect: +4 ##### Re: 3U Maths Question Thread « Reply #2033 on: May 20, 2017, 02:39:17 pm » +1 I forgot a negative #### beau77bro • Forum Obsessive • Posts: 249 • Respect: +4 ##### Re: 3U Maths Question Thread « Reply #2034 on: May 20, 2017, 02:49:37 pm » +1 Happy to say I finally got it #### RuiAce • ATAR Notes HSC Lecturer • Moderator • ATAR Notes Legend • Posts: 4847 • I don't write essays. I write proofs. • Respect: +381 ##### Re: 3U Maths Question Thread « Reply #2035 on: May 20, 2017, 03:01:33 pm » +1 Oh wow. I can't believe I forgot the possibility of something like this. I owe some people a big apology here. Here's a diagram to explain what was actually going on. I had to zoom in to make it obvious that there was actually a tangent going on Since the quartic is reducible to a quadratic, the quadratic formula can be used to avoid factorisation Interested in Mathematics Tutoring? Message for details! ATAR: 98.60 Currently studying: Bachelor of Science (Advanced Mathematics)/Bachelor of Science (Computer Science) @ UNSW Formerly studying: Bachelor of Actuarial Studies/Bachelor of Science (Advanced Mathematics) #### beau77bro • Forum Obsessive • Posts: 249 • Respect: +4 ##### Re: 3U Maths Question Thread « Reply #2036 on: May 20, 2017, 03:06:14 pm » 0 wait rui how did u work out f(x) was there a shortcut? #### RuiAce • ATAR Notes HSC Lecturer • Moderator • ATAR Notes Legend • Posts: 4847 • I don't write essays. I write proofs. • Respect: +381 ##### Re: 3U Maths Question Thread « Reply #2037 on: May 20, 2017, 03:12:58 pm » 0 wait rui how did u work out f(x) was there a shortcut? Nah not really. I'd stick to the discriminant method. Interested in Mathematics Tutoring? Message for details! ATAR: 98.60 Currently studying: Bachelor of Science (Advanced Mathematics)/Bachelor of Science (Computer Science) @ UNSW Formerly studying: Bachelor of Actuarial Studies/Bachelor of Science (Advanced Mathematics) #### bsdfjnlkasn • Forum Obsessive • Posts: 317 • Respect: +8 ##### Re: 3U Maths Question Thread « Reply #2038 on: May 20, 2017, 03:41:18 pm » 0 Hey there, I'm pretty sure that this is an easy probability question and I somehow managed to stumble to the correct answer but I would probably prefer a method by which to get the answer as I don't think my guessing is a good way to start and continue with the topic haha 50 tagged fish were released into a dam known to contain fish. Later a sample of thirty fish was netted from this dam, of which eight were found to be tagged. Estimate the total number of fish in the dam just prior to the sample of thirty being removed. The answer is 187 or 188 Thank you! #### RuiAce • ATAR Notes HSC Lecturer • Moderator • ATAR Notes Legend • Posts: 4847 • I don't write essays. I write proofs. • Respect: +381 ##### Re: 3U Maths Question Thread « Reply #2039 on: May 20, 2017, 03:57:23 pm » +1 Hey there, I'm pretty sure that this is an easy probability question and I somehow managed to stumble to the correct answer but I would probably prefer a method by which to get the answer as I don't think my guessing is a good way to start and continue with the topic haha 50 tagged fish were released into a dam known to contain fish. Later a sample of thirty fish was netted from this dam, of which eight were found to be tagged. Estimate the total number of fish in the dam just prior to the sample of thirty being removed. The answer is 187 or 188 Thank you! Ironically this question appears from just Google searching the first sentence. $\text{Let }x\text{ be the total number of fish in the dam}\\ \textit{just prior to the sample of 30 being removed}\\ \text{i.e. after 50 tagged, before 30 removed}\\ \text{This question is then just a ratio analysis question}$ $\text{At any time, the ratio }\frac{\text{No. of tagged fish}}{\text{No. of untagged fish}}\text{ is equal (for this estimate)}\\ \text{Hence, }\frac{8}{22} = \frac{50}{x-50}\\ \text{The solution is }x=187.5$ Interested in Mathematics Tutoring? Message for details! ATAR: 98.60 Currently studying: Bachelor of Science (Advanced Mathematics)/Bachelor of Science (Computer Science) @ UNSW Formerly studying: Bachelor of Actuarial Studies/Bachelor of Science (Advanced Mathematics)	2017-08-19 07:13:53	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 6, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7571930289268494, "perplexity": 5408.942576923666}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886105326.6/warc/CC-MAIN-20170819070335-20170819090335-00431.warc.gz"}
https://cs.stackexchange.com/questions/75473/is-there-an-algorithm-that-given-a-pda-m-decides-if-there-exists-a-string-in?noredirect=1	# Is there an algorithm that given a PDA $M$ decides if there exists a string in $L(M)$ with a suitable decomposition? I am looking for an algorithm that given a PDA $M$ decides whether there exists $w \in L(M)$ for which there exists a decomposition $w = uvxyz$ that satisfies $\|vy\| \geq 1$ and $uv^ixy^iz \in L(M)$ for every $i \geq 0$. • It appears you may have created multiple accounts. I encourage you to merge them. Also, I encourage you to register your account so you can retain access to it. This will allow you to post comments under an answer requesting clarification. (Those comments should not be posted in the 'answer' box.) – D.W. May 17 '17 at 17:11 In a context-free language $L$, by the Pumping Lemma, there exists a $p > 0$ such that every $w \in L$ such that $\|w\| \ge p$ admits a decomposition $w = uvxyz$ such that $uv^ixy^iz \in L$ for all $i \geq 0$. If $L$ is infinite, it has words of length $\ge p$, and you can pick any of them. If $L$ is finite then no word satisfies the condition (this is not a contradiction with the pumping lemma, because the pumping lemma gives a length $p$ which is more than the length of any of the words in $L$). It is decidable if a context-free grammar generates an infinite language (see e.g. Is it decidable whether a given context free grammar generates an infinite number of strings?). Therefore the algorithm is On input $\langle M \rangle$, where $M$ is a PDA 1. Convert $\langle M \rangle$ to the description of an equivalent CFG $\langle G \rangle$ 2. Check if $L(G)$ is infinite	2019-12-08 04:04:51	{"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": 0.7627089023590088, "perplexity": 132.56341207106288}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540504338.31/warc/CC-MAIN-20191208021121-20191208045121-00407.warc.gz"}
https://academy.vertabelo.com/course/ms-sql-group-by-extensions/rollup/rollup/rollup-with-coalesce	Introduction GROUP BY – Recap ROLLUP Summary ## Instruction Perfect! The last thing we'll show you in this part is how to get rid of those nasty NULL values in higher grouping levels. To that end, we'll use the COALESCE() function. SELECT COALESCE(FullName, N'All Contestants') AS FullName, COALESCE(Category, N'All Categories') AS Category, Week, AVG(Score) AS AvgScore FROM ContestScore GROUP BY ROLLUP(FullName, Category), Week; COALESCE() takes as many arguments as you wish and returns the first element that is not NULL. For instance, COALESCE(FullName, 'All Contestants') will produce either the respective FullName value or the string 'All Contestants' if FullName is NULL. Naturally, you can use any other text value instead of 'All Contestants'. ## Exercise Run the template query and see how it works.	2018-12-15 22:28:43	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.21561826765537262, "perplexity": 5261.838343493708}, "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-2018-51/segments/1544376827137.61/warc/CC-MAIN-20181215222234-20181216004234-00406.warc.gz"}
https://brilliant.org/problems/star-waltz/	# Star Waltz Two stars orbit their common center of mass as shown in the diagram below. The masses of the two stars are $$3M$$ and $$M$$. The distance between the stars is $$d$$. Determine the period of orbit for the star of mass $$3M$$. Note: This problem appeared on the 2009 F = ma exam. ×	2019-01-18 02:33:14	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.8047937154769897, "perplexity": 141.41729398786057}, "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/1547583659654.11/warc/CC-MAIN-20190118005216-20190118031216-00569.warc.gz"}
https://socratic.org/questions/how-can-i-read-a-c-13-nmr-spectrum	# How can I read a C^13 NMR spectrum? Feb 4, 2017 How many signals do you see? #### Explanation: If you see $\text{2,3,4,5}$ signals, there are $\text{2,3,4,5}$ different $\text{^13"C}$ environments. This may seem glib, but you can usually use symmetry to tell you how many signals you should observe in the $\text{^13"C{"""^1"H}}$ $\text{NMR}$ spectrum, and thereby decide a likely molecular structure. Methane (when dissolved in a suitable solvent) should reasonably gives rise to one signal in its ""^13"C"{""^1"H"} $\text{NMR}$ spectrum. Why? Because there is only one chemical environment. Ethane, again should give rise to the ONE signal; this will occur at a different chemical shift to that of methane. Clearly, the methyl groups are symmetric, and should respond equally to radiation. When we go to propane, however, ${\text{H"_3"CCH"_2"CH}}_{3}$, the spectrum should give rise to TWO signals. Why? Because there are 2 chemical environments: $\text{(i) methyl}$, ${\text{CH}}_{3}$; and "(ii) methylene", ${\text{CH}}_{2.}$ $\text{Butane}$, again should give rise to 2 signals. You will have to tell me why. $\text{Pentane,}$ ${\text{H"_3"C""{(CH"_2")"_3"}CH}}_{3}$ on the other hand, should give rise to 3 signals. I cannot hope to tell you all you need to know about $\text{NMR spectroscopy}$ in a few sentences (for a start, I don't know everything), but for the moment forget about WHERE the absorptions come (this is something the experiment tells you), and consider HOW MANY absorptions there should be. Aspects of symmetry will help you here; it will also help you in relation to the $\text{^1"H}$ $\text{NMR spectrum}$. So how many signals in the ""^13"C"{""^1"H"} $\text{NMR}$ spectrum should be seen for: $\left(i\right)$ $\text{H"_3"CCO"_2"H}$, $\left(i i\right)$, $\text{H"_3"CCH"_2"Cl}$, $\left(i i i\right)$ $\text{H"_3"CCH"_2"C"-="N}$, and $\left(i v\right)$, ${\text{H"_3"CCH"_2"CH"_2"CH"_2"CH}}_{3}$?	2023-03-29 16:28:50	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 28, "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": 0.8063941597938538, "perplexity": 695.5747778247122}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1679296949009.11/warc/CC-MAIN-20230329151629-20230329181629-00762.warc.gz"}
https://proofwiki.org/wiki/Category:Relational_Closures	Category:Relational Closures This category contains results about Relational Closures. Definitions specific to this category can be found in Definitions/Relational Closures. Let $A$ be a class. Let $\RR \subseteq A \times A$ be a relation. Let $x$ be a small class that is a subset of $A$. The relational closure of $x$ is the smallest small class containing $x$ that is also $\RR$-transitive. Pages in category "Relational Closures" The following 6 pages are in this category, out of 6 total.	2020-07-08 22:43:55	{"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": 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": 0.7935725450515747, "perplexity": 949.9515229629984}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655897707.23/warc/CC-MAIN-20200708211828-20200709001828-00214.warc.gz"}
https://mlr-org.com/posts/2017-03-22-usemlrmbotooptimizeviacommandline/index.html	Tutorial on using mlrMBO from the command line R r-bloggers Author Jakob Richter Published March 22, 2017 Many people who want to apply Bayesian optimization want to use it to optimize an algorithm that is not implemented in R but runs on the command line as a shell script or an executable. We recently published mlrMBO on CRAN. As a normal package it normally operates inside of R, but with this post I want to demonstrate how mlrMBO can be used to optimize an external application. At the same time I will highlight some issues you can likely run into. First of all we need a bash script that we want to optimize. This tutorial will only run on Unix systems (Linux, OSX etc.) but should also be informative for windows users. The following code will write a tiny bash script that uses bc to calculate $$sin(x_1-1) + (x_1^2 + x_2^2)$$ and write the result “hidden” in a sentence (The result is 12.34!) in a result.txt text file. ### The bash script # write bash script lines = '#!/bin/bash fun () { x1=$1 x2=$2 command="(s($x1-1) + ($x1^2 + $x2^2))" result=$(bc -l <<< $command) } echo "Start calculation." fun$1 $2 echo "The result is$result!" > "result.txt" echo "Finish calculation." ' writeLines(lines, "fun.sh") # make it executable: system("chmod +x fun.sh") ### Running the script from R Now we need a R function that starts the script, reads the result from the text file and returns it. library(stringi) runScript = function(x) { command = sprintf("./fun.sh %f %f", x[['x1']], x[['x2']]) error.code = system(command) if (error.code != 0) { } # the pattern matches 12 as well as 12.34 and .34 # the ?: makes the decimals a non-capturing group. result = stri_match_first_regex(result, pattern = "\\d*(?:\\.\\d+)?(?=\\!)") as.numeric(result) } This function uses stringi and regular expressions to match the result within the sentence. Depending on the output different strategies to read the result make sense. XML files can usually be accessed with XML::xmlParse, XML::getNodeSet, XML::xmlAttrs etc. using XPath queries. Sometimes the good old read.table() is also sufficient. If, for example, the output is written in a file like this: value1 = 23.45 value2 = 13.82 You can easily use source() like that: EV = new.env() eval(expr = {a = 1}, envir = EV) as.list(EV) source(file = "result.txt", local = EV) res = as.list(EV) rm(EV) which will return a list with the entries $value1 and $value2. ### Define bounds, wrap function. To evaluate the function from within mlrMBO it has to be wrapped in smoof function. The smoof function also contains information about the bounds and scales of the domain of the objective function defined in a ParameterSet. library(mlrMBO) ## Warning message: 'mlr' is in 'maintenance-only' mode since July 2019. Future development will only happen ## in 'mlr3' (<https://mlr3.mlr-org.com>). Due to the focus on 'mlr3' there might be uncaught bugs meanwhile ## in {mlr} - please consider switching. ## Warning: no DISPLAY variable so Tk is not available # Defining the bounds of the parameters: par.set = makeParamSet( makeNumericParam("x1", lower = -3, upper = 3), makeNumericParam("x2", lower = -2.5, upper = 2.5) ) # Wrapping everything in a smoof function: fn = makeSingleObjectiveFunction( id = "fun.sh", fn = runScript, par.set = par.set, has.simple.signature = FALSE ) # let's see if the function is working des = generateGridDesign(par.set, resolution = 3) des$y = apply(des, 1, fn) des ## x1 x2 y ## 1 -3 -2.5 16.006802 ## 2 0 -2.5 5.408529 ## 3 3 -2.5 16.159297 ## 4 -3 0.0 9.756802 ## 5 0 0.0 0.841471 ## 6 3 0.0 9.909297 ## 7 -3 2.5 16.006802 ## 8 0 2.5 5.408529 ## 9 3 2.5 16.159297 If you run this locally, you will see that the console output generated by our shell script directly appears in the R-console. This can be helpful but also annoying. ### Redirecting output If a lot of output is generated during a single call of system() it might even crash R. To avoid that I suggest to redirect the output into a file. This way no output is lost and the R console does not get flooded. We can simply achieve that by replacing the command in the function runScript from above with the following code: # console output file output_1490030005_1.1_2.4.txt output_file = sprintf("output_%i_%.1f_%.1f.txt", as.integer(Sys.time()), x[['x1']], x[['x2']]) # redirect output with ./fun.sh 1.1 2.4 > output.txt # alternative: ./fun.sh 1.1 2.4 > /dev/null to drop it command = sprintf("./fun.sh %f %f > %s", x[['x1']], x[['x2']], output_file) ### Start the Optimization Now everything is set so we can proceed with the usual MBO setup: ctrl = makeMBOControl() ctrl = setMBOControlInfill(ctrl, crit = crit.ei) ctrl = setMBOControlTermination(ctrl, iters = 10) configureMlr(show.info = FALSE, show.learner.output = FALSE) run = mbo(fun = fn, control = ctrl) ## Computing y column(s) for design. Not provided. ## [mbo] 0: x1=-1.35; x2=0.815 : y = 1.77 : 0.0 secs : initdesign ## [mbo] 0: x1=1.93; x2=-0.485 : y = 4.76 : 0.0 secs : initdesign ## [mbo] 0: x1=-2.61; x2=-1.66 : y = 9.98 : 0.0 secs : initdesign ## [mbo] 0: x1=-0.223; x2=0.239 : y = 0.833 : 0.0 secs : initdesign ## [mbo] 0: x1=0.373; x2=2.22 : y = 4.48 : 0.0 secs : initdesign ## [mbo] 0: x1=0.763; x2=-0.825 : y = 1.03 : 0.0 secs : initdesign ## [mbo] 0: x1=2.38; x2=1.31 : y = 8.39 : 0.0 secs : initdesign ## [mbo] 0: x1=-1.8; x2=-2.42 : y = 8.77 : 0.0 secs : initdesign ## [mbo] 1: x1=0.18; x2=-0.478 : y = 0.47 : 0.0 secs : infill_ei ## [mbo] 2: x1=-0.118; x2=-0.806 : y = 0.236 : 0.0 secs : infill_ei ## [mbo] 3: x1=0.242; x2=-1.39 : y = 1.31 : 0.0 secs : infill_ei ## [mbo] 4: x1=-0.5; x2=-0.462 : y = 0.535 : 0.0 secs : infill_ei ## [mbo] 5: x1=-0.109; x2=-0.64 : y = 0.474 : 0.0 secs : infill_ei ## [mbo] 6: x1=0.161; x2=-0.893 : y = 0.0784 : 0.0 secs : infill_ei ## [mbo] 7: x1=-3; x2=2.5 : y = 16 : 0.0 secs : infill_ei ## [mbo] 8: x1=-0.866; x2=-0.878 : y = 0.564 : 0.0 secs : infill_ei ## [mbo] 9: x1=-0.555; x2=0.939 : y = 0.189 : 0.0 secs : infill_ei ## [mbo] 10: x1=0.15; x2=0.806 : y = 0.0787 : 0.0 secs : infill_ei # The resulting optimal configuration: run$x ## $x1 ## [1] 0.1609242 ## ##$x2 ## [1] -0.8925002 # The best reached value: run\$y ## [1] 0.07842672 ### Execute the R script from a shell Also you might not want to bothered having to start R and run this script manually so what I would recommend is saving all above as an R-script plus some lines that write the output in a JSON file like this: library(jsonlite) write_json(run[c("x","y")], "mbo_res.json") Let’s assume we saved all of that above as an R-script under the name runMBO.R (actually it is available as a gist). Then you can simply run it from the command line: Rscript runMBO.R As an extra the script in the gist also contains a simple handler for command line arguments. In this case you can define the number of optimization iterations and the maximal allowed time in seconds for the optimization. You can also define the seed to make runs reproducible: Rscript runMBO.R iters=20 time=10 seed=3 If you want to build a more advanced command line interface you might want to have a look at docopt. ### Clean up To clean up all the files generated by this script you can run: file.remove("result.txt") file.remove("fun.sh") file.remove("mbo_res.json") output.files = list.files(pattern = "output_\\d+_[0-9_.-]+\\.txt") file.remove(output.files)	2022-12-05 20:22:52	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.418026864528656, "perplexity": 3913.442099479428}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1669446711045.18/warc/CC-MAIN-20221205200634-20221205230634-00813.warc.gz"}
https://hal.archives-ouvertes.fr/hal-01141801	# Reflected BSDEs when the obstacle is not right-continuous and optimal stopping Abstract : In the first part of the paper, we study reflected backward stochastic differential equations (RBSDEs) with lower obstacle which is assumed to be right upper-semicontinuous but not necessarily right-continuous. We prove existence and uniqueness of the solutions to such RBSDEs in appropriate Banach spaces. The result is established by using some tools from the general theory of processes such as Mertens decomposition of optional strong (but not necessarily right-continuous) supermartingales, some tools from optimal stopping theory, as well as an appropriate generalization of Itô's formula due to Gal'chouk and Lenglart. In the second part of the paper, we provide some links between the RBSDE studied in the first part and an optimal stopping problem in which the risk of a financial position $\xi$ is assessed by an $f$-conditional expectation $\mathcal{E}^f(\cdot)$ (where $f$ is a Lipschitz driver). We characterize the "value function" of the problem in terms of the solution to our RBSDE. Under an additional assumption of left upper-semicontinuity on $\xi$, we show the existence of an optimal stopping time. We also provide a generalization of Mertens decomposition to the case of strong $\mathcal{E}^f$-supermartingales. Keywords : Type de document : Pré-publication, Document de travail 2015 Domaine : https://hal.archives-ouvertes.fr/hal-01141801 Contributeur : Miryana Grigorova <> Soumis le : lundi 13 avril 2015 - 17:37:59 Dernière modification le : mardi 11 octobre 2016 - 14:10:29 Document(s) archivé(s) le : lundi 14 septembre 2015 - 07:55:31 ### Fichiers RBSDEs with discontinuous barr... Fichiers produits par l'(les) auteur(s) ### Identifiants • HAL Id : hal-01141801, version 1 • ARXIV : 1504.06094 ### Citation Miryana Grigorova, Peter Imkeller, Elias Offen, Youssef Ouknine, Marie-Claire Quenez. Reflected BSDEs when the obstacle is not right-continuous and optimal stopping. 2015. <hal-01141801> Consultations de la notice ## 160 Téléchargements du document	2017-04-23 19:48:42	{"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": 0.6463187336921692, "perplexity": 1994.0181152798555}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917118743.41/warc/CC-MAIN-20170423031158-00612-ip-10-145-167-34.ec2.internal.warc.gz"}
https://www.zbmath.org/authors/?q=rv%3A17206	# zbMATH — the first resource for mathematics ## Loran, Farhang Compute Distance To: Author ID: loran.farhang Published as: Loran, Farhang; Loran, F. External Links: Google Scholar · ORCID Documents Indexed: 28 Publications since 2001 Reviewing Activity: 23 Reviews all top 5 all top 5 #### Serials 6 Physics Letters. B 4 International Journal of Modern Physics A 3 Classical and Quantum Gravity 3 Journal of Mathematical Physics 3 Journal of High Energy Physics 2 Modern Physics Letters A 1 Communications in Mathematical Physics 1 Nuclear Physics. B 1 Annals of Physics 1 Advances in Theoretical and Mathematical Physics 1 Journal of Physics A: Mathematical and Theoretical 1 Proceedings of the Royal Society of London. A. Mathematical, Physical and Engineering Sciences all top 5 #### Fields 22 Quantum theory (81-XX) 7 Relativity and gravitational theory (83-XX) 5 Mechanics of particles and systems (70-XX) 2 Differential geometry (53-XX) 1 Algebraic geometry (14-XX) 1 Topological groups, Lie groups (22-XX) 1 Several complex variables and analytic spaces (32-XX) 1 Partial differential equations (35-XX) 1 Functional analysis (46-XX) 1 Optics, electromagnetic theory (78-XX) 1 Statistical mechanics, structure of matter (82-XX) #### Citations contained in zbMATH Open 19 Publications have been cited 73 times in 63 Documents Cited by Year 5D extremal rotating Black holes and CFT duals. Zbl 1172.83341 Loran, Farhang; Soltanpanahi, Hesam 2009 Classification of constraints using the chain-by-chain method. Zbl 1042.81089 2002 Beyond logarithmic corrections to Cardy formula. Zbl 1214.81224 Loran, Farhang; Sheikh-Jabbari, M. M.; Vincon, Massimiliano 2011 Non-chiral 2d CFT with integer energy levels. Zbl 1390.81220 Ashrafi, M.; Loran, F. 2016 Gravity/CFT correspondence for three-dimensional Einstein gravity with a conformal scalar field. Zbl 1262.83010 Hasanpour, M.; Loran, F.; Razaghian, H. 2013 Abelian subset of second class constraints. Zbl 1008.22009 Loran, F. 2003 Comment on “Constraint quantization of open string in background $$B$$ field and noncommutative D-brane”. Zbl 0997.81123 Loran, F. 2002 $$\varphi^{4}$$-model and holography in four dimensions. Zbl 1247.81256 Loran, Farhang 2005 Gauge fixing in the chain-by-chain method. Zbl 1026.81035 2002 Non-abelianizable first class constraints. Zbl 1079.81064 Loran, Farhang 2005 Unidirectional invisibility and non-reciprocal transmission in two and three dimensions. Zbl 1371.78039 2016 Orientifolded locally AdS$$_{3}$$ geometries. Zbl 1207.83043 Loran, F.; Sheikh-Jabbari, M. M. 2011 Exact solution of the two-dimensional scattering problem for a class of $$\delta$$-function potentials supported on subsets of a line. Zbl 1397.81424 2018 $$\phi^4$$ model on a circle. Zbl 1273.81149 Loran, F. 2007 Holographic anomaly in 3D $$f$$(Ric) gravity. Zbl 1277.83089 Loran, Farhang 2013 Abelianization of first class constraints. Zbl 0999.81051 Loran, F. 2002 Fubini vacua as a classical de Sitter vacua. Zbl 1198.83023 Loran, F. 2007 Fermionic one-particle states in curved spacetimes. Zbl 1395.83034 Loran, Farhang 2018 Chiral fermions on 2D curved space-times. Zbl 1372.81137 Loran, Farhang 2017 Exact solution of the two-dimensional scattering problem for a class of $$\delta$$-function potentials supported on subsets of a line. Zbl 1397.81424 2018 Fermionic one-particle states in curved spacetimes. Zbl 1395.83034 Loran, Farhang 2018 Chiral fermions on 2D curved space-times. Zbl 1372.81137 Loran, Farhang 2017 Non-chiral 2d CFT with integer energy levels. Zbl 1390.81220 Ashrafi, M.; Loran, F. 2016 Unidirectional invisibility and non-reciprocal transmission in two and three dimensions. Zbl 1371.78039 2016 Gravity/CFT correspondence for three-dimensional Einstein gravity with a conformal scalar field. Zbl 1262.83010 Hasanpour, M.; Loran, F.; Razaghian, H. 2013 Holographic anomaly in 3D $$f$$(Ric) gravity. Zbl 1277.83089 Loran, Farhang 2013 Beyond logarithmic corrections to Cardy formula. Zbl 1214.81224 Loran, Farhang; Sheikh-Jabbari, M. M.; Vincon, Massimiliano 2011 Orientifolded locally AdS$$_{3}$$ geometries. Zbl 1207.83043 Loran, F.; Sheikh-Jabbari, M. M. 2011 5D extremal rotating Black holes and CFT duals. Zbl 1172.83341 Loran, Farhang; Soltanpanahi, Hesam 2009 $$\phi^4$$ model on a circle. Zbl 1273.81149 Loran, F. 2007 Fubini vacua as a classical de Sitter vacua. Zbl 1198.83023 Loran, F. 2007 $$\varphi^{4}$$-model and holography in four dimensions. Zbl 1247.81256 Loran, Farhang 2005 Non-abelianizable first class constraints. Zbl 1079.81064 Loran, Farhang 2005 Abelian subset of second class constraints. Zbl 1008.22009 Loran, F. 2003 Classification of constraints using the chain-by-chain method. Zbl 1042.81089 2002 Comment on “Constraint quantization of open string in background $$B$$ field and noncommutative D-brane”. Zbl 0997.81123 Loran, F. 2002 Gauge fixing in the chain-by-chain method. Zbl 1026.81035 2002 Abelianization of first class constraints. Zbl 0999.81051 Loran, F. 2002 all top 5 #### Cited by 88 Authors 9 Loran, Farhang 7 Monemzadeh, Majid 6 Shirzad, Ahmad 4 Compère, Geoffrey 3 Ebrahimi, Aghileh S. 3 Mostafazadeh, Ali 3 Naji, Jalil 3 Sheikh-Jabbari, Mohammad Mahdi 2 Das, Diptarka 2 Dehghani, Mehdi 2 Nejad, Salman Abarghouei 2 Peng, Jun-Jin 2 Simón, Joan 2 Wu, Shuangqing 2 Yavartanoo, Hossein 1 Afshar, Hamid Reza 1 Alba, Vincenzo 1 Alishahiha, Mohsen 1 Bagchi, Arjun 1 Băleanu, Dumitru I. 1 Brehm, Enrico M. 1 Bui, Hai Viet 1 Bytsenko, Andrei A. 1 Calabrese, Pasquale 1 Chen, Bin 1 Chen, Deyou 1 Dariescu, Ciprian 1 Dariescu, Marina-Aura 1 Datta, Shouvik 1 Defterli, Ozlem 1 Deng, Xijun 1 Duplij, Steven 1 Dymarsky, Anatoly 1 E, Yan 1 Erdmenger, Johanna 1 Flory, Mario 1 Golchin, Hanif 1 Grumiller, Daniel 1 Hajian, Kamal 1 Hartman, Thomas E. 1 Heshmatian, S. 1 Iliesiu, Luca V. 1 Jin, Shuo 1 Jing, Jian 1 Johnstone, Maria 1 Koohsarian, Y. 1 La Fuente, Anton De 1 Lee, Bum-Hoon 1 Lee, Wonwoo 1 Li, Xi 1 Long, Zhengwen 1 Lou, Yijun 1 Lusanna, Luca 1 Mahanta, Ratul 1 Maharana, Anshuman 1 Mazáč, Dalimil 1 Mojiri, Mohsen 1 Naseh, Ali 1 Newrzella, Max-Niklas 1 Nikoofard, Vahid 1 Oh, Changheon 1 Plakhov, A. Yu 1 Qaemmaqami, Mohammad M. 1 Raman, Madhusudhan 1 Ramezani-Arani, Reza 1 Rastelli, Leonardo 1 Ro, Daeho 1 Roshchina, Vera 1 Saadat, Hassan 1 Saha, Amartya 1 Sarvi, Fahimeh 1 Seraj, Ali 1 Shapere, Alfred D. 1 Song, Wei 1 Sramadi, S. 1 Stelea, Cristian 1 Strominger, Andrew Eben 1 Sundrum, Raman 1 Tian, Lijun 1 Tonni, Erik 1 Tureanu, Anca 1 Turiaci, Gustavo J. 1 Wang, Peng 1 Wu, Houwen 1 Yang, Haitang 1 Yeom, Dong-Han 1 Zhang, Jia-ju 1 Zodinmawia all top 5 #### Cited in 19 Serials 18 Journal of High Energy Physics 6 Modern Physics Letters A 6 International Journal of Modern Physics A 5 International Journal of Theoretical Physics 4 Advances in High Energy Physics 3 General Relativity and Gravitation 3 Journal of Mathematical Physics 3 Physics Letters. B 2 Nuclear Physics. B 2 Living Reviews in Relativity 2 International Journal of Geometric Methods in Modern Physics 2 Journal of Physics A: Mathematical and Theoretical 1 Applied Mathematics and Computation 1 Mathematical and Computer Modelling 1 Annals of Physics 1 Journal of Dynamical and Control Systems 1 Czechoslovak Journal of Physics 1 Journal of Statistical Mechanics: Theory and Experiment 1 The European Physical Journal C. Particles and Fields all top 5 #### Cited in 19 Fields 47 Quantum theory (81-XX) 34 Relativity and gravitational theory (83-XX) 12 Mechanics of particles and systems (70-XX) 5 Information and communication theory, circuits (94-XX) 3 Nonassociative rings and algebras (17-XX) 3 Dynamical systems and ergodic theory (37-XX) 2 Partial differential equations (35-XX) 2 Statistics (62-XX) 2 Optics, electromagnetic theory (78-XX) 2 Classical thermodynamics, heat transfer (80-XX) 2 Statistical mechanics, structure of matter (82-XX) 1 Combinatorics (05-XX) 1 Number theory (11-XX) 1 Special functions (33-XX) 1 Integral transforms, operational calculus (44-XX) 1 Functional analysis (46-XX) 1 Convex and discrete geometry (52-XX) 1 Global analysis, analysis on manifolds (58-XX) 1 Astronomy and astrophysics (85-XX)	2021-09-21 01:38:59	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.35657721757888794, "perplexity": 10067.985828519017}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1631780057131.88/warc/CC-MAIN-20210921011047-20210921041047-00261.warc.gz"}
https://brilliant.org/discussions/thread/buoyant-force/	This note has been used to help create the Fluid Mechanics wiki Buoyant force (or buoyancy) is the net force on a body caused by the pressure differences in the surrounding medium caused by gravity. Buoyant force acts through the centre of gravity of the displaced fluid (the centre of buoyancy). If the centre of buoyancy is not on the same vertical line as the centre of gravity, there will be a torque (a turning force). Archimedes' Principle: The buoyant force on a body is vertical, and is equal and opposite to the weight of the fluid displaced by the body. Archimedes' principle: From Archimedes' principle we see that the buoyant force on an object is equal to the weight of the fluid it displaces. Why is this so? Let us imagine a cube of any substance of side length h. Fluid around the cube exerts pressure on each side of the cube. On the perpendicular faces the force cancels out, but on the top and bottom faces there is a pressure differential due to the variation of pressure with depth in fluid. If we call the depth of the top face xt and the depth of the bottom face xb, then the pressure differential across the cube can be written as: ΔP=ρ{f}g(x{b}−x{t})=ρ{f}gh Since, x{b}−x{t}=h Since pressure is a force per unit area the pressure differential must be the same as the buoyant force divided by the area of one of the cube's faces. In other words we can write the buoyant force as being the pressure differential multiplied by the area of one face of the cube. F_{B}=(ΔP)h^{2} F{B}=(ρ{f}gh)h^{2}=ρ{f}V{g}=M_{fg} This shows that by considering pressure differentials we have arrived at the same result Archimedes' principle tells us we should arrive at. Alternatively, consider the forces which would be on the displaced fluid if it were still there …the only forces are the weight of the displaced fluid, and the pressure from the surrounding fluid …if the fluid is in equilibrium, then these forces must be equal and opposite …and the pressure from the surrounding fluid on the body is the same as it would be on the displaced fluid. Centre of buoyancy: The buoyant force acts through the centre of buoyancy. The centre of buoyancy of a body is at the centre of gravity of the fluid displaced (for a fluid of uniform density, that is at the centroid of the displaced volume). Water, even at great depths, may be taken to be of uniform density. If the body is totally immersed in a fluid of uniform density, it is at the same point, whatever the orientation of the body. Note that the centre of gravity of the body may change if the body is not rigid (for example, if it contains shifting cargo). If the body also is of uniform density, it is at the centre of gravity (the centre of mass) of the body. If the body is floating at the surface, the shape of the volume of fluid displaced will change as the orientation changes, and so the centre of buoyancy will change. For stability, the centre of buoyancy should be above the centre of gravity. Total immersion ("underwater"): If the body is totally immersed, then the volume of fluid displaced is equal to the volume of the body, and so the buoyant force (measured vertically upward) is that volume times the density of the fluid minus the average density of the body: F{b} = V{g}(ρ{fluid} − ρ{body}) = mg(1 − ρ{body}/ρ{fluid}) and the acceleration upward will be: g(1 − ρ{body}/ρ{fluid}) Obviously, if the body has lesser average density than the fluid, that acceleration will be positive, and the body will rise. And if it has greater average density, that acceleration will be negative, and the body will sink. Partial immersion ("floating") of a uniform body: If a body of uniform density is floating, in equilibrium, only partially immersed, at the surface of the fluid, then the volume of fluid displaced is less than the volume of the body, and so the proportion submerged equals the ratio of the densities. A ship is not of uniform density (its volume is mostly filled with air), and so there is no such simple relationship. Strictly, a ship is totally immersed, in water and air, but the weight of the displaced air is so small, compared with the weight of the displaced water, that it can be ignored. If such a floating body is pushed down by an external force, the volume displaced will increase, and so the body will feel an increased buoyant force. This increased force, together with the weight of the body, provides a restoring force, tending to bring the body back to its equilibrium position. If the body has a uniform vertical cross-section, this restoring force will be proportional to depth, and so the body will bob up and down in simple harmonic motion (shm). If a floating body is pushed sideways by an external force, the volume displaced will depend on the shape of the body … the stability of the body will be greater if the cross-section is nearer an arc of a circle. Note by Agnibha Sen 3 years, 11 months ago MarkdownAppears as italics or _italics_ italics bold or __bold__ bold - bulleted- list • bulleted • list 1. numbered2. list 1. numbered 2. list Note: you must add a full line of space before and after lists for them to show up correctly paragraph 1paragraph 2 paragraph 1 paragraph 2 [example link](https://brilliant.org)example link > This is a quote This is a quote # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" MathAppears as Remember to wrap math in $$...$$ or $...$ to ensure proper formatting. 2 \times 3 $$2 \times 3$$ 2^{34} $$2^{34}$$ a_{i-1} $$a_{i-1}$$ \frac{2}{3} $$\frac{2}{3}$$ \sqrt{2} $$\sqrt{2}$$ \sum_{i=1}^3 $$\sum_{i=1}^3$$ \sin \theta $$\sin \theta$$ \boxed{123} $$\boxed{123}$$	2018-04-26 15:44:26	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.8606694936752319, "perplexity": 412.6532116531249}, "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-2018-17/segments/1524125948285.62/warc/CC-MAIN-20180426144615-20180426164615-00310.warc.gz"}
https://oreillymedia.github.io/Using_SVG/guide/units.html	Using SVG with CSS3 and HTML5 — Supplementary Material Example code and online extras for the O'Reilly Media book by Amelia Bellamy-Royds, Kurt Cagle, and Dudley Storey. # Units for Measurements The length units you can use to specify positions and coordinates in SVG can be separated into absolute units, which have explicit conversion factors, and relative units, which depend on other properties of the layout. As discussed in the text, the newer CSS units should be used with caution for SVG. Percentage units have special meaning depending on the attribute or style property. However, most SVG length values use a common interpretation of percentages, that depends on whether the length is horizontal, vertical, or other. Units are also defined for angle measurements and time measurements. ## Absolute Length Units# The following absolute length units are defined in CSS 2 and SVG 1 and 1.1. They should be supported in nearly all SVG implementations. However, the relationship between the px unit and the other absolute units was defined in CSS 2.1; in older software, the relationship between px and other units will fluctuate according to the screen resolution. px Pixel units, directly equivalent to SVG user units. • For print, a px should be equal to 1/96th of an inch. • For screens, a px should represent approximately the same distance in the user’s field of view (the same visual angle) as 1/96th of an inch at arm’s length. • The px unit can be adjusted to represent an even number of actual screen pixels. in Inches. • 1in = 96px or user units • 1in = 2.54cm • In modern software, inches and all other absolute units will be adjusted to match the px unit. cm Centimeters. • 1cm ≅ 37.795px or user units mm Millimeters. • 1mm ≅ 3.7795px or user units pt Points. • 1pt ≅ 1.3333px or user units (1px = 0.75pt) • 1pt = 1/72in pc Picas. • 1pc = 16px or user units • 1pc = 1/6in ## Font-Relative Length Units# The following units describe lengths proportional to the size and properties of the font set on the element. They are defined in CSS 2 and SVG 1 and 1.1, and should be well supported for SVG. However, some SVG viewers do not apply a reasonable default font size for SVG, so test carefully—font-relative units may be unusually small or large if you don’t specify font-size yourself! em Em units. Equivalent to the computed font-size in effect for an element. ex Ex units. Equivalent to the height of a lower-case letter in the font (and font-size) in effect for an element. If the font doesn’t include lower-case letters, or doesn’t include the metadata about the ex-height, then 1ex = 0.5em. The following additional font-relative units are defined in the CSS 3 Values and Units specification; they may not be supported correctly in some SVG software. ch Character units. Equivalent to the width of the 0 (zero) character in the font and font-size in effect for an element. rem Root-em units. The font size of the root element in the document, regardless of any settings for the current element. When styles defined with relative lengths are inherited, it is the computed value of the length that is inherited, not the relationship to the current font. In other words, the value is not affected if the child element has a different font or font-size. ## Viewport Length Units# The CSS 3 Values and Units specification also defines viewport units, which are relative to the viewport used to draw the entire document. Similar to the media queries discussed in Chapter 3, the size of the document region for images and embedded objects is different than for inline SVG. Viewport units also may not be well supported in SVG. vw Viewport width units. 1% of the viewport width. vh Viewport height units. 1% of the viewport height. vmin Viewport minimum units. 1% of the viewport width or height, whichever is smaller. vmax Viewport maximum units. 1% of the viewport width or height, whichever is larger. ## Percentage Lengths# In SVG, most percentages are calculated relative to the height and width of the nearest ancestor element that defines a coordinate system. As described in Chapter 8, the height and width refer to the scaled viewBox graphic region, not the available space. Percentages for width or x-position are relative to the coordinate system width, percentages for height or y-position are relative to the coordinate system height, and percentages for all other properties are calculated as follows: ## Angle Units# In SVG 1, there were no angle units. Angles were always specified only as numbers, and implicitly meant degrees. The angles in CSS hsl() color notation work the same way. (In contrast, in JavaScript trigonometry, angles are always implicitly radians.) CSS Values and Units Level 3 adds angle units. They are used in the CSS transform syntax and CSS linear gradients. SVG 2 adopts them for other angle-related properties and attributes—but for best support, continue to use unitless degrees for most SVG attributes. Positive angles are normally clockwise in both CSS and SVG (unless a negative scale has created a mirror reflection). In most CSS contexts, 0° is straight up. In most SVG contexts, 0° is pointing right, towards the positive x-axis. You can usually use values less than 0 or greater than a full circle. When animating and transitioning angle values, this allows you to control the direction of transition and the number of times something rotates. deg Degrees. We use the symbol version (°) in the book text. • 360deg is a full-circle turn • 90deg is a right-angle rad • 1rad is the angle such that a circular arc has the same path length as its radius. grad Gradians. (A rarely-used metric degree unit.) • 400grad is a full-circle turn turn Turns. • 1turn is a full-circle • 0.25turn is a right-angle ## Time Units# SVG 1 used time units from SMIL for the SVG/SMIL animation elements. CSS uses a more limited set of time units for animations and transitions. Most JavaScript time-related methods use milliseconds (implicitly, as a number). However, the SVG/SMIL and CSS animation DOM methods use seconds. The following units can be used in both CSS animations and SVG/SMIL animations: s Seconds. ms Milliseconds. • 1000ms = 1s The following units and formats are also valid in SVG/SMIL: min Minutes. • 1min = 60s h Hours. • 1h = 60min = 3600s mm:ss Minutes and seconds. hh:mm:ss Hours, minutes, and seconds. • These are both implicit units based on the number format with : • The minutes and seconds parts must always have 2 digits, between 00 and 59 • The seconds part can optionally have a decimal part, like 00:02.5 for 2.5s In time-related SVG/SMIL attributes, numbers without units are treated as an amount in seconds.	2022-05-20 04:57:52	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.5200645923614502, "perplexity": 3451.0733513921746}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1652662531352.50/warc/CC-MAIN-20220520030533-20220520060533-00397.warc.gz"}
http://mathoverflow.net/revisions/119423/list	(Addressing only the title question.) There is a short proof avoiding Peter-Weyl and the theory of compact operators. It is due to Nachbin and is reproduced in Hewitt-Ross, Abstract Harmonic Analysis 1, p. 344. A slight simplification of it runs as follows: Pick a unit vector $v$ in your representation space $V$. Schur's lemma gives $$\int_G gv(gv,\cdot)dg = \lambda1 \tag 1$$ where $\lambda = \int_G\|(v,gv)\|^2dg>0$ (sandwiching (1) with $(v,\cdot v)$). Now let $W\subset V$ be finite-dimensional, and write $E=E^2$ for the orthogonal projection $V\to W$. We get $$\int_GEgv(gv,E\cdot)dg = \lambda E, \tag 2$$ whence, taking traces in (2), $\lambda\dim(W)=\int_G\|\|Egv\|\|^2dg\leqslant\operatorname{vol}(G)$. Thus, the dimension of any finite-dimensional subspace is bounded, as was to be shown. (Addressing only the title question.) There is a short proof avoiding Peter-Weyl and the theory of compact operators. It is due to Nachbin and reproduced in Hewitt-Ross, Abstract Harmonic Analysis 1, p. 344. A slight simplification of it runs as follows: Pick a unit vector $v$ in your representation space $V$. Schur's lemma gives $$\int_G gv(gv,\cdot)dg = \lambda1 \tag 1$$ where $\lambda = \int_G\|(v,gv)\|^2dg>0$ (sandwiching (1) with $(v,\cdot v)$). Now let $W\subset V$ be finite-dimensional, and write $E=E^2$ for the orthogonal projection $V\to W$. We get $$\int_GEgv(gv,E\cdot)dg = \lambda E, \tag 2$$ whence, taking traces in (2), $\lambda\dim(W)=\int_G\|\|Egv\|\|^2dg\leqslant\operatorname{vol}(G)$. Thus, the dimension of any finite-dimensional subspace is bounded, as was to be shown.	2013-05-25 06:07:51	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9733263850212097, "perplexity": 211.01230707837962}, "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-2013-20/segments/1368705559639/warc/CC-MAIN-20130516115919-00009-ip-10-60-113-184.ec2.internal.warc.gz"}
https://mohammadshariatmadari.com/rh7iqb8u/cmb-power-spectrum-and-inflation-4b2ff5	CMB power spectrum from the world's combined data, including the recent WMAP satellite results (Hinshaw et al. Much of the recent progress in cosmology has come from studying the power spectrum of the cosmic microwave background (CMB). Dodelson 2003 . Based on vanilla inflation, most theorists thought that the flatness Moreover, the history of the early universe, its early years and the trigger that initiates the current epoch are all easily resolved. In our previous work, we presented an alternative explanation for the matter power spectrum based on nonperturbative … Power spectra always play an important role in the theory of inflation. spectrum yield r < 0.71 at the 95% confidence level. Topological defects are predicted by most inflationary theories involving symmetry breaking in the early Universe. A power spectrum plot of the CMB yields five curves that are from left to right recognized as dark energy, observable matter, dark matter, matter without identification, and finally visible matter. 4.8. contribution to the CMB power spectrum. Liddle & Lyth 2000). have begun to constrain models of the inflaton potential (Table 1 and 10 comments. CMB data n Status of natural inflation ... n The observed B-mode power spectrum is well- fit by a lensed-Lambda CDM + tensor theoretical model with tensor/scalar ratio r = 0.20+0.07-0.05 , with r = 0 disfavored at 7.0 sigma. One such feature is the well known suppression of the $$\ell =2$$ moment of the CMB power spectrum observed both by Planck [ 14 ] and by the Wilkinson Microwave Anisotropy Probe ( WMAP ) [ 16 ]. Inflationary models and the new ekpyrotic models make different The constants of nature are the same everywhere. Viability of Slow-Roll Inflation in Light of the Non-Zero k min Measured in the CMB Power Spectrum. 2003 002 ()] have suggested that an initial period of kinetic energy domination in single field inflation may explain the lack of CMB power at large angular scales. Fig.2: Angular power spectrum of CMB temperature fluctuations. It moves the Abstract: In this paper, we investigate some consequences of multiple-field inflation for the cosmic microwave background radiation (CMB). MQ resolves the same values using only the classical notions described by the Standard Model. This thread is archived. The observational scorecard of inflation is mixed. We use Nambu-Goto and field theory simulations for cosmic strings and domain walls respectively, and we determine the power spectra they produce with a modified Einstein-Boltzmann solver sourced by unequal time correlators from components of the energy-momentum tensor of the defects. We use Nambu-Goto and field theory simulations for cosmic strings and domain walls respectively, and we determine the power spectra they produce with a modified Einstein-Boltzmann solver sourced by unequal time correlators from components of the energy-momentum tensor of the defects. 1. temperatures, flat geometry and seeds of structure need an explanation. To characterize the pure power law primordial spectrum, one considers two parameters: n S and the amplitude. be extracted by measuring the angular power spectrum of the CMB anisotropies. whether the Universe could have been other than what it is. Inflationary "zoology" and the CMB. Even so, additional parameters, such as cosmic defects, are still allowed by current observational data. Angular power spectrum of the FastICA CMB component from BEAST data Donzelli, S., et.al., 2006, MNRAS, 369, 441D ADS / astro-ph. have the opposite. Measurements of CMB polarization over the next five years CMB data [7] moderately favor a 10% contribution from strings to the temperature power spectrum measured at multipole ‘ 10 with a corresponding spectral index of primordial scalar perturbations ns ’ 1. 1 - score one for inflation. This power is called the "l = 1" contribution to the power spectrum. In particular, the ability to reproduce the galaxy matter power spectrum P ( k ) and the CMB temperature angular power spectrum C l ’s to high accuracy is often considered a triumph of inflation. Moreover, the history of the early universe, its early years and the trigger that initiates the current epoch are all easily resolved. The fluctuations in the temperature of the cosmic microwave background (CMB) are a snapshot of the distribution of matter at a much later cosmic epoch than inflation, as they date back to 380,000 years after inflation ended. No CMB experiment is sensitive to this entire range of angular scale. inflaton potential is so flat." flatness problem. The latest results from the Planck satellite confirmed that the inflationary paradigm with the $\Lambda$CDM six-parameter model provides a very good description of the observed structures in the Universe. Dodelson 2003 . If these uniformities need no explanation then why should the uniform Browse our catalogue of tasks and access state-of-the-art solutions. contrary. No code available yet. best. A possible such feature was identified in the power spectrum of galaxy clustering in the automated plate measurement (APM)Q2 survey at the scale k~0.1hMpc -1 and it was shown that the secondary acoustic peaks in the power spectrum of the cosmic microwave background (CMB)Q2 anisotropy should consequently be suppressed. Models of inflation usually consist of choosing a form for the potential In particular, the ability to reproduce the galaxy matter power spectrum and the CMB temperature angular power spectrum coefficients to high accuracy is often considered a triumph of inflation. The results do not rule out either single or multi field models of slow-roll inflation. Imagine you could pluck it once in a while to add more vibration modes incoherently. Minute, random quantum fluctuations in the structure of the Universe that were present at the moment when inflation started, were amplified up to cosmologically large scales during inflation. (1997b) have noted that according to our current understanding of the unification of fundamental interactions, there should have been phase transitions associated with spontaneous symmetry breaking during the inflationary era. respect to is Is this first group more fundamental than the second? one for inflation. Inflation doesn't solve the fine-tuning problem. We derive expressions for the amplitudes, the spectral indices and the derivatives of the indices of the CMB power spectrum in the context of a very general multiple-field theory of slow-roll inflation, where the field metric can be non-trivial. After all, the electron mass is the same everywhere. solutions can be devised, whose explanatory power depends on On the APM power spectrum and the CMB anisotropy: evidence for a phase transition during inflation? - score one for the observers. The spherical-harmonic multipole number, , is conjugate to the separation angle . models Quantum Inflation & CMB Formation: using MQ we may approach the evolution of a quantum singularity using classical mechanics to resolve the quantity, age, present density and temperature of the CMB. single broad peak in the CMB power spectrum . The ini-tial power spectrum we assume is written as Ps =Askn−1 (1) 1 + i αi exp − (k−kci)2 2σ2 i, where As and nare the amplitudeof primordialﬂuctu-ation and the spectral index as usually assumed. Much of the recent progress in cosmology has come from studying the power spectrum of the cosmic microwave background (CMB). It is commonly assumed by astronomers that inflation predicts a scale-invariant ‘Harrison-Zeldovich’H−Z spectrum of scalar density perturbations, with n = 1; for example this is a standard input in calculations of the expected large-scale structure (LSS) and … CMB data n Status of natural inflation ... n The observed B-mode power spectrum is well- fit by a lensed-Lambda CDM + tensor theoretical model with tensor/scalar ratio r = 0.20+0.07-0.05 , with r = 0 disfavored at 7.0 sigma. This is not the only way that the string can vibrate. All values match to the same precision as our best measur This leads to a prediction for So far most of inflation's predictions have been retrodictions - explaining The Universe has now been measured to be flat to … The CMB Power Spectrum is traditionally interpreted using well-known principles of light with respect to a model of events identified as Lambda CDM. The inflation may not be due to a scalar field This in turn reveals the amount ofenergy emitted by different sized "ripples" of sound echoing through the early matter ofthe universe. modes if they exist at a detectable level. This may have resulted in the breaking of scale-invariance of the primordial density perturbation for brief periods. New comments cannot be posted and votes cannot be cast. Galactic foreground contribution to the BEAST CMB Anisotropy Maps Mejía, Jorge, et.al., 2005, ApJS, 158, 109M ADS / astro-ph. We propose a phenomenological ‘three-shape’ model, based on the fundamental shapes we have observed by studying the halo model that are also present in the simulations. Rev. "Why is the Universe so flat? The general principle seems to be that Power spectra always play an important role in the theory of inflation. CMB frequency spectrum; Ending Inflation; Component separation and the Planck maps ; CMB power spectra, likelihood, and cosmological parameters; CMB lensing; copy_of_erc18b2_f4._scaled.png; Figure1_scaled.png; Figure1_scaled2.png; erc18b2_f4._scaled2.png; Figure2_scaled2.png; copy_of_erc18b2_f4._scaled2.png; erc18b2_f5_scaled2.png; Large Scale Structure and Precision … The new ekpyrotic models make different predictions about the status of the primordial spectrum, one considers parameters... Features of this pre-recombination fluid implications for models of Slow-Roll inflation in Light of the hot and cold in. 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Power spectra ( ns > 1 ), flat geometry and seeds of all structure in CMB...	2021-05-06 22:07:06	{"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": 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": 0.7420994639396667, "perplexity": 1397.3157743798618}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988763.83/warc/CC-MAIN-20210506205251-20210506235251-00348.warc.gz"}
https://www.embeddedrelated.com/showarticle/646.php	# Slew Rate Limiters: Nonlinear and Proud of It! October 6, 2014 I first learned about slew rate limits when I was in college. Usually the subject comes up when talking about the nonideal behavior of op-amps. In order for the op-amp output to swing up and down quickly, it has to charge up an internal capacitor with a transistor circuit that’s limited in its current capability. So the slew rate limit $\frac{dV}{dt} = \frac{I_{\rm max}}{C}$. And as long as the amplitude and frequency aren’t too high, you won’t notice it. But try to output a square wave? BAH HA HA HA HA! Those nice fast edges that you’d like to see will slow down. import numpy as np import matplotlib.pyplot as plt def ratelimit(x,t,rlim): def helper(): y = x[0] tprev = t[0] for (xi,ti) in zip(x,t): dy = xi - y dt = ti - tprev y += np.clip(dy,-rlimdt,rlimdt) tprev = ti yield y return np.array(list(helper())) t = np.arange(0,2,0.001) sqwave = np.mod(t,1) > 0.5 maxslew = 10.0 out = ratelimit(sqwave, t, maxslew) plt.plot(t,sqwave,':',t,out) plt.ylim(-0.1,1.1) plt.legend(['in','out'],'best') We don’t like slew-rate limits in hardware because they represent distortion. Nonlinearity! Look what happens if we put double the amplitude of a square wave with a slew-rate limit: plt.plot(t,sqwave,':',t,out,t,sqwave2,':',t,ratelimit(sqwave2,t,maxslew)) plt.ylim(-0.2,2.2) plt.legend(['in 1','out 1','in 2','out 2'],'best') Double the amplitude, and you double the time needed to slew from one value to the other. That’s a nonlinear effect: if slew-rate limiting were linear, doubling the input would double the output, and hence double the rate of change of the output. No dice, Q.E.D. Slew-rate limiting isn’t only something that shows up in op-amps. It can occur in any dynamic system — electrical, mechanical, pneumatic, hydraulic, chemical, etc. — where the rate of change of the system state is constrained. It happens a lot with actuators. And there’s not much else to say about actuators. You get what you get. And sometimes we even add our own slew rate limits to a system. ## Adding a Slew Rate Limit On Purpose Now why would we want to add a slew rate on purpose? We don’t like this nonlinearity when we’re stuck with it; why would we want to add it ourselves? If we want a square wave, let’s get a square wave, not this lame trapezoidal thing. Except for one thing: sometimes we have systems where there are costs or constraints that relate to how fast something changes. And that something may not have a nice built-in slew rate limit. Blah blah blah, boring generalization. Mea culpa. Okay, here’s a specific example: Suppose you are closing a door. It takes you what, maybe a second from start to finish? You probably don’t even think about all the trajectory planning your brain does automatically when you go to close the door. Now close it twice as fast. Half a second. Okay, no big deal, it just feels faster than normal. How about five times faster: 0.2 seconds. Could you do it? Maybe. But look what happens: • The door closing will be louder. • The door closing will vibrate more, and maybe damage something in the door hinge or the door frame. • You will have to use 5× the force you normally would. • When the door closes, you might hurt your hand from the impact. • You will have less precise control of closing the door. • You could lose your balance. • You might pull or strain a muscle. You might not be able to close the door that fast; the force you can exert on the door may be inherently slew-rate limited by your muscle strength. That’s inherent slew-rate limiting in an actuator. But if that’s not the case, then one of the preceding effects might occur, and these represent costs or consequences that occur before reaching the inherent slew rate limit. The only thing preventing these effects is your ability to control how you close a door. This happens all the time with muscle movement. You almost never try to command a step trajectory in the way your body moves; instead, it’s a smooth. continuous movement. In electrical systems the analogy is to charge up a capacitor, where faster rates of voltage change require higher currents. In mechanical systems the analogy is to move something with inertia, where faster rates of velocity change require higher forces. And sometimes if we were to command a step in voltage change on a capacitor, or velocity change in an inertia, instead of a nice stable limit coming from our actuator, we could cause a component to overheat or break or not behave in a controlled manner. So instead, we want our system to change in a more moderate manner, and we can do that by adding our own slew rate limit. ## Slew-Rate Limiting Algorithms for Digital Controllers There are two main methods to limit the slew rate of a system using a digital controller. Let’s say we have a second-order system with a transfer function of $H(s) = \frac{{\omega_n}^2}{s^2 + 2\zeta\omega_n s + {\omega_n}^2}$. Now I haven’t written my own article on second-order systems yet, so bear with me, and don’t get discouraged by all these zetas (ζζζζζζζζζ!!!!) that show up. ### Method 1: Limit the rate of the input command This one’s pretty easy. We just constrain the input command to our control system so its rate of change is less than some value R. In a digital system, we could do that like this: int16_t srlimit_update(int16_t x_raw_input, int16_t x_prev_limit_out, int16_t maxchange) { // 1. Compute change in input int16_t delta = x_raw_input - x_prev_limit_out; // 2. Limit this change to within acceptable limits if (delta > maxchange) delta = maxchange; if (delta < -maxchange) delta = -maxchange; // 3. Apply the limited change to // compute the new adjusted output. // Use this as the next value // of x_prev_limit_out return x_prev_limit_out + delta; } If we call srlimit_update() regularly at a timestep dt, then we should use maxchange = R * dt. Except there’s one bug here. This method does not work properly if x_raw_input and x_limit_out are more than 32767 counts apart. If so, we’ll run into overflow problems and the value of delta will not be properly computed. The simplest way of fixing it is to promote the input operands to 32-bit numbers: int16_t srlimit_update(int16_t x_raw_input, int16_t x_prev_limit_out, int16_t maxchange) { // 1. Compute change in input int32_t delta = (int32_t)x_raw_input - x_prev_limit_out; // 2. Limit this change to within acceptable limits if (delta > maxchange) delta = maxchange; if (delta < -maxchange) delta = -maxchange; // 3. Apply the limited change to // compute the new adjusted output. // Use this as the next value // of x_prev_limit_out return x_prev_limit_out + (int16_t)delta; } Now, how do we see how this works in practice? ### Interlude: Best approaches for simulating linear systems The Python code we used earlier (including the ratelimit() function I wrote) lets us limit a step. There are a few approaches in Python to simulate linear systems: 1. Simulate it manually using differential-equation solving techniques. In this case you have to do everything yourself. 2. Use scipy.signal.lti() and scipy.signal.lsim(). Here scipy will simulate a linear system given an arbitrary timeseries input, and the output is only as accurate as you can model the input signal 3. Use scipy.signal.lti() and be clever and use the lti class’s methods, like step() and impulse(), to compute exact answers for specific times. I’ll show all three of these techniques. The lsim() approach is probably easiest, so let’s start there. Here’s how to create LTI objects for a given rational transfer function, and how to plot a step response: from scipy.signal import lsim, lti # ignore the math, I'll cover it in the upcoming # article on second-order systems def zeta_from_ov(ov): '''Given an amount of overshoot, returns the value of zeta that causes that overshoot in a second-order system with no zeros.''' lnov = np.log(ov) return -lnov/np.sqrt(np.pi2 + lnov2) zeta = zeta_from_ov(0.4) print "zeta = %.6f" % zeta wn = 2np.pi # Create an LTI object by passing in # arrays of numerator and denominator coefficients H = lti([wnwn],[1, 2zetawn, wnwn]) t,y = H.step() plt.plot(t,y) print "max numerical value of step: %.6f" % np.max(y) zeta = 0.279998 max numerical value of step: 1.399979 And here’s how to use lsim(). First we need to construct a test input. We could use the ratelimit() function I wrote earlier in this article, but I like closed-form solutions, and there is one for a rate-limited step. The reason I like closed-form solutions, rather than a numerical timeseries, is that we can write a waveform as a function of any time t, and evaluate it at any time we like, essentially giving us “infinite” precision in time, whereas a numerical timeseries is inherently discrete-time, even if the timesteps are variable. Chew on that for a while, and read on: def ratelimitedstep(slewrate, initval=0.0, finalval=1.0): T_end = (finalval - initval) 1.0 / slewrate A = (finalval-initval) / 2.0 / T_end B = (finalval+initval) / 2.0 def f(t): return A(np.abs(t) - np.abs(t-T_end)) + B return f trange = [-0.5,4] f2 = ratelimitedstep(2.0, initval=3.0, finalval = 5.0) f0_5 = ratelimitedstep(0.5, initval=1.0, finalval = 2.5) f5 = ratelimitedstep(5) t = np.arange(trange[0],trange[1],0.01) plt.plot(t,f2(t), t,f0_5(t),t,f5(t)) plt.xlim(trange) plt.legend(('ratelim=2.0','ratelim=0.5','ratelim=5'),'best') Now let’s put it all together: t = np.arange(-1,5,0.001) f = ratelimitedstep(2.0) t2, y, _ = lsim(H,f(t),t) tstep, ystep = H.step() plt.plot(tstep,ystep,t2,y) plt.plot(t,t>0,':',t,f(t),':') plt.legend(('step response','response to ratelim step'),'best',fontsize=10) Tada! Now that’s interesting; the overshoot has been reduced. We’ll come back to this idea in a bit, but first let’s handle the other two approaches. Here’s an example using both the forward Euler, midpoint, and Runge-Kutta methods, as well as the step() function of scipy’s LTI objects: import scipy.signal def secondordersystem(wn,zeta,method): def fderiv(x,u): ''' derivative function for 2nd-order system state variable x has two components: x[0] = output x[1] = derivative of output d2x/dt2 = (u-x)wn^2 - 2zetawn(dx/dt) ''' return (x[1], (u-x[0])wnwn - 2zetawnx[1]) def project(x,dxdt,dt): '''project a state vector x forward by dx/dt * dt''' return (x[0]+dtdxdt[0], x[1]+dtdxdt[1]) def solver(t,u,fderiv,stepper): x = (0,0) # initial state tprev = t[0] uprev = u[0] for tval, uval in zip(t,u): dt = tval - tprev dxdt = stepper(fderiv,uprev,uval,x,dt) x = project(x,dxdt,dt) yield x[0] tprev = tval uprev = uval def forward_euler_step(fderiv,uprev,u,x,dt): return fderiv(x,u) def midpoint_method_step(fderiv,uprev,u,x,dt): dx1 = fderiv(x,u) xmid = project(x,dx1,dt/2) return fderiv(xmid,u) def runge_kutta_step(fderiv,uprev,u,x,dt): dx1 = fderiv(x,u) x1 = project(x,dx1,dt/2) dx2 = fderiv(x1,u) x2 = project(x,dx2,dt/2) dx3 = fderiv(x2,u) x3 = project(x,dx3,dt) dx4 = fderiv(x3,u) dxdt = ((dx1[0]+2dx2[0]+2dx3[0]+dx4[0])/6, (dx1[1]+2dx2[1]+2dx3[1]+dx4[1])/6) return dxdt if method == 'midpoint': stepper = midpoint_method_step elif method == 'euler': stepper = forward_euler_step elif method == 'runge': stepper = runge_kutta_step else: raise ValueError("method must be 'midpoint' or 'euler' or 'runge'") def sim(t,u): return np.array(list(solver(t,u,fderiv,stepper))) return sim for timestep in [0.05, 0.01]: t = np.arange(-1,5,timestep) # add more points around the discontinuity t = np.union1d(t, np.arange(-5,5,0.01)timestep) sos1 = secondordersystem(wn,zeta,'euler') sos2 = secondordersystem(wn,zeta,'midpoint') sos3 = secondordersystem(wn,zeta,'runge') u = (t>0)1.0 yeuler = sos1(t,u) ymid = sos2(t,u) yrunge = sos3(t,u) def step2(H, t): '''make up for the fact that scipy step functions don't handle start times other than zero >:( ''' return np.hstack((t[t<0]0, H.step(T=t[t>=0])[1])) y = step2(H,t) plt.figure(figsize=(8,6)) plt.subplot(4,1,1) plt.plot(t,yeuler, t,ymid, t,yrunge, t,y) plt.title('Simulation, timestep=%.3f' % timestep) for (i,approx) in enumerate([(yeuler,'Euler'),(ymid,'Midpoint'),(yrunge,'Runge-Kutta')]): plt.subplot(4,1,i+2) yapprox, approxname = approx plt.plot(t,yapprox-y) plt.ylabel('Error: %s' % approxname) Woohoo! We get close to the expected value with step(), and the error term behaviors are what they should be, with Euler error being the largest, and midpoint and Runge-Kutta error decreasing more rapidly as a function of the timestep. There’s a subtlety here, though. I had to vary the timestep and decrease it near the discontinuity of the step input. Without this change, the timestep is constant, and when I tried this, the midpoint and Runge-Kutta solvers didn’t work much better than Euler, because the error introduced at the discontinuity persisted in the feedback state. In addition to having to get all these details right, any of these approaches runs slower, since it happens within the Python environment, whereas lsim() and step() can use optimized implementations. So I’m going to abandon the manual approach and focus on the other two: • The lsim() approach is to create a rate-limited step waveform and use lsim() to find the response to it. • The step() approach is to integrate the transfer function and use the step() function to find two time-shifted ramp responses, which when subtracted yield the response to a ramp-limited step. Let’s use both and see how close they get: def rampresponse(H, t, t0): '''make up for the fact that scipy step functions don't handle start times other than zero >:( ''' # Create a transfer function H (1/s) # but we can't use 0 as the new coefficient because it # doesn't work with the step() code, so come close H_integrated = lti(H.num, np.hstack([H.den, 1e-9])) return np.hstack((t[t<t0]0, H_integrated.step(T=t[t>=t0])[1])) def ramplimstepresponse(H,ratelim,t): '''ramp-limited step with rate limit R = response to the difference between ramps r(t) - r(t-1/R) ''' return (rampresponse(H,t,0) - rampresponse(H,t,1.0/ratelim))ratelim t = np.arange(-1,5,0.01) ramprate = 2.0 # method 1: lsim() f = ratelimitedstep(ramprate) t1, y1, _ = lsim(H,f(t),t) # method 2: step() y2 = ramplimstepresponse(H,ramprate,t) plt.subplot(211) plt.plot(t,y1,t,y2) plt.subplot(212) plt.plot(t,y1-y2) plt.ylabel('error') Which approach is better, and why do we see an error between the two? I don’t know which approach is better. I’m more confident in staying with lsim() becase it’s more straightforward and doesn’t involve hacks to create an approximate ramp response. I don’t know if the error is because of numerical issues in the calculations. Or maybe it has to do with the discontinuities in the beginning and end of the ramp, which combine with the inherent discrete time nature of the lsim() function. That’s the end of our interlude. ### Analyzing the overshoot Let’s look at the overshoot in our system for different ramp-limited steps with the same ramp rate but different step sizes. trange=[-0.5,6] t = np.arange(trange[0],trange[1],0.01) ratelim = 1.8 plt.figure(figsize=(8,6)) for usize in [0.5, 0.9, 0.95, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0]: Trampend = usize/ratelim f = ratelimitedstep(ratelim/usize) u = f(t)usize _, y, _ = lsim(H,u,t) # find maximum: # let's calculate this faster by using close-together points # that are in the neighborhood of the maximum i = np.argmax(y) ymax = y[i] tmax = t[i] t2 = np.union1d(t, tmax+np.arange(-0.1,0.1,0.0005)) u2 = f(t2)usize _, y2, _ = lsim(H,u2,t2) i = np.argmax(y2) ymax = y2[i] tmax = t2[i] print "step size=%5.2f, abs overshoot=%.4f" % (usize, ymax-usize) plt.plot(t,u,':',t2,y2,tmax,ymax,'.') plt.xlim(trange) plt.ylim(0,ymax1.05) step size= 0.50, abs overshoot=0.1757 step size= 0.90, abs overshoot=0.2331 step size= 0.95, abs overshoot=0.2334 step size= 1.00, abs overshoot=0.2323 step size= 2.00, abs overshoot=0.1427 step size= 3.00, abs overshoot=0.1755 step size= 4.00, abs overshoot=0.1638 step size= 5.00, abs overshoot=0.1677 step size= 6.00, abs overshoot=0.1664 step size= 7.00, abs overshoot=0.1668 step size= 8.00, abs overshoot=0.1667 (0, 8.5750331075081245) That’s interesting. I see three things to note: • The slew rate of the output can be greater than the slew rate of the input. (In fact, it turns out that the maximum output slew rate is 1.4 times the input slew rate, where the 1.4 comes from the step response overshoot of 0.4.) This means that if you want to use an underdamped system, and you want to keep the output slew rate below an absolute limit, you must derate that limit before using it as a slew rate limit of the system input. • Except for short steps comparable to the system response time, the absolute overshoot is about the same. It settles down to about 0.1667 for long steps. The peak overshoot is about 0.2334, which is 1.400 times as much. There’s that number again! • The output reaches a constant following error below the input. The following error for a unit ramp (with slope of 1) has nonzero steady-state error of $2\zeta/\omega_n$, and we’ll see why in that upcoming article on second order systems. So if we have a ramp rate of R, we should expect steady-state ramp error of $2\zeta R/\omega_n$. If you go through a bunch of grungy algebra, the end result is that the overshoot OVSRL of a 2nd-order system with an input of a slew-rate limited step, if the ramp time is long, is proportional to the slew rate: $$OVSRL = \frac{SR}{\omega_n}e^{-\zeta\frac{\pi/2 + \sin^{-1}\zeta}{\sqrt{1-\zeta^2}}}$$ So we can decrease the overshoot by decreasing the slew rate. The overshoot of a step response with no slew rate limit is $OV = e^{-\frac{\pi\zeta}{\sqrt{1-\zeta^2}}}$. Just as a double-check: def ovsr(wn,zeta,slewrate): return slewrate/wnnp.exp(-zeta(np.pi/2 + np.arcsin(zeta))/np.sqrt(1-zeta2)) ovsr(wn,zeta,ratelim) 0.16679199263977906 Bingo! So, given ζ and ωn, we can predict the step reponse overshoot OV, we can predict the overshoot OVSRL for long ramp inputs, we can predict the maximum overshoot (= (1+OV)× OVSRL) for any ramp input, we can predict the maximum output slew rate (= (1+OV) × input slew rate) for any ramp input. How about that! ### Method #2: Rate-limit the input using output feedback Rate-limiting based purely on the input command is fairly straightforward, as we’ve seen. In systems with slow dynamics where the output takes a while to change, and our sensing of the output value is low-noise, there’s another method: Limit the input command to within ± Δ of the output. This actually uses the inherent dynamics of the output to help slew-rate limit the input. We found earlier that if we have an input ramp rate of R, we should expect steady-state ramp error, which is the steady-state difference between input and output during the ramp, of $2\zeta R/\omega_n$. The trick here is to work backwards: we start with that difference, so if we tweak the input to constrain the error between input and output to be at most Δ, then we can expect an input ramp rate R, and therefore choose $\Delta = 2\zeta R/\omega_n$. This approach also has the advantage that if the output gets “stuck” for some reason, the input command will wait for it. In C this would look something like the function shown below. It’s actually identical to the first C function srlimit_update(), except that we feed in the system output rather than the previous input, and the value of delta is different. int16_t srlimit_update2(int16_t x_raw_input, int16_t system_output, int16_t delta) { // Constrain the input to be within some value "delta" // of the system output. // 1. Compute difference between raw input and output int32_t diff = (int32_t)x_raw_input - system_output; // 2. Limit this difference to within acceptable limits if (diff > delta) diff = delta; if (diff < -delta) diff = -delta; // 3. Apply the limited difference to return system_output + (int16_t)diff; } We can try simulating this too, but to do so, we need to use the manual differential equation solver approach, since there is now feedback from the output to the input, and we therefore have a nonlinear system, which neither lsim() nor step() can help us with. def secondordersystem_feedback(wn,zeta): '''2nd-order system simulator. u_limited = f(t, u_raw,x) is an input here. We use Runge-Kutta. ''' def fderiv(x,u): ''' derivative function for 2nd-order system state variable x has two components: x[0] = output x[1] = derivative of output d2x/dt2 = (u-x)wn^2 - 2zetawn(dx/dt) ''' return (x[1], (u-x[0])wnwn - 2zetawnx[1]) def project(x,dxdt,dt): '''project a state vector x forward by dx/dt * dt''' return (x[0]+dtdxdt[0], x[1]+dtdxdt[1]) def solver(t,u_raw,f,fderiv,stepper): x = (0,0) # initial state tprev = t[0] uprev = 0 for tval, urawval in zip(t,u_raw): dt = tval - tprev u = f(t,urawval,x[0]) dxdt = stepper(fderiv,uprev,u,x,dt) x = project(x,dxdt,dt) yield (u,x[0],x[1]) tprev = tval uprev = u def runge_kutta_step(fderiv,uprev,u,x,dt): dx1 = fderiv(x,u) x1 = project(x,dx1,dt/2) dx2 = fderiv(x1,u) x2 = project(x,dx2,dt/2) dx3 = fderiv(x2,u) x3 = project(x,dx3,dt) dx4 = fderiv(x3,u) dxdt = ((dx1[0]+2dx2[0]+2dx3[0]+dx4[0])/6, (dx1[1]+2dx2[1]+2dx3[1]+dx4[1])/6) return dxdt def sim(t,u_raw,f): '''returns a 3-column matrix: column 0 is the limited value u column 1 is the output x column 2 is the derivative dx/dt ''' return np.array(list(solver(t,u_raw,f,fderiv,runge_kutta_step))) return sim def slewrate_method2(delta): '''limit u to x +/- delta''' def f(t, u, x): if u > x+delta: u = x+delta elif u < x-delta: u = x-delta return u return f Okay, let’s use it: t = np.arange(-1,5,0.01) # add more points around the discontinuity t = np.union1d(t, np.arange(-5,5,0.01)0.01) u_raw = (t>0)1.0 R = 0.8 delta = 2zetaR/wn fslewer = slewrate_method2(delta) sim = secondordersystem_feedback(wn,zeta) ux = sim(t,u_raw,fslewer) u = ux[:,0] x = ux[:,1] dxdt = ux[:,2] import matplotlib.gridspec as gridspec gs = gridspec.GridSpec(2, 1, height_ratios=[2,1]) fig = plt.figure(figsize=(8,6)) ax = plt.subplot(gs[0]) ax.fill_between(t, x+delta, x-delta, facecolor='gray', alpha=0.08) ax.plot(t,u_raw,':',t,u,t,x) ax.legend(('u_raw (raw input)','u (input)','x (output)'),'lower right') ax.set_ylim(0,1.2) ax.set_xlabel('time (s)') ax.text(3,x[-1]-delta,'$\\Delta = %.3f$' % delta, fontsize=16) ax2 = plt.subplot(gs[1]) ax2.plot(t,dxdt) ax2.legend(('dx/dt',),'upper right') Woot! We controlled the maximum output slew rate to $\pm R = 0.8$ by limiting our input so that the difference between input and output is within $\pm\Delta = 2R\zeta/\omega_n = 0.071$. Now let’s compare and see what both methods look like: trange = [-0.5,5] t0 = np.arange(trange[0],trange[1],0.01) R = 0.8 for t_unclamp in [0, 0.5]: # add more points around the discontinuity t = np.union1d(t0, t_unclamp + np.arange(-5,5,0.01)0.01) u_raw = (t>0)1.0 # method 1 of slew-rate limiting: # derate to R/(1+overshoot) f = ratelimitedstep(R/1.400) u1 = f(t) u1_unclamped = u1 * (t>=t_unclamp) Hderiv = lti([wnwn,0],[1,2zetawn,wnwn]) _, x1, _ = lsim(H,u1_unclamped,t) _, dx1dt, _ = lsim(Hderiv,u1_unclamped,t) # method 2 delta = 2zetaR/wn fslewer = slewrate_method2(delta) sim = secondordersystem_feedback(wn,zeta) u2 = ux[:,0] x2 = ux[:,1] dx2dt = ux[:,2] import matplotlib.gridspec as gridspec gs = gridspec.GridSpec(2, 1, height_ratios=[2,1]) fig = plt.figure(figsize=(8,6)) ax = plt.subplot(gs[0]) ax.fill_between(t, x2+delta, x2-delta, facecolor='gray', alpha=0.08) ax.plot(t,u_raw,':', t,u1,'black',t,x1,'red', t,u2,'blue',t,x2,'green') ax.legend(('u_raw (raw input)', 'u_1 (input)','x_1 (output)', 'u_2 (input)','x_2 (output)'),'lower right') ax.set_ylim(0,1.2) ax.set_xlabel('time (s)') ax.text(3,x[-1]-delta,'$\\Delta = %.3f$' % delta, fontsize=16) title = 'Comparison between two slew-rate limiting methods' if t_unclamp > 0: title += ',\n output unclamp at t=%.1f' % t_unclamp ax.set_title(title) ax.set_xlim(trange) ax2 = plt.subplot(gs[1]) ax2.plot(t,dx1dt,t,dx2dt) ax2.set_xlim(trange) ax2.legend(('dx_1/dt','dx_2/dt'),'upper right') Okay, what are we looking at here? The first graph shows that both methods keep the output slew rate within ±R. But the second method (constrain the input to the output ±Δ) finishes faster. The second graph shows what happens if we clamp the output somehow to zero until t=0.5, and then unclamp it. The first method gets no feedback from the output and goes ahead and starts ramping up at t=0. At t=0.5, all of a sudden the output is unclamped and ZOOM! it has to catch up with the input ramp that has already started. In this case it settles earlier than the output of the second method. But we have failed to keep the output slew rate below the desired limit, which was the whole point of limiting the input. The second method works fine: it just waits patiently until the output starts to change, and then transitions smoothly towards its final value. ### Which method is better? I generally like the constrain-input-within-a-band-around-the-output approach better (method #2), if I know the output sensing is accurate and low-noise. If there is noise, a little bit of filtering (or even slew-rate limiting!) can reduce this noise to a reasonable value. I suppose it’s possible to combine both methods, but there are some tricky corner cases where you have to give priority to one method over the other, for example if the raw input is pulled one way and a disturbance pulls the output another way, then the raw input and the output are far enough apart that you can’t satisfy both methods simultaneously. Both of these methods have large portions of the initial rampup where the output rate of change is less than the allowed maximum. There’s probably yet another way to apply a slew rate to a system, by using feedback to push the input system as hard as we can, while keeping the output slew rate below the allowed maximum. But I’m skeptical whether it’s stable and robust and is not sensitive to noise. ## Wrapup Slew-rate limiting is a nonlinear behavior. In some systems it exists even though we don’t want it. In other systems, there’s not an inherent slew-rate limit, but we have good reason to add one: it keeps the system state within more well-behaved limits. Two well-known methods of slew-rate limiting are very similar. Given a raw input waveform $u_0(t)$, we can do one of two things: 1. Try to move the system input towards $u_0$, but constrain the input to within a band around the previous value of the input. (Feedforward slew rate limit.) 2. Try to move the system input towards $u_0$, but constrain the input to within a band around the system output. (Feedback slew rate limit.) In general the second method is better, as long as the output feedback sensing is accurate and low-noise. We can use scipy.signal and its lti() and lsim() functions to try these methods out in Python. For the feedback slew rate limit technique, we have to use a differential equation solver to simulate it. That’s all for now. One of these days I’ll get around to writing up my article on second-order systems, and you’ll hear more about overshoot. Previous post by Jason Sachs: You Will Make Mistakes Next post by Jason Sachs: Important Programming Concepts (Even on Embedded Systems) Part III: Volatility To post reply to a comment, click on the 'reply' button attached to each comment. To post a new comment (not a reply to a comment) check out the 'Write a Comment' tab at the top of the comments.	2019-10-14 01:46:35	{"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": 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": 0.5585246086120605, "perplexity": 3134.003937822623}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986648481.7/warc/CC-MAIN-20191014003258-20191014030258-00538.warc.gz"}

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