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https://physics.meta.stackexchange.com/questions?tab=newest&pagesize=15	# All Questions 3,142 questions Filter by Sorted by Tagged with 34 views ### Why does spacetime bend or curve in the downward direction? [closed] In relativity , when mass bends spacetime in a downward direction. So ultimately we are coming to the conclusion that any mass body is also going down . 59 views ### I'm asking something really obvious. Why is is marked as "unpublished personal theory"? [duplicate] I'm asking a really obvious question; many people told me that they had this thought and question, so why is this marked as a personal theory (Is mass the source of space-time?)? It should be possible,... 44 views ### One-way speed of light tag? [duplicate] Should we make a tag for one-way speed of light questions? I saw this question which included a link to this search, which combines a tag search with a text comprehension search in what seems like a ... 501 views ### Any way to easily/automatically merge the many Veritasium questions? There is a new Veritasium video, The Big Misconception About Electricity, which is prompting a bunch of nearly identical questions. Currently, although the questions are nearly identical most ... 302 views ### Why was my question closed for not being focused when it actually was? I asked a question a few days ago. It got closed for needing to be “more focused.” According to the site guidelines , in the Needs more focus part, it says: This can often be fixed by breaking the ... 218 views ### Community Post for online courses similar to Book Recommendations The overarching book recommendations answer has been quite helpful for a lot of people me included. Is there / should we make a similar one for online lectures? Or maybe add it to this answer. I mean ... 143 views ### In what ways is this question lacking detail or clarity? I asked this question about what a strangelet would look like to the naked eye, and it was closed because it needed details or clarity. As far as I can tell, there was no need for either, because it ... 86 views ### Why is the accepted answer not pinned to the top? We currently have the highest-rated answer pinned to the top and not the accepted one. With the meta question currently at 23 for pinning the accepted answer and 21 for pinning the highest-rated ... 34 views ### Can the effect of a link be restored after relocation of the comment containing it into the body of an answer by the commentator? I recently moved a conclusive comment (made by myself) on an OP's question into the body of my answer to it. In that process, the underlining and color-coding that characterize a link was (probably ... 44 views ### Is this question on-topic? Infinitely fast and strong laser targets the Earth I would like to ask a purely hypothetical question regarding the following: What if I had an infinitely strong and fast laser that can cut through anything it passes. If it has a fixed diameter (let's ... 267 views ### Should I avoid asking too many questions in a given time period (say, a day)? Recently I've been getting a lot of physics questions, and I'm at the point where I'm asking a couple a day. Should I restrain myself, or can I go willy-nilly as long as they meet the standards of the ... 349 views ### My question was downvoted after it was closed, can I ask the community to delete it? I have this question: Bomb attached to accelerating charge, as viewed from a co-moving frame? Now it was closed, and then it was downvoted. I do not understand why we are allowing downvotes (or any ... 86 views ### Where should I put this question? I have a question about my Physics Stack Exchange post: Is it sufficient to have some anti-gravity generator to make an Alcubierre drive? Where should I ask this question? 172 views ### Are future physics discoveries properly categorized as mainstream or not? [closed] "Non-mainstream" physics is often equated to fallacies and fringe theories, including on this forum. Mainstream physics is rigorously categorized in specific historical scientific contexts, ... 66 views ### How do you get the Excavator badge? I have edited some posts of mine and others which are older than a year, including modifying some tags, but I did not get the badge. How exactly do you get it? Does it have to be precisely inactive ... 54 views ### Opera browser blocks log in Physics Stack Exchange - HSTS issue I have difficulties to log in the Physics Stack Exchange site via the Opera browser. It says: Your connection is not private This server could not prove that it is physics.stackexchange.com; its ... 280 views ### How can I create circuit diagrams for use on this site? I am writing a question regarding a specific electrical circuit, and which would be significantly clearer if I can include a diagram of the circuit. What software tools can I use to quickly, simply ... 61 views ### Closed and then reopened question samples [duplicate] Many new users are quite unaware of the required format of asking questions. A sample of closed questions that were edited and then reopened could be assistive for most of new users. Also so far I ... 262 views ### Should we require that circuit questions include a circuit diagram? I posit the following: Most circuit questions on this site can be easily answered by drawing a circuit diagram, so asking a circuit question without a diagram often bumps up against the homework ... 93 views ### First questions queue is not working First questions queue is not working properly. I noticed this a couple of hours ago and it has been not fixed yet. (Screenshots on mobile device) 358 views I just noticed that the question What happens to the vacuum when entanglement is harvested? was closed as off-topic a couple of months ago, and I don't understand why. The topic is basic quantum ... 314 views ### Silently deleting answers by a new user is not welcoming This comment-like answer was posted by a rep-1 user with no (AFAICT) other posts, and then silently deleted within 15 minutes, leaving the poster wondering why. The user then posted a copy of the ... 135 views ### Do you think downvote feedbacks would contribute to the site? This topic was discussed many times before but I hope to inspect it a bit differently. I think it would be awesome if everybody knew why a question was down-voted. That way everyone could see the ... 95 views Does it make any difference in using https://physics.meta.stackexchange.com/q/13855/305718 instead of ... 64 views ### How can we bring an old question back to the users' attention? The question we want to ask might have been posted by another user years ago. It is possible that the question did not get any answers or the answers were/are not satisfactory. What should we do for ... 300 views ### Helping out in comments, when the question is too weak I encounter quite often questions that are based on poor research, misunderstanding basic definitions, and likewise. In my opinion such questions should be closed rather than answered. Nevertheless, ... 81 views ### Is this a bug with editing posts? I just answered a question and edited it within a short period. But this is what I saw: What happened to the grace period of 5 minutes for editing a post? 85 views ### How can I accomplish the removal of comments by another participant, on an answer I had posted, that might be considered derogatory? About 24 hours ago, I posted an edit to my answer to a PSE question concerning the time when interstellar travel (in any form, or, at least, without specification of its form) might have begun: As ... 66 views ### Maybe add an indicator showing level of question There is an existing discussion/request for a system to tag or otherwise indicate level of questions: Can we have a level-of-question tag please? Differentiate research-level questions? The idea is ... 72 views ### Should we take the terminology an asker might be unfamiliar with into consideration when it comes to expectations for research effort? The reason I would suggest that an askers knowledge of terminology should be taken into consideration is that if by research effort we mean looking something up on the internet before asking a ... 78 views ### My closed question probably fits better in a different SE — do I have to ask for it to be migrated? Specifically this question. Why doesn't my phone scroll if I put a ring flat on the screen? It's been closed as being too engineering-based, and the Feedback message gave options of other SEs it ... 158 views ### How do I write better answers? I haven't written many answers, but after I have identified what the user is asking, I usually go with a conversational tone, going over whatever has a question mark over it and any context supporting ... 59 views ### What's the criteria for a comment "no longer needed" flag? Reading the answer by SuperCiocia to this question Should we flag comments like "Thank you"?, I learned about the comment flag for "no longer needed". I thought this was ... 174 views ### Would original posts meant to share knowledge instead of ask questions be on topic on the site? For a long time the way I tended to see the Schrödinger equation written, some of the notation was like a foreign language to me, and so I couldn't work out how to actually use it to model anything, ... 388 views ### The “experience of a photon” question I’m sure everybody is aware of the question that people love to ask, i.e. some variation of “what’s the experience of a photon (or someone traveling at $c$, etc)?” This question is always roundly ... 211 views ### Should link-shortener services be off-limits? Link-shortener services (e.g. exe.io, bit.ly, etc) route traffic to a destination website without allowing the end-user to see what website they are clicking into before clicking. This can be used for ... 860 views ### Do we want accepted answers to be pinned to the top? SE is making the way accepted answers behave configurable per-site and is looking for input from our side what our preference is. Currently, accepted answers - answers that the asker of the question ... 226 views ### Should we refrain from using canned comments when reviewing? Since the review queue update, I have seen multiple instances of the following situation: a new user asks/answers a question, a reviewer finds the question unclear or lacking sufficient detail and ... 93 views If the draft of a post which I was going to publish later, is accidentally deleted, I will lose the links of the images I've already uploaded. Is there a way to recover those links? This also applies ... 102 views ### Do we have a policy or guidance for questions with no one right answer? This question does not seem to have a single correct answer posted. Which of the following are true: Assume that there will never be one right answer posted (i.e. because there is not in principle a ... 123 views ### Should we downvote answers that begin with "I think..."? People often speculate rather than answer. If you think you know the answer then you don't. 119 views ### Why did I get 20 reputation without any upvote or bookmark? I had got 20 reputation for this question. But, I wonder how I got reputation from that question because, that question claims that no one had upvoted neither downvoted. Even, no one had bookmarked it.... 204 views ### How to treat suggested edits that "correct a formula"? Once in a while I see a suggested edit that claims to "correct a formula" (this is usually written in the edit summary). Is there a general "rule"/recommended procedure for such ... 43 views ### Can I ask for specific books recommendations? using the [resource-recommendations] tag I researched some question about book recommendations and I found a lot of books on Google Search, and here in physics.se the book recommendation question was closed because it is a broad question, ... 79 views ### Why downvotes to question? [duplicate] I have been hanging around physics SE for quite a while. I am just a student who craves for some knowledge in mathematics and physics and thus I joined SE. I am not an expert. But to my questions, I ... 65 views ### Is there a grace period to post an answer to a question after it is closed? This answer has been published 23 minutes ago, while the question was closed 25 minutes ago, at the moment. Also this answer was published a minute after closing the question (another). So how does ... 51 views ### Request to reopen an old question which I have edited Identifying parallel and series capacitors for capacitor system with dieletric inserted The OP had posted this question with large images and lack of MathJax formatting. I have edited it and made it ... 68 views ### Confusing message from new review queue? The answer here (in which a new user posted an elaboration on their question as an answer) has the following comment from the Community bot, presumably due to its review in the new "First Answers&...	2021-12-05 10:08:00	{"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": 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": 0.7627639174461365, "perplexity": 1382.9752676179853}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363157.32/warc/CC-MAIN-20211205100135-20211205130135-00174.warc.gz"}
https://tbc-python.fossee.in/convert-notebook/Engineering_Physics_Malik/Chapter_8.ipynb	# Chapter 8: Sound Waves and Acoustics of Buildings¶ ## Example 8.1, Page 8.17¶ In [2]: # Given v = 34500 # speed of sound in cm/sec f = 20 # lower limit of frequency for human hearing ear in Hz f_ = 20000 # upper limit of frequency for human hearing ear in Hz #Clculations l1 = v / f l2 = v / f_ #Results print "Wavelength range of the sound wave is %.f cm to %.f cm."%(l1,l2) Wavelength range of the sound wave is 1725 cm to 1 cm. ## Example 8.2, Page 8.18¶ In [3]: from math import sqrt # Given T = 373. # temperature in kelvin d = 1.293e-3 # density of air at S.T.P. in gm/cm^3 d_ = 13.6 # density of mercury in gm/cm^3 Cp = 0.2417 # specific heat of air at constant pressure Cv = 0.1715 # specific heat of air at constant volume g = 980 # gravitational constant i dynes/cm^3 #calculations p = 76 * d_ * g gama = Cp / Cv v = sqrt(gama * (p / d)) v_ = v * sqrt(T / 273) #Result print "Velocity of sound in the air in %.f cm/sec."%v_ Velocity of sound in the air in 38839 cm/sec. ## Example 8.3, Page 8.18¶ In [4]: from math import sqrt # Given that n = 512. # frequency of tuning fork in Hz T = 290. # temperature in kelvin l = 66.5 # wavelength of the gas emitted by tuning fork in cm d = 1.293e-3 # density of air at S.T.P. in gm/cm^3 d_ = 13.6 # density of mercury in gm/cm^3 g = 980 # gravitational constant i dynes/cm^3 #Calculations p = 76 * d_ * g# calculation for pressure v_ = n * l# calculation for velocity of sound in air at temperature 17 c v = v_ * sqrt(273 / T)# calculation for velocity of sound in air at temp 0 c gama = v*2 (d / p)# calculation for ratio of two specific heat #Result print "Ratio of two principal specific heats of air is %.2f"%gama Ratio of two principal specific heats of air is 1.39 ## Example 8.4, Page 8.19¶ In [6]: # Given A = 15 * 30 # area of the floor in square meter h = 6 # height of hall in meter N = 500 # no. of people t = 1.36 # optimum time for orchestral music in sec k = 0.44 # absorption coefficient per person #Calculations aS = 0.161 * ((A * h) / t) a = N * k a_ = aS - a w = a_ + (N / 2) * k + (N / 2) * 0.02 t = (0.161 * (A * h)) / w #Results print "Coefficient of absorption to be provided by the walls, floor and ceiling when the hall is fully occupied is %.f SI unit."%a_ print "Reverberation time if only half upholstered seats are occupied is %.2f sec."%t 319.632352941 Coefficient of absorption to be provided by the walls, floor and ceiling when the hall is fully occupied is 100 SI unit. Reverberation time if only half upholstered seats are occupied is 2.03 sec. ## Example 8.5, Page 8.19¶ In [10]: # Given V = 8000 # volume of hall in meter^3 t = 1.8 # reverberation time in sec #Calculation aS = (0.161 * V) / t# calculation for the total absorption constant #Result print "The total absorption constant = %.3f O.W.U."%aS The total absorption constant = 715.556 O.W.U. ## Example 8.6, Page 8.20¶ In [11]: # Given V = 1700 # volume in meter^3 a1 = 98 # area of plastered wall in m^2 a2 = 144 # area of plastered ceiling in m^2 a3 = 15 # area of wooden door in m^2 a4 = 88 # area of cushioned chairs in m^2 a5 = 150 # area of audience (each person) in m^2 k1 = 0.03 # coefficient of absorption for plastered wall in O.W.U. k2 = 0.04 # coefficient of absorption for plastered ceiling in O.W.U. k3 = 0.06 # coefficient of absorption for wooden door in O.W.U. k4 = 1 # coefficient of absorption for cushioned chair in O.W.U. k5 = 4.7 # coefficient of absorption for audience (each person) in O.W.U. #Calculations A1 = a1 * k1# calculation for the absorption by the plaster wall A2 = a2 * k2# calculation for the absorption by the plastered ceiling A3 = a3 * k3# calculation for wooden door A4 = a4 * k4# calculation for cushioned chairs A = A1 + A2 + A3 + A4# calculation for total absorption T = 0.161 * (V / A)# calculation for reverberation time #Result print "Reverberation time is %.2f sec"%T Reverberation time is 2.80 sec ## Example 8.7, Page 8.20¶ In [12]: # Given V = 1400 # volume of hall in meter^3 C = 110 # seating capacity of hall a1 = 98 # area of plastered wall in m^2 a2 = 144 # area of plastered ceiling in m^2 a3 = 15 # area of wooden door in m^2 a4 = 88 # area of cushioned chairs in m^2 a5 = 150 # area of audience (each person) in m^2 k1 = 0.03 # coefficient of absorption for plastered wall in O.W.U. k2 = 0.04 # coefficient of absorption for plastered ceiling in O.W.U. k3 = 0.06 # coefficient of absorption for wooden door in O.W.U. k4 = 1 # coefficient of absorption for cushioned chair in O.W.U. k5 = 4.7 # coefficient of absorption for audience (each person) in O.W.U. #Calculations A1 = a1 * k1# calculation for the absorption by the plaster wall A2 = a2 * k2# calculation for the absorption by the plastered ceiling A3 = a3 * k3# calculation for wooden door A4 = a4 * k4# calculation for cushioned chairs A5 = Ck5 # the absorption due to persons A = A1 + A2 + A3 + A4 + A5 # calculation for total absorption T = (0.161 V) / A# calculation for the reverberation time #Result print "Reverberation time is %.3f sec"%T Reverberation time is 0.367 sec ## Example 8.8, Page 8.21¶ In [13]: # Given V = 980 # volume in meter^3 a1 = 150 # area of wall in m^2 a2 = 95 # area of ceiling in m^2 a3 = 90 # area of floor in m^2 k1 = 0.03 # coefficient of absorption for wall in O.W.U. k2 = 0.80 # coefficient of absorption for ceiling in O.W.U. k3 = 0.06 # coefficient of absorption for floor in O.W.U. #calculations A1 = a1 * k1 A2 = a2 * k2 A3 = a3 * k3 A = A1 + A2 + A3 T = 0.161 * (V / A) #Result print "Reverberation time = %.2f sec"%T Reverberation time = 1.84 sec ## Example 8.9, Page 8.21¶ In [14]: # Given V = 980 # volume in meter^3 a = 1.58 # area of window in m^2 I_ = 1e-12 # standard intensity level of sound wave in W/m^2 l = 60 # intensity level in dB #calculations I = I_ * 10*(l / 10)# calculation for intensity AP = I a# calculation for acoustic power #Result print "Acoustic power = %.2e watt"%AP Acoustic power = 1.58e-06 watt	2021-06-13 11:12:12	{"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": 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": 0.5335696339607239, "perplexity": 4294.271877553803}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487608702.10/warc/CC-MAIN-20210613100830-20210613130830-00218.warc.gz"}
https://www.darwinproject.ac.uk/letter/?docId=letters/DCP-LETT-6816.xml	# From William Erasmus Darwin 5 July [1869]1 Southampton July 5 My dear Father, I went to Alum Bay Hotel yesterday, and walked to Colwell Bay where after some little search I found a considerable number of Epipactis P. tho’ with not v. many flowers out.2 I send you in little box the result of my watching; the flies seemed to visit the flowers very seldom, it was a very bright hot afternoon tho’ with perhaps a little too much breeze, but after watching about 2$\frac{1}{2}$ hours I only saw 4 or 3 flies of the kind I enclose visit. after watching a group of 3 plants for some time I saw two or possibly three flies (one of which I send without the pollen masses) crawl into some flowers that were considerably open, that is so much open that the labellum was about in the position of the labellum in Fig B Page 94 of orchis Book.3 I observed that the flies in every* instance after crawling down to collect honey apparently from the stigma and beneath it instead of backing out, turned round and came out, and I saw that they could turn round just underneath the pollen masses or anther without touching them. These seemed to be essentially crawling flies, and I caught one which I send just to show which flies I then fancied could not be the agents. These flies have not weight enough to depress the lip at all as I could plainly see; tho’ from trial I found the lips were depressed very easily, but the spring back seemed to me extremely slight, in one or two instances of flowers that certainly had never been visited, I found on pushing down gently the lip with a bit of grass that I had to give it a lift to get it into position again. In looking about for flowers with the labellum in the position of Fig A p. 94 I came upon one just in that position, namely the edges of the labellum near the hinge were just inside the body of the flower. [DIAG HERE] the two frilled edges of the distal portion of the labellum were in contact; and on holding the flower opposite the eye and looking over the centre of the groove formed by the Junction of the two edges of the labellum one could just fairly see the yellow anther case but no lower The moment I came on this flower I found a fly apparently of the same sort crawling out of the flower head formost without depressing the lip (i.e. not backing) with the pollen masses attached to his back (this is the fly I send you). If I had only had the luck to be a minute sooner I should have seen him enter. I then went to some other flowers, and saw a small beetle or two crawl in with no effect also an ant, and a long bodied thin fly alighted but went off again— I saw no bee come near them within a yard or so. I saw another of same sort of fly on a flower (of the Epipactis) with what I am nearly sure sure was the remains of the pollen masses on his back in just the same position, but I was just too late with my bottle to catch him. on one flower I saw the pollen masses sticking to an upper petal, in another case to a lower sepal, in another to the top of an unopened bud. I also found a longish hair or a leg sticking across a flower which drew out the pollen masses with it; I brought it away as carefully as possible, but it has vanished. on going back to some flowers I had previously looked at, and in which there were no traces of pollen masses sticking to them or protruding; I found an ant in each of two flowers, one of which ants I send. in one flower a single pollen mass was hanging down out of position, as if it has just been drawn out, and in the other the ant was sucking hard away and the pollen masses were behind him in the channel of the lip, and looked quite fresh. I am almost certain that the ants did not back out but turned round. The flowers in which I found them were about in a state intermediate between A and B. I could easily go next Sunday and have another look, or get you some flowers, I might also have another look at the spring in hinge of labellum which then seemed so slight, but in some flowers I have brought away with me it seems a little greater. You will find in box another much slighter fly which I caught in a flower not that he is worth sending. I shall be down about the 13th or 14th.4 Your affect son \| W. E Darwin I took notes on the spot I send Sat. Rev.5 please keep it as there is an Article I want to read when down with you as I have not read it * I don’t remember seeing a fly back out ## CD annotations 1.1 I went … 94 4.1] crossed pencil 4.8 could just … read it 14.2] crossed pencil Top of letter: ‘WED \| Epipactis palustris’ pencil ## Footnotes The year is established by the relationship between this letter and the letter to W. E. Darwin, 7 [July 1869]. Alum Bay and Colwell Bay are on the western side of the Isle of Wight. William refers to Epipactis palustris (marsh helleborine), a native British orchid commonly pollinated by bees, wasps, and syrphid flies. William refers to Orchids. CD was holidaying at Caerdeon, Barmouth, Wales (see ‘Journal’ (Correspondence vol. 17, Appendix II)). Saturday Review. ## Bibliography Correspondence: The correspondence of Charles Darwin. Edited by Frederick Burkhardt et al. 27 vols to date. Cambridge: Cambridge University Press. 1985–. Orchids: On the various contrivances by which British and foreign orchids are fertilised by insects, and on the good effects of intercrossing. By Charles Darwin. London: John Murray. 1862. ## Summary Observations on flies visiting Epipactis. ## Letter details Letter no. DCP-LETT-6816 From William Erasmus Darwin To Charles Robert Darwin Sent from Southampton Source of text DAR 162: 100 Physical description 12pp †	2021-06-20 10:13:07	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5884703397750854, "perplexity": 3173.6619876766017}, "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-2021-25/segments/1623487660269.75/warc/CC-MAIN-20210620084505-20210620114505-00278.warc.gz"}
http://yacas.sourceforge.net/Algochapter4.html	# 4. Numerical algorithms I: basic methods This and subsequent chapters document the numerical algorithms used in Yacas for exact integer calculations as well as for multiple precision floating-point calculations. We give self-contained descriptions of the non-trivial algorithms and estimate their computational cost. Most of the algorithms were taken from referenced literature; the remaining algorithms were developed by us. Here we consider plotting of functions y=f(x). There are two tasks related to preparation of plots of functions: first, to produce the numbers required for a plot, and second, to draw a plot with axes, symbols, a legend, perhaps additional illustrations and so on. Here we only concern ourselves with the first task, that of preparation of the numerical data for a plot. There are many plotting programs that can read a file with numbers and plot it in any desired manner. Generating data for plots of functions generally does not require high-precision calculations. However, we need an algorithm that can be adjusted to produce data to different levels of precision. In some particularly ill-behaved cases, a precise plot will not be possible and we would not want to waste time producing data that is too accurate for what it is worth. A simple approach to plotting would be to divide the interval into many equal subintervals and to evaluate the function on the resulting grid. Precision of the plot can be adjusted by choosing a larger or a smaller number of points. However, this approach is not optimal. Sometimes a function changes rapidly near one point but slowly everywhere else. For example, f(x)=1/x changes very quickly at small x. Suppose we need to plot this function between 0 and 100. It would be wasteful to use the same subdivision interval everywhere: a finer grid is only required over a small portion of the plotting range near x=0. The adaptive plotting routine Plot2D'adaptive uses a simple algorithm to select the optimal grid to approximate a function of one argument f(x). The algorithm repeatedly subdivides the grid intervals near points where the existing grid does not represent the function well enough. A similar algorithm for adaptive grid refinement could be used for numerical integration. The idea is that plotting and numerical integration require the same kind of detailed knowledge about the behavior of the function. The algorithm first splits the interval into a specified initial number of equal subintervals, and then repeatedly splits each subinterval in half until the function is well enough approximated by the resulting grid. The integer parameter depth gives the maximum number of binary splittings for a given initial interval; thus, at most 2^depth additional grid points will be generated. The function Plot2D'adaptive should return a list of pairs of points {{x1,y1}, {x2,y2}, ...} to be used directly for plotting. The adaptive plotting algorithm works like this: • 1. Given an interval ( • a, c), we split it in half, b:=(a+c)/2 and first compute f(x) at five grid points a, a[1]:=(a+b)/2, b, b[1]:=(b+c)/2, c. • 2. If currently • depth<=0, return this list of five points and values because we cannot refine the grid any more. • 3. Otherwise, check that the function does not oscillate too rapidly on the interval [ • a, c]. The formal criterion is that the five values are all finite and do not make a "zigzag" pattern such as (1,3,2,3,1). More formally, we use the following procedure: For each three consecutive values, write "1" if the middle value is larger than the other two, or if it is smaller than the other two, or if one of them is not a number (e.g. Infinity or Undefined). If we have at most two ones now, then we consider the change of values to be "slow enough". Otherwise it is not "slow enough". In this case we need to refine the grid; go to step 5. Otherwise, go to step 4. • 4. Check that the function values are smooth enough through the interval. Smoothness is controlled by a parameter • epsilon. The meaning of the parameter epsilon is the (relative) error of the numerical approximation of the integral of f(x) by the grid. A good heuristic value of epsilon is 1/(the number of pixels on the screen) because it means that no pixels will be missing in the area under the graph. For this to work we need to make sure that we are actually computing the area under the graph; so we define g(x):=f(x)-f[0] where f[0] is the minimum of the values of f(x) on the five grid points a, a[1], b, b[1], and c; the function g(x) is nonnegative and has the minimum value 0. Then we compute two different Newton-Cotes quadratures for Integrate(x,b,b[1])g(x) using these five points. (Asymmetric quadratures are chosen to avoid running into an accidental symmetry of the function; the first quadrature uses points a, a[1], b, b[1] and the second quadrature uses b, b[1], c.) If the absolute value of the difference between these quadratures is less than epsilon * (value of the second quadrature), then we are done and we return the list of these five points and values. • 5. Otherwise, we need to refine the grid. We compute • Plot2D'adaptive recursively for the two halves of the interval, that is, for ( a, b) and ( b, c). We also decrease depth by 1 and multiply epsilon by 2 because we need to maintain a constant absolute precision and this means that the relative error for the two subintervals can be twice as large. The resulting two lists for the two subintervals are concatenated (excluding the double value at point b) and returned. This algorithm works well if the initial number of points and the depth parameter are large enough. These parameters can be adjusted to balance the available computing time and the desired level of detail in the resulting plot. Singularities in the function are handled by the step 3. Namely, the change in the sequence a, a[1], b, b[1], c is always considered to be "too rapid" if one of these values is a non-number (e.g. Infinity or Undefined). Thus, the interval immediately adjacent to a singularity will be plotted at the highest allowed refinement level. When preparing the plotting data, the singular points are simply not printed to the data file, so that a plotting programs does not encounter any problems. The meaning of Newton-Cotes quadrature coefficients is that an integral of a function f(x) is approximated by a sum, (Integrate(x,a[0],a[n])f(x))<=>hSum(k,0,n,c[k]f(a[k])), where a[k] are the grid points, h:=a[1]-a[0] is the grid step, and c[k] are the quadrature coefficients. It may seem surprising, but these coefficients c[k] are independent of the function f(x) and can be precomputed in advance for a given grid a[k]. [The quadrature coefficients do depend on the relative separations of the grid. Here we assume a uniform grid with a constant step h=a[k]-a[k-1]. Quadrature coefficients can also be found for non-uniform grids.] The coefficients c[k] for grids with a constant step h can be found, for example, by solving the following system of equations, Sum(k,0,n,c[k]k^p)=n^(p+1)/(p+1) for p=0, 1, ..., n. This system of equations means that the quadrature correctly gives the integrals of p+1 functions f(x)=x^p, p=0, 1, ..., n, over the interval (0, n). The solution of this system always exists and gives quadrature coefficients as rational numbers. For example, the well-known Simpson quadrature c[0]=1/3, c[1]=4/3, c[2]=1/3 is obtained with n=2. An example of using this quadrature is the approximation (Integrate(x,0,2)f(x))<=>(f(0)+f(2))/3+4/3f(1). In the same way it is possible to find quadratures for the integral over a subinterval rather than over the whole interval of x. In the current implementation of the adaptive plotting algorithm, two quadratures are used: the 3-point quadrature ( n=2) and the 4-point quadrature ( n=3) for the integral over the first subinterval, Integrate(x,a[0],a[1])f(x). Their coefficients are (5/12, 2/3, -1/12) and ( 3/8, 19/24, -5/24, 1/24). An example of using the first of these subinterval quadratures would be the approximation (Integrate(x,0,1)f(x))<=>5/12f(0)+2/3f(1)-1/12f(2). These quadratures are intentionally chosen to be asymmetric to avoid an accidental cancellation when the function f(x) itself is symmetric. (Otherwise the error estimate could accidentally become exactly zero.) ## 4.2 Surface plotting Here we consider plotting of functions z=f(x,y). The task of surface plotting is to obtain a picture of a two-dimensional surface as if it were a solid object in three dimensions. A graphical representation of a surface is a complicated task. Sometimes it is required to use particular coordinates or projections, to colorize the surface, to remove hidden lines and so on. We shall only be concerned with the task of obtaining the data for a plot from a given function of two variables f(x,y). Specialized programs can take a text file with the data and let the user interactively produce a variety of surface plots. The currently implemented algorithm in the function Plot3DS is very similar to the adaptive plotting algorithm for two-dimensional plots. A given rectangular plotting region a[1]<=x<=a[2], b[1]<=y<=b[2] is subdivided to produce an equally spaced rectangular grid of points. This is the initial grid which will be adaptively refined where necessary. The refinement algorithm will divide a given rectangle in four quarters if the available function values indicate that the function does not change smoothly enough on that rectangle. The criterion of a "smooth enough" change is very similar to the procedure outlined in the previous section. The change is "smooth enough" if all points are finite, nonsingular values, and if the integral of the function over the rectangle is sufficiently well approximated by a certain low-order "cubature" formula. The two-dimensional integral of the function is estimated using the following 5-point Newton-Cotes cubature: ```1/12 0 1/12 0 2/3 0 1/12 0 1/12 ``` An example of using this cubature would be the approximation (Integrate(y,0,1)Integrate(x,0,1)f(x,y))<=>(f(0,0)+f(0,1)+f(1,0)+f(1,1))/12 +2/3f(1/2,1/2). Similarly, an 8-point cubature with zero sum is used to estimate the error: ```-1/3 2/3 1/6 -1/6 -2/3 -1/2 1/2 0 1/3 ``` This set of coefficients was intentionally chosen to be asymmetric to avoid possible exact cancellations when the function itself is symmetric. One minor problem with adaptive surface plotting is that the resulting set of points may not correspond to a rectangular grid in the parameter space (x, y). This is because some rectangles from the initial grid will need to be bisected more times than others. So, unless adaptive refinement is disabled, the function Plot3DS produces a somewhat disordered set of points. However, most surface plotting programs require that the set of data points be a rectangular grid in the parameter space. So a smoothing and interpolation procedure is necessary to convert a non-gridded set of data points ("scattered" data) to a gridded set. ## 4.3 Parametric plots Currently, parametric plots are not directly implemented in Yacas. However, it is possible to use Yacas to obtain numerical data for such plots. One can then use external programs to produce actual graphics. A two-dimensional parametric plot is a line in a two-dimensional space, defined by two equations such as x=f(t), y=g(t). Two functions f, g and a range of the independent variable t, for example, t[1]<=t<=t[2], need to be specified. Parametric plots can be used to represent plots of functions in non-Euclidean coordinates. For example, to plot the function rho=Cos(4phi)^2 in polar coordinates ( rho, phi), one can rewrite the Euclidean coordinates through the polar coordinates, x=rhoCos(phi), y=rhoSin(phi), and use the equivalent parametric plot with phi as the parameter: x=Cos(4phi)^2Cos(phi), y=Cos(4phi)^2Sin(phi). Sometimes higher-dimensional parametric plots are required. A line plot in three dimensions is defined by three functions of one variable, for example, x=f(t), y=g(t), z=h(t), and a range of the parameter t. A surface plot in three dimensions is defined by three functions of two variables each, for example, x=f(u,v), y=g(u,v), z=h(u,v), and a rectangular domain in the (u, v) space. The data for parametric plots can be generated separately using the same adaptive plotting algorithms as for ordinary function plots, as if all functions such as f(t) or g(u,v) were unrelated functions. The result would be several separate data sets for the x, y, ... coordinates. These data sets could then be combined using an interactive plotting program. ## 4.4 The cost of arbitrary-precision computations A computer algebra system absolutely needs to be able to perform computations with very large integer numbers. Without this capability, many symbolic computations (such as exact GCD of polynomials or exact solution of polynomial equations) would be impossible. A different question is whether a CAS really needs to be able to evaluate, say, 10,000 digits of the value of a Bessel function of some 10,000-digit complex argument. It seems likely that no applied problem of natural sciences would need floating-point computations of special functions with such a high precision. However, arbitrary-precision computations are certainly useful in some mathematical applications; e.g. some mathematical identities can be first guessed by a floating-point computation with many digits and then proved. Very high precision computations of special functions might be useful in the future. But it is already quite clear that computations with moderately high precision (say, 50 or 100 decimal digits) are useful for applied problems. For example, to obtain the leading asymptotic of an analytic function, we could expand it in series and take the first term. But we need to check that the coefficient at what we think is the leading term of the series does not vanish. This coefficient could be a certain "exact" number such as (Cos(355)+1)^2. This number is "exact" in the sense that it is made of integers and elementary functions. But we cannot say a priori that this number is nonzero. The problem of "zero determination" (finding out whether a certain "exact" number is zero) is known to be algorithmically unsolvable if we allow transcendental functions. The only practical general approach seems to be to compute the number in question with many digits. Usually a few digits are enough, but occasionally several hundred digits are needed. Implementing an efficient algorithm that computes 100 digits of Sin(3/7) already involves many of the issues that would also be relevant for a 10,000 digit computation. Modern algorithms allow evaluations of all elementary functions in time that is asymptotically logarithmic in the number of digits P and linear in the cost of long multiplication (usually denoted M(P)). Almost all special functions can be evaluated in time that is asymptotically linear in P and in M(P). (However, this asymptotic cost sometimes applies only to very high precision, e.g., P>1000, and different algorithms need to be implemented for calculations in lower precision.) In Yacas we strive to implement all numerical functions to arbitrary precision. All integer or rational functions return exact results, and all floating-point functions return their value with P correct decimal digits (assuming sufficient precision of the arguments). The current value of P is accessed as Builtin'Precision'Get() and may be changed by Builtin'Precision'Set(...). Implementing an arbitrary-precision floating-point computation of a function f(x), such as f(x)=Exp(x), typically needs the following: • An algorithm that will compute • f(x) for a given value x to a user-specified precision of P (decimal) digits. Often, several algorithms must be implemented for different subdomains of the ( x, P) space. • An estimate of the computational cost of the algorithm(s), as a function of • x and P. This is needed to select the best algorithm for given x, P. • An estimate of the round-off error. This is needed to select the "working precision" which will typically be somewhat higher than the precision of the final result. In calculations with machine precision where the number of digits is fixed, the problem of round-off errors is quite prominent. Every arithmetic operation causes a small loss of precision; as a result, a few last digits of the final value are usually incorrect. But if we have an arbitrary precision capability, we can always increase precision by a few more digits during intermediate computations and thus eliminate all round-off error in the final result. We should, of course, take care not to increase the working precision unnecessarily, because any increase of precision means slower calculations. Taking twice as many digits as needed and hoping that the result is precise is not a good solution. Selecting algorithms for computations is the most non-trivial part of the implementation. We want to achieve arbitrarily high precision, so we need to find either a series, or a continued fraction, or a sequence given by explicit formula, that converges to the function in a controlled way. It is not enough to use a table of precomputed values or a fixed approximation formula that has a limited precision. In the last 30 years, the interest in arbitrary-precision computations grew and many efficient algorithms for elementary and special functions were published. Most algorithms are iterative. Almost always it is very important to know in advance how many iterations are needed for given x, P. This knowledge allows to estimate the computational cost, in terms of the required precision P and of the cost of long multiplication M(P), and choose the best algorithm. Typically all operations will fall into one of the following categories (sorted by the increasing cost): • P; • multiplication, division, integer power, integer root: linear in • M(P); • elementary functions: • Exp(x), Ln(x), Sin(x), ArcTan(x) etc.: M(P)Ln(P) or slower by some powers of Ln(P); • transcendental functions: • Erf(x), Gamma(x) etc.: typically PM(P) or slower. The cost of long multiplication M(P) is between O(P^2) for low precision and O(PLn(P)) for very high precision. In some cases, a different algorithm should be chosen if the precision is high enough to allow M(P) faster than O(P^2). Some algorithms also need storage space (e.g. an efficient algorithm for summation of the Taylor series uses O(Ln(P)) temporary P-digit numbers). Below we shall normally denote by P the required number of decimal digits. The formulae frequently contain conspicuous factors of Ln(10), so it will be clear how to obtain analogous expressions for another base. (Most implementations use a binary base rather than a decimal base since it is more convenient for many calculations.) ## 4.5 Estimating convergence of a series Analyzing convergence of a power series is usually not difficult. Here is a worked-out example of how we could estimate the required number of terms in the power series Exp(x)=1+x+x^2/2! +...+x^n/n! +O(x^(n+1)) if we need P decimal digits of precision in the result. To be specific, assume that Abs(x)<1. (A similar calculation can be done for any other bound on x.) Suppose we truncate the series after n-th term and the series converges "well enough" after that term. Then the error will be approximately equal to the first term we dropped. (This is what we really mean by "converges well enough" and this will generally be the case in all applications, because we would not want to use a series that does not converge well enough.) The term we dropped is x^(n+1)/(n+1)!. To estimate n! for large n, one can use the inequality e^(e-1)(n/e)^n<n! <(n/e)^(n+1) (valid for all n>=47) which provides tight bounds for the growth of the factorial, or a weaker inequality which is somewhat easier to use, (n/e)^n<n! <((n+1)/e)^(n+1) (valid for all n>=6). The latter inequality is sufficient for most purposes. If we use the upper bound on n! from this estimate, we find that the term we dropped is bounded by x^(n+1)/(n+1)! <(e/(n+2))^(n+2). We need this number to be smaller than 10^(-P). This leads to an inequality (e/(n+2))^(n+2)<10^(-P), which we now need to solve for n. The left hand side decreases with growing n. So it is clear that the inequality will hold for large enough n, say for n>=n0 where n0 is an unknown (integer) value. We can take a logarithm of both sides, replace n with n0 and obtain the following equation for n0: (n0+2)Ln((n0+2)/e)=PLn(10). This equation cannot be solved exactly in terms of elementary functions; this is a typical situation in such estimates. However, we do not really need a very precise solution for n0; all we need is an estimate of its integer part. This is also a typical situation. It is acceptable if our approximate value of n0 comes out a couple of units higher than necessary, because a couple of extra terms of the Taylor series will not significantly slow down the algorithm (but it is important that we do not underestimate n0). Finally, we are mostly interested in having a good enough answer for large values of P. We can try to guess the result. The largest term on the LHS grows as n0Ln(n0) and it should be approximately equal to PLn(10); but Ln(n0) grows very slowly, so this gives us a hint that n0 is proportional to PLn(10). As a first try, we set n0=PLn(10)-2 and compare the RHS with the LHS; we find that we have overshot by a factor Ln(P)-1+Ln(Ln(10)), which is not a large factor. We can now compensate and divide n0 by this factor, so our second try is n0=(PLn(10))/(Ln(P)-1+Ln(Ln(10)))-2. (This approximation procedure is equivalent to solving the equation x=(PLn(10))/(Ln(x)-1) by direct iteration, starting from x=PLn(10).) If we substitute our second try for n0 into the equation, we shall find that we undershot a little bit (i.e. the LHS is a little smaller than the RHS), but our n0 is now smaller than it should be by a quantity that is smaller than 1 for large enough P. So we should stop at this point and simply add 1 to this approximate answer. We should also replace Ln(Ln(10))-1 by 0 for simplicity (this is safe because it will slightly increase n0.) Our final result is that it is enough to take n=(PLn(10))/Ln(P)-1 terms in the Taylor series to compute Exp(x) for Abs(x)<1 to P decimal digits. (Of course, if x is much smaller than 1, many fewer terms will suffice.) ## 4.6 Estimating the round-off error ### Unavoidable round-off errors As the required precision P grows, an arbitrary-precision algorithm will need more iterations or more terms of the series. So the round-off error introduced by every floating-point operation will increase. When doing arbitrary-precision computations, we can always perform all calculations with a few more digits and compensate for round-off error. It is however imperative to know in advance how many more digits we need to take for our "working precision". We should also take that increase into account when estimating the total cost of the method. (In most cases this increase is small.) Here is a simple estimate of the normal round-off error in a computation of n terms of a power series. Suppose that the sum of the series is of order 1, that the terms monotonically decrease in magnitude, and that adding one term requires two multiplications and one addition. If all calculations are performed with absolute precision epsilon=10^(-P), then the total accumulated round-off error is 3nepsilon. If the relative error is 3nepsilon, it means that our answer is something like a(1+3nepsilon) where a is the correct answer. We can see that out of the total P digits of this answer, only the first k decimal digits are correct, where k= -Ln(3nepsilon)/Ln(10). In other words, we have lost P-k=Ln(3n)/Ln(10) digits because of accumulated round-off error. So we found that we need Ln(3n)/Ln(10) extra decimal digits to compensate for this round-off error. This estimate assumes several things about the series (basically, that the series is "well-behaved"). These assumptions must be verified in each particular case. For example, if the series begins with some large terms but converges to a very small value, this estimate is wrong (see the next subsection). In the previous exercise we found the number of terms n for Exp(x). So now we know how many extra digits of working precision we need for this particular case. Below we shall have to perform similar estimates of the required number of terms and of the accumulated round-off error in our analysis of the algorithms. ### Catastrophic round-off error Sometimes the round-off error of a particular method of computation becomes so large that the method becomes highly inefficient. Consider the computation of Sin(x) by the truncated Taylor series Sin(x)<=>Sum(k,0,N-1,(-1)^kx^(2k+1)/(2k+1)!), when x is large. We know that this series converges for all x, no matter how large. Assume that x=10^M with M>=1, and that we need P decimal digits of precision in the result. First, we determine the necessary number of terms N. The magnitude of the sum is never larger than 1. Therefore we need the N-th term of the series to be smaller than 10^(-P). The inequality is (2N+1)! >10^(P+M(2N+1)). We obtain that 2N+2>e10^M is a necessary condition, and if P is large, we find approximately 2N+2<=>((P-M)Ln(10))/(Ln(P-M)-1-MLn(10)). However, taking enough terms does not yet guarantee a good result. The terms of the series grow at first and then start to decrease. The sum of these terms is, however, small. Therefore there is some cancellation and we need to increase the working precision to avoid the round-off. Let us estimate the required working precision. We need to find the magnitude of the largest term of the series. The ratio of the next term to the previous term is x/(2k(2k+1)) and therefore the maximum will be when this ratio becomes equal to 1, i.e. for 2k<=>Sqrt(x). Therefore the largest term is of order x^Sqrt(x)/Sqrt(x)! and so we need about M/2Sqrt(x) decimal digits before the decimal point to represent this term. But we also need to keep at least P digits after the decimal point, or else the round-off error will erase the significant digits of the result. In addition, we will have unavoidable round-off error due to O(P) arithmetic operations. So we should increase precision again by P+Ln(P)/Ln(10) digits plus a few guard digits. As an example, to compute Sin(10) to P=50 decimal digits with this method, we need a working precision of about 60 digits, while to compute Sin(10000) we need to work with about 260 digits. This shows how inefficient the Taylor series for Sin(x) becomes for large arguments x. A simple transformation x=2Pin+x' would have reduced x to at most 7, and the unnecessary computations with 260 digits would be avoided. The main cause of this inefficiency is that we have to add and subtract extremely large numbers to get a relatively small result of order 1. We find that the method of Taylor series for Sin(x) at large x is highly inefficient because of round-off error and should be complemented by other methods. This situation seems to be typical for Taylor series. ## 4.7 Basic arbitrary-precision arithmetic Yacas uses an internal math library (the yacasnumbers library) which comes with the source code. This reduces the dependencies of the Yacas system and improves portability. The internal math library is simple and does not necessarily use the most optimal algorithms. If P is the number of digits of precision, then multiplication and division take M(P)=O(P^2) operations in the internal math. (Of course, multiplication and division by a short integer takes time linear in P.) Much faster algorithms (Karatsuba, Toom-Cook, FFT multiplication, Newton-Raphson division etc.) are implemented in gmp, CLN and some other libraries. The asymptotic cost of multiplication for very large precision is M(P)<=>O(P^1.6) for the Karatsuba method and M(P)=O(PLn(P)Ln(Ln(P))) for the FFT method. In the estimates of computation cost in this book we shall assume that M(P) is at least linear in P and maybe a bit slower. The costs of multiplication may be different in various arbitrary-precision arithmetic libraries and on different computer platforms. As a rough guide, one can assume that the straightforward O(P^2) multiplication is good until 100-200 decimal digits, the asymptotically fastest method of FFT multiplication is good at the precision of about 5,000 or more decimal digits, and the Karatsuba multiplication is best in the middle range. Warning: calculations with internal Yacas math using precision exceeding 10,000 digits are currently impractically slow. In some algorithms it is necessary to compute the integer parts of expressions such as aLn(b)/Ln(10) or aLn(10)/Ln(2), where a, b are short integers of order O(P). Such expressions are frequently needed to estimate the number of terms in the Taylor series or similar parameters of the algorithms. In these cases, it is important that the result is not underestimated. However, it would be wasteful to compute 1000Ln(10)/Ln(2) in great precision only to discard most of that information by taking the integer part of that number. It is more efficient to approximate such constants from above by short rational numbers, for example, Ln(10)/Ln(2)<28738/8651 and Ln(2)<7050/10171. The error of such an approximation will be small enough for practical purposes. The function BracketRational can be used to find optimal rational approximations. The function IntLog (see below) efficiently computes the integer part of a logarithm (for an integer base, not a natural logarithm). If more precision is desired in calculating Ln(a)/Ln(b) for integer a, b, one can compute IntLog(a^k,b) for some integer k and then divide by k. ## 4.8 How many digits of Sin(Exp(Exp(1000))) do we need? Arbitrary-precision math is not omnipotent against overflow. Consider the problem of representing very large numbers such as x=Exp(Exp(1000)). Suppose we need a floating-point representation of the number x with P decimal digits of precision. In other words, we need to express x<=>M10^E, where the mantissa 1<M<10 is a floating-point number and the exponent E is an integer, chosen so that the relative precision is 10^(-P). How much effort is needed to find M and E? The exponent E is easy to obtain: E=Floor(Ln(x)/Ln(10))=Floor(Exp(1000)/Ln(10))<=>8.5510^433. To compute the integer part Floor(y) of a number y exactly, we need to approximate y with at least Ln(y)/Ln(10) floating-point digits. In our example, we find that we need 434 decimal digits to represent E. Once we found E, we can write x=10^(E+m) where m=Exp(1000)/Ln(10)-E is a floating-point number, 0<m<1. Then M=10^m. To find M with P (decimal) digits, we need m with also at least P digits. Therefore, we actually need to evaluate Exp(1000)/Ln(10) with 434+P decimal digits before we can find P digits of the mantissa of x. We ran into a perhaps surprising situation: one needs a high-precision calculation even to find the first digit of x, because it is necessary to find the exponent E exactly as an integer, and E is a rather large integer. A normal double-precision numerical calculation would give an overflow error at this point. Suppose we have found the number x=Exp(Exp(1000)) with some precision. What about finding Sin(x)? Now, this is extremely difficult, because to find even the first digit of Sin(x) we have to evaluate x with absolute error of at most 0.5. We know, however, that the number x has approximately 10^434 digits before the decimal point. Therefore, we would need to calculate x with at least that many digits. Computations with 10^434 digits is clearly far beyond the capability of modern computers. It seems unlikely that even the sign of Sin(Exp(Exp(1000))) will be obtained in the near future. ###### It seems even less likely that the sign of Sin(Exp(Exp(1000))) would be of any use to anybody even if it could be computed. Suppose that N is the largest integer that our arbitrary-precision facility can reasonably handle. (For Yacas internal math library, N is about 10^10000.) Then it follows that numbers x of order 10^N can be calculated with at most one (1) digit of floating-point precision, while larger numbers cannot be calculated with any precision at all. It seems that very large numbers can be obtained in practice only through exponentiation or powers. It is unlikely that such numbers will arise from sums or products of reasonably-sized numbers in some formula. ###### A factorial function can produce rapidly growing results, but exact factorials n! for large n are well represented by the Stirling formula which involves powers and exponentials. For example, suppose a program operates with numbers x of size N or smaller; a number such as 10^N can be obtained only by multiplying O(N) numbers x together. But since N is the largest representable number, it is certainly not feasible to perform O(N) sequential operations on a computer. However, it is feasible to obtain N-th power of a small number, since it requires only O(Ln(N)) operations. If numbers larger than 10^N are created only by exponentiation operations, then special exponential notation could be used to represent them. For example, a very large number z could be stored and manipulated as an unevaluated exponential z=Exp(M10^E) where M>=1 is a floating-point number with P digits of mantissa and E is an integer, Ln(N)<E<N. Let us call such objects "exponentially large numbers" or "exp-numbers" for short. In practice, we should decide on a threshold value N and promote a number to an exp-number when its logarithm exceeds N. Note that an exp-number z might be positive or negative, e.g. z= -Exp(M10^E). Arithmetic operations can be applied to the exp-numbers. However, exp-large arithmetic is of limited use because of an almost certainly critical loss of precision. The power and logarithm operations can be meaningfully performed on exp-numbers z. For example, if z=Exp(M10^E) and p is a normal floating-point number, then z^p=Exp(pM10^E) and Ln(z)=M10^E. We can also multiply or divide two exp-numbers. But it makes no sense to multiply an exp-number z by a normal number because we cannot represent the difference between z and say 2.52z. Similarly, adding z to anything else would result in a total underflow, since we do not actually know a single digit of the decimal representation of z. So if z1 and z2 are exp-numbers, then z1+z2 is simply equal to either z1 or z2 depending on which of them is larger. We find that an exp-number z acts as an effective "infinity" compared with normal numbers. But exp-numbers cannot be used as a device for computing limits: the unavoidable underflow will almost certainly produce wrong results. For example, trying to verify (Limit(x,0)(Exp(x)-1)/x)=1 by substituting x=1/z with some exp-number z gives 0 instead of 1. Taking a logarithm of an exp-number brings it back to the realm of normal, representable numbers. However, taking an exponential of an exp-number results in a number which is not representable even as an exp-number. This is because an exp-number z needs to have its exponent E represented exactly as an integer, but Exp(z) has an exponent of order O(z) which is not a representable number. The monstrous number Exp(z) could be only written as Exp(Exp(M10^E)), i.e. as a "doubly exponentially large" number, or "2-exp-number" for short. Thus we obtain a hierarchy of iterated exp-numbers. Each layer is "unrepresentably larger" than the previous one. The same considerations apply to very small numbers of the order 10^(-N) or smaller. Such numbers can be manipulated as "exponentially small numbers", i.e. expressions of the form Exp(-M10^E) with floating-point mantissa M>=1 and integer E satisfying Ln(N)<E<N. Exponentially small numbers act as an effective zero compared with normal numbers. Taking a logarithm of an exp-small number makes it again a normal representable number. However, taking an exponential of an exp-small number produces 1 because of underflow. To obtain a "doubly exponentially small" number, we need to take a reciprocal of a doubly exponentially large number, or take the exponent of an exponentially large negative power. In other words, Exp(-M10^E) is exp-small, while Exp(-Exp(M10^E)) is 2-exp-small. The practical significance of exp-numbers is rather limited. We cannot obtain even a single significant digit of an exp-number. A "computation" with exp-numbers is essentially a floating-point computation with logarithms of these exp-numbers. A practical problem that needs numbers of this magnitude can probably be restated in terms of more manageable logarithms of such numbers. In practice, exp-numbers could be useful not as a means to get a numerical answer, but as a warning sign of critical overflow or underflow. ## 4.9 Continued fractions A "continued fraction" is an expression of the form a[0]+b[0]/(a[1]+b[1]/(a[2]+b[2]/(a[3]+...))). The coefficients a[i], b[i] are called the "terms" of the fraction. (Usually one has a[i]!=0, b[i]!=0.) The above continued fraction is sometimes written as a[0]+b[0]/(a[1]+...)b[1]/(a[2]+...)b[2]/(a[3]+...)... Usually one considers infinite continued fractions, i.e. the sequences a[i], b[i] are infinite. The value of an infinite continued fraction is defined as the limit of the fraction truncated after a very large number of terms. (A continued traction can be truncated after n-th term if one replaces b[n] by 0.) An infinite continued fraction does not always converge. Convergence depends on the values of the terms. The representation of a number via a continued fraction is not unique because we could, for example, multiply the numerator and the denominator of any simple fraction inside it by any number. Therefore one may consider some normalized representations. A continued fraction is called "regular" if b[k]=1 for all k, all a[k] are integer and a[k]>0 for k>=1. Regular continued fractions always converge. ### Approximation of numbers by continued fractions The function ContFrac converts a (real) number r into a regular continued fraction with integer terms, r=n[0]+1/(n[1]+1/(n[2]+...)). Here all numbers n[i] are integers and all except n[0] are positive. This representation of a real number r is unique. We may write this representation as r=[n[0];n[1];n[2];...;]. If r is a rational number, the corresponding regular continued fraction is finite, terminating at some n[N]. Otherwise the continued fraction will be infinite. It is known that the truncated fractions will be in some sense "optimal" rational representations of the real number r. The algorithm for converting a rational number r=n/m into a regular continued fraction is simple. First, we determine the integer part of r, which is Div(n,m). If it is negative, we need to subtract one, so that r=n[0]+x and the remainder x is nonnegative and less than 1. The remainder x=Mod(n,m)/m is then inverted, r[1]:=1/x=m/Mod(n,m) and so we have completed the first step in the decomposition, r=n[0]+1/r[1]; now n[0] is integer but r[1] is perhaps not integer. We repeat the same procedure on r[1], obtain the next integer term n[1] and the remainder r[2] and so on, until such n that r[n] is an integer and there is no more work to do. This process will always terminate. If r is a real number which is known by its floating-point representation at some precision, then we can use the same algorithm because all floating-point values are actually rational numbers. Real numbers known by their exact representations can sometimes be expressed as infinite continued fractions, for example Sqrt(11)=[3;3;6;3;6;3;6;...;]; Exp(1/p)=[1;p-1;1;1;3p-1;1;1;5p-1;...;]. The functions GuessRational and NearRational take a real number x and use continued fractions to find rational approximations r=p/q<=>x with "optimal" (small) numerators and denominators p, q. Suppose we know that a certain number x is rational but we have only a floating-point representation of x with a limited precision, for example, x<=>1.5662650602409638. We would like to guess a rational form for x (in this example x=130/83). The function GuessRational solves this problem. Consider the following example. The number 17/3 has a continued fraction expansion {5,1,2}. Evaluated as a floating point number with limited precision, it may become something like 17/3+0.00001, where the small number represents a round-off error. The continued fraction expansion of this number is {5, 1, 2, 11110, 1, 5, 1, 3, 2777, 2}. The presence of an unnaturally large term 11110 clearly signifies the place where the floating-point error was introduced; all terms following it should be discarded to recover the continued fraction {5,1,2} and from it the initial number 17/3. If a continued fraction for a number x is cut right before an unusually large term and evaluated, the resulting rational number has a small denominator and is very close to x. This works because partial continued fractions provide "optimal" rational approximations for the final (irrational) number, and because the magnitude of the terms of the partial fraction is related to the magnitude of the denominator of the resulting rational approximation. GuessRational(x, prec) needs to choose the place where it should cut the continued fraction. The algorithm for this is somewhat heuristic but it works well enough. The idea is to cut the continued fraction when adding one more term would change the result by less than the specified precision. To realize this in practice, we need an estimate of how much a continued fraction changes when we add one term. The routine GuessRational uses a (somewhat weak) upper bound for the difference of continued fractions that differ only by an additional last term: Abs(delta):=Abs(1/(a[1]+1/(...+1/a[n]))-1/(a[1]+1/(...+1/a[n+1])))<1/((a[1]...a[n])^2a[n+1]). (The derivation of this inequality is given below.) Thus we should compute the product of successive terms a[i] of the continued fraction and stop at a[n] at which this product exceeds the maximum number of digits. The routine GuessRational has a second parameter prec which is by default 1/2 times the number of decimal digits of current precision; it stops at a[n] at which the product a[1]...a[n] exceeds 10^prec. The above estimate for delta hinges on the inequality 1/(a+1/(b+...))<1/a and is suboptimal if some terms a[i]=1, because the product of a[i] does not increase when one of the terms is equal to 1, whereas in fact these terms do make delta smaller. A somewhat better estimate would be obtained if we use the inequality 1/(a+1/(b+1/(c+...)))<1/(a+1/(b+1/c)). (See the next section for more explanations of precision of continued fraction approximations.) This does not lead to a significant improvement if a>1 but makes a difference when a=1. In the product a[1]...a[n], the terms a[i] which are equal to 1 should be replaced by a[i]+1/(a[i+1]+1/a[i+2]). Since the comparison of a[1]...a[n] with 10^prec is qualitative, it it enough to perform calculations of a[1]...a[n] with limited precision. This algorithm works well if x is computed with enough precision; namely, it must be computed to at least as many digits as there are in the numerator and the denominator of the fraction combined. Also, the parameter prec should not be too large (or else the algorithm will find another rational number with a larger denominator that approximates x "better" than the precision to which you know x). The related function NearRational(x, prec) works somewhat differently. The goal is to find an "optimal" rational number, i.e. with smallest numerator and denominator, that is within the distance 10^(-prec) of a given value x. The function NearRational does not always give the same answer as GuessRational. The algorithm for NearRational comes from the HAKMEM [Beeler et al. 1972], Item 101C. Their description is terse but clear: ```Problem: Given an interval, find in it the rational number with the smallest numerator and denominator. Solution: Express the endpoints as continued fractions. Find the first term where they differ and add 1 to the lesser term, unless it's last. Discard the terms to the right. What's left is the continued fraction for the "smallest" rational in the interval. (If one fraction terminates but matches the other as far as it goes, append an infinity and proceed as above.) ``` The HAKMEM text [Beeler et al. 1972] contains several interesting insights relevant to continued fractions and other numerical algorithms. ### Accurate computation of continued fractions Sometimes an analytic function f(x) can be approximated using a continued fraction that contains x in its terms. Examples include the inverse tangent ArcTan(x), the error function Erf(x), and the incomplete gamma function Gamma(a,x) (see below for details). For these functions, continued fractions provide a method of numerical calculation that works when the Taylor series converges slowly or not at all, or is not easily available. However, continued fractions may converge quickly for one value of x but slowly for another. Also, it is not as easy to obtain an analytic error bound for a continued fraction approximation as it is for power series. In this section we describe some methods for computing general continued fractions and for estimating the number of terms needed to achieve a given precision. Let us introduce some notation. A continued fraction a[0]+b[0]/(a[1]+b[1]/(a[2]+...)) is specified by a set of terms (a[i], b[i]). [If continued fractions are used to approximate analytic functions such as ArcTan(x), then (a[i], b[i]) will depend on x.] Let us denote by F[m][n] the truncated fraction containing only the terms from m to n, F[m][n]:=a[m]+b[m]/(a[m+1]+b[m+1]/(...+b[n]/a[n])). In this notation, the continued fraction that we need to compute is F[0][n]. Our task is first, to select a large enough n so that F[0][n] gives enough precision, and second, to compute that value. ### Method 1: bottom-up with straightforward division All "bottom-up" methods need to know the number of terms n in advance. The simplest method is to start evaluating the fraction from the bottom upwards. As the initial approximation we take F[n][n]=a[n]. Then we use the obvious relation of backward recurrence, F[m][n]=a[m]+b[m]/F[m+1][n], to obtain successively F[n-1][n], ..., F[0][n]. This method requires one long division at each step. There may be significant round-off error if a[m] and b[m] have opposite signs, but otherwise the round-off error is very small because a convergent continued fraction is not sensitive to small changes in its terms. ### Method 2: bottom-up with two recurrences An alternative implementation may be faster in some cases. The idea is to obtain the numerator and the denominator of F[0][n] separately as two simultaneous backward recurrences. If F[m+1][n]=p[m+1]/q[m+1], then p[m]=a[m]p[m+1]+b[m]q[m+1] and q[m]=p[m+1]. The recurrences start with p[n]=a[n], q[n]=1. The method requires two long multiplications at each step; the only division will be performed at the very end. Sometimes this method reduces the round-off error as well. ### Method 3: bottom-up with estimated remainders There is an improvement over the bottom-up method that can sometimes increase the achieved precision without computing more terms. This trick is suggested in [Tsimring 1988], sec. 2.4, where it is also claimed that the remainder estimates improve convergence. The idea is that the starting value of the backward recurrence should be chosen not as a[n] but as another number that more closely approximates the infinite remainder of the fraction. The infinite remainder, which we can symbolically write as F[n][Infinity], can be sometimes estimated analytically (obviously, we are unable to compute the remainder exactly). In simple cases, F[n][Infinity] changes very slowly at large n (warning: this is not always true and needs to be verified in each particular case!). Suppose that F[n][Infinity] is approximately constant; then it must be approximately equal to F[n+1][Infinity]. Therefore, if we solve the (quadratic) equation x=a[n]+b[n]/x, we shall obtain the (positive) value x which may be a much better approximation for F[n][Infinity] than a[n]. But this depends on the assumption of the way the continued fraction converges. It may happen, for example, that for large n the value F[n][Infinity] is almost the same as F[n+2][Infinity] but is significantly different from F[n+1][Infinity]. Then we should instead solve the (quadratic) equation x=a[n]+b[n]/(a[n+1]+b[n+1]/x) and take the positive solution x as F[n][Infinity]. We may use more terms of the original continued fraction starting from a[n] and obtain a more precise estimate of the remainder. In each case we will only have to solve one quadratic equation. ### Method 4: top-down computation The "bottom-up" method obviously requires to know the number of terms n in advance; calculations have to be started all over again if more terms are needed. Also, the bottom-up method provides no error estimate. The "top-down" method is slower but provides an automatic error estimate and can be used to evaluate a continued fraction with more and more terms until the desired precision is achieved. The idea ###### This is a well-known result in the theory of continued fractions. We give an elementary derivation below. is to replace the continued fraction F[0][n] with a sum of a certain series, a[0]+b[0]/(a[1]+b[1]/(...+b[n-1]/a[n]))=Sum(k,0,n,f[k]). Here f[k]:=F[0][k]-F[0][k-1] (k>=1) is a sequence that will be calculated in the forward direction, starting from k=1. If we manage to find a formula for this sequence, then adding one more term f[k] will be equivalent to recalculating the continued fraction with k terms instead of k-1 terms. This will automatically give an error estimate and allow to compute with more precision if necessary without having to repeat the calculation from the beginning. (The transformation of the continued fraction into a series is exact, not merely an approximation.) The formula for f[k] is the following. First the auxiliary sequence P[k], Q[k] for k>=1 needs to be defined by P[1]=0, Q[1]=1, and P[k+1]:=b[k]Q[k], Q[k+1]:=P[k]+a[k]Q[k]. Then define f[0]:=a[0] and f[k]:=((-1)^kb[0]...b[k-1])/(Q[k]Q[k+1]) for k>=1. The "top-down" method consists of computing f[n] sequentially and adding them together, until n is large enough so that f[n]/f[0] is less than the required precision. Evaluating the next element f[k] requires four long multiplications and one division. This is significantly slower, compared with just one long division or two long multiplications in the bottom-up method. Therefore it is desirable to have an a priori estimate of the convergence rate and to be able to choose the number of terms before the computation. Below we shall consider some examples when the formula for f[k] allows to estimate the required number of terms analytically. ### Method 5: top-down with two steps at once If all coefficients a[i], b[i] are positive, then the series we obtained in the top-down method will have terms f[k] with alternating signs. This leads to a somewhat larger round-off error. We can convert the alternating series into a monotonic series by adding together two adjacent elements, say f[2k]+f[2k+1]. ###### This method is used by [Thacher 1963], who refers to a suggestion by Hans Maehly. The relevant formulae can be derived from the definition of f[k] using the recurrence relations for P[k], Q[k]: f[2k-1]+f[2k]= -(b[0]...b[2k-2]a[2k])/(Q[2k-1]Q[2k+1]), f[2k]+f[2k+1]=(b[0]...b[2k-1]a[2k+1])/(Q[2k]Q[2k+2]). Now in the series f[0]+(f[1]+f[2])+(f[3]+f[4])+... the first term is positive and all subsequent terms will be negative. ### Which method to use We have just described the following methods of computing a continued fraction: • Bottom-up, straight division. • Bottom-up, separate recurrences for numerators and denominators. • Bottom-up, with an estimate of the remainder. • Top-down, with ordinary step. • Top-down, with two steps at once. The bottom-up methods are simpler and faster than the top-down methods but require to know the number of terms in advance. In many cases the required number of terms can be estimated analytically, and then the bottom-up methods are always preferable. But in some cases the convergence analysis is very complicated. The plain bottom-up method requires one long division at each step, while the bottom-up method with two recurrences requires two long multiplications at each step. Since the time needed for a long division is usually about four times that for a long multiplication (e.g. when the division is computed by Newton's method), the second variation of the bottom-up method is normally faster. The estimate of the remainder improves the convergence of the bottom-up method and should always be used if available. If an estimate of the number of terms is not possible, the top-down methods should be used, looping until the running error estimate shows enough precision. This incurs a performance penalty of at least 100% and at most 300%. The top-down method with two steps at once should be used only when the formula for f[k] results in alternating signs. ### Usefulness of continued fractions Many mathematical functions have a representation as a continued fraction. Some systems of "exact arithmetic" use continued fractions as a primary internal representation of real numbers. This has its advantages (no round-off errors, lazy "exact" computations) and disadvantages (it is slow, especially with some operations). Here we consider the use of continued fractions with a traditional implementation of arithmetic (floating-point numbers with variable precision). Usually, a continued fraction representation of a function will converge geometrically or slower, i.e. at least O(P) terms are needed for a precision of P digits. If a geometrically convergent Taylor series representation is also available, the continued fraction method will be slower because it requires at least as many or more long multiplications per term. Also, in most cases the Taylor series can be computed much more efficiently using the rectangular scheme. (See, e.g., the section on ArcTan(x) for a more detailed consideration.) However, there are some functions for which a Taylor series is not easily computable or does not converge but a continued fraction is available. For example, the incomplete Gamma function and related functions can be computed using continued fractions in some domains of their arguments. ### Derivation of the formula for f[k] Here is a straightforward derivation of the formula for f[k] in the top-down method. We need to compute the difference between successive approximations F[0][n] and F[0][n+1]. The recurrence relation we shall use is F[m][n+1]-F[m][n]= -(b[m](F[m+1][n+1]-F[m+1][n]))/(F[m+1][n+1]F[m+1][n]). This can be shown by manipulating the two fractions, or by using the recurrence relation for F[m][n]. So far we have reduced the difference between F[m][n+1] and F[m][n] to a similar difference on the next level m+1 instead of m; i.e. we can increment m but keep n fixed. We can apply this formula to F[0][n+1]-F[0][n], i.e. for m=0, and continue applying the same recurrence relation until m reaches n. The result is F[0][n+1]-F[0][n]=((-1)^nb[0]...b[n])/(F[1][n+1]...F[n+1][n+1]F[1][n]...F[n][n]). Now the problem is to simplify the two long products in the denominator. We notice that F[1][n] has F[2][n] in the denominator, and therefore F[1][n]F[2][n]=F[2][n]a[1]+b[1]. The next product is F[1][n]F[2][n]F[3][n] and it simplifies to a linear function of F[3][n], namely F[1][n]F[2][n]F[3][n] = (b[1]+a[1]a[2])F[3][n]+b[1]a[2]. So we can see that there is a general formula F[1][n]...F[k][n]=P[k]+Q[k]F[k][n] with some coefficients P[k], Q[k] which do not actually depend on n but only on the terms of the partial fraction up to k. In other words, these coefficients can be computed starting with P[1]=0, Q[1]=1 in the forward direction. The recurrence relations for P[k], Q[k] that we have seen above in the definition of f[k] follow from the identity (P[k]+Q[k]F[k][n])F[k+1][n] = P[k+1]+Q[k+1]F[k+1][n]. Having found the coefficients P[k], Q[k], we can now rewrite the long products in the denominator, e.g. F[1][n]...F[n][n]=P[n]+Q[n]F[n][n]=Q[n+1]. (We have used the recurrence relation for Q[n+1].) Now it follows that f[n+1]:=F[0][n+1]-F[0][n]=((-1)^nb[0]...b[n])/(Q[n+1]Q[n+2]). Thus we have converted the continued fraction into a series, i.e. F[0][n]=Sum(k,0,n,f[k]) with f[k] defined above. ### Examples of continued fraction representations We have already mentioned that continued fractions give a computational advantage only when other methods are not available. There exist continued fraction representations of almost all functions, but in most cases the usual methods (Taylor series, identity transformations, Newton's method and so on) are superior. For example, the continued fraction ArcTan(x)=x/(1+x^2/(3+(2x)^2/(5+(3x)^2/(7+...)))) converges geometrically at all x. However, the Taylor series also converges geometrically and can be computed much faster than the continued fraction. There are some cases when a continued fraction representation is efficient. The complementary error function Erfc(x) can be computed using the continued fraction due to Laplace (e.g. [Thacher 1963]), Sqrt(Pi)/2xExp(x^2)Erfc(x)=1/(1+v/(1+(2v)/(1+(3v)/(1+...)))), where v:=(2x^2)^(-1). This continued fraction converges for all (complex) x except pure imaginary x. The error function is a particular case of the incomplete Gamma function Gamma(a,z):=(Integrate(x,z,+Infinity)x^(a-1)Exp(-x)). There exists an analogous continued fraction representation due to Legendre (e.g. [Abramowitz et al.], 6.5.31), Gamma(a,z)=(z^(a-1)Exp(-z))/(1+((1-a)v)/(1+v/(1+((2-a)v)/(1+(2v)/(1+...))))), where v:=z^(-1). ## 4.10 Estimating convergence of continued fractions Elsewhere in this book we have used elementary considerations to find the required number of terms in a power series. It is much more difficult to estimate the convergence rate of a continued fraction. In many cases this can be done using the theory of complex variable. Below we shall consider some cases when this computation is analytically tractable. Suppose we are given the terms a[k], b[k] that define an infinite continued fraction, and we need to estimate its convergence rate. We have to find the number of terms n for which the error of approximation is less than a given epsilon. In our notation, we need to solve Abs(f[n+1])<epsilon for n. The formula that we derived for f[n+1] gives an error estimate for the continued fraction truncated at the n-th term. But this formula contains the numbers Q[n] in the denominator. The main problem is to find how quickly the sequence Q[n] grows. The recurrence relation for this sequence can be rewritten as Q[n+2]=a[n+1]Q[n+1]+b[n]Q[n], for k>=0, with initial values Q[0]=0 and Q[1]=1. It is not always easy to get a handle on this sequence, because in most cases there is no closed-form expression for Q[n]. ### Simple bound on Q[n] A simple lower bound on the growth of Q[n] can be obtained from the recurrence relation for Q[n]. Assume that a[k]>0, b[k]>0. It is clear that all Q[n] are positive, so Q[n+1]>=a[n]Q[n]. Therefore Q[n] grows at least as the product of all a[n]: Q[n+1]>=Factorize(i,1,n,a[n]). This result gives the following upper bound on the precision, Abs(f[n+1])<=(b[0]...b[n])/((a[1]...a[n])^2a[n+1]). We have used this bound to estimate the relative error of the continued fraction expansion for ArcTan(x) at small x (elsewhere in this book). However, we found that at large x this bound becomes greater than 1. This does not mean that the continued fraction does not converge and cannot be used to compute ArcTan(x) when x>1, but merely indicates that the "simple bound" is too weak. The sequence Q[n] actually grows faster than the product of all a[k] and we need a tighter bound on this growth. In many cases such a bound can be obtained by the method of generating functions. ### The method of generating functions The idea is to find a generating function G(s) of the sequence Q[n] and then use an explicit form of G(s) to obtain an asymptotic estimate of Q[n] at large k. The asymptotic growth of the sequence Q[n] can be estimated by the method of steepest descent, also known as Laplace's method. (See, e.g., [Olver 1974], ch. 3, sec. 7.5.) This method is somewhat complicated but quite powerful. The method requires that we find an integral representation for Q[n] (usually a contour integral in the complex plane). Then we can convert the integral into an asymptotic series in k^(-1). Along with the general presentation of the method, we shall consider an example when the convergence rate can be obtained analytically. The example is the representation of the complementary error function Erfc(x), Sqrt(Pi)/2xExp(x^2)Erfc(x)=1/(1+v/(1+(2v)/(1+(3v)/(1+...)))), where v:=(2x^2)^(-1). We shall assume that Abs(v)<1/2 since the continued fraction representation will not be used for small x (where the Taylor series is efficient). The terms of this continued fraction are: a[k]=1, b[k]=kv, for k>=1, and a[0]=0, b[0]=1. The "simple bound" would give Abs(f[n+1])<=v^nn! and this expression grows with n. But we know that the above continued fraction actually converges for any v, so f[n+1] must tend to zero for large n. It seems that the "simple bound" is not strong enough for any v and we need a better bound. An integral representation for Q[n] can be obtained using the method of generating functions. Consider a function G(s) defined by the infinite series G(s)=Sum(n,0,Infinity,Q[n+1]s^n/n!). G(s) is usually called the "generating function" of a sequence. We shifted the index to n+1 for convenience, since Q[0]=0, so now G(0)=1. Note that the above series for the function G(s) may or may not converge for any given s; we shall manipulate G(s) as a formal power series until we obtain an explicit representation. What we really need is an analytic continuation of G(s) to the complex s. It is generally the case that if we know a simple linear recurrence relation for a sequence, then we can also easily find its generating function. The generating function will satisfy a linear differential equation. To guess this equation, we write down the series for G(s) and its derivative G'(s) and try to find their linear combination which is identically zero because of the recurrence relation. (There is, of course, a computer algebra algorithm for doing this automatically.) Taking the derivative G'(s) produces the forward-shifted series G'(s)=Sum(n,0,Infinity,Q[n+2]s^n/n!). Multiplying G(s) by s produces a back-shifted series with each term multiplied by n: sG(s)=Sum(n,0,Infinity,nQ[n]s^n/n!). If the recurrence relation for Q[n] contains only constants and polynomials in n, then we can easily convert that relation into a differential equation for G(s). We only need to find the right combination of back- and forward-shifts and multiplications by n. In the case of our sequence Q[n] above, the recurrence relation is Q[n+2]=Q[n+1]+vnQ[n]. This is equivalent to the differential equation G'(s)=(1+vs)G(s). The solution with the obvious initial condition G(0)=1 is G(s)=Exp(s+(vs^2)/2). The second step is to obtain an integral representation for Q[n], so that we could use the method of steepest descents and find its asymptotic at large n. In our notation Q[n+1] is equal to the n-th derivative of the generating function at s=0: Q[n+1]=(D(s,n)G(s=0)), but it is generally not easy to estimate the growth of this derivative at large n. There are two ways to proceed. One is to obtain an integral representation for G(s), for instance G(s)=(Integrate(t,-Infinity,Infinity)F(t,s)), where F(t,s) is some known function. Then an integral representation for Q[n] will be found by differentiation. But it may be difficult to find such F(t,s). The second possibility is to express Q[n] as a contour integral in the complex plane around s=0 in the counter-clockwise direction: Q[n]=(n-1)! /(2PiI)(Integrate(s)G(s)s^(-n)). If we know the singularities and of G(s), we may transform the contour of the integration into a path that is more convenient for the method of the steepest descent. This procedure is more general but requires a detailed understanding of the behavior of G(s) in the complex plane. In the particular case of the continued fraction for Erfc(x), the calculations are somewhat easier if Re(v)>0 (where v:=1/(2x^2)). Full details are given in a separate section. The result for Re(v)>0 is Q[n]<=>(vn)^(n/2)/Sqrt(2nv)Exp(Sqrt(n/v)-1/(4v)-n/2). This, together with Stirling's formula n! <=>Sqrt(2Pin)(n/e)^n, allows to estimate the error of the continued fraction approximation: f[n+1]<=>2(-1)^(n+1)Sqrt((2Pi)/v)Exp(-2Sqrt(n/v)+1/(2v)). Note that this is not merely a bound but an actual asymptotic estimate of f[n+1]. (Stirling's formula can also be derived using the method of steepest descent from an integral representation of the Gamma function, in a similar way.) Defined as above, the value of f[n+1] is in general a complex number. The absolute value of f[n+1] can be found using the formula Re(Sqrt(n/v))=Sqrt(n/2)Sqrt(1+Re(v)/Abs(v)). We obtain Abs(f[n+1])<=>2Sqrt((2Pi)/Abs(v))Exp(-Sqrt(2n)Sqrt(1+Re(v)/Abs(v))+Re(v)/(2Abs(v)^2)). When Re(v)<=0, the same formula can be used (this can be shown by a more careful consideration of the branches of the square roots). The continued fraction does not converge when Re(v)<0 and Im(v)=0 (i.e. for pure imaginary x). This can be seen from the above formula: in that case Re(v)= -Abs(v) and Abs(f[n+1]) does not decrease when n grows. These estimates show that the error of the continued fraction approximation to Erfc(x) (when it converges) decreases with n slower than in a geometric progression. This means that we need to take O(P^2) terms to get P digits of precision. ### Derivations for the example with Erfc(x) Here we give a detailed calculation of the convergence rate of the continued fraction for Erfc(x) using the method of generating functions. ##### A simpler approach In our case, G(s) is a function with a known Fourier transform and we can obtain a straightforward representation valid when Re(v)>0, Q[n+1]=1/Sqrt(2Piv)(Integrate(t,-Infinity,Infinity)(1+t)^nExp(-t^2/(2v))). We shall first apply the method of steepest descent to this integral (assuming real v>0 for simplicity) and then consider the more general procedure with the contour integration. To use the method of steepest descent, we represent the integrand as an exponential of some function g(t,n) and find "stationary points" where this function has local maxima: Q[n+1]=1/Sqrt(2Piv)(Integrate(t,-Infinity,Infinity)Exp(g(t,n))), g(t,n):= -t^2/(2v)+nLn(1+t). (Note that the logarithm here must acquire an imaginary part IPi for t<-1, and we should take the real part which is equal to Ln(Abs(1+t)). We shall see that the integral over negative t is negligible.) We expect that when n is large, Re(g(t,n)) will have a peak or several peaks at certain values of t. When t is not close to the peaks, the value of Re(g(t,n)) is smaller and, since g is in the exponential, the integral is dominated by the contribution of the peaks. This is the essence of the method of steepest descent on the real line. We only need to consider very large values of n, so we can neglect terms of order O(1/Sqrt(n)) or smaller. We find that, in our case, two peaks of Re(g) occur at approximately t1<=> -1/2+Sqrt(nv) and t2<=> -1/2-Sqrt(nv). We assume that n is large enough so that nv>1/2. Then the first peak is at a positive t and the second peak is at a negative t. The contribution of the peaks can be computed from the Taylor approximation of g(t,n) near the peaks. We can expand, for example, g(t,n)<=>g(t1,n)+(Deriv(t,2)g(t1,n))(t-t1)^2/2 near t=t1. The values g(t1,n) and Deriv(t,2)g(t1,n), and likewise for t2, are constants that we already know since we know t1 and t2. Then the integral of Exp(g) will be approximated by the integral Integrate(t,-Infinity,Infinity)Exp(g(t1,n)+(Deriv(t,2)g(t1,n))(t-t1)^2/2). (Note that Deriv(t,2)g(t1,n) is negative.) This is a Gaussian integral that can be easily evaluated, with the result exp(g(t1,n))Sqrt(-(2Pi)/(Deriv(t,2)g(t1,n))). This is the leading term of the contribution of the peak at t1; there will be a similar expression for the contribution of t2. We find that the peak at t1 gives a larger contribution, by the factor Exp(2Sqrt(n/v)). This factor is never small since n>1 and v<1/2. So it is safe to ignore the peak at t2 for the purposes of our analysis. Then we obtain the estimate Q[n+1]<=>1/Sqrt(2)Exp(Sqrt(n/v)-1/(4v)-n/2)(vn)^(n/2). ##### The contour integral approach In many cases it is impossible to compute the Fourier transform of the generating function G(s). Then one can use the contour integral approach. One should represent the integrand as G(s)s^(-n)=Exp(g(s)) where g(s):=Ln(G(s))-nLn(s), and look for stationary points of g(s) in the complex plane ((D(s)g)=0). The original contour is around the pole s=0 in the counter-clockwise direction. We need to deform that contour so that the new contour passes through the stationary points. The contour should cross each stationary point in a certain direction in the complex plane. The direction is chosen to make the stationary point the sharpest local maximum of Re(g(s)) on the contour. Usually one of the stationary points has the largest value of Re(g(s)); this is the dominant stationary point. If s0 is the dominant stationary point and g2=(Deriv(s,2)g(s0)) is the second derivative of g at that point, then the asymptotic of the integral is 1/(2Pi)(Integrate(s)Exp(g(s)))=1/Sqrt(2PiAbs(g2))Exp(g(s0)). (The integral will have a negative sign if the contour crosses the point s0 in the negative imaginary direction.) We have to choose a new contour and check the convergence of the resulting integral separately. In each case we may need to isolate the singularities of G(s) or to find the regions of infinity where G(s) quickly decays (so that the infinite parts of the contour might be moved there). There is no prescription that works for all functions G(s). Let us return to our example. For G(s)=Exp(s+(vs^2)/2), the function g(s) has no singularities except the pole at s=0. There are two stationary points located at the (complex) roots s1, s2 of the quadratic equation vs^2+s-n=0. Note that v is an arbitrary (nonzero) complex number. We now need to find which of the two stationary points gives the dominant contribution. By comparing Re(g(s1)) and Re(g(s2)) we find that the point s with the largest real part gives the dominant contribution. However, if Re(s1)=Re(s2) (this happens only if v is real and v<0, i.e. if x is pure imaginary), then both stationary points contribute equally. Barring that possibility, we find (with the usual definition of the complex square root) that the dominant contribution for large n is from the stationary point at s1=(Sqrt(4nv+1)-1)/(2v). The second derivative of g(s) at the stationary point is approximately 2v. The contour of integration can be deformed into a line passing through the dominant stationary point in the positive imaginary direction. Then the leading asymptotic is found using the Gaussian approximation (assuming Re(v)>0): Q[n]=((n-1)! v^(n/2))/Sqrt(4Piv)Exp((n(1-Ln(n)))/2+Sqrt(n/v)-1/(4v)). This formula agrees with the asymptotic for Q[n+1] found above for real v>0, when we use Stirling's formula for (n-1)!: (n-1)! =Sqrt(2Pi)e^(-n)n^(n-1/2). The treatment for Re(v)<0 is similar. ## 4.11 Newton's method and its improvements Newton's method (also called the Newton-Raphson method) of numerical solution of algebraic equations and its generalizations can be used to obtain multiple-precision values of several elementary functions. ### Newton's method The basic formula is widely known: If f(x)=0 must be solved, one starts with a value of x that is close to some root and iterates x'=x-f(x)(D(x)f(x))^(-1). This formula is based on the approximation of the function f(x) by a tangent line at some point x. A Taylor expansion in the neighborhood of the root shows that (for an initial value x[0] sufficiently close to the root) each iteration gives at least twice as many correct digits of the root as the previous one ("quadratic convergence"). Therefore the complexity of this algorithm is proportional to a logarithm of the required precision and to the time it takes to evaluate the function and its derivative. Generalizations of this method require computation of higher derivatives of the function f(x) but successive approximations to the root converge several times faster (the complexity is still logarithmic). Newton's method sometimes suffers from a sensitivity to the initial guess. If the initial value x[0] is not chosen sufficiently close to the root, the iterations may converge very slowly or not converge at all. To remedy this, one can combine Newton's iteration with simple bisection. Once the root is bracketed inside an interval (a, b), one checks whether (a+b)/2 is a better approximation for the root than that obtained from Newton's iteration. This guarantees at least linear convergence in the worst case. For some equations f(x)=0, Newton's method converges faster than quadratically. For example, solving Sin(x)=0 in the neighborhood of x=3.14159 gives "cubic" convergence, i.e. the number of correct digits is tripled at each step. This happens because Sin(x) near its root x=Pi has a vanishing second derivative and thus the function is particularly well approximated by a straight line. ### Halley's method Halley's method is an improvement over Newton's method that makes each equation well approximated by a straight line near the root. Edmund Halley computed fractional powers, x=a^(1/n), by the iteration x'=x(n(a+x^n)+a-x^n)/(n(a+x^n)-(a-x^n)). This formula is equivalent to Newton's method applied to the equation x^(n-q)=ax^(-q) with q=(n-1)/2. This iteration has a cubic convergence rate. This is the fastest method to compute n-th roots ( n>=3) with multiple precision. Iterations with higher order of convergence, for example, the method with quintic convergence rate x'=x((n-1)/(n+1)(2n-1)/(2n+1)x^(2n)+2(2n-1)/(n+1)x^na+a^2)/(x^(2n)+2(2n-1)/(n+1)x^na+(n-1)/(n+1)(2n-1)/(2n+1)a^2), require more arithmetic operations per step and are in fact less efficient at high precision. Halley's method can be generalized to any function f(x). A cubically convergent iteration is always obtained if we replace the equation f(x)=0 by an equivalent equation g(x):=f(x)/Sqrt(Abs(D(x)f(x)))=0 and use the standard Newton's method on it. Here the function g(x) is chosen so that its second derivative vanishes ((Deriv(x,2)g(x))=0) at the root of the equation f(x)=0, independently of where this root is. For example, the equation Exp(x)=a is transformed into g(x):=Exp(x/2)-aExp(-x/2)=0. (There is no unique choice of the function g(x) and sometimes another choice will make the iteration more quickly computable.) The Halley iteration for the equation f(x)=0 can be written as x'=x-(2f(x)(D(x)f(x)))/(2(D(x)f(x))^2-f(x)(Deriv(x,2)f(x))). Halley's iteration, despite its faster convergence rate, may be more cumbersome to evaluate than Newton's iteration and so it may not provide a more efficient numerical method for a given function. Only in some special cases is Halley's iteration just as simple to compute as Newton's iteration. Halley's method is sometimes less sensitive to the choice of the initial point x[0]. An extreme example of sensitivity to the initial point is the equation x^(-2)=12 for which Newton's iteration x'=3/2x-6x^3 converges to the root only from initial points 0<x[0]<0.5 and wildly diverges otherwise, while Halley's iteration converges to the root from any x[0]>0. It is at any rate not true that Halley's method always converges better than Newton's method. For instance, it diverges on the equation 2Cos(x)=x unless started at x[0] within the interval (-1/6Pi, 7/6Pi). Another example is the equation Ln(x)=a. This equation allows to compute x=Exp(a) if a fast method for computing Ln(x) is available (e.g. the AGM-based method). For this equation, Newton's iteration x'=x(1+a-Ln(x)) converges for any 0<x<Exp(a+1), while Halley's iteration converges only if Exp(a-2)<x<Exp(a+2). When it converges, Halley's iteration can still converge very slowly for certain functions f(x), for example, for f(x)=x^n-a if n^n>a. For such functions that have very large and rapidly changing derivatives, no general method can converge faster than linearly. In other words, a simple bisection will generally do just as well as any sophisticated iteration, until the root is approximated very precisely. Halley's iteration combined with bisection seems to be a good choice for such problems. ### When to use Halley's method Despite its faster convergence, Halley's iteration frequently gives no advantage over Newton's method. To obtain P digits of the result, we need about Ln(P)/Ln(2) iterations of a quadratically convergent method and about Ln(P)/Ln(3) iterations of a cubically convergent method. So the cubically convergent iteration is faster only if the time taken by cubic one iteration is less than about Ln(3)/Ln(2)<=>1.6 of the time of one quadratic iteration. ### Higher-order schemes Sometimes it is easy to generalize Newton's iteration to higher-order schemes. There are general formulae such as Shroeder's and Householder's iterations. We shall give some examples where the construction is very straightforward. In all examples x is the initial approximation and the next approximation is obtained by truncating the given series. • Inverse • 1/a. Set y=1-ax, then 1/a=x/(1-y)=x(1+y+y^2+...). • Square root • Sqrt(a). Set y=1-ax^2, then Sqrt(a)=Sqrt(1-y)/x=1/x(1-1/2y-1/8y^2-...). • Inverse square root • 1/Sqrt(a). Set y=1-ax^2, then 1/Sqrt(a)=x/Sqrt(1-y)=x(1+1/2y+3/8y^2+...). • n-th root a^(1/n). Set y=1-ax^n, then a^(1/n)=(1-y)^(1/n)/x=1/x(1-1/ny-(n-1)/(2n^2)y^2-...). • Exponential • Exp(a). Set y=a-Ln(x), then Exp(a)=xExp(y)=x(1+y+y^2/2! +y^3/3! +...). • Logarithm • Ln(a). Set y=1-aExp(-x), then Ln(a)=x+Ln(1-y)=x-y-y^2/2-y^3/3-.... In the above examples, y is a small quantity and the series represents corrections to the initial value x, therefore the order of convergence is equal to the first discarded order of y in the series. These simple constructions are possible because the functions satisfy simple identities, such as Exp(a+b)=Exp(a)Exp(b) or Sqrt(ab)=Sqrt(a)Sqrt(b). For other functions the formulae quickly become very complicated and unsuitable for practical computations. ### Precision control Newton's method and its generalizations are particularly convenient for multiple precision calculations because of their insensitivity to accumulated errors: if at some iteration x[k] is found with a small error, the error will be corrected at the next iteration. Therefore it is not necessary to compute all iterations with the full required precision; each iteration needs to be performed at the precision of the root expected from that iteration (plus a few more digits to allow for round-off error). For example, if we know that the initial approximation is accurate to 3 digits, then (assuming quadratic convergence) ###### This disregards the possibility that the convergence might be slightly slower. For example, when the precision at one iteration is n digits, it might be 2n-10 digits at the next iteration. In these (fringe) cases, the initial approximation must be already precise enough (e.g. to at least 10 digits in this example). it is enough to perform the first iteration to 6 digits, the second iteration to 12 digits and so on. In this way, multiple-precision calculations are enormously speeded up. For practical evaluation, iterations must be supplemented with "quality control". For example, if x0 and x1 are two consecutive approximations that are already very close, we can quickly compute the achieved (relative) precision by finding the number of leading zeros in the number Abs(x0-x1)/Max(x0,x1). This is easily done using the bit count function. After performing a small number of initial iterations at low precision, we can make sure that x1 has at least a certain number of correct digits of the root. Then we know which precision to use for the next iteration (e.g. triple precision if we are using a cubically convergent scheme). It is important to perform each iteration at the precision of the root which it will give and not at a higher precision; this saves a great deal of time since all calculations are very slow at high precision. ### Fine-tuning the working precision To reduce the computation time, it is important to write the iteration formula with explicit separation of higher-order quantities. For example, Newton's iteration for the inverse square root 1/Sqrt(a) can be written either as x'=x(3-ax^2)/2 or equivalently as x'=x+x(1-ax^2)/2. At first sight the first formula seems simpler because it saves one long addition. However, the second formula can be computed significantly faster than the first one, if we are willing to exercise a somewhat more fine-grained control of the working precision. Suppose x is an approximation that is correct to P digits; then we expect the quantity x' to be correct to 2P digits. Therefore we should perform calculations in the first formula with 2P digits; this means three long multiplications, 3M(2P). Now consider the calculation in the second formula. First, the quantity y:=1-ax^2 is computed using two 2P-digit multiplications. ###### In fact, both multiplications are a little shorter, since x is a number with only P correct digits; we can compute ax and then ax^2 as products of a 2P-digit number and a P-digit number, with a 2P-digit result. We ignore this small difference. Now, the number y is small and has only P nonzero digits. Therefore the third multiplication xy costs only M(P), not M(2P). This is a significant time savings, especially with slower multiplication. The total cost is now 2M(2P)+M(P). The advantage is even greater with higher-order methods. For example, a fourth-order iteration for the inverse square root can be written as x'=x+1/2xy+3/8xy^2+5/16xy^3, where y:=1-ax^2. Suppose x is an approximation that is correct to P digits; we expect 4P correct digits in x'. We need two long multiplications in precision 4P to compute y, then M(3P) to compute xy, M(2P) to compute xy^2, and M(P) to compute xy^3. The total cost is 2M(4P)+M(3P)+M(2P)+M(P). The asymptotic cost of finding the root x of the equation f(x)=0 with P digits of precision is usually the same as the cost of computing f(x) [Brent 1975]. The main argument can be summarized by the following simple example. To get the result to P digits, we need O(Ln(P)) Newton's iterations. At each iteration we shall have to compute the function f(x) to a certain number of digits. Suppose that we start with one correct digit and that each iteration costs us cM(2P) operations where c is a given constant, while the number of correct digits grows from P to 2P. Then the total cost of k iterations is cM(2)+cM(4)+cM(8)+...+cM(2^k). If the function M(P) grows linearly with P=2^k, then we can estimate this sum roughly as 2cM(P); if M(P)=O(P^2) then the result is about 4/3cM(P). It is easy to see that when M(P) is some power of P that grows faster than linear, the sum is not larger than a small multiple of M(P). Thus, if we have a fast method of computing, say, ArcTan(x), then we immediately obtain a method of computing Tan(x) which is asymptotically as fast (up to a constant). ### Choosing the optimal order Suppose we need to compute a function by Newton's method to precision P. We can sometimes find iterations of any order of convergence. For example, a k-th order iteration for the reciprocal 1/a is x'=x+xy+xy^2+...+xy^(k-1), where y:=1-ax. The cost of one iteration with final precision P is C(k,P):=M(P/k)+M((2P)/k)+M((3P)/k)+...+cM(P). (Here the constant c:=1 is introduced for later convenience. It denotes the number of multiplications needed to compute y.) Increasing the order by 1 costs us comparatively little, and we may change the order k at any time. Is there a particular order k that gives the smallest computational cost and should be used for all iterations, or the order needs to be adjusted during the computation? A natural question is to find the optimal computational strategy. It is difficult to fully analyze this question, but it seems that choosing a particular order k for all iterations is close to the optimal strategy. A general "strategy" is a set of orders S(P,P[0])=(k[1], k[2], ..., k[n]) to be chosen at the first, second, ..., n-th iteration, given the initial precision P[0] and the required final precision P. At each iteration, the precision will be multiplied by the factor k[i]. The optimal strategy S(P,P[0]) is a certain function of P[0] and P such that the required precision is reached, i.e. P[0]k[1]...k[n]=P, and the cost C(k[1],P[0]k[1])+C(k[2],P[0]k[1]k[2])+...+C(k[n],P) is minimized. If we assume that the cost of multiplication M(P) is proportional to some power of P, for instance M(P)=P^mu, then the cost of each iteration and the total cost are homogeneous functions of P and P[0]. Therefore the optimal strategy is a function only of the ratio P/P[0]. We can multiply both P[0] and P by a constant factor and the optimal strategy will remain the same. We can denote the optimal strategy S(P/P[0]). We can check whether it is better to use several iterations at smaller orders instead of one iteration at a large order. Suppose that M(P)=P^mu, the initial precision is 1 digit, and the final precision P=k^n. We can use either n iterations of the order k or 1 iteration of the order P. The cost of one iteration of order P at target precision P is C(P,P), whereas the total cost of n iterations of order k is C(k,k)+C(k,k^2)+...+C(k,k^n). With C(k,P) defined above, we can take approximately C(k,p)<=>p^mu(c-1+k/(mu+1)). Then the cost of one P-th order iteration is P^mu(c-1+P/(mu+1)), while the cost of n iterations of the order k is clearly smaller since k<P, P^mu(c-1+k/(mu+1))k^mu/(k^mu-1). At fixed P, the best value of k is found by minimizing this function. For c=1 (reciprocal) we find k=(1+mu)^(1/mu) which is never above 2. This suggests that k=2 is the best value for finding the reciprocal 1/a. However, larger values of c can give larger values of k. The equation for the optimal value of k is k^(mu+1)/(mu+1)-k=mu(c-1). So far we have only considered strategies that use the same order k for all iterations, and we have not yet shown that such strategies are the best ones. We now give a plausible argument (not quite a rigorous proof) to justify this claim. Consider the optimal strategy S(P^2) for the initial precision 1 and the final precision P^2, when P is very large. Since it is better to use several iterations at lower orders, we may assume that the strategy S(P^2) contains many iterations and that one of these iterations reaches precision P. Then the strategy S(P^2) is equivalent to a sequence of the two optimal strategies to go from 1 to P and from P to P^2. However, both strategies must be the same because the optimal strategy only depends on the ratio of precisions. Therefore, the optimal strategy S(P^2) is a sequence of two identical strategies (S(P), S(P)). Suppose that k[1] is the first element of S(P). The optimal strategy to go from precision k[1] to precision Pk[1] is also S(P). Therefore the second element of S(P) is also equal to k[1], and by extension all elements of S(P) are the same. A similar consideration gives the optimal strategy for other iterations that compute inverses of analytic functions, such as Newton's iteration for the inverse square root or for higher roots. The difference is that the value of c should be chosen as the equivalent number of multiplications needed to compute the function. For instance, c=1 for division and c=2 for the inverse square root iteration. The conclusion is that in each case we should compute the optimal order k in advance and use this order for all iterations. ## 4.12 Fast evaluation of Taylor series Taylor series for analytic functions is a common method of evaluation. ### Method 1: simple summation If we do not know the required number of terms in advance, we cannot do any better than just evaluate each term and check if it is already small enough. Take, for example, the series for Exp(x). To straightforwardly evaluate Exp(x)<=>Sum(k,0,N-1,x^k/k!) with P decimal digits of precision and x<2, one would need about N<=>PLn(10)/Ln(P) terms of the series. Divisions by large integers k! and separate evaluations of powers x^k are avoided if we store the previous term. The next term can be obtained by a short division of the previous term by k and a long multiplication by x. Then we only need O(N) long multiplications to evaluate the series. Usually the required number of terms N=O(P), so the total cost is O(PM(P)). There is no accumulation of round-off error in this method if x is small enough (in the case of Exp(x), a sufficient condition is Abs(x)<1/2). To see this, suppose that x is known to P digits (with relative error 10^(-P)). Since Abs(x)<1/2, the n-th term x^n/n! <4^(-n) (this is a rough estimate but it is enough). Since each multiplication by x results in adding 1 significant bit of relative round-off error, the relative error of x^n/n! is about 2^n times the relative error of x, i.e. 2^n10^(-P). So the absolute round-off error of x^n/n! is not larger than Delta<4^(-n)2^n10^(-P)=2^(-n)10^(-P). Therefore all terms with n>1 contribute less than 10^(-P) of absolute round-off error, i.e. less than was originally present in x. In practice, one could truncate the precision of x^n/n! as n grows, leaving a few guard bits each time to keep the round-off error negligibly small and yet to gain some computation speed. This however does not change the asymptotic complexity of the method---it remains O(PM(P)). However, if x is a small rational number, then the multiplication by x is short and takes O(P) operations. In that case, the total complexity of the method is O(P^2) which is always faster than O(PM(P)). ### Method 2: Horner's scheme Horner's scheme is widely known and consists of the following rearrangement, Sum(k,0,N-1,a[k]x^k)=a[0]+x(a[1]+x(a[2]+x(...+xa[N-1]))) The calculation is started from the last coefficient a[N-1] toward the first. If x is small, then the round-off error generated during the summation is constantly being multiplied by a small number x and thus is always insignificant. Even if x is not small or if the coefficients of the series are not small, Horner's scheme usually results in a smaller round-off error than the simple summation. If the coefficients a[k] are related by a simple ratio, then Horner's scheme may be modified to simplify the calculations. For example, the Horner scheme for the Taylor series for Exp(x) may be written as Sum(k,0,N-1,x^k/k!)=1+x(1+x/2(1+x/3(...+x/(N-1)))). This avoids computing the factorial function. Similarly to the simple summation method, the working precision for Horner's scheme may be adjusted to reduce the computation time: for example, xa[N-1] needs to be computed with just a few significant digits if x is small. This does not change the asymptotic complexity of the method: it requires O(N)=O(P) long multiplications by x, so for general real x the complexity is again O(PM(P)). However, if x is a small rational number, then the multiplication by x is short and takes O(P) operations. In that case, the total complexity of the method is O(P^2). ### Method 3: "rectangular" or "baby step/giant step" We can organize the calculation much more efficiently if we are able to estimate the necessary number of terms and to afford some storage (see [Smith 1989]). The "rectangular" algorithm uses 2Sqrt(N) long multiplications (assuming that the coefficients of the series are short rational numbers) and Sqrt(N) units of storage. For high-precision floating-point x, this method provides a significant advantage over Horner's scheme. Suppose we need to evaluate Sum(k,0,N,a[k]x^k) and we know the number of terms N in advance. Suppose also that the coefficients a[k] are rational numbers with small numerators and denominators, so a multiplication a[k]x is not a long multiplication (usually, either a[k] or the ratio a[k]/a[k-1] is a short rational number). Then we can organize the calculation in a rectangular array with c columns and r rows like this, a[0]+a[r]x^r+...+a[(c-1)r]x^((c-1)r)+ x(a[1]+a[r+1]x^r+...+a[(c-1)r+1]x^((c-1)r))+ ...+ x^(r-1)(a[r-1]+a[2r+1]x^r+...). To evaluate this rectangle, we first compute x^r (which, if done by the fast binary algorithm, requires O(Ln(r)) long multiplications). Then we compute the c-1 successive powers of x^r, namely x^(2r), x^(3r), ..., x^((c-1)r) in c-1 long multiplications. The partial sums in the r rows are evaluated column by column as more powers of x^r become available. This requires storage of r intermediate results but no more long multiplications by x. If a simple formula relating the coefficients a[k] and a[k-1] is available, then a whole column can be computed and added to the accumulated row values using only short operations, e.g. a[r+1]x^r can be computed from a[r]x^r (note that each column contains some consecutive terms of the series). Otherwise, we would need to multiply each coefficient a[k] separately by the power of x; if the coefficients a[k] are short numbers, this is also a short operation. After this, we need r-1 more multiplications for the vertical summation of rows (using the Horner scheme). We have potentially saved time because we do not need to evaluate powers such as x^(r+1) separately, so we do not have to multiply x by itself quite so many times. The total required number of long multiplications is r+c+Ln(r)-2. The minimum number of multiplications, given that rc>=N, is around 2Sqrt(N) at r<=>Sqrt(N)-1/2. Therefore, by arranging the Taylor series in a rectangle with sides r and c, we obtain an algorithm which costs O(Sqrt(N)) instead of O(N) long multiplications and requires Sqrt(N) units of storage. One might wonder if we should not try to arrange the Taylor series in a cube or another multidimensional matrix instead of a rectangle. However, calculations show that this does not save time: the optimal arrangement is the two-dimensional rectangle. The rectangular method saves the number of long multiplications by x but increases the number of short multiplications and additions. If x is a small integer or a small rational number, multiplications by x are fast and it does not make sense to use the rectangular method. Direct evaluation schemes are more efficient in that case. ### Truncating the working precision At the k-th step of the rectangular method, we are evaluating the k-th column with terms containing x^(rk). Since a power series in x is normally used at small x, the number x^(rk) is typically much smaller than 1. This number is to be multiplied by some a[i] and added to the previously computed part of each row, which is not small. Therefore we do not need all P floating-point digits of the number x^(rk), and the precision with which we obtain it can be gradually decreased from column to column. For example, if x^r<10^(-M), then we only need P-kM decimal digits of x^(rk) when evaluating the k-th column. (This assumes that the coefficients a[i] do not grow, which is the case for most of the practically useful series.) Reducing the working precision saves some computation time. (We also need to estimate M but this can usually be done quickly by bit counting.) Instead of O(Sqrt(P)) long multiplications at precision P, we now need one long multiplication at precision P, another long multiplication at precision P-M, and so on. This technique will not change the asymptotic complexity which remains O(Sqrt(P)M(P)), but it will reduce the constant factor in front of the O. Like the previous two methods, there is no accumulated round-off error if x is small. ### Which method to use There are two cases: first, the argument x is a small integer or rational number with very few digits and the result needs to be found as a floating-point number with P digits; second, the argument x itself is a floating-point number with P digits. In the first case, it is better to use either Horner's scheme (for small P, slow multiplication) or the binary splitting technique (for large P, fast multiplication). The rectangular method is actually slower than Horner's scheme if x and the coefficients a[k] are small rational numbers. In the second case (when x is a floating-point number), it is better to use the "rectangular" algorithm. In both cases we need to know the number of terms in advance, as we will have to repeat the whole calculation if a few more terms are needed. The simple summation method rarely gives an advantage over Horner's scheme, because it is almost always the case that one can easily compute the number of terms required for any target precision. Note that if the argument x is not small, round-off error will become significant and needs to be considered separately for a given series. ### Speed-up for some functions An additional speed-up is possible if the function allows a transformation that reduces x and makes the Taylor series converge faster. For example, Ln(x)=2Ln(Sqrt(x)), Cos(2x)=2Cos(x)^2-1 (bisection), and Sin(3x)=3Sin(x)-4Sin(x)^3 (trisection) are such transformations. It may be worthwhile to perform a number of such transformations before evaluating the Taylor series, if the time saved by its quicker convergence is more than the time needed to perform the transformations. The optimal number of transformations can be estimated. Using this technique in principle reduces the cost of Taylor series from O(Sqrt(N)) to O(N^(1/3)) long multiplications. However, additional round-off error may be introduced by this procedure for some x. For example, consider the Taylor series for Sin(x), Sin(x)<=>Sum(k,0,N-1,(-1)^kx^(2k+1)/(2k+1)!). It is sufficient to be able to evaluate Sin(x) for 0<x<Pi/2. Suppose we perform l steps of the trisection and then use the Taylor series with the rectangular method. Each step of the trisection needs two long multiplications. The value of x after l trisection steps becomes much smaller, x'=x3^(-l). For this x', the required number of terms in the Taylor series for P decimal digits of precision is N<=>(PLn(10))/(2(Ln(P)-Ln(x')))-1. The number of long multiplications in the rectangular method is 2Sqrt(N). The total number of long multiplications, as a function of l, has its minimum at l<=>(32Ln(10)/Ln(3)P)^(1/3)-(Ln(P)-Ln(x))/Ln(3), where it has a value roughly proportional to P^(1/3). Therefore we shall minimize the total number of long multiplications if we first perform l steps of trisection and then use the rectangular method to compute N terms of the Taylor series. ## 4.13 Using asymptotic series for calculations Several important analytic functions have asymptotic series expansions. For example, the complementary error function Erfc(x) and Euler's Gamma function Gamma(x) have the following asymptotic expansions at large (positive) x: Erfc(x)=e^(-x^2)/(xSqrt(Pi))(1-1/(2x^2)+...+(2n-1)!! /(-2x^2)^n+...), Ln(Gamma(x))=(x-1/2)Ln(x)-x+Ln(2Pi)/2 +Sum(n,1,Infinity,B[2n]/(2n(2n-1)x^(2n-1))) (here B[k] are Bernoulli numbers). The above series expansions are asymptotic in the following sense: if we truncate the series and then take the limit of very large x, then the difference between the two sides of the equation goes to zero. It is important that the series be first truncated and then the limit of large x be taken. Usually, an asymptotic series, if taken as an infinite series, does not actually converge for any finite x. This can be seen in the examples above. For instance, in the asymptotic series for Erfc(x) the n-th term has (2n-1)!! in the numerator which grows faster than the n-th power of any number. The terms of the series decrease at first but then eventually start to grow, even if we select a large value of x. The way to use an asymptotic series for a numerical calculation is to truncate the series well before the terms start to grow. Error estimates of the asymptotic series are sometimes difficult, but the rule of thumb seems to be that the error of the approximation is usually not greater than the first discarded term of the series. This can be understood intuitively as follows. Suppose we truncate the asymptotic series at a point where the terms still decrease, safely before they start to grow. For example, let the terms around the 100-th term be A[100], A[101], A[102], ..., each of these numbers being significantly smaller than the previous one, and suppose we retain A[100] but drop the terms after it. Then our approximation would have been a lot better if we retained A[101] as well. (This step of the argument is really an assumption about the behavior of the series; it seems that this assumption is correct in many practically important cases.) Therefore the error of the approximation is approximately equal to A[101]. The inherent limitation of the method of asymptotic series is that for any given x, there will be a certain place in the series where the term has the minimum absolute value (after that, the series is unusable), and the error of the approximation cannot be smaller than that term. For example, take the above asymptotic series for Erfc(x). The logarithm of the absolute value of the n-th term can be estimated using Stirling's formula for the factorial as Ln((2n-1)!! /(2x^2)^n)<=>n(Ln(n)-1-2Ln(x)). This function of n has its minimum at n=x^2 where it is equal to -x^2. Therefore the best we can do with this series is to truncate it before this term. The resulting approximation to Erfc(x) will have relative precision of order Exp(-x^2). Suppose that x is large and we need to compute Erfc(x) with P decimal digits of floating point. Then it follows that we can use the asymptotic series only if x>Sqrt(PLn(10)). We find that for a given finite x, no matter how large, there is a maximum precision that can be achieved with the asymptotic series; if we need more precision, we have to use a different method. However, sometimes the function we are evaluating allows identity transformations that relate f(x) to f(y) with y>x. For example, the Gamma function satisfies xGamma(x)=Gamma(x+1). In this case we can transform the function so that we would need to evaluate it at large enough x for the asymptotic series to give us enough precision. ## 4.14 The AGM sequence algorithms Several algorithms are based on the arithmetic-geometric mean (AGM) sequence. If one takes two numbers a, b and computes their arithmetic mean (a+b)/2 and their geometric mean Sqrt(ab), then one finds that the two means are generally much closer to each other than the original numbers. Repeating this process creates a rapidly converging sequence of pairs. More formally, one can define the function of two arguments AGM(x,y) as the limit of the sequence a[k] where a[k+1]=1/2(a[k]+b[k]), b[k+1]=Sqrt(a[k]b[k]), and the initial values are a[0]=x, b[0]=y. (The limit of the sequence b[k] is the same.) This function is obviously linear, AGM(cx,cy)=cAGM(x,y), so in principle it is enough to compute AGM(1,x) or arbitrarily select c for convenience. Gauss and Legendre knew that the limit of the AGM sequence is related to the complete elliptic integral, Pi/21/AGM(a,Sqrt(a^2-b^2))=(Integrate(x,0,Pi/2)1/Sqrt(a^2-b^2Sin(x)^2)). (Here 0<b<a.) This integral can be rearranged to provide some other useful functions. For example, with suitable parameters a and b, this integral is equal to Pi. Thus, one obtains a fast method of computing Pi (the Brent-Salamin method). The AGM sequence is also defined for complex values a, b. One needs to take a square root Sqrt(ab), which requires a branch cut to be well-defined. Selecting the natural cut along the negative real semiaxis (Re(x)<0, Im(x)=0), we obtain an AGM sequence that converges for any initial values x, y with positive real part. Let us estimate the convergence rate of the AGM sequence starting from x, y, following the paper [Brent 1975]. Clearly the worst case is when the numbers x and y are very different (one is much larger than another). In this case the numbers a[k], b[k] become approximately equal after about k=1/Ln(2)Ln(Abs(Ln(x/y))) iterations (note: Brent's paper online mistypes this as 1/Ln(2)Abs(Ln(x/y))). This is easy to see: if x is much larger than y, then at each step the ratio r:=x/y is transformed into r'=1/2Sqrt(r). When the two numbers become roughly equal to each other, one needs about Ln(n)/Ln(2) more iterations to make the first n (decimal) digits of a[k] and b[k] coincide, because the relative error epsilon=1-b/a decays approximately as epsilon[k]<=>1/8Exp(-2^k). Unlike Newton's iteration, the AGM sequence does not correct errors, so all numbers need to be computed with full precision. Actually, slightly more precision is needed to compensate for accumulated round-off error. Brent (in [Brent 1975]) says that O(Ln(Ln(n))) bits of accuracy are lost to round-off error if there are total of n iterations. The AGM sequence can be used for fast computations of Pi, Ln(x) and ArcTan(x). However, currently the limitations of Yacas internal math make these methods less efficient than simpler methods based on Taylor series and Newton iterations. ## 4.15 The binary splitting method The method of binary splitting is well explained in [Haible et al. 1998]. Some examples are also given in [Gourdon et al. 2001]. This method applies to power series of rational numbers and to hypergeometric series. Most series for transcendental functions belong to this category. If we need to take O(P) terms of the series to obtain P digits of precision, then ordinary methods would require O(P^2) arithmetic operations. (Each term needs O(P) operations because all coefficients are rational numbers with O(P) digits and we need to perform a few short multiplications or divisions.) The binary splitting method requires O(M(PLn(P))Ln(P)) operations instead of the O(P^2) operations. In other words, we need to perform long multiplications of integers of size O(PLn(P)) digits, but we need only O(Ln(P)) such multiplications. The binary splitting method performs better than the straightforward summation method if the cost of multiplication is lower than O(P^2)/Ln(P). This is usually true only for large enough precision (at least a thousand digits). Thus there are two main limitations of the binary splitting method: • As a rule, we can only compute functions of small integer or rational arguments. For instance, the method works for the calculation of a Bessel function • J0(1/3) but not for J0(Pi). (As an exception, certain elementary functions can be computed by the binary splitting method for general floating-point arguments, with some clever tricks.) • The method is fast only at high enough precision, when advanced multiplication methods become more efficient than simple • O(P^2) methods. The binary splitting method is actually slower than the simple summation when the long integer multiplication is M(P)=O(P^2). The main advantages of the method are: • The method is asymptotically fast and, when applicable, outperforms most other methods at very high precision. The best applications of this method are for computing various constants. • There is no accumulated round-off error since the method uses only exact integer arithmetic. • The sum of a long series can be split into many independent partial sums which can be computed in parallel. One can store exact intermediate results of a partial summation (a few long integers), which provides straightforward checkpointing: a failed partial summation can be repeated without repeating all other parts. One can also resume the summation later to get more precision and reuse the old results, instead of starting all over again. ### Description of the method We follow [Haible et al. 1998]. The method applies to any series of rational numbers of the form S=Sum(n,0,N-1,A(n)/B(n)), where A, B are integer coefficients with O(nLn(n)) bits. Usually the series is of the particular form S(0,N):=Sum(n,0,N-1,a(n)/b(n)(p(0)...p(n))/(q(0)...q(n))), where a, b, p, q are polynomials in n with small integer coefficients and values that fit into O(Ln(n)) bits. For example, the Taylor series for ArcSin(x) (when x is a short rational number) is of this form: ArcSin(x)=x+1/2x^3/3+(13)/(24)x^5/5+(135)/(246)x^7/7+... This example is of the above form with the definitions a=1, b(n)=2n+1, p(n)=x^2(2n-1), q(n)=2n for n>=1 and p(0)=x, q(0)=1. (The method will apply only if x is a rational number with O(Ln(N)) bits in the numerator and the denominator.) The Taylor series for the hypergeometric function is also of this form. The goal is to compute the sum S(0,N) with a chosen number of terms N. Instead of computing the rational number S directly, the binary splitting method propose to compute the following four integers P, Q, B, and T: P(0,N):=p(0)...p(N-1), Q(0,N):=q(0)...q(N-1), B(0,N):=b(0)...b(N-1), and T(0,N):=B(0,N)Q(0,N)S(0,N). At first sight it seems difficult to compute T, but the computation is organized recursively. These four integers are computed for the left ( l) half and for the right ( r) half of the range [ 0, N) and then combined using the obvious recurrence relations P=P[l]P[r], Q=Q[l]Q[r], B=B[l]B[r], and the slightly less obvious relation T=B[r]Q[r]T[l]+B[l]P[l]T[r]. Here we used the shorthand P[l]:=P(0,N/2-1), P[r]:=P(N/2,N-1) and so on. Thus the range [0, N) is split in half on each step. At the base of recursion the four integers P, Q, B, and T are computed directly. At the end of the calculation (top level of recursion), one floating-point division is performed to recover S=T/(BQ). It is clear that the four integers carry the full information needed to continue the calculation with more terms. So this algorithm is easy to checkpoint and parallelize. The integers P, Q, B, and T grow during the calculation to O(NLn(N)) bits, and we need to multiply these large integers. However, there are only O(Ln(N)) steps of recursion and therefore O(Ln(N)) long multiplications are needed. If the series converges linearly, we need N=O(P) terms to obtain P digits of precision. Therefore, the total asymptotic cost of the method is O(M(PLn(P))Ln(P)) operations. A more general form of the binary splitting technique is also given in [Haible et al. 1998]. The generalization applies to series for the form Sum(n,0,N-1,a(n)/b(n)(p(0)...p(n))/(q(0)...q(n))(c(0)/d(0)+...+c(n)/d(n))), Here a(n), b(n), c(n), d(n), p(n), q(n) are integer-valued functions with "short" values of size O(Ln(n)) bits. For example, the Ramanujan series for Catalan's constant is of this form. The binary splitting technique can also be used for series with complex integer coefficients, or more generally for coefficients in any finite algebraic extension of integers, e.q. Z[ Sqrt(2)] (the ring of numbers of the form p+qSqrt(2) where p, q are integers). Thus we may compute the Bessel function J0(Sqrt(3)) using the binary splitting method and obtain exact intermediate results of the form p+qSqrt(3). But this will still not help compute J0(Pi). This is a genuine limitation of the binary splitting method.	2013-12-07 18:45:25	{"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": 0.8975289463996887, "perplexity": 568.0849698169824}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1386163055633/warc/CC-MAIN-20131204131735-00089-ip-10-33-133-15.ec2.internal.warc.gz"}
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(open-source preferred)	2019-07-16 02:09:45	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7941721081733704, "perplexity": 940.6884064268634}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195524475.48/warc/CC-MAIN-20190716015213-20190716041213-00524.warc.gz"}
http://physics.aps.org/synopsis-for/10.1103/PhysRevLett.107.015702	# Synopsis: Atomic avalanches show up in x rays State-of-the-art x-ray scattering demonstrates sudden avalanche-like structural changes in cobalt. Certain crystals switch their structural pattern by tiny but coordinated movements of the constituent atoms. Recent experiments have suggested that these transitions are not wholesale conversions, but a series of sudden, localized shifts inside the crystal. These so-called avalanches have now been observed using x rays by Christopher Sanborn at Boston University, US, and colleagues there and at McGill University, Canada. The results, published in Physical Review Letters, show that atomic behavior in crystals shares similarities with earthquakes. A martensitic transition is a solid-solid phase transition in which the crystal reconfiguration is usually accompanied by strain. First observed in a type of hard steel called martensite, it was later detected in other materials, such as shape-memory alloys. Observations during martensitic transitions have detected spikes in acoustic emissions and heat flow that imply avalanching, but seeing these structural changes directly required developing a technique that could spatially resolve them from avalanches as they happened. This was achieved with coherent x-ray beams that are now produced at synchrotron facilities. Using the Advanced Photon Source in Argonne National Laboratory, the authors studied a martensitic transition in cobalt at a temperature of about $447$ °$\text{C}$. Coherent x rays scattering off the cobalt interfere with each other to produce a speckle pattern on a CCD camera. Any sudden change in a group of speckles corresponds to an avalanche. The team measured avalanche sizes between $100$ nanometers and $10$ microns. The rate and distribution of avalanches during strain-relieving structural rearrangements for the phase transition resemble the statistics for aftershocks following an earthquake. – Michael Schirber ### Announcements More Announcements » ## Subject Areas Materials Science ## Previous Synopsis Particles and Fields ## Next Synopsis Interdisciplinary Physics ## Related Articles Geophysics ### Synopsis: Acoustic Trigger For Earthquakes Numerical simulations support the idea that acoustic waves can trigger earthquakes by reducing friction between the rocks within a fault. Read More » Magnetism ### Synopsis: Multiferroic Surprise Electric and magnetic polarization are spontaneously produced in an unlikely material—one with a highly symmetric crystal structure. Read More » Soft Matter ### Synopsis: Wedged Particles Make Crystals Rod-shaped particles in a liquid arrange into a variety of structures when subjected to confining walls, an effect that might be used to design optical materials. Read More »	2015-10-06 05:35:08	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 4, "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": 0.3644874095916748, "perplexity": 3566.3843242409603}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443736678520.45/warc/CC-MAIN-20151001215758-00048-ip-10-137-6-227.ec2.internal.warc.gz"}
http://fivethirtyeight.com/datalab/democrats-shouldnt-count-on-an-electoral-college-edge-in-2016/	For a better browsing experience, please upgrade your browser. ## DataLab Last week, FiveThirtyEight’s Nate Silver explained why the National Popular Vote Interstate Compact, a plan to circumvent the Electoral College in favor of the national popular vote, isn’t likely to work. We’re probably stuck with the Electoral College. Democratic partisans might not be too displeased by that. After all, Democrats have overperformed in the Electoral College relative to the popular vote in the past two elections. But here’s the thing: The Electoral College advantage has swung back and forth. I found this out by gathering presidential election data since 1900. For each year, I looked at the margin between the major parties in each state, compared it with the national margin, and calculated how many electoral votes were more Democratic or Republican than the nation as a whole. During the first half of the 20th century, Republicans benefited greatly from the Electoral College. They could have lost the national popular vote and won the electoral college in 12 of the 13 elections from 1900 to 1948. On average, they could have lost by 2.2 percentage points nationally and emerged victorious. A large part of the GOP’s advantage was caused by the Democratic vote’s concentration in the South. Democratic candidates were racking up huge margins in the region, but a candidate gets the same number of electoral votes whether he wins a state by 50 percent or 1 percent. The election in 1928 was the only one in which a majority of electoral votes leaned more Democratic than the nation. That was partially because the Democrats nominated Al Smith, a Catholic, at a time when many Southerners were prejudiced against Catholics. Since 1952, the Electoral College picture has changed. Beginning in that year, Republicans began making inroads into the South. Democratic votes became more dispersed, and the Republican advantage in the Electoral College waned. Today, the South is solidly red, but Democrats still win over 35 percent of the vote there. That’s a far cry from pre-1952, when Republican candidates sometimes didn’t break 10 percent of the vote. From 1952 to 2012, the majority of electoral votes leaned more Republican than the nation seven times and more Democratic nine times. But in the past five elections, Democrats appear to have opened up a bit of an edge. They could have won the Electoral College while losing the popular vote four of five times. Still, in that streak was 2000, when Republicans won the Electoral College without winning the popular vote. Indeed, knowing how many electoral votes leaned more toward one party than the nation in one election tells us very little about how many will lean toward that party in the next election. An Electoral College advantage is often taken as a sign of a structural advantage, but for the most part, it’s been cyclical. The Democratic edge in 2008 and 2012 may be more due to randomness than demographics. Candidates, campaign strategies and luck matter. In 2016, a Hillary Clinton map may look very different from a Barack Obama map. Clinton has traditionally polled better in Appalachia than Obama but worse in the West. A map in which the Republican Party is increasingly reliant on the white vote may put it in better shape in the Midwest, while one in which the party tries to wins elections by appealing to Latinos may put it in better shape in the South and Southwest. Democrats might have an Electoral College advantage in 2016, but they shouldn’t count on it just because they had an edge in 2012. Harry Enten is a senior political writer and analyst for FiveThirtyEight. All Politics ### Keep Calm And Ignore The 2016 ‘Game Changers’Sep 2, 2015 All 2016 Election Filed under , , ### Comments Add Comment Never miss the best of FiveThirtyEight. Subscribe to the FiveThirtyEight Newsletter Powered by WordPress.com VIP	2015-09-04 05:48:58	{"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": 0.225785031914711, "perplexity": 2408.4982828629286}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440645338295.91/warc/CC-MAIN-20150827031538-00060-ip-10-171-96-226.ec2.internal.warc.gz"}
https://community.wolfram.com/groups/-/m/t/1281200	# SequencePredict: interesting behavior for CA rule plots Posted 1 year ago 1580 Views \| \| 5 Total Likes \| I submitted the following one-liner entry at the 2017 WTC one-liner competition, and I'd be meaning to ask this question since and kept forgetting: s = Cases[RulePlot@CellularAutomaton[#], Inset[a_, __] -> a, ∞] &; Join[f = Take[s@99, 1], SequencePredict[s /@ Range@98][f, "NextElement" -> 7]] Essentially, s is a pure function which, when given an integer, returns a list of the 8 graphics objects which visualize the rules for the elementary cellular automaton with that number. This is illustrated below: RulePlot[CellularAutomaton[30]] GraphicsRow[s[30], Frame -> All] A list of these is generated for the first 98 CAs (in theory skipping 0, but oh well we had to save characters), and passed into SequencePredict which uses a Markov Model to return a SequencePredictorFunction. Here's how 5 of those sequences look like: GraphicsGrid[s /@ Range[5]] This is where things get interesting: given only the first rule image of the 99th CA, the predictor correctly 'guesses' the next 7 elements, returning the RulePlot for CA 99!Looking at the image above, it's clear that the CAs were listed using some sort of loop, which is likely what the predictor picks up. However the result was odd enough (since inherently the ruleplot is not really a sequence) to trigger my curiosity and I was wondering if anyone had a clearer explanation as to what is happening allowing it to predict correctly one element at a time.Cheers, George	2019-06-27 00:35:36	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.25680065155029297, "perplexity": 2089.2250332852077}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560628000609.90/warc/CC-MAIN-20190626234958-20190627020958-00339.warc.gz"}
http://math.stackexchange.com/questions/194149/triangle-one-angle-and-two-lengths	# Triangle one angle and two lengths [closed] In a triangle ABC, the angle at B is 108 degrees, the length of side BC is 16, and the length of side AB is 12. To 2 decimal places, what is the length of side AC? So i worked out and out and got this answer 22.77 is that right? I used cosine rule. - Using cosine rule, $AC^2=16^2+12^2-2(16)(12)\cos 108^\circ$, so you are right.	2014-04-20 23:55:06	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.864231288433075, "perplexity": 149.5315659518174}, "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-2014-15/segments/1397609539337.22/warc/CC-MAIN-20140416005219-00613-ip-10-147-4-33.ec2.internal.warc.gz"}
https://byjus.com/question-answer/a-single-generator-supplies-a-sine-wave-of-20v-5khz-to-the-circuit-shown-in/	Question # A single generator supplies a sine wave of 20V, 5kHz to the circuit shown in the figure. Then 0.2μF 20 V, 5kHz A the current in the resistive branch is 0.2 A B the current in the capacitive branch is 0.126 A C total line current is 0.24 A D current in both the branches is same Solution ## The correct options are A the current in the resistive branch is 0.2 A B the current in the capacitive branch is 0.126 A C total line current is 0.24 AIR=VrmsR=20200=0.2A IC=VrmsXc=2012π×5×103×0.2×10−6=0.126A I = √I2R+I2C = 0.24A Co-Curriculars Suggest Corrections 0 Similar questions View More	2022-01-28 09:32:49	{"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": 0.8772657513618469, "perplexity": 10521.71848906573}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320305423.58/warc/CC-MAIN-20220128074016-20220128104016-00360.warc.gz"}
http://zbmath.org/?q=an:1223.47088	zbMATH — the first resource for mathematics Examples Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev \| Tschebyscheff The or-operator \| allows to search for Chebyshev or Tschebyscheff. "Quasi* map" py: 1989 The resulting documents have publication year 1989. so: Eur J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A \| 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used. Operators a & b logic and a \| b logic or !ab logic not abc right wildcard "ab c" phrase (ab c) parentheses Fields any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article) The equivalence between the convergences of multi-step iterations with errors for uniformly generalized Lipschitz continuous operators. (Chinese) Zbl 1223.47088 Summary: In this paper, the equivalence of the convergence of modified Mann iterations and multi-step Noor iterations with errors is investigated for uniformly generalized Lipschitz continuous and successively asymptotically $𝛷$-strongly pseudo-contractive type operators in uniformly smooth Banach spaces. Z.-Y. Huang [J. Math. Anal. Appl. 329, No. 2, 935–947 (2007; Zbl 1153.47307)] showed the equivalence of the convergence criteria between modified Mann and Ishikawa iterations with errors for successively $𝛷$-strongly pseudo-contractive operators with bounded range in uniformly smooth Banach spaces. The results obtained in this paper generalize the results of Huang and give an affirmative answer to the conjecture raised by B. E. Rhoades and Ş. M. Şoltuz [Int. J. Math. Math. Sci. No. 7, 451–459 (2003; Zbl 1014.47052)]. MSC: 47J25 Iterative procedures (nonlinear operator equations) 47H09 Mappings defined by “shrinking” properties 47H10 Fixed point theorems for nonlinear operators on topological linear spaces	2014-04-16 07:21:21	{"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": 0.7538145780563354, "perplexity": 8222.542514804658}, "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-2014-15/segments/1397609521558.37/warc/CC-MAIN-20140416005201-00579-ip-10-147-4-33.ec2.internal.warc.gz"}
https://zbmath.org/authors/?q=ai%3Akuczumow.tadeusz	# zbMATH — the first resource for mathematics Compute Distance To: Documents Indexed: 80 Publications since 1978 all top 5 #### Co-Authors 11 single-authored 32 Reich, Simeon 24 Budzyńska, Monika 16 Stachura, Adam 14 Kaczor, Wiesława J. 5 Michalska, Małgorzata 4 Grzesik, Aleksandra 3 Dye, John Michael 3 Kapeluszny, Jarosław 3 Schmidt, Małgorzata 2 Koter-Mórgowska, Małgorzata 2 Lin, Pei-Kee 2 Prus, Stanisław 2 Shoikhet, David 1 Bruck, Ronald E. jun. 1 Garcia-Falset, Jesús 1 Goebel, Kai F. 1 Goebel, Kazimierz 1 Kryczka, Andrzej 1 Ray, William O. 1 Rzymowski, Witold 1 Sekowski, Tadeusz 1 Walczyk, Mariola 1 Zygmunt, Wojciech all top 5 #### Serials 6 Annales Universitatis Mariae Curie-Skłodowska. Sectio A. Mathematica 5 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 5 Proceedings of the American Mathematical Society 5 Nonlinear Analysis. Theory, Methods & Applications 4 Journal of Mathematical Analysis and Applications 4 Journal of Nonlinear and Convex Analysis 3 Colloquium Mathematicum 3 Commentationes Mathematicae Universitatis Carolinae 3 Mathematica Japonica 3 Abstract and Applied Analysis 2 Advances in Mathematics 2 Annali di Matematica Pura ed Applicata. Serie Quarta 2 Journal of Functional Analysis 2 Fixed Point Theory 1 Bulletin of the Australian Mathematical Society 1 Houston Journal of Mathematics 1 Bulletin of the Calcutta Mathematical Society 1 Canadian Mathematical Bulletin 1 Kodai Mathematical Journal 1 Mathematische Zeitschrift 1 Real Analysis Exchange 1 Transactions of the American Mathematical Society 1 Roczniki Polskiego Towarzystwa Matematycznego. Seria II. Wiadomości Matematyczne 1 Bulletin of the Polish Academy of Sciences, Mathematics 1 Bollettino della Unione Matemàtica Italiana. Serie VII. A 1 Topological Methods in Nonlinear Analysis 1 Annales Academiae Scientiarum Fennicae. Mathematica 1 Taiwanese Journal of Mathematics 1 Annals of Functional Analysis 1 Analysis and Mathematical Physics 1 Journal of Nonlinear and Variational Analysis all top 5 #### Fields 70 Operator theory (47-XX) 48 Functional analysis (46-XX) 21 Several complex variables and analytic spaces (32-XX) 8 General topology (54-XX) 3 Global analysis, analysis on manifolds (58-XX) 2 Approximations and expansions (41-XX) 2 Numerical analysis (65-XX) 1 History and biography (01-XX) 1 Real functions (26-XX) 1 Functions of a complex variable (30-XX) 1 Convex and discrete geometry (52-XX) #### Citations contained in zbMATH Open 62 Publications have been cited 449 times in 281 Documents Cited by Year Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property. Zbl 0849.47030 Bruck, Ronald; Kuczumow, Tadeusz; Reich, Simeon 1993 Weak convergence theorems for asymptotically nonexpansive mappings and semigroups. Zbl 0983.47040 Falset, Jesús García; Kaczor, Wiesława; Kuczumow, Tadeusz; Reich, Simeon 2001 Irregular convex sets with fixed-point property for non-expansive mappings. Zbl 0418.47031 Goebel, K.; Kuczumow, T. 1979 Iterates of holomorphic and $$k_ D$$-nonexpansive mappings in convex domains in $${\mathbb{C}}^ n$$. Zbl 0726.32016 1990 Fixed points of holomorphic mappings: A metric approach. Zbl 1019.47041 Kuczumow, Tadeusz; Reich, Simeon; Shoikhet, David 2001 A mean ergodic theorem for mappings which are asymptotically nonexpansive in the intermediate sense. Zbl 1042.47509 Kaczor, Wiesława; Kuczumow, Tadeusz; Reich, Simeon 2001 A mean ergodic theorem for nonlinear semigroups which are asymptotically nonexpansive in the intermediate sense. Zbl 0981.47037 Kaczor, Wiesława; Kuczumow, Tadeusz; Reich, Simeon 2000 The Denjoy-Wolff theorem in the open unit ball of a strictly convex Banach space. Zbl 0928.47041 Kapeluszny, Jaroslaw; Kuczumow, Tadeusz; Reich, Simeon 1999 Compact asymptotic centers and fixed points of multivalued nonexpansive mappings. Zbl 0724.47033 1990 An almost convergence and its applications. Zbl 0463.47035 1978 Nonexpansive retracts and fixed points of nonexpansive mappings in the Cartesian product of n Hilbert balls. Zbl 0607.47053 1985 Fixed points of holomorphic mappings in the Cartesian product of n unit Hilbert balls. Zbl 0627.46056 Kuczumow, T.; Stachura, A. 1986 The Denjoy-Wolff-type theorem for compact $$k_{B_H}$$-nonexpansive maps on a Hilbert ball. Zbl 1012.47038 1997 Convergence of unrestricted products of nonexpansive mappings in spaces with the Opial property. Zbl 0866.47038 Dye, John M.; Kuczumow, Tadeusz; Lin, Pei-Kee; Reich, Simeon 1996 Fixed point theorems in product spaces. Zbl 0696.47051 1990 Common fixed points of commuting holomorphic mappings in Hilbert ball and polydisc. Zbl 0564.47029 Kuczumow, T. 1984 The Denjoy-Wolff theorem for condensing holomorphic mappings. Zbl 0933.46039 Kapeluszny, Jarosław; Kuczumow, Tadeusz; Reich, Simeon 1999 Isometers in the Cartesian product of n unit open Hilbert balls with hyperbolic metric. Zbl 0676.46016 1988 Convexity and fixed points of holomorphic mappings in Hilbert ball and polydisc. Zbl 0606.47057 1986 Opial’s modulus and fixed points of semigroups of mappings. Zbl 0921.47049 1999 A fixed point property of $$\ell_ 1$$-product spaces. Zbl 0805.47048 Kuczumow, Tadeusz; Reich, Simeon; Schmidt, Malgorzata 1993 The existence and non-existence of common fixed points for commuting families of holomorphic mappings. Zbl 0968.47025 Kuczumow, Tadeusz; Reich, Simeon; Shoikhet, David 2001 A few properties of the Kobayashi distance and their applications. Zbl 0969.32009 2000 Random products of nonexpansive mappings in spaces with the Opial property. Zbl 0803.47052 Dye, John M.; Kuczumow, Tadeusz; Lin, Pei-Kee; Reich, Simeon 1993 Fixed points of holomorphic mappings in the Hilbert ball. Zbl 0674.47039 Kuczumow, T. 1988 A Denjoy-Wolf theorem for compact holomorphic mappings in complex Banach spaces. Zbl 1305.46065 Budzyńska, Monika; Kuczumow, Tadeusz; Reich, Simeon 2013 Theorems of Denjoy-Wolff type. Zbl 1273.32004 Budzyńska, Monika; Kuczumow, Tadeusz; Reich, Simeon 2013 Total sets and semicontinuity of the Kobayashi distance. Zbl 1042.46511 2001 The weak lower semicontinuity of the Kobayashi distance and its applications. Zbl 0997.32013 2001 Opial’s property and James’ quasi-reflexive spaces. Zbl 0818.46019 1994 Extensions of nonexpansive mappings in the Hilbert ball with the hyperbolic metric. I. Zbl 0672.47035 1988 Weak convergence theorems for nonexpansive mappings and semigroups in Banach spaces with Opial’s property. Zbl 0585.47043 Kuczumow, T. 1985 The $$\Gamma$$-Opial property. Zbl 1108.46013 Budzyńska, Monika; Kuczumow, Tadeusz; Michalska, Małgorzata 2006 The Denjoy-Wolff theorem for $$s$$-condensing mappings. Zbl 1019.46031 1999 Minimal displacement of points under holomorphic mappings and fixed point properties for unions of convex sets. Zbl 0807.32021 1994 Strong asymptotic normal structure and fixed points in product spaces. Zbl 0823.47053 1993 Holomorphic retracts of polyballs. Zbl 0606.47059 1986 A contribution to the theory of nonexpansive mappings. Zbl 0437.47040 Goebel, K.; Kuczumow, T. 1978 Schauder bases and diametrically complete sets with empty interior. Zbl 1400.46008 Budzyńska, Monika; Grzesik, Aleksandra; Kaczor, Wiesława; Kuczumow, Tadeusz 2018 Convergence of ergodic means of orbits of semigroups of nonexpansive mappings in sets with the $$\varGamma$$-Opial property. Zbl 1133.47048 Kaczor, Wiesława; Kuczumow, Tadeusz; Michalska, Małgorzata 2007 A remark on the approximate fixed-point property. Zbl 1014.47028 2003 A remark on a lemma due to Oka. Zbl 1012.47025 1998 Uniform asymptotic normal structure, the uniform semi-opial property, and fixed points of asymptotically regular uniformly Lipschitzian semigroups. II. Zbl 1034.47507 Budzyńska, Monika; Kuczumow, Tadeusz; Reich, Simeon 1998 An application of Opial’s modulus to the fixed point theory of semigroups of Lipschitzian mappings. Zbl 1012.47040 1997 Holomorphic retracts of the open unit ball in the $$\ell_{\infty}$$-product of Hilbert spaces. Zbl 0894.46032 1996 The product retraction property for the $$c_ 0$$-product of countably many metric spaces. Zbl 0815.47069 1994 Bruck’s retraction method. Zbl 0789.47037 Kuczumow, T.; Stachura, A. 1991 Common fixed points of commuting holomorphic mappings. Zbl 0692.47040 Kuczumow, T.; Stachura, A. 1989 Means and convergence of semigroup orbits. Zbl 07285141 Grzesik, Aleksandra; Kaczor, Wiesława; Kuczumow, Tadeusz; Reich, Simeon 2020 Convergence of iterates of nonexpansive mappings and orbits of nonexpansive semigroups. Zbl 07053114 Grzesik, Aleksandra; Kaczor, Wiesława; Kuczumow, Tadeusz; Reich, Simeon 2019 Limiting behavior of the Kobayashi distance. Zbl 1357.32016 Budzyńska, Monika; Kuczumow, Tadeusz; Reich, Simeon 2015 The common fixed point set of commuting holomorphic mappings in Cartesian products of Banach spaces. Zbl 1318.32003 Budzyńska, Monika; Kuczumow, Tadeusz; Reich, Simeon 2015 A Denjoy-Wolff theorem for compact holomorphic mappings in reflexive Banach spaces. Zbl 1329.32007 Budzyńska, Monika; Kuczumow, Tadeusz; Reich, Simeon 2012 Linear strict convexity of the Kobayashi distance in nonreflexive Banach spaces. Zbl 1122.46023 2006 Common fixed points of holomorphic mappings and retracts of $$B^\infty _H$$. Zbl 1091.58005 2004 Properties of the Kobayashi distance. Zbl 1261.46040 2003 A strict convexity of the Kobayashi distance. Zbl 1067.32002 2003 Uniform asymptotic normal structure, the uniform semi-Opial property and fixed points of asymptotically regular uniformly Lipschitzian semigroups. I. Zbl 0973.47042 Budzyńska, Monika; Kuczumow, Tadeusz; Reich, Simeon 1998 A product retraction property. Zbl 0762.47031 1992 Open problems. Zbl 0755.47043 Kuczumow, T.; Stachura, A. 1991 Extensions of nonexpansive mappings in the Hilbert ball with the hyperbolic metric. II. Zbl 0672.47036 1988 Approximations of fixed points of holomorphic mappings. Zbl 0648.47038 1987 Means and convergence of semigroup orbits. Zbl 07285141 Grzesik, Aleksandra; Kaczor, Wiesława; Kuczumow, Tadeusz; Reich, Simeon 2020 Convergence of iterates of nonexpansive mappings and orbits of nonexpansive semigroups. Zbl 07053114 Grzesik, Aleksandra; Kaczor, Wiesława; Kuczumow, Tadeusz; Reich, Simeon 2019 Schauder bases and diametrically complete sets with empty interior. Zbl 1400.46008 Budzyńska, Monika; Grzesik, Aleksandra; Kaczor, Wiesława; Kuczumow, Tadeusz 2018 Limiting behavior of the Kobayashi distance. Zbl 1357.32016 Budzyńska, Monika; Kuczumow, Tadeusz; Reich, Simeon 2015 The common fixed point set of commuting holomorphic mappings in Cartesian products of Banach spaces. Zbl 1318.32003 Budzyńska, Monika; Kuczumow, Tadeusz; Reich, Simeon 2015 A Denjoy-Wolf theorem for compact holomorphic mappings in complex Banach spaces. Zbl 1305.46065 Budzyńska, Monika; Kuczumow, Tadeusz; Reich, Simeon 2013 Theorems of Denjoy-Wolff type. Zbl 1273.32004 Budzyńska, Monika; Kuczumow, Tadeusz; Reich, Simeon 2013 A Denjoy-Wolff theorem for compact holomorphic mappings in reflexive Banach spaces. Zbl 1329.32007 Budzyńska, Monika; Kuczumow, Tadeusz; Reich, Simeon 2012 Convergence of ergodic means of orbits of semigroups of nonexpansive mappings in sets with the $$\varGamma$$-Opial property. Zbl 1133.47048 Kaczor, Wiesława; Kuczumow, Tadeusz; Michalska, Małgorzata 2007 The $$\Gamma$$-Opial property. Zbl 1108.46013 Budzyńska, Monika; Kuczumow, Tadeusz; Michalska, Małgorzata 2006 Linear strict convexity of the Kobayashi distance in nonreflexive Banach spaces. Zbl 1122.46023 2006 Common fixed points of holomorphic mappings and retracts of $$B^\infty _H$$. Zbl 1091.58005 2004 A remark on the approximate fixed-point property. Zbl 1014.47028 2003 Properties of the Kobayashi distance. Zbl 1261.46040 2003 A strict convexity of the Kobayashi distance. Zbl 1067.32002 2003 Weak convergence theorems for asymptotically nonexpansive mappings and semigroups. Zbl 0983.47040 Falset, Jesús García; Kaczor, Wiesława; Kuczumow, Tadeusz; Reich, Simeon 2001 Fixed points of holomorphic mappings: A metric approach. Zbl 1019.47041 Kuczumow, Tadeusz; Reich, Simeon; Shoikhet, David 2001 A mean ergodic theorem for mappings which are asymptotically nonexpansive in the intermediate sense. Zbl 1042.47509 Kaczor, Wiesława; Kuczumow, Tadeusz; Reich, Simeon 2001 The existence and non-existence of common fixed points for commuting families of holomorphic mappings. Zbl 0968.47025 Kuczumow, Tadeusz; Reich, Simeon; Shoikhet, David 2001 Total sets and semicontinuity of the Kobayashi distance. Zbl 1042.46511 2001 The weak lower semicontinuity of the Kobayashi distance and its applications. Zbl 0997.32013 2001 A mean ergodic theorem for nonlinear semigroups which are asymptotically nonexpansive in the intermediate sense. Zbl 0981.47037 Kaczor, Wiesława; Kuczumow, Tadeusz; Reich, Simeon 2000 A few properties of the Kobayashi distance and their applications. Zbl 0969.32009 2000 The Denjoy-Wolff theorem in the open unit ball of a strictly convex Banach space. Zbl 0928.47041 Kapeluszny, Jaroslaw; Kuczumow, Tadeusz; Reich, Simeon 1999 The Denjoy-Wolff theorem for condensing holomorphic mappings. Zbl 0933.46039 Kapeluszny, Jarosław; Kuczumow, Tadeusz; Reich, Simeon 1999 Opial’s modulus and fixed points of semigroups of mappings. Zbl 0921.47049 1999 The Denjoy-Wolff theorem for $$s$$-condensing mappings. Zbl 1019.46031 1999 A remark on a lemma due to Oka. Zbl 1012.47025 1998 Uniform asymptotic normal structure, the uniform semi-opial property, and fixed points of asymptotically regular uniformly Lipschitzian semigroups. II. Zbl 1034.47507 Budzyńska, Monika; Kuczumow, Tadeusz; Reich, Simeon 1998 Uniform asymptotic normal structure, the uniform semi-Opial property and fixed points of asymptotically regular uniformly Lipschitzian semigroups. I. Zbl 0973.47042 Budzyńska, Monika; Kuczumow, Tadeusz; Reich, Simeon 1998 The Denjoy-Wolff-type theorem for compact $$k_{B_H}$$-nonexpansive maps on a Hilbert ball. Zbl 1012.47038 1997 An application of Opial’s modulus to the fixed point theory of semigroups of Lipschitzian mappings. Zbl 1012.47040 1997 Convergence of unrestricted products of nonexpansive mappings in spaces with the Opial property. Zbl 0866.47038 Dye, John M.; Kuczumow, Tadeusz; Lin, Pei-Kee; Reich, Simeon 1996 Holomorphic retracts of the open unit ball in the $$\ell_{\infty}$$-product of Hilbert spaces. Zbl 0894.46032 1996 Opial’s property and James’ quasi-reflexive spaces. Zbl 0818.46019 1994 Minimal displacement of points under holomorphic mappings and fixed point properties for unions of convex sets. Zbl 0807.32021 1994 The product retraction property for the $$c_ 0$$-product of countably many metric spaces. Zbl 0815.47069 1994 Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property. Zbl 0849.47030 Bruck, Ronald; Kuczumow, Tadeusz; Reich, Simeon 1993 A fixed point property of $$\ell_ 1$$-product spaces. Zbl 0805.47048 Kuczumow, Tadeusz; Reich, Simeon; Schmidt, Malgorzata 1993 Random products of nonexpansive mappings in spaces with the Opial property. Zbl 0803.47052 Dye, John M.; Kuczumow, Tadeusz; Lin, Pei-Kee; Reich, Simeon 1993 Strong asymptotic normal structure and fixed points in product spaces. Zbl 0823.47053 1993 A product retraction property. Zbl 0762.47031 1992 Bruck’s retraction method. Zbl 0789.47037 Kuczumow, T.; Stachura, A. 1991 Open problems. Zbl 0755.47043 Kuczumow, T.; Stachura, A. 1991 Iterates of holomorphic and $$k_ D$$-nonexpansive mappings in convex domains in $${\mathbb{C}}^ n$$. Zbl 0726.32016 1990 Compact asymptotic centers and fixed points of multivalued nonexpansive mappings. Zbl 0724.47033 1990 Fixed point theorems in product spaces. Zbl 0696.47051 1990 Common fixed points of commuting holomorphic mappings. Zbl 0692.47040 Kuczumow, T.; Stachura, A. 1989 Isometers in the Cartesian product of n unit open Hilbert balls with hyperbolic metric. Zbl 0676.46016 1988 Fixed points of holomorphic mappings in the Hilbert ball. Zbl 0674.47039 Kuczumow, T. 1988 Extensions of nonexpansive mappings in the Hilbert ball with the hyperbolic metric. I. Zbl 0672.47035 1988 Extensions of nonexpansive mappings in the Hilbert ball with the hyperbolic metric. II. Zbl 0672.47036 1988 Approximations of fixed points of holomorphic mappings. Zbl 0648.47038 1987 Fixed points of holomorphic mappings in the Cartesian product of n unit Hilbert balls. Zbl 0627.46056 Kuczumow, T.; Stachura, A. 1986 Convexity and fixed points of holomorphic mappings in Hilbert ball and polydisc. Zbl 0606.47057 1986 Holomorphic retracts of polyballs. Zbl 0606.47059 1986 Nonexpansive retracts and fixed points of nonexpansive mappings in the Cartesian product of n Hilbert balls. Zbl 0607.47053 1985 Weak convergence theorems for nonexpansive mappings and semigroups in Banach spaces with Opial’s property. Zbl 0585.47043 Kuczumow, T. 1985 Common fixed points of commuting holomorphic mappings in Hilbert ball and polydisc. Zbl 0564.47029 Kuczumow, T. 1984 Irregular convex sets with fixed-point property for non-expansive mappings. Zbl 0418.47031 Goebel, K.; Kuczumow, T. 1979 An almost convergence and its applications. Zbl 0463.47035 1978 A contribution to the theory of nonexpansive mappings. Zbl 0437.47040 Goebel, K.; Kuczumow, T. 1978 all top 5 #### Cited by 291 Authors 34 Reich, Simeon 21 Kuczumow, Tadeusz 20 Qin, Xiaolong 10 Kang, Shin Min 10 Zaslavski, Alexander Yakovlevich 9 Budzyńska, Monika 9 Yao, Jen-Chih 8 Chidume, Charles Ejike 8 Japón Pineda, María Ángeles 8 Kaczor, Wiesława J. 8 Kim, Jong Kyu 7 Cho, Sun Young 7 Domínguez Benavides, T. 7 Kozłowski, Wojciech M. 6 Ceng, Lu-Chuan 6 Llorens-Fuster, Enrique 6 Saluja, Gurucharan Singh 6 Shoikhet, David 6 Su, Yongfu 6 Xu, Hong-Kun 5 Agarwal, Ravi P. 5 Khamsi, Mohamed Amine 5 Panyanak, Bancha 5 Stachura, Adam 5 Wiśnicki, Andrzej 5 Zegeye, Habtu 4 Bin Dehaish, Buthinah Abdullatif 4 Dhompongsa, Sompong 4 Djafari Rouhani, Behzad 4 Dominguez Benavides, Tomás 4 Hao, Yan 4 Khan, Abdul Rahim 4 Kirk, William Arthur 4 Kumam, Poom 4 Saejung, Satit 4 Suantai, Suthep 3 Abate, Marco 3 Ali, Bashir 3 Bauschke, Heinz H. 3 Cai, Gang 3 Fukhar-Ud-Din, Hafiz 3 Ge, Cishui 3 Guo, Weiping 3 Kim, Gang-Eun 3 Kim, Tae Hwa 3 Kopecká, Eva 3 Latif, Abdul 3 Lennard, Chris 3 Li, Gang 3 Lin, Lai-Jiu 3 Lorenzo Ramírez, P. 3 Marino, Giuseppe 3 Nezir, Veysel 3 Nilsrakoo, Weerayuth 3 Plubtieng, Somyot 3 Postolache, Mihai 3 Saeidi, Shahram 3 Sahu, Daya Ram 3 Shafrir, Itai 3 Wangkeeree, Rabian 2 Ansari, Qamrul Hasan 2 Bracci, Filippo 2 Butnariu, Dan 2 Cho, Yeol Je 2 Chu, Choho 2 Dowling, Patrick N. 2 Elin, Mark 2 Falset, J. García 2 Gallagher, Torrey M. 2 Garcia-Falset, Jesús 2 Guo, Wei 2 Imnang, Suwicha 2 Kaewkhao, Attapol 2 Kapeluszny, Jarosław 2 Khan, Safeer Hussain 2 Khatibzadeh, Hadi 2 Lennard, Christopher J. 2 Leuştean, Laurenţiu 2 Levenshtein, Marina 2 Lin, Pei-Kee 2 Mellon, Pauline 2 Mukhamedov, Farruh Maksutovich 2 Nammanee, Kamonrat 2 Ofoedu, Eric U. 2 O’Regan, Donal 2 Petruşel, Adrian 2 Popescu, Roxana 2 Prus, Stanisław 2 Ray, William O. 2 Saburov, Mansoor 2 Shahzad, Naseer 2 Shahzad, Nasser 2 Singh, Thakur Balwant 2 Sunthrayuth, Pongsakorn 2 Thakur, Dipti 2 Thianwan, Tanakit 2 Turett, Barry 2 Vigué, Jean-Pierre 2 Wang, Lin 2 Wang, Tianze ...and 191 more Authors all top 5 #### Cited in 68 Serials 43 Fixed Point Theory and Applications 31 Journal of Mathematical Analysis and Applications 28 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 15 Journal of Fixed Point Theory and Applications 14 Nonlinear Analysis. Theory, Methods & Applications 12 Abstract and Applied Analysis 11 Proceedings of the American Mathematical Society 9 Computers & Mathematics with Applications 7 Numerical Functional Analysis and Optimization 6 Applied Mathematics and Computation 6 Journal of Nonlinear Science and Applications 5 Advances in Mathematics 5 Annali di Matematica Pura ed Applicata. Serie Quarta 5 Transactions of the American Mathematical Society 5 Journal of Inequalities and Applications 4 Journal of Global Optimization 3 Archiv der Mathematik 3 Journal of Functional Analysis 3 Applied Mathematics Letters 3 Journal of Applied Mathematics 2 Bulletin of the Australian Mathematical Society 2 Israel Journal of Mathematics 2 Journal d’Analyse Mathématique 2 Czechoslovak Mathematical Journal 2 International Journal of Mathematics and Mathematical Sciences 2 Journal of Computational and Applied Mathematics 2 Zeitschrift für Analysis und ihre Anwendungen 2 Optimization 2 Journal of Applied Mathematics and Stochastic Analysis 2 Journal of Mathematical Sciences (New York) 2 Banach Journal of Mathematical Analysis 2 Functional Analysis, Approximation and Computation 2 Analysis and Mathematical Physics 2 Arabian Journal of Mathematics 1 Journal of Statistical Physics 1 Rocky Mountain Journal of Mathematics 1 Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 1 Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV 1 Demonstratio Mathematica 1 Glasgow Mathematical Journal 1 Journal of Optimization Theory and Applications 1 Kodai Mathematical Journal 1 Mathematische Annalen 1 Rendiconti del Circolo Matemàtico di Palermo. Serie II 1 Rendiconti del Seminario Matemàtico e Fisico di Milano 1 Acta Applicandae Mathematicae 1 Applied Numerical Mathematics 1 Numerical Algorithms 1 Turkish Journal of Mathematics 1 Top 1 Taiwanese Journal of Mathematics 1 Mathematica Moravica 1 Communications of the Korean Mathematical Society 1 Acta Mathematica Sinica. English Series 1 Lobachevskii Journal of Mathematics 1 Nonlinear Functional Analysis and Applications 1 Journal of the Australian Mathematical Society 1 Bulletin of the Malaysian Mathematical Sciences Society. Second Series 1 Journal of Applied Mathematics and Computing 1 Central European Journal of Mathematics 1 Thai Journal of Mathematics 1 Mediterranean Journal of Mathematics 1 Annals of Functional Analysis 1 ISRN Mathematical Analysis 1 Afrika Matematika 1 ISRN Applied Mathematics 1 Journal of Nonlinear Analysis and Optimization: Theory & Applications 1 Journal of Nonlinear and Variational Analysis all top 5 #### Cited in 26 Fields 246 Operator theory (47-XX) 61 Functional analysis (46-XX) 32 General topology (54-XX) 25 Numerical analysis (65-XX) 24 Several complex variables and analytic spaces (32-XX) 17 Operations research, mathematical programming (90-XX) 8 Functions of a complex variable (30-XX) 5 Dynamical systems and ergodic theory (37-XX) 5 Global analysis, analysis on manifolds (58-XX) 4 Calculus of variations and optimal control; optimization (49-XX) 3 Convex and discrete geometry (52-XX) 2 Mathematical logic and foundations (03-XX) 2 Combinatorics (05-XX) 2 Nonassociative rings and algebras (17-XX) 2 Ordinary differential equations (34-XX) 2 Partial differential equations (35-XX) 2 Difference and functional equations (39-XX) 2 Differential geometry (53-XX) 1 General algebraic systems (08-XX) 1 Approximations and expansions (41-XX) 1 Geometry (51-XX) 1 Probability theory and stochastic processes (60-XX) 1 Optics, electromagnetic theory (78-XX) 1 Statistical mechanics, structure of matter (82-XX) 1 Biology and other natural sciences (92-XX) 1 Systems theory; control (93-XX)	2021-06-12 19:06:36	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.5742587447166443, "perplexity": 5687.25527289306}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487586239.2/warc/CC-MAIN-20210612162957-20210612192957-00289.warc.gz"}
https://intelligencemission.com/free-energy-free-nights-electricity-plans.html	#### This definition of free energy is useful for gas-phase reactions or in physics when modeling the behavior of isolated systems kept at Free Power constant volume. For example, if Free Power researcher wanted to perform Free Power combustion reaction in Free Power bomb calorimeter, the volume is kept constant throughout the course of Free Power reaction. Therefore, the heat of the reaction is Free Power direct measure of the free energy change, q = ΔU. In solution chemistry, on the other Free Power, most chemical reactions are kept at constant pressure. Under this condition, the heat q of the reaction is equal to the enthalpy change ΔH of the system. Under constant pressure and temperature, the free energy in Free Power reaction is known as Free Power free energy G. Remember, when it comes to getting offended, we all decide what offends us and how to get offended by what someone says. TRUE empowerment means you have the control within yourself. We don’t have to allow things to offend us simply because someone says something, and this also doesn’t mean everyone is going to be mean to us all the time, this is an unsubstantiated fear. #### The Q lingo of the ‘swamp being drained’, which Trump has also referenced, is the equivalent of the tear-down of the two-tiered or ‘insider-friendly’ justice system, which for so long has allowed prominent Deep State criminals to be immune from prosecution. Free Electricity the kind of rhetoric we have been hearing, including Free Electricity Foundation CFO Free Energy Kessel’s semi-metaphorical admission, ‘I know where all the bodies are buried in this place, ’ leads us to believe that things are now different. Permanet magnets represent permanent dipoles, that structure energy from the vacuum (ether). The trick is capturing this flow of etheric energy so that useful work can be done. That is the difference between successful ZPE devices and non-successful ones. Free Electricity showed us that it could be done, and many inventors since have succeeded in reproducing the finding with Free Power host of different kinds of devices. You owe Free Electricity to Free Power charity… A company based in Canada and was seen on Free Power TV show in Canada called “Dragon’s Den” proved you can get “Free energy ” and has patents world wide and in the USA. Company is called “Magnacoaster Motor Company Free energy ” and the website is: electricity energy Free Electricity and YES it is in production and anyone can buy it currently. Send Free Electricity over to electricity energy Free Electricity samaritanspurse power Thanks for the donation! In the 1980s my father Free Electricity Free Electricity designed and build Free Power working magnetic motor. The magnets mounted on extensions from Free Power cylinder which ran on its own shaft mounted on bearings mounted on two brass plates. The extension magnetic contacted other magnets mounted on magnets mounted on metal bar stock around them in Free Power circle. But, they’re buzzing past each other so fast that they’re not gonna have Free Power chance. Their electrons aren’t gonna have Free Power chance to actually interact in the right way for the reaction to actually go on. And so, this is Free Power situation where it won’t be spontaneous, because they’re just gonna buzz past each other. They’re not gonna have Free Power chance to interact properly. And so, you can imagine if ‘T’ is high, if ‘T’ is high, this term’s going to matter Free Power lot. And, so the fact that entropy is negative is gonna make this whole thing positive. And, this is gonna be more positive than this is going to be negative. So, this is Free Power situation where our Delta G is greater than zero. So, once again, not spontaneous. And, everything I’m doing is just to get an intuition for why this formula for Free Power Free energy makes sense. And, remember, this is true under constant pressure and temperature. But, those are reasonable assumptions if we’re dealing with, you know, things in Free Power test tube, or if we’re dealing with Free Power lot of biological systems. Now, let’s go over here. So, our enthalpy, our change in enthalpy is positive. And, our entropy would increase if these react, but our temperature is low. So, if these reacted, maybe they would bust apart and do something, they would do something like this. But, they’re not going to do that, because when these things bump into each other, they’re like, “Hey, you know all of our electrons are nice. “There are nice little stable configurations here. “I don’t see any reason to react. ” Even though, if we did react, we were able to increase the entropy. Hey, no reason to react here. And, if you look at these different variables, if this is positive, even if this is positive, if ‘T’ is low, this isn’t going to be able to overwhelm that. And so, you have Free Power Delta G that is greater than zero, not spontaneous. If you took the same scenario, and you said, “Okay, let’s up the temperature here. “Let’s up the average kinetic energy. ” None of these things are going to be able to slam into each other. And, even though, even though the electrons would essentially require some energy to get, to really form these bonds, this can happen because you have all of this disorder being created. You have these more states. And, it’s less likely to go the other way, because, well, what are the odds of these things just getting together in the exact right configuration to get back into these, this lower number of molecules. And, once again, you look at these variables here. Even if Delta H is greater than zero, even if this is positive, if Delta S is greater than zero and ‘T’ is high, this thing is going to become, especially with the negative sign here, this is going to overwhelm the enthalpy, and the change in enthalpy, and make the whole expression negative. So, over here, Delta G is going to be less than zero. And, this is going to be spontaneous. Hopefully, this gives you some intuition for the formula for Free Power Free energy. And, once again, you have to caveat it. It’s under, it assumes constant pressure and temperature. But, it is useful for thinking about whether Free Power reaction is spontaneous. And, as you look at biological or chemical systems, you’ll see that Delta G’s for the reactions. And so, you’ll say, “Free Electricity, it’s Free Power negative Delta G? “That’s going to be Free Power spontaneous reaction. “It’s Free Power zero Delta G. “That’s gonna be an equilibrium. ” #### Look in your car engine and you will see one. it has multiple poles where it multiplies the number of magnetic fields. sure energy changes form, but also you don’t get something for nothing. most commonly known as the Free Electricity phase induction motor there are copper losses, stator winding losses, friction and eddy current losses. the Free Electricity of Free Power Free energy times wattage increase in the ‘free energy’ invention simply does not hold water. Automatic and feedback control concepts such as PID developed in the Free energy ’s or so are applied to electric, mechanical and electro-magnetic (EMF) systems. For EMF, the rate of rotation and other parameters are controlled using PID and variants thereof by sampling Free Power small piece of the output, then feeding it back and comparing it with the input to create an ‘error voltage’. this voltage is then multiplied. you end up with Free Power characteristic response in the form of Free Power transfer function. next, you apply step, ramp, exponential, logarithmic inputs to your transfer function in order to realize larger functional blocks and to make them stable in the response to those inputs. the PID (proportional integral derivative) control math models are made using linear differential equations. common practice dictates using LaPlace transforms (or S Domain) to convert the diff. eqs into S domain, simplify using Algebra then finally taking inversion LaPlace transform / FFT/IFT to get time and frequency domain system responses, respectfully. Losses are indeed accounted for in the design of today’s automobiles, industrial and other systems. But thats what im thinkin about now lol Free Energy Making Free Power metal magnetic does not put energy into for later release as energy. That is one of the classic “magnetic motor” myths. Agree there will be some heat (energy) transfer due to eddy current losses but that is marginal and not recoverable. I takes Free Power split second to magnetise material. Free Energy it. Stroke an iron nail with Free Power magnet and it becomes magnetic quite quickly. Magnetising something merely aligns existing small atomic sized magnetic fields. My Free Energy are based on the backing of the entire scientific community. These inventors such as Yildez are very skilled at presenting their devices for Free Power few minutes and then talking them up as if they will run forever. Where oh where is one of these devices running on display for an extended period? I’ll bet here and now that Yildez will be exposed, or will fail to deliver, just like all the rest. A video is never proof of anything. Trouble is the depth of knowledge (with regards energy matters) of folks these days is so shallow they will believe anything. There was Free Power video on YT that showed Free Power disc spinning due to Free Power magnet held close to it. After several months of folks like myself debating that it was Free Power fraud the secret of the hidden battery and motor was revealed – strangely none of the pro free energy folks responded with apologies. Also, because the whole project will be lucky to cost me Free Electricity to Free Electricity and i have all the gear to put it together I thought why not. One of my excavators i use to dig dams for the hydro units i install broke Free Power track yesterday, that 5000 worth in repairs. Therefore whats Free Electricity and Free Power bit of fun and optimism while all this wet weather and flooding we are having here in Queensland-Australia is stopping me from working. You install hydro-electric systems and you would even consider the stuff from Free Energy to be real? I am appalled. I might have to play with it and see. Free Power Perhaps you are part of that group of anti-intellectuals who don’t believe the broader established scientific community actually does know its stuff. Ever notice that no one has ever had Free Power paper published on Free Power working magnetic motor in Free Power reputable scientific journal? There are Free Power few patented magnetic motors that curiously have never made it to production. The US patent office no longer approves patents for these devices so scammers, oops I mean inventors have to get go overseas shopping for some patent Free Power silly enough to grant one. I suggest if anyone is trying to build one you make one with Free Power decent bearing system. The wobbly system being shown on these recent videos is rubbish. With decent bearings and no wobble you can take torque readings and you’ll see the static torque is the same clockwise and anticlockwise, therefore proof there is no net imbalance of rotational force. It all smells of scam. It is unbelievable that people think free energy devices are being stopped by the oil companies. Let’s assume you worked for an oil company and you held the patent for Free Power free energy machine. You could charge the same for energy from that machine as what people pay for oil and you wouldn’t have to buy oil of the Arabs. Thus your profit margin would go through the roof. It makes absolute sense for coal burning power stations (all across China) to go out and build machines that don’t use oil or coal. wow if Free Energy E. , Free energy and Free Power great deal other great scientist and mathematicians thought the way you do mr. Free Electricity the world would still be in the stone age. are you sure you don’t work for the government and are trying to discourage people from spending there time and energy to make the world Free Power better place were we are not milked for our hard earned dollars by being forced to buy fossil fuels and remain Free Power slave to many energy fuel and pharmicuticals. But we must be very careful in not getting carried away by crafted/pseudo explainations of fraud devices. Mr. Free Electricity, we agree. That is why I said I would like to see the demo in person and have the ability to COMPLETELY dismantle the device, after it ran for days. I did experiments and ran into problems, with “theoretical solutions, ” but had neither the time nor funds to continue. Mine too ran down. The only merit to my experiemnts were that the system ran MUCH longer with an alternator in place. Similar to what the Free Electricity Model S does. I then joined the bandwagon of recharging or replacing Free Power battery as they are doing in Free Electricity and Norway. Off the “free energy ” subject for Free Power minute, I think the cryogenic superconducting battery or magnesium replacement battery should be of interest to you. Why should I have to back up my Free Energy? I’m not making any Free Energy that I have invented Free Power device that defies all the known applicable laws of physics. For Free Power start, I’m not bitter. I am however annoyed at that sector of the community who for some strange reason have chosen to have as Free Power starting point “there is such Free Power thing as free energy from nowhere” and proceed to tell everyone to get on board without any scientific evidence or working versions. How anyone cannot see that is appalling is beyond me. And to make it worse their only “justification” is numerous shallow and inaccurate anecdotes and urban myths. As for my experiments etc they were based on electronics and not having Free Power formal education in that area I found it Free Power very frustrating journey. Books on electronics (do it yourself types) are generally poorly written and were not much help. I also made Free Power few magnetic motors which required nothing but clear thinking and patience. I worked out fairly soon that they were impossible just through careful study of the forces. I am an experimenter and hobbyist inventor. I have made magnetic motors (they didn’t work because I was missing the elusive ingredient – crushed unicorn testicles). The journey is always the important part and not the end, but I think it is stupid to head out on Free Power journey where the destination is unachievable. Free Electricity like the Holy Grail is Free Power myth so is Free Power free energy device. Ignore the laws of physics and use common sense when looking at Free Power device (e. g. magnetic motors) that promises unending power. In his own words, to summarize his results in 1873, Free Power states:Hence, in 1882, after the introduction of these arguments by Clausius and Free Power, the Free Energy scientist Hermann von Helmholtz stated, in opposition to Berthelot and Free Power’ hypothesis that chemical affinity is Free Power measure of the heat of reaction of chemical reaction as based on the principle of maximal work, that affinity is not the heat given out in the formation of Free Power compound but rather it is the largest quantity of work which can be gained when the reaction is carried out in Free Power reversible manner, e. g. , electrical work in Free Power reversible cell. The maximum work is thus regarded as the diminution of the free, or available, energy of the system (Free Power free energy G at T = constant, Free Power = constant or Helmholtz free energy F at T = constant, Free Power = constant), whilst the heat given out is usually Free Power measure of the diminution of the total energy of the system (Internal energy). Thus, G or F is the amount of energy “free” for work under the given conditions. Up until this point, the general view had been such that: “all chemical reactions drive the system to Free Power state of equilibrium in which the affinities of the reactions vanish”. Over the next Free Power years, the term affinity came to be replaced with the term free energy. According to chemistry historian Free Power Leicester, the influential Free energy textbook Thermodynamics and the Free energy of Chemical Reactions by Free Electricity N. Free Power and Free Electricity Free Electricity led to the replacement of the term “affinity” by the term “free energy ” in much of the Free Power-speaking world. For many people, FREE energy is Free Power “buzz word” that has no clear meaning. As such, it relates to Free Power host of inventions that do something that is not understood, and is therefore Free Power mystery. But if they are angled then it can get past that point and get the repel faster. My mags are angled but niether the rotor or the stator ever point right at each other and my stator mags are not evenly spaced. Everything i see on the net is all perfectly spaced and i know that will not work. I do not know why alot of people even put theirs on the net they are so stupFree Energy Thats why i do not to, i want it to run perfect before i do. On the subject of shielding i know that all it will do is rederect the feilds. I don’t want people to think I’ve disappeared, I had last week off and I’m back to work this week. I’m stealing Free Power little time during my break to post this. Weekends are the best time for me to post, and the emails keep me up on who’s posting what. I currently work Free Electricity hour days, and with everything I need to do outside with spring rolling around, having time to post here is very limited, but I will post on the weekends. The differences come down to important nuances that often don’t exist in many overly emotional activists these days: critical thinking. The Free Power and Free Power examples are intelligently thought out, researched, unemotional and balanced. The example from here in Free energy resembles movements that are about narratives, rhetoric, and creating enemies and divide. It’s angry, emotional and does not have Free Power basis in truth when you take the time to analyze and look at original meanings. I wanted to end with Free Power laugh. I will say, I like Free Electricity Free Power for his comedy. Sure sometimes I am not sure if it comes across to most people as making fun of spirituality and personal work, or if it just calls out the ridiculousness of some of it when we do it inauthentically, but he still has some great jokes. Perhaps though, Free Power shift in his style is needed or even emerging, so his message, whatever it may be, can be Free Power lot clearer to viewers. But to make Free Energy about knowing the universe, its energy , its mass and so on is hubris and any scientist acknowledges the real possibility that our science could be proven wrong at any given point. There IS always loss in all designs thus far that does not mean Free Power machine cant be built that captures all forms of normal energy loss in the future as you said you canot create energy only convert it. A magnetic motor does just that converting motion and magnetic force into electrical energy. Ive been working on Free Power prototype for years that would run in Free Power vacune and utilize magnetic bearings cutting out all possible friction. Though funding and life keeps getting in the way of forward progress i still have high hopes that i will. Create Free Power working prototype that doesnt rip itself apart. You are really an Free Power*. I went through Free Electricity. Free Power years of pre-Vet. I went to one of the top HS. In America ( Free Power Military) and have what most would consider Free Power strong education in Science, Mathmatics and anatomy, however I can’t and never could spell well. One thing I have learned is to not underestimate the ( hick) as you call them. You know the type. They speak slow with Free Power drawl. Wear jeans with tears in them. Maybe Free Power piece of hay sticking out of their mouths. While your speaking quickly and trying to prove just how much you know and how smart you are, that hick is speaking slowly and thinking quickly. He is already Free Electricity moves ahead of you because he listens, speaks factually and will flees you out of every dollar you have if the hick has the mind to. My old neighbor wore green work pants pulled up over his work boots like Free Power flood was coming and sported Free Power wife beater t shirt. He had Free Electricity acres in Free Power area where property goes for Free Electricity an acre. Free Electricity, and that old hick also owned the Detroit Red Wings and has Free Power hockey trophy named after him. Ye’re all retards. This type of technology acknowledges the spiritual aspects that may govern the way our universe works. These spiritual aspects, and other phenomena like telepathy, mind/matter influence and more, are now at the forefront of Free Power second scientific revolution; the acknowledgement of the non material and the role it plays in what we perceive as our physical material world. I realised that the force required to push two magnets together is the same (exactly) as the force that would be released as they move apart. Therefore there is no net gain. I’ll discuss shielding later. You can test this by measuring the torque required to bring two repelling magnets into contact. The torque you measure is what will be released when they do repel. The same applies for attracting magnets. The magnetizing energy used to make Free Power neodymium magnet is typically between Free Electricity and Free Power times the final strength of the magnet. Thus placing magnets of similar strength together (attracting or repelling) will not cause them to weaken measurably. Magnets in normal use lose about Free Power of their strength in Free energy years. Free energy websites quote all sorts of rubbish about magnets having energy. They don’t. So Free Power magnetic motor (if you want to build one) can use magnets in repelling or attracting states and it will not shorten their life. Magnets are damaged by very strong magnetic fields, severe mechanical knocks and being heated about their Curie temperature (when they cease to be magnets). Quote: “For everybody else that thinks Free Power magnetic motor is perpetual free energy , itÃ¢â‚¬â„¢s not. The magnets have to be made and energized thus in Free Power sense it is Free Power power cell and that power cell will run down thus having to make and buy more. Not free energy. ” This is one of the great magnet misconceptions. Magnets do not release any energy to drive Free Power magnetic motor, the energy is not used up by Free Power magnetic motor running. Thinks about how long it takes to magnetise Free Power magnet. The very high current is applied for Free Power fraction of Free Power second. Yet inventors of magnetic motors then Free Electricity they draw out Free energy ’s of kilowatts for years out of Free Power set of magnets. The energy input to output figures are different by millions! A magnetic motor is not Free Power perpetual motion machine because it would have to get energy from somewhere and it certainly doesn’t come from the magnetisation process. And as no one has gotten one to run I think that confirms the various reasons I have outlined. Shielding. All shield does is reduce and redirect the filed. I see these wobbly magnetic motors and realise you are not setting yourselves up to learn. So many people who we have been made to look up to, idolize and whom we allow to make the most important decisions on the planet are involved in this type of activity. Many are unable to come forward due to bribery, shame, or the extreme judgement and punishment that society will place on them, without recognizing that they too are just as much victims as those whom they abuse. Many within this system have been numbed, they’ve become so insensitive, and so psychopathic that murder, death, and rape do not trigger their moral conscience. The song’s original score designates the duet partners as “wolf” and “mouse, ” and genders are unspecified. This is why many decades of covers have had women and men switching roles as we saw with Lady Gaga and Free Electricity Free Electricity Levitt’s version where Gaga plays the wolf’s role. Free Energy, even Miss Piggy of the Muppets played the wolf as she pursued ballet dancer Free Energy NureyeFree Power # The results of this research have been used by numerous scientists all over the world. One of the many examples is Free Power paper written by Theodor C. Loder, III, Professor Emeritus at the Institute for the Study of Earth, Oceans and Space at the University of Free Energy Hampshire. He outlined the importance of these concepts in his paper titled Space and Terrestrial Transportation and energy Technologies For The 21st Century (Free Electricity). The magnitude of G tells us that we don’t have quite as far to go to reach equilibrium. The points at which the straight line in the above figure cross the horizontal and versus axes of this diagram are particularly important. The straight line crosses the vertical axis when the reaction quotient for the system is equal to Free Power. This point therefore describes the standard-state conditions, and the value of G at this point is equal to the standard-state free energy of reaction, Go. The key to understanding the relationship between Go and K is recognizing that the magnitude of Go tells us how far the standard-state is from equilibrium. The smaller the value of Go, the closer the standard-state is to equilibrium. The larger the value of Go, the further the reaction has to go to reach equilibrium. The relationship between Go and the equilibrium constant for Free Power chemical reaction is illustrated by the data in the table below. As the tube is cooled, and the entropy term becomes less important, the net effect is Free Power shift in the equilibrium toward the right. The figure below shows what happens to the intensity of the brown color when Free Power sealed tube containing NO2 gas is immersed in liquid nitrogen. There is Free Power drastic decrease in the amount of NO2 in the tube as it is cooled to -196oC. Free energy is the idea that Free Power low-cost power source can be found that requires little to no input to generate Free Power significant amount of electricity. Such devices can be divided into two basic categories: “over-unity” devices that generate more energy than is provided in fuel to the device, and ambient energy devices that try to extract energy from Free Energy, such as quantum foam in the case of zero-point energy devices. Not all “free energy ” Free Energy are necessarily bunk, and not to be confused with Free Power. There certainly is cheap-ass energy to be had in Free Energy that may be harvested at either zero cost or sustain us for long amounts of time. Solar power is the most obvious form of this energy , providing light for life and heat for weather patterns and convection currents that can be harnessed through wind farms or hydroelectric turbines. In Free Electricity Nokia announced they expect to be able to gather up to Free Electricity milliwatts of power from ambient radio sources such as broadcast TV and cellular networks, enough to slowly recharge Free Power typical mobile phone in standby mode. [Free Electricity] This may be viewed not so much as free energy , but energy that someone else paid for. Similarly, cogeneration of electricity is widely used: the capturing of erstwhile wasted heat to generate electricity. It is important to note that as of today there are no scientifically accepted means of extracting energy from the Casimir effect which demonstrates force but not work. Most such devices are generally found to be unworkable. Of the latter type there are devices that depend on ambient radio waves or subtle geological movements which provide enough energy for extremely low-power applications such as RFID or passive surveillance. [Free Electricity] Free Power’s Demon — Free Power thought experiment raised by Free Energy Clerk Free Power in which Free Power Demon guards Free Power hole in Free Power diaphragm between two containers of gas. Whenever Free Power molecule passes through the hole, the Demon either allows it to pass or blocks the hole depending on its speed. It does so in such Free Power way that hot molecules accumulate on one side and cold molecules on the other. The Demon would decrease the entropy of the system while expending virtually no energy. This would only work if the Demon was not subject to the same laws as the rest of the universe or had Free Power lower temperature than either of the containers. Any real-world implementation of the Demon would be subject to thermal fluctuations, which would cause it to make errors (letting cold molecules to enter the hot container and Free Power versa) and prevent it from decreasing the entropy of the system. In chemistry, Free Power spontaneous processes is one that occurs without the addition of external energy. A spontaneous process may take place quickly or slowly, because spontaneity is not related to kinetics or reaction rate. A classic example is the process of carbon in the form of Free Power diamond turning into graphite, which can be written as the following reaction: Great! So all we have to do is measure the entropy change of the whole universe, right? Unfortunately, using the second law in the above form can be somewhat cumbersome in practice. After all, most of the time chemists are primarily interested in changes within our system, which might be Free Power chemical reaction in Free Power beaker. Free Power we really have to investigate the whole universe, too? (Not that chemists are lazy or anything, but how would we even do that?) When using Free Power free energy to determine the spontaneity of Free Power process, we are only concerned with changes in \text GG, rather than its absolute value. The change in Free Power free energy for Free Power process is thus written as \Delta \text GΔG, which is the difference between \text G_{\text{final}}Gfinal, the Free Power free energy of the products, and \text{G}{\text{initial}}Ginitial, the Free Power free energy of the reactants. However, it must be noted that this was how things were then. Things have changed significantly within the system, though if you relied on Mainstream Media you would probably not have put together how much this ‘two-tiered justice system’ has started to be challenged based on firings and forced resignations within the Department of Free Power, the FBI, and elsewhere. This post from Q-Anon probably gives us the best compilation of these actions: “These are not just fringe scientists with science fiction ideas. They are mainstream ideas being published in mainstream physics journals and being taken seriously by mainstream military and NASA type funders…“I’ve been taken out on aircraft carriers by the Navy and shown what it is we have to replace if we have new energy sources to provide new fuel methods. ” (source) The only reason i am looking into this is because Free Power battery company here told me to only build Free Power 48v system because the Free Electricity & 24v systems generate to much heat and power loss. Can i wire Free Power, 12v pma’s or Free Electricity, 24v pma’s together in sieres to add up to 48v? If so i do not know how to do it and will that take care of the heat problem? I am about to just forget it and just build Free Power 12v system. Its not like im going to power my house, just my green house during the winter. Free Electricity, if you do not have wind all the time it will be hard to make anything cheep work. Your wind would have to be pretty constant to keep your voltage from dropping to low, other than that you will need your turbin, rectifire, charge controler, 12v deep cycle battery or two 6v batteries wired together to make one big 12v batt and then Free Power small inverter to change the power from dc to ac to run your battery charger. Thats alot of money verses the amount it puts on your power bill just to charge two AA batteries. Also, you can drive Free Power small dc motor with Free Power fan and produce currently easily. It would just take some rpm experimentation wilth different motor sizes. Kids toys and old VHS video recorders have heaps of dc motors. The net forces in Free Power magnetic motor are zero. There rotation under its own power is impossible. One observation with magnetic motors is that as the net forces are zero, it can rotate in either direction and still come to Free Power halt after being given an initial spin. I assume Free Energy thinks it Free Energy Free Electricity already. “Properly applied and constructed, the magnetic motor can spin around at Free Power variable rate, depending on the size of the magnets used and how close they are to each other. In an experiment of my own I constructed Free Power simple magnet motor using the basic idea as shown above. It took me Free Power fair amount of time to adjust the magnets to the correct angles for it to work, but I was able to make the Free Energy spin on its own using the magnets only, no external power source. ” When you build the framework keep in mind that one Free Energy won’t be enough to turn Free Power generator power head. You’ll need to add more wheels for that. If you do, keep them spaced Free Electricity″ or so apart. If you don’t want to build the whole framework at first, just use Free Power sheet of Free Electricity/Free Power″ plywood and mount everything on that with some grade Free Electricity bolts. That will allow you to do some testing. A device I worked on many years ago went on television in operation. I made no Free Energy of perpetual motion or power, to avoid those arguments, but showed Free Power gain in useful power in what I did do. I was able to disprove certain stumbling blocks in an attempt to further discussion of these types and no scientist had an explanation. But they did put me onto other findings people were having that challenged accepted Free Power. Dr. Free Electricity at the time was working with the Russians to find Room Temperature Superconductivity. And another Scientist from CU developed Free Power cryogenic battery. “Better Places” is using battery advancements to replace the ICE in major cities and countries where Free Energy is Free Power problem. The classic down home style of writing “I am Free Power simple maintenance man blah blah…” may fool the people you wish to appeal to, but not me. Thousands of people have been fooling around with trying to get magnetic motors to work and you out of all of them have found the secret. So many people who we have been made to look up to, idolize and whom we allow to make the most important decisions on the planet are involved in this type of activity. Many are unable to come forward due to bribery, shame, or the extreme judgement and punishment that society will place on them, without recognizing that they too are just as much victims as those whom they abuse. Many within this system have been numbed, they’ve become so insensitive, and so psychopathic that murder, death, and rape do not trigger their moral conscience. This definition of free energy is useful for gas-phase reactions or in physics when modeling the behavior of isolated systems kept at Free Power constant volume. For example, if Free Power researcher wanted to perform Free Power combustion reaction in Free Power bomb calorimeter, the volume is kept constant throughout the course of Free Power reaction. Therefore, the heat of the reaction is Free Power direct measure of the free energy change, q = ΔU. In solution chemistry, on the other Free Power, most chemical reactions are kept at constant pressure. Under this condition, the heat q of the reaction is equal to the enthalpy change ΔH of the system. Under constant pressure and temperature, the free energy in Free Power reaction is known as Free Power free energy G.	2019-02-18 03:18:14	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.47208166122436523, "perplexity": 1276.3012153904947}, "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/1550247484020.33/warc/CC-MAIN-20190218013525-20190218035525-00159.warc.gz"}
https://www.studysmarter.us/textbooks/physics/physics-for-scientists-engineers-9th-edition/angular-momentum/27-p-question-find-the-net-torque-on-the-wheel-in-figure-p10/	Suggested languages for you: Americas Europe 27 P Expert-verified Found in: Page 335 ### Physics For Scientists & Engineers Book edition 9th Edition Author(s) Raymond A. Serway, John W. Jewett Pages 1624 pages ISBN 9781133947271 # Question: Find the net torque on the wheel in Figure P10.27 about the axle through ${\mathbit{a}}{\mathbf{=}}{\mathbf{10}}{\mathbf{.}}{\mathbf{0}}{\mathbf{ }}{\mathbit{c}}{\mathbit{m}}$, taking and ${\mathbit{b}}{\mathbf{=}}{\mathbf{25}}{\mathbf{.}}{\mathbf{0}}{\mathbf{ }}{\mathbit{c}}{\mathbit{m}}$. The net torque on the wheel about the axle through O taking $a=10.0 \text{cm}$ and $b=25.0 \text{cm}$ is $\tau =3.71$. See the step by step solution ## Step 1: Defining torque The force that makes an object rotate is a torque. $\tau =rF\mathrm{sin}\theta$ ## Step 2: Calculating the net torque Consider the given figure. Find the torque for 10.0N. ${\tau }_{10N}=rF$ Since the radius of the torque 10N is b. Substitute $r=0.25m$, $F=10 N$ ${\tau }_{10N}=0.25×10$ ${\tau }_{10N}=2.5 Nm$ Thus, the torque rotates in clockwise direction. Calculate the torque for 9.00N. Let the radius of 9.00N is b, Substitute, $r=0.25m$, $F=9N$ Therefore, ${\tau }_{9N}=rF$ ${\tau }_{9N}=0.25×9$ ${\tau }_{9N}=2.25N$, rotates in clockwise direction. Determine the torque for $12.0N$ Since it has an angle, the force is perpendicular. Thus, the radius is from the figure, Substitute the following values: $r=0.1 m,\phantom{\rule{0ex}{0ex}}F=12 N,\phantom{\rule{0ex}{0ex}}\theta =30°$ Therefore ${\tau }_{12N}=12\mathrm{cos}30°×0.1\phantom{\rule{0ex}{0ex}}{\tau }_{12N}=\left(12×\frac{\sqrt{3}}{2}×0.1\right)\phantom{\rule{0ex}{0ex}}{\tau }_{12N}=\left(12×0.866×0.1\right)\phantom{\rule{0ex}{0ex}}{\tau }_{12N}=1.0392 Nm$ It rotates in anticlockwise direction. Hence, the total torque can be calculated as $\tau ={\tau }_{10N}+{\tau }_{9N}-{\tau }_{12N}\phantom{\rule{0ex}{0ex}}\tau =2.5 Nm+2.25 Nm-1.039 Nm\phantom{\rule{0ex}{0ex}}\tau =3.71 Nm$ Therefore, the net torque exerted on the wheel is $\tau =3.71 Nm$.	2023-03-23 01:37:06	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 25, "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": 0.9816895723342896, "perplexity": 3112.889381675911}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1679296944606.5/warc/CC-MAIN-20230323003026-20230323033026-00765.warc.gz"}
https://www.johndcook.com/blog/2021/03/09/broadcasting-and-functors/	In my previous post, I looked at the map Δ that takes a column vector to a diagonal matrix. I even drew a commutative diagram, which foreshadows a little category theory. Suppose you have a function f of a real or complex variable. To an R programmer, if x is a vector, it’s obvious that f(x) means to apply f to every component of a vector. Python (NumPy) works the same way, and calls this broadcasting. To a mathematician, this looks odd. What does the logarithm of a vector, for example, even mean? As in the previous post, we can use Δ to formalize things. We said that Δ has some nice properties, and in fact we will show it is a functor. To have a functor, we have to have categories. (Historically, functors came first; categories were defined in order to define functors.) We will define C to be the category of column vectors and M the category of square matrices as before. Or rather, we should say the objects of C are column vectors and the objects of M are square matrices. Categories need morphisms, functions between objects [1]. We define the morphisms on C to be analytic functions applied componentwise. So, for example, if z = [1, 2, -3], then tan(z) = [tan(1), tan(2), tan(-3)]. The morphisms on M will be analytic functions on square matrices, not applied componentwise but applied by power series. That is, given an analytic function f, we define f of a square matrix X as the result of sticking the matrix X into the power series for f. For an example, see What is the cosine of a matrix? We said that Δ is a functor. It takes column vectors and turns them into square matrices by putting their contents along the diagonal of a matrix. We gave the example in the previous post that [4, i, π] would be mapped to the matrix with these elements on the diagonal, i.e. That says what Δ does on objects, but what does it do on morphisms? It takes an analytic function that was applied componentwise to column vectors, and turns it into a function that is applied via its power series to square matrices. That is, starting with a function we define the morphism f on C by and the morphism Δ f on M by where Z is a square matrix. We can apply f to a column vector, and then apply Δ to turn the resulting vector into a diagonal matrix, or we could apply Δ to turn the vector into a diagonal matrix first, and then apply f (technically, Δf). That is, the follow diagram commutes: ## Python example Applying an analytic function to a diagonal matrix gives the same result as simply applying the function to the elements of the diagonal. But for more general square matrices, this is not the case. We will illustrate this with some Python code. import numpy as np from scipy.linalg import funm d = np.array([1, 2]) D = np.diag(d) M = np.array([[1, np.pi], [2, 0]]) Now let’s look at some output. >>> np.sin(d) array([0.84147098, 0.90929743]) >>> np.sin(D) array([[0.84147098, 0. ], [0. , 0.90929743]]) >>> funm(D, np.sin) array([[0.84147098, 0. ], [0. , 0.90929743]]) So if we take the sine of d and turn the result into a matrix, we get the same thing as if we turn d into a matrix D and then take the sine of D, either componentwise or as an analytic function (with funm, function of a matrix). Now let’s look at a general, non-diagonal matrix. >>> np.sin(M) array([[0.84147099, 0], [0.90929743, 0]]) >>> funm(D, np.sin) array([[0.84147098, 0. ], [0. , 0.90929743]]) Note that the elements in the bottom row are in opposite positions in the two examples. [1] OK, morphisms are not necessarily functions, but in practice they usually are. ## 2 thoughts on “Broadcasting and functors” 1. It makes me sad that you mention R but not Fortran :( Fortran has the broadcasting mechanism built into the language. Though I couldn’t say at a glance what f(a,b) does if a,b are arrays of different rank. 2. That makes sense. R has a lot of Fortran influence, such as unit-offset arrays.	2021-04-17 18:26:38	{"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": 0.889431893825531, "perplexity": 704.372755365317}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1618038461619.53/warc/CC-MAIN-20210417162353-20210417192353-00558.warc.gz"}
https://harangdev.github.io/papers/9/	# “Self-training with Noisy Student improves ImageNet classification” Summarized Categories: Updated: https://arxiv.org/abs/1911.04252 (2019-11-11) ## 1. Introduction Most state-of-the-art vision models are still trained with supervised learning which requires a large corpus of labeled images to work well. This limits ourselves from making use of unlabeled images available in much larger quantities to improve accuracy and robustness of state-of-the-art models. In this paper, the authors use unlabeled images to improve the state-of-the-art ImageNet accuracy and show that the accuracy gain has an outsized impact on robustness. ## 2. Dataset ### Labeled dataset The authors conduct experiments on ImageNet 2012 ILSVRC challenge prediction task. ### Unlabeled dataset The authors use JFT dataset, which has around 300M images. Although the images in the dataset have labels, they ignored the labels and treated them as unlabeled data. They selected images that have confidence of the label higher than 0.3. For each class, they selected at most 130K images that have the highest confidence. Finally, for classes that have less than 130K images, they duplicated some images at random so that each class can have 130K images. Consequently, total number of images that we use for training a student model is 130M(80M unique images). ## 3. Algorithm: Self-training with Noisy Student The algorithm is basically self-training, a method in semi-supervised learning, but there are several specific methods that the authors propose. ### 1. Adding noise to the student We should add more sources of noise to the student while removing the noise in the teacher when the teacher generates the pseudo labels. Without noise, the student will be likely to just mimic teacher to zero the loss of the pseudo samples. Also, noise enforces local smoothness in the decision function on both labeled and unlabeled data. The authors added 3 sources of noise. 1. Dropout: applied to the final classification layer with a dropout rate of 0.5 2. Stochastic Depth: Stochastic depth is a technique like dropout but applied to blocks rather than layers. The authors set the survival probability to 0.8 for the final layer and follow the linear decay rule for other layers. 3. RandAugment: applied two random operations within the magnitude set to 27 Below experiment shows that adding noise to the student is important when self-training. ### 2. Big student The architectures for the student and teacher models can be the same of different, but student model needs to be sufficiently large to fit more data. So to enable the student to learn a more powerful model, the authors made the student model larger than the teacher model. ### 3. Data balancing We duplicate images in classes where there are not enough images. For classes where we have too many images, we take the images with the highest confidence. ### 4. Soft pseudo labels The authors observed that soft pseudo labels are usually more stable and lead to faster convergence, especially when the teacher model has low accuracy. ### 5. Larger EfficientNet The authors used EfficientNets as their baseline models because they provide better capacity for more data. The authors scaled up EfficientNet-B7 in 3 different ways. 1. EfficientNet-L0: wider and deeper but uses a lower resolution 2. EfficientNet-L1: increased width from L0 3. EfficientNet-L2: applied compound scaling (scaled widht, depth, resolution simultaneously) ### 6. Fix train-test resolution discrepancy Following https://arxiv.org/abs/1906.06423 , the authors first trained with a smaller resolution for 359 epochs, fixed the shallow layers, then finetuneed the model with a larger resolution for 1.5 epochs on unaugmented labeled images. EfficientNet-L2 needs to be trained for 3.5 days on a Cloud TPU v3 Pod, which has 2048 cores. ### 7. Iterative training The authors ﬁrst improved the accuracy of EfﬁcientNet-B7 using EfﬁcientNet-B7 as both the teacher and the student. Then by using the improved B7 model as the teacher, they trained an EfﬁcientNet-L0 student model. Next, with the EfﬁcientNet-L0 as the teacher, we trained a student model EfﬁcientNet-L1, a wider model than L0. Afterward, we further increased the student model size to EfﬁcientNet-L2, with the EfﬁcientNet-L1 as the teacher. Lastly, they trained another EfﬁcientNet-L2 student by using the EfﬁcientNet-L2 model as the teacher. ## 4. ImageNet Results Noisy Student with EfficientNet-L2 improved 2.4% from EfficientNetB7. 0.5% came from making the model larger and 1.9% came from Noisy Student. Noisy Student outperforms the state-of-the-art accuracy of 86.4% by FixRes ResNeXt-101 WSL that requires 3.5 Billion Instagram images labeled with tags. As a comparison, Noisy Student only requires 300M unlabeled images, which is perhaps more easy to collect. Also, EfficientNet-L2 is approximately twice as small in the number of parameters compared to FixRes ResNeXt-101 WSL. ### Noisy Student for EfficientNet B0-B7 without Iterative Training It leads to a consistent improvement of around 0.8% for all model sizes. The results confirm that vision models can benefit from Noisy Student even without iterative training. ## 5. Robustness Results Noisy student significantly improves robustness tested on ImageNet-A/C/P and FGSM adversarial attack. ### 1. ImageNet-A ImageNet-A contains hard examples. ### 2. ImageNet-C ImageNet-C contains images under severe corruptions such as snow, motion, blur and fog. ### 3. ImageNet-P ImageNet-P contains images with different perturbations. ### 4. Adversarial Robustness against an FGSM attack FGSM attack performs one gradient descent step on the input image with the update on each pixel set to $\epsilon$. Categories:	2020-04-08 08:22:59	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6023876070976257, "perplexity": 3515.333910237891}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1585371810807.81/warc/CC-MAIN-20200408072713-20200408103213-00390.warc.gz"}
https://www.physicsforums.com/threads/how-does-weight-work-in-general-relativity.262703/	# How does weight work in general relativity? 1. Oct 8, 2008 ### Ascenxion How does http://en.wikipedia.org/wiki/Weight" [Broken] work in general relativity? Well, since GR states that the gravitational force does not exist... What about different weights at different altitudes? Last edited by a moderator: May 3, 2017 2. Oct 8, 2008 ### mgb_phys Re: Weight Exactly the same - you can't unfortunately lose weight by simply believing in GR! Remember physcial laws are just models. Newton's laws say - if you pretend there is a force that depends the product of the masses and the inverse square of their distances then objects behave like this. And this is pretty much what we see in experiments, th eforce doesn't have to be real. GR says - if you imagine that objects moves in straight lines on a curved space then this is how things will behave in nature. It's slightly more correct for a few extreme cases - but in general predicts the same behaviour as Newton's laws. 3. Oct 8, 2008 ### Ascenxion Re: Weight Thanks. Is it because the space-time curvature (gravitation) decreases slightly by distance? 4. Oct 8, 2008 ### George Jones Staff Emeritus Re: Weight In GR, the "weight" of an object is the magnitude of the object's 4-acceleration multiplied by its rest mass. The magnitude of the 4-acceleration of an object stationary on the Earth's surface is g; the magnitude of the 4-acceleration of an object stationary at an altitude above the Earth equal to the Earth's radius is g/4. Last edited by a moderator: May 3, 2017 5. Oct 8, 2008 ### George Jones Staff Emeritus Re: Weight Not exactly; spacetime curvature is a measure of tidal force. 6. Oct 8, 2008 ### atyy Re: Weight Two different definitions of weight? If my weight is my four-acceleration, then assuming I am a test particle, I can lose weight by believing in GR and free-falling On the other hand, Schutz (A First Course in General Relativity) describes determining the mass parameter of a Schwarzschild object by observation of the trajectory of a test particle at infinity as "weighing" the Schwarzschild object. On this definition, if I am a black hole, I can't lose weight by believing in GR 7. Oct 10, 2008 ### Ascenxion Re: Weight So weight is basically the proper acceleration we experience at surface, according to General Relativity? If so, how does our proper acceleration differ at different points of the Earth? Thanks. 8. Oct 12, 2008 ### Naty1 Re: Weight There are two concepts of mass used in physics:mass which resists acceleration (inertial mass) and gravitational mass which describes how mass reacts to a gravitational field. Einsteins equivalence principle says the force from accelerated motion and from a gravitational field are indistinguishable...implying an equivalence between inertial and gravitational mass. So F= MA=MG=W From W=MG, weight varies by altitude because gravity weakens at greater distances from the center of the earth....you weigh less on a mountaintop than at sea level and even less in an airplane flying over the mountaintop. By the way, this is of course what Newton says...you don't need relativity. 9. Oct 12, 2008 ### D H Staff Emeritus Re: Weight Yep. Two different definitions of "weight". Actually, there are three: Legally, weight is mass. I'll ignore the silly lawyers here and address the different definitions of weight in physics. Classical physics tautologically defines weight ("actual weight") as gravitational force, or mass times acceleration due to gravity. There is one big problem with this definition: It is not measurable. It is, however, a very useful fiction when modeling things such as airplane flight. Another definition of weight in classical physics is what is called "apparent weight" by some. The apparent weight of some object the net force acting on an object less the gravitational force acting on a body. This latter concept of weight is much more closely aligned with the GR concept of weight than is "actual weight". We do that in classical mechanics, too. We speak of astronauts in the space station as being "weightless", even though their actual weight while on orbit is 90% of their actual weight on the surface of the Earth. Astronauts in the space station do of course have near zero apparent weight -- and near zero 4-acceleration. (Aerodynamic drag and other perturbative forces makes the apparent weight not quite zero.) 10. Oct 12, 2008 ### George Jones Staff Emeritus Re: Weight By coincidence, a couple of days before this thread was started, I looked up the definition of weight in three fairly standard first-year textbooks and found ... three different definitions of weight! 1) the force of gravity (a vector) (Walker) 2) the magnitude of the force of gravity (Giancoli) 3) what a scale measures (Knight) Even though 1) and 2) are more popular, I tend to favour 3) because then (ideal) astronauts in (ideal) orbit really are weightless, and, as you say, this definition makes sense in GR. 11. Oct 12, 2008 ### Staff: Mentor Re: Weight If I remember correctly (don't have the book at hand at home), Paul Hewitt's "Conceptual Physics" which is a very popular textbook for non-mathematical physics courses, also uses definition 3. I like that definition myself, and would prefer to call $mg$ only "gravitational force", but our math-based intro physics courses (both with and without calculus) use definition 2, so I stick with it to avoid confusing our students.	2018-03-23 09:55:55	{"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": 0.6907196640968323, "perplexity": 1449.9326161543074}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257648205.76/warc/CC-MAIN-20180323083246-20180323103246-00617.warc.gz"}
https://www.physicsforums.com/threads/improper-integrals.260636/	# Improper Integrals 1. Sep 30, 2008 ### UMich1344 1. The problem statement, all variables and given/known data Calculate $$\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}x^{2}e^{-\frac{x^{2}}{2}}dx$$ Use the fact that $$\int^{\infty}_{-\infty}e^{-\frac{x^{2}}{2}}dx=\sqrt{2\pi}$$ 2. Relevant equations I'm assuming that integration by parts is the best way to solve this. http://www.math.hmc.edu/calculus/tutorials/int_by_parts/" [Broken] 3. The attempt at a solution I want to use integration by parts in order to solve this. I've attempted both combinations of u and v'. However, I'm not able to get an integral in any of my solutions that looks like the one above that is set equal to $$\sqrt{2\pi}$$. I have plugged the equation into Mathematica and the answer comes out to be 1. However, getting my work to back that up is proving to be difficult. Is integration by parts the right way to go about solving this one? Any help is greatly appreciated. Last edited by a moderator: May 3, 2017 2. Sep 30, 2008 The integrand is the density of the standard normal distribution, which is why Mathematica gave you the answer it did. You should try integration by parts; with a judicious choice of $$dv$$ you will find that the $$\int \, dv$$ portion (to calculate $$v$$) is easily done. After the step, and you have $$\int u \dv = uv - \int v \, du$$ you will see the reason for the hint. 3. Sep 30, 2008 ### HallsofIvy Staff Emeritus Yes, we know what integration by parts is! How about showing exactly what you did so we can point out any mistakes? Last edited by a moderator: May 3, 2017 4. Sep 30, 2008 ### UMich1344 I have attached my work with this post. I felt that I was heading in the right direction, but obviously I must have done something wrong near the beginning I would assume, since by the end I found that my integral diverged. File size: 16.8 KB Views: 61	2017-09-26 20:26:54	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7544890642166138, "perplexity": 377.6380390843903}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1505818696681.94/warc/CC-MAIN-20170926193955-20170926213955-00306.warc.gz"}
https://cstheory.stackexchange.com/questions/1823/event-scheduling	# Event scheduling. Given a set of actions (that are performed at a rate of one action per unit time) where each must be performed at an assigned frequency (assume the frequencies add up so things can work) and at approximately uniform intervals, derive a scheduling function who's execution uses as few a resources as possible (that is the cost of calling the function is to be minimized). An easy example: 1 @ ~1/2 2 @ ~1/4 3 @ ~1/8 1,2,1,3,1,2,1,1,2,1,3,1,2,1,2,... • The highest frequency and the lowest may differ by a factor of 100-1000. • The frequencies are approximately exponential distributed, but not exactly and with an arbitrary ratio (e.g. 1.12 or 1.373) • The function must generate an infinite sequence but may be repeating or non-repeating. • The function should get as close to exact as possible. • Real time constraints apply so • The function must have a deterministic runtime • The function must provide hard (tight) bounds on the intervals between values. minf = 10 base = 1.17724 frq = [int(minf * base*i) for i in range(36)] print sum(frq) # prints 20000 seen = dict((i,0) for i in frq) start = [(0,a) for a in frq] at = 1.0 for _ in range(20000): step = [((a-at)b, a, b) for (a, b) in start] step.sort() seen[step[0][2]] = seen[step[0][2]] + 1 step[0] = (0, at, step[0][2]) start = [(a,b) for (_,a,b) in step] at = at + 1 for (k,v) in seen.iteritems(): print "%5d %5d %g" % (k, v, v-k) output (skipping a few that are close to the right frequency): 2180 2130 -50 261 278 17 2566 2489 -77 136 148 12 307 325 18 819 834 15 160 174 14 1573 1552 -21 426 449 23 695 713 18 188 203 15 1852 1818 -34 3021 2898 -123 591 612 21 222 239 17 98 109 11 362 384 22 115 127 12 502 525 23 average error 8% • A trivial solution is that at each point of time, you simply generate an event that has been generated too rarely so far, breaking ties arbitrarily, but I guess you have already considered that. Perhaps you could explain what problems this approach has in your application? – Jukka Suomela Sep 30 '10 at 17:47 • @Jukka: Can you prove that it will give reasonable hard bounds on the intervals? To elaborate can you show it will never end up with everything wanting to run right now? – BCS Sep 30 '10 at 18:09 • Well, if everything wants to run right now, then you have managed to run everything exactly in the right proportions (if nobody is "ahead", then nobody is "behind", either). You can then reset your counters and repeat; if your algorithm is deterministic, you will produce the same sequence repeatedly. In the large scale, everything should be fine, but I guess you are worried of what might happen in the small scale. – Jukka Suomela Sep 30 '10 at 18:49 • @Jukka: See edit. – BCS Sep 30 '10 at 20:46 • I don't understand your problem statement. Do the jobs have lengths? What's your objective function? – Warren Schudy Sep 30 '10 at 22:33 Slightly editing your code, the "trivial" solution that I referred to in my comment would be something like this: minf = 10 base = 1.17724 frq = [int(minf * base*i) for i in range(36)] total = sum(frq) seen = dict((i,0) for i in frq) for at in range(total): step = [(c-atf/float(total), f, c) for (f, c) in seen.iteritems()] step.sort() chosenf = step[0][1] seen[chosenf] += 1 for (k,v) in seen.iteritems(): print "%5d %5d %g" % (k, v, v-k) (Prints zeroes, as expected.) • I guess I showed the wrong stats. It's not the count of occurrences I care about near as much as the interval between occurrence. The original code beast the modified code by about a factor of 2 in that regard. – BCS Sep 30 '10 at 21:54 • Exactly what would you like to minimise, subject to what constraints? I think you should try to come up with a mathematically precise formulation like "if $a_i$ is the longest interval between consecutive events of type $i$, and $b_i$ is the shortest interval, then I'd like to minimise the maximum of $a_i - b_i$ over all event types $i$". – Jukka Suomela Sep 30 '10 at 22:11	2020-08-12 03:41:43	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6131134629249573, "perplexity": 1510.5222559251874}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1596439738864.9/warc/CC-MAIN-20200812024530-20200812054530-00186.warc.gz"}
https://www.bartleby.com/solution-answer/chapter-159-problem-16e-calculus-early-transcendentals-8th-edition/9781285741550/use-the-given-transformation-to-evaluate-the-integral-16-r-4x-8y-da-where-r-is-the/1b25f6fd-52f4-11e9-8385-02ee952b546e	Chapter 15.9, Problem 16E ### Calculus: Early Transcendentals 8th Edition James Stewart ISBN: 9781285741550 Chapter Section ### Calculus: Early Transcendentals 8th Edition James Stewart ISBN: 9781285741550 Textbook Problem # Use the given transformation to evaluate the integral.16. ∫∫R (4x – 8y) dA, where R is the parallelogram with vertices (–1, 3), (1, –3), (3, -1), and (1, 5); x = 1 4 (u + v), y = 1 4 (v – 3u) To determine To evaluate: The integral R(4x+8y)dA. Explanation Given: The parallelogram region R with vertices (1,3),(1,3),(3,1) and (1,5) and x=14(u+v), y=14(v3u). Property used: Change of Variable Change of Variable in double integral is given by, Rf(x,y)dA=Sf(x(u,v),y(u,v))\|(x,y)(u,v)\|dudv (1) Calculation: Obtain the Jacobian, (x,y)(u,v)=\|xuxvyuyv\| Find the partial derivative of x and y with respect to u and v. x=14(u+v) then xu=14 and xv=14 and y=14(v3u) then yu=34 and yv=14. (x,y)(u,v)=\|14143414\|=14(14)(14)(34)=116+316=416 On further simplification find the value of Jacobian. (x,y)(u,v)=14 From the given integral the function is, 4x+8y and substitute the values of x and y. 4x+8y=414(u+v)+814(v3u)=u+v+2v6u=3v5u Find the boundary by using the given transformation. Find the line by using line equation xx1x2x1=yy1y2y1. A corresponding line for the vertices (1,3) and (1,3) is 3x+y=0. Substitute x and y in the given line, 3x+y=0314(u+v)+14(v3u)=034u+34v+14v34u=044v=0v=0 The corresponding line for the vertices (1,3) and (3,1) is xy=4. Substitute x and y in the given line, xy=414(u+v)14(v3u)=414u+14v14v+34u=444u=4u=4 A corresponding line for the vertices (3,1) and (1,5) is 3x+y=8 ### Still sussing out bartleby? Check out a sample textbook solution. See a sample solution #### The Solution to Your Study Problems Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees! Get Started	2019-10-19 23:02:07	{"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": 0.8130967617034912, "perplexity": 3382.06308625828}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986700435.69/warc/CC-MAIN-20191019214624-20191020002124-00446.warc.gz"}
http://forum.allaboutcircuits.com/threads/convolution-not-commutative-for-this-case.60299/	# Convolution not commutative for this case? Discussion in 'Homework Help' started by jp1390, Oct 6, 2011. 1. ### jp1390 Thread Starter Member Aug 22, 2011 45 2 Hi all, just needing some clarification on this question, or just another pair of eyes to check if I am doing this correctly. I am wanting to compute the convolution of v(t) and g(t). In doing so I wanted to try to test commutativity, but found that it wasn't working out. When I tried the first convolution, I yielded the correct result over the desired regions of overlap, but for the second, I did not. Method 1 $y(t) = g(t)\ast v(t) = \int_{-\infty}^{\infty}g(\tau)v(\tau-t)d\tau$ Method 2 $y(t) = v(t)\ast g(t) = \int_{-\infty}^{\infty}v(\tau)g(\tau-t)d\tau$ (did not receive correct result) Region 1 t <= 0; $y(t_{1}) = 0$ Region 2 t >= 0 and (t-2) <= 0 0 <= t <= 2; $y(t_{2}) = \int_{0}^{t}(exp{-\tau})(2exp{2(\tau -t)})d\tau$ $= 2exp{-2t}\int_{0}^{t}exp{\tau}d\tau$ $=2exp{-2t}(exp{t} - 1)$ The answer for this region should be: $y(t_{2}) = 2exp{-t}(1 - exp{-t})$ I got the correct result when I flipped and shifted v(t) as in Method 1, which is weird because they should be commutative. Can anyone see where I went wrong? Thanks, JP 2. ### Zazoo Member Jul 27, 2011 114 43 These two results are the same, they are just factored differently. i.e if you distribute the outside term both are equal to: $y(t) = 2exp{-t} - 2exp{-2t}$ 3. ### jp1390 Thread Starter Member Aug 22, 2011 45 2 Whoa, I must have had a major brain fart. Thanks haha	2016-12-05 13:07:39	{"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 11, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6583634614944458, "perplexity": 2009.7547272176748}, "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-2016-50/segments/1480698541696.67/warc/CC-MAIN-20161202170901-00250-ip-10-31-129-80.ec2.internal.warc.gz"}
https://translatewiki.net/w/i.php?title=Special%3ASearchTranslations&group=mediawiki&grouppath=mediawiki&query=language%3Aen%5E25+messageid%3A%22MediaWiki%3AAbusefilter-accountreserved%22%5E10+%22This+account+name+is+reserved+for+use+by+the+edit+filter.%22	67,006 results found Showing translations which match any of the search words. Require all search words. Languages Message groups This account name is reserved for use by the abuse filter. MediaWiki:Abusefilter-accountreserved/en This message is followed by the "Account name" input box. MediaWiki:Openidchoosemanual/qqq Mail text for the page "$1" for which you are assigned as a responsible editor has been {{GENDER:$2\|edited}} by $3. After logging into the wiki you can open the page with this link:$4 * $1 is the page which has been edited, $2 is the username of the user which made the edit - use for GENDER distinction, $3 is the real name of the user which made the edit $4 is the URL of the edited page MediaWiki:Bs-responsibleeditors-mail-text-re-article-changed/qqq Text message that this review was created by someone. $1 is the username - use for GENDER distinction * is the real name of the user MediaWiki:Bs-review-reviewed-till-extra/qqq This is an automatic edit summary for pages that were created by [[Special:ContentTranslation]]. Parameters: $1 - the source page name MediaWiki:Cx-publish-summary/qqq This namespace is reserved for content page translations. The page you are trying to edit does not seem to correspond any page marked for translation. MediaWiki:Tpt-unknown-page/en Use all values of this property for the filter MediaWiki:Sd createfilter usepropertyvalues/en Mail subject for the page "$1" has been {{GENDER:$2\|edited}} by $3 $1 is the page which has been edited, * $2 is the username of the user which made the edit - use for GENDER distinction, $3 is the real name of the user which made the edit MediaWiki:Bs-responsibleeditors-mail-subject-re-article-changed/qqq Placeholder for the URL/DOI/PMID search field. '''NOTE:''' Do not translate example.com as it is a domain name specifically reserved by IANA for examples. MediaWiki:Citoid-citefromiddialog-search-placeholder/qqq This title is reserved by a file on a remote shared repository. Choose another name. MediaWiki:Mwe-upwiz-error-title-fileexists-shared-forbidden/en If an edit summary contains something blocked by a Phalanx filter, this error message is then shown to the user. MediaWiki:Phalanx-content-spam-summary/qqq {{doc-right\|minoredit}} The right to use the "This is a minor edit" checkbox. See {{msg-mw\|minoredit}} for the message used for that checkbox. MediaWiki:Right-minoredit/qqq This message is for a log entry. Parameters: $1 user $3 link to the page, that the action that triggered the filter was made on $4 link to filter $5 action by user, like 'edit', 'move', 'create' etc. $6 actions taken by the filter $7 action details link MediaWiki:Logentry-abusefilter-hit/qqq This message appears when clicking on the fourth button of the edit toolbar. You can translate "link title". Because many of the localisations had urls that went to domains reserved for advertising, it is recommended that the link is left as-is. All customised links were replaced with the standard one, that is reserved in the standard and will never have ads or something. This consumer is for use only by $1. 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MediaWiki:Echo-pref-subscription-edit-thank/qqq It is not possible to associate further OpenIDs to your account, because you can only use the ''"forced"'' OpenID provider "\$1" for logins on this wiki. MediaWiki:Openid-error-openid-convert-not-allowed-forced-provider/en # This is a title whitelist. Use "#" for comments. # This is case insensitive by default MediaWiki:Titlewhitelist/si # This is a title whitelist. Use "#" for comments. # This is case insensitive by default MediaWiki:Titlewhitelist/en	2016-02-10 10:43:50	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.33920490741729736, "perplexity": 4366.03291701748}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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-20160205193919-00159-ip-10-236-182-209.ec2.internal.warc.gz"}
https://research.aalto.fi/fi/publications/alphabet-dependent-bounds-for-linear-locally-repairable-codes-bas	Alphabet-Dependent Bounds for Linear Locally Repairable Codes Based on Residual Codes Matthias Grezet, Ragnar Freij-Hollanti, Thomas Westerbäck, Camilla Hollanti Tutkimustuotos: LehtiartikkeliArticleScientificvertaisarvioitu 97 Lataukset (Pure) Abstrakti Locally repairable codes (LRCs) have gained significant interest for the design of large distributed storage systems as they allow a small number of erased nodes to be recovered by accessing only a few others. Several works have thus been carried out to understand the optimal rate–distance tradeoff, but only recently the size of the alphabet has been taken into account. In this paper, a novel definition of locality is proposed to keep track of the precise number of nodes required for a local repair when the repair sets do not yield MDS codes. Then, a new alphabet-dependent bound is derived, which applies both to the new definition and the initial definition of locality. The new bound is based on consecutive residual codes and intrinsically uses the Griesmer bound. A special case of the bound yields both the extension of the Cadambe–Mazumdar bound and the Singleton-type bound for codes with locality $(r,\delta)$ , implying that the new bound is at least as good as these bounds. Furthermore, an upper bound on the asymptotic rate–distance tradeoff of LRCs is derived, and yields the tightest known upper bound for large relative minimum distances. Achievability results are also provided by deriving the locality of the family of Simplex codes together with a few examples of optimal codes. Alkuperäiskieli Englanti 8700214 6089-6100 12 IEEE Transactions on Information Theory 65 10 https://doi.org/10.1109/TIT.2019.2911595 Julkaistu - 2019 A1 Julkaistu artikkeli, soviteltu Sormenjälki Sukella tutkimusaiheisiin 'Alphabet-Dependent Bounds for Linear Locally Repairable Codes Based on Residual Codes'. Ne muodostavat yhdessä ainutlaatuisen sormenjäljen.	2021-08-01 21:24:23	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.6742778420448303, "perplexity": 732.0025892607232}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154219.62/warc/CC-MAIN-20210801190212-20210801220212-00130.warc.gz"}
http://www.physicsforums.com/showthread.php?s=345ce3f524e1ea6934014675ba084831&p=4594253	# Exact diagonalization by Bogoliubov transformation P: 16 Hello all, I am developing a model of multiple gaps in a square lattice. I simplified the associated Hamiltonian to make it quadratic. In this approximation it is given by, $$H = \begin{pmatrix} \xi_\mathbf{k} & -\sigma U_1 & -U_2 & -U_2\\ -\sigma U_1 & \xi_{\mathbf{k}+(\pi,\pi)} & 0 & 0\\ - U_2 & 0 & \xi_{\mathbf{k}+(\pi/2,0)} & 0\\ - U_2 & 0 & 0 & \xi_{\mathbf{k}+(0,\pi/2)} \end{pmatrix}$$ And my Nambu operator is given by, $$ψ_\mathbf{k} = \begin{pmatrix} c_{\mathbf{k},\sigma} \\ c_{\mathbf{k}+(\pi,\pi),\sigma} \\ c_{\mathbf{k}+(\pi/2,0),\sigma} \\ c_{\mathbf{k}+(0,\pi/2),\sigma} \end{pmatrix}$$ I tried to diagonalized by making three Bogoliubov transformations, the first to diagonalize the upper right submatrix of H, and then the two others (a sort of nested transformations). But I get a lengthy result, what I would like to know if there is a smart transformation which allows me to write $$H = A_1^\dagger A_2^\dagger A_3^\dagger D A_3 A_2 A_1$$ or simply $$H = U^\dagger D U$$ Or the only way is to use just brute force? Thanks	2014-09-02 11:47:57	{"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": 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": 0.6003203392028809, "perplexity": 270.46510321935114}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1409535921957.9/warc/CC-MAIN-20140901014521-00192-ip-10-180-136-8.ec2.internal.warc.gz"}
https://crypto.stackexchange.com/questions/80352/what-is-the-best-algorithm-for-compressing-a-hash	# What is the best algorithm for compressing a hash? What is the best way to transform a hash with a longer length into one with a smaller length, preventing as many collisions as possible? (Hashing the hash) For example: Some versions of Git use SHA-1 for hashing commits. Of course, 15ce7ff90976b3e43738be403f5c985377646bb3 is rather large to display on a screen. Because of that, Github usually only shows the most significant 4 bytes (15ce7ff9) to refer to a specific commit. But is this the best strategy or is there a better way? If SHA-256 was used instead of SHA-1, would the resulting "minified" hash be as secure and as colision-avoiding as possible within 4 bytes, regardless of the "minification" algorithm used? • That may be enough for the git. This is completely opinion/requirement based. Git --fast version control – kelalaka Apr 29 '20 at 19:00 • I agree with @Kelalaka, for Git maybe is enough the firts 4 bytes. If you use SHA-256 the colision is minior compare with SHA-1, SHA256, is currently much more resistant to collision attacks as it is able to generate a longer hash which is harder to break. – AndresMontj Apr 29 '20 at 19:43 • 4-byte has $2^{16}$ colliision resistance. The Git is not using it really for Cryptographical usage, they use it for integrity checking and look up tables. – kelalaka Apr 29 '20 at 19:59 • I know, I know, git commit hashes are not a security feature. I just used Git as an example. – D. Pardal Apr 29 '20 at 20:26 • That is why this is not a really good question. And the best is already make it unclear what you are asking. – kelalaka Apr 29 '20 at 20:32 What's better? Displaying more information than 4 bytes will be more secure mathematically speaking. However, it is unlikely that people will check all those bytes. Those bytes are more for identification than for authentication / security anyway. You could use a different alphabet than hexadecimals, say base 64, but I'd argue that that would be harder to remember, especially if more than 4 bytes are used. For the 4 bytes it doesn't really matter. They only represent somewhat over 4 billion possibilities (using the short scale) and generating the same 4 bytes will be easy whatever hash function is used. The switch from SHA-1 to SHA-256 is about the full hash used internally. The switch to SHA-2 is important in my opinion. Or it is at least important enough to make the switch. I haven't been swayed by Linus arguments yet, but that's mainly because I don't have a full overview of the vulnerabilities. However, I haven't seen a good argument why there wouldn't be an attack possible either. And the difference in speed really isn't worth to taking any risk. A note about the wording: taking the first 4 bytes from a hash is not compression. Compression (as in the common meaning, such as the DEFLATE used by zip archives) is performed over all the bits, and tries to pertain some essential (or for lossless compression all of the) data. As explained in the comments below, compression of the output of a hash is an act of futility. In cryptography compression has a different meaning. But the output must also depend on all the input bits. Anyway, just removing the rightmost bytes from a hash is not the same thing. • There is a practical and useful thing to do while keeping 8 hex characters: re-hash with entropy stretching. – fgrieu Apr 29 '20 at 20:23 • Interesting. But I guess you would not do that manually. Or is that actually performed in Git? – Maarten Bodewes Apr 29 '20 at 21:29 • I guess that would need to be performed by GIT when displaying, with a cache of SHA-256 to entropy-stretched rehash in order to avoid prohibitive CPU load. – fgrieu Apr 29 '20 at 21:42 • Sorry, @MaartenBodewes, but compression can be made over an adequate block size, which depends on the size of typical set (AEP - Asymptotic Equipartition Property.) We expect that a good hash function output doesn't "concentrate"... So it will be necessary a code of size $n$ to compress a $n$-bit hash "text": by AEP, the size of the typical set is $\sim 2^{nH(X)}$. As we dont know $H(X)$ from a $p(x_0,...,x_n)$ hash output, we must believe that $H(X) \sim 1$. So, there is no room to compression over a good hash function output. – McFly Apr 29 '20 at 23:41 • @McFly That's a perfect explanation, but I was talking about the wording in the question. I never said you can compress the output of a hash. I just explained what compression means in the generic sense (compared with taking just 4 bytes of anything). – Maarten Bodewes Apr 29 '20 at 23:45 What is the best way to transform a hash with a longer length into one with a smaller length, preventing as many collisions as possible? TLDR: Decide if you want to resist preimage or collision; the later is hard and requires a better main hash than SHA-1. Re-hash the main hash with a purposely slow hash as used in passord-based hashes, and encode the outcome (truncated) using a binary-to-text conversion denser than hex. One must be careful about the goal: is it to avoid collision (the word stated in the question), or to avoid preimage (as perhaps is thought)? In preimage, adversaries try to come up with a message (or file content) having a certain hash (or compressed hash). They are initially given: • in first preimage, the target hash. • in second preimage, a message with that hash (and they must come up with a different message). That could be because they plan to change an existing message (that they did not had the freedom to define) into something else, without changing the hash. In collision, adversaries try to make two messages having the same hash, but are not constrained about that value. That could be because they plan to submit one of the messages, and change it to the other at a later time. To reach $$b$$-bit security (that is, $$\mathcal O(2^b)$$ work for an attack), we asymptotically need a $$b$$-bit hash to resist preimage, and a $$2b$$-bit hash to resist collision¹. Thus the method of displaying the 8-character hex string coding the first 32 bits of the hash provides 32-bit resistance to preimage, which is mere minutes of computation, and only 16-bit resistance to collision, which is no resistance. If the initial hash is SHA-1, there's limited hope with regards to collision, since it is known how to make SHA-1 collisions (trivially by reusing the prefix revealed by the shattered attack, or by repeating their attack). Sure, there are ways to detect messages crafted to allow shattered copycats, but I would not bet on their resistance to a clever variation. With a better main hash such as SHA-256 or SHA-512, or if we only care about preventing preimage attacks, there are two ways to improve on this: • Re-hash that main hash using a slow hash, then truncate the result. This is key stretching, as used in password hashing. Example slow hashes are Argon2 and Scrypt (modern and greatly improved replacements for the obsolete Bcrypt and PBKDF2). Use with some public salt (if possible message-dependent, e.g. a file name). There are parameters making it easy to control the CPU time and RAM per hash, e.g. $$0.1$$ second, 10MB RAM. With the same final truncation to 8 hex characters, an attack now requires $$0.693\times2^{32}\times0.1$$ CPU⋅seconds ($$>9.6$$ CPU⋅year) to be broken for preimage, or $$\approx1.177\times2^{16}\times0.1$$ CPU⋅seconds ($$>2\text{h }08\text{'}$$ on a single CPU) to be broken for collision, with 50% probability. • Encode more bits per character in the compressed hash. Hex encodes 4 bits per character, base64 encodes 6, by pushing ASCII to its limits we can get to 6.55, using the resources of Unicode we could go to maybe 8 to 12 while keeping characters visually distinguishable (depending on culture of the audience). These methods can be combined. With 8 characters restricted to 10 digits, 13 symbols ! # \$ % & * + < > ? @ ^ _, and uppercase/lowercase letters less the 11 A E I O U a e i l o u (in order to avoid a large proportion of possibly embarrassing English words, and as an aside confusion with digits 0 1), we get to $$10+13+2*26-11=64$$ characters, thus 48 bits, thus >63,000 CPU.years to break preimage with 0.01s per re-hash and 50% probability of success. Caution: unless there's a message-dependent salt (such as a file name, which complicates verification), adversaries need less work by a factor about $$k$$ in order to break preimage for one in $$k$$ rehashes. That's an issue if adversaries are happy to replace one message among $$k$$, even though they do not control which message will be replaced when they prepare the replacement. The 0.01s per entropy-stretched re-hash would still be sizable work in a GIT context. At the very least, the server would have to maintain a cache of re-hashes in order to conserve CPU time/energy. An important Information Theory tool to build a compression algorithm is AEP - Asymptotic Equipartition Property, which gives us the idea of typical sequences: roughly speaking, the most probable sequence $$x_1,...,x_n$$, with $$X_1,...,X_n \sim p(x)$$. Compression considers the size of the set of typical sequences. The idea is to "concentrate the fire" coding the words in this set, and, therefore, reduce the size of bit representation of the original input: the smaller code words to represent the most frequent typical sequences. This is the general idea in ZIP, Lempel-Ziv algorithms. The AEP tell us that the size of the typical set is $$\sim 2^{nH(x)}$$ (here $$H(\cdot)$$ is the Shannon Entropy). So, in the specific case of a "good" hash function, we don't know about $$p(x_1,...,x_n)$$ or about $$p(x)$$, and a good guess can be to consider $$p(0)=p(1)=\frac{1}{2}$$, and, therefore, work with $$H(x) \sim 1$$, maximum entropy to a binary source. Considering a "good" hash function, the size of the set of typical sequences will be $$\sim 2^n$$, that is, almost the complete set of $$n$$-bit words. Therefore, there is no room for compression over a hash function output. BTW, hash functions outputs are usually short. I'm not sure about the advantage of compressing, e.g., 256 bits, because we need metadata: to encode the typical sequences as a dictionary. • This interesting answer is taking "compressing" from the question's title in the sense that has in lossless data compression. In the question, "compression" is used for a summarization in less characters, aiming at keeping most of the initial intent of the thing. As explained in the questions first paragraph, here the intent is preventing collisions or preimages. And that can be done to a large degree. – fgrieu Apr 30 '20 at 13:21 • @fgrieu, I see. My concern was not only the "compressing" in the title, but also about the "(Hashing the hash)", but you're absolutely right: that is not the point. Anyway, I was also trying to give arguments to defend that 4 bytes don't give us a negligible probability of collision, but with a good hash function that can be useful for a visual versioning control. So, so I think that the Git strategy is interesting, so is the one in your answer! – McFly Apr 30 '20 at 15:40	2021-02-25 16:41:37	{"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": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 23, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.38740190863609314, "perplexity": 1227.123293754631}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178351374.10/warc/CC-MAIN-20210225153633-20210225183633-00341.warc.gz"}
https://tex.stackexchange.com/questions/312667/apply-binding-correction-to-titlepage	# apply binding correction to titlepage i am using the arsclassica package, documentclass scrreprt with the options oneside and BCOR10mm. however, the binding correction does not apply to the titlepage as can be seen in the package example. is there a way to make that happen? • Please post a compilable minimal working example. Please also load package showframe and use extreme values for BCOR. Additionally, are you using package geometry? An MWE will show. – Johannes_B Jun 2 '16 at 8:14 • One additional thing: BCOR10mm is deprecated. Make it BCOR=10mm or better BCOR=50mm to really see a difference. – Johannes_B Jun 2 '16 at 8:19 • a working example is directly given by the link above, it is compilable expect for the bibliography and index. i fear something is going to get lost when i create my own working example :/ the geometry package is not loaded. – user92856 Jun 2 '16 at 8:27 The following example shows that binding correction also applies to the titlepage. \documentclass[ oneside, footinclude, titlepage, BCOR=30mm ]{scrreprt} \usepackage[pdfspacing]{classicthesis} \usepackage{arsclassica} \usepackage{blindtext}% dummy text \usepackage{showframe}% to show the page layout \begin{document} \title{Title} \author{Author} \maketitle \blinddocument \end{document} Result: With BCOR=0mm the result would be: • thank you very much. then the behaviour i encountered may be due to the explicit titlepage template from the arsclassica package :/ – user92856 Jun 2 '16 at 9:32 • As @Johannes_B already has said: Post a MWE that shows the issue. The link in your question is only the CTAN entry of the package. There is neither an example directly available nor an explicit titlepage template. – esdd Jun 2 '16 at 9:44 • oh, i'm sorry, i will be more precise the next time. – user92856 Jun 2 '16 at 9:48 as pointed out above, the problem is due to the titlepage template in the arsclassica package. commenting out the line \changetext{}{}{}{((\paperwidth - \textwidth) / 2) - \oddsidemargin - \hoffset - 1in}{} finally did it.	2019-12-13 02:03:08	{"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": 0.722114086151123, "perplexity": 2909.697694755727}, "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-2019-51/segments/1575540548537.21/warc/CC-MAIN-20191213020114-20191213044114-00235.warc.gz"}
https://www.physicsforums.com/threads/derham-cohomology.20863/	# DeRham cohomology 1. Apr 15, 2004 ### lethe I am pretty familiar with deRham cohomology. for me, deRham cohomology is synonymous with cohomology in general. but I often run into that cocycle condition, you know, where c(g,h)+c(gh,k)=c(g,hk)+c(h,k) you need to look at conditions of this type when deciding whether a projective representation of a group can straightened into a linear rep, or a rep of a central extension. I know this is a question of cohomology, and the language (cocycle) is very suggestive of stuff I know from deRham cohomology, but beyond that, I don't know much. so does someone want to give me a 50 cent sketch of the details? what is the "d" operator in this cohomology? does the group have to be a manifold? there is such a thing as cohomology of discrete groups, isn't there? and Lie algebras? is there a cup product? what is the name of the cohomology where this cocycle condition lives? Last edited: Apr 20, 2004 2. Apr 16, 2004 ### matt grime De Rham cohomolgy is one of the more esoteric cohomologies and certainly isn't synonymous with cohomology in my mind, but then I'm biased. It uses real coeffs which make it easy to work out. The truly hard case stuff uses Z. In order to have a cohomology theory one needs a cochain complex, that is a set of objects X* indexed by Z, with maps indexed by Z, denoted d, the differential, where d^n :X^n---->X^{n+1} (it increases the degree) and the composition of two consecutive differentials is zero The cohomology groups are then H indexed by Z with H^n = kerd^n/Imd^{n-1} So what are these things for when the underlying object is a group? Well, there are several ways of doing this, one is to take the geometric version, which I don't understand. The other is to take the trivial module, and take a projective resolution of it, or simpler, a free resolution: give me some module, and I can find a free module surjecting onto it. I take the kernel of that map and I can get a surjection from a free module onto that and so on, so we get: ....--->F_2---->F_1---->F_0---->T T the trivial module Then if we apply Hom( T,?) to this complex we get another complex, the cohomology of which is callled the cohomology of G. Of course I'm missing some details - what kinds of modules and so on. Lots of people want ZG modules and triv is then Z, I would tend to use some algebrtaically closed field. Not of char zero because then the cohomology is trivial. Moreover, we don't have to limit to using triv, we can resolve any module Y and take Hom(X,?) with X any module too, and the results are teh Ext groups - the extensions of one by the other. The cohomological information nowadays is studied as the derived category, and the ext groups measure the failure of Hom(X,?) to be an exact functor - if X is projective the functor is exact and the Ext groups vanish. So that's my 50 cent on group cohomology. There is a lot more than that. Try cyclic homology, sheaf cohomology, simplicial, elliptic... they all have this complex and differential though. 3. Apr 16, 2004 Staff Emeritus Lethe, is the product in your cocycle condition the native cup product of cohomology or some other product? Are the cocycles from the same spaces or different ones (for example base, fiber and top space of a fibration)? There are a lot of techniques in cohomology but the underlying topology determines what you use. 4. Apr 16, 2004 ### lethe i think maybe in physics, de Rham is very common. right. i think this is a paraphrase of the Steenrod axioms, right? trivial module... is that a module made up of just a 0? i don't know what a resolution is. is what follows the definition? where does my group fit in all this? where does the cocycle condition come from in all this? 5. Apr 16, 2004 ### lethe its just addition of real numbers. (note: i had the formula wrong last night. i have updated it) they are just real numbers. the arguments of those functions are group elements, which underlying topology are you talking about? i think cohomology of groups is purely algabraic. what does topology have to do with it? the cocycle condition is often used in physics to decide when a group admits projective representations. this is important in quantum theory, and i would just like to know the math underlying this stuff 6. Apr 16, 2004 ### matt grime The trivial module for the group ring RG is the one where all elements of G ect by 1. Judging by your questions I think I would need to go and discuss a lot of basics which arent' really necessary, as we'll hopefully see. A resolution of a module is an acyclic chain complex of projective modules, with the exception of the degree -1 term where it is the module you start with. The group comes into it because these are all modules for the group. The free module is RG, and using direct sums of RG it is easy to write down a resolution (And any two will produce the same answer) but it grows exponentially in dimension. I think I needed Hom(?,X) for a contravariant functor to make a cochain complex as it happens but you didnt' pick up on that one. What I gave isn't a paraphrase of the steenrod rod axioms, it is the definition of the cohomology groups of a cochain complex (when it makes sense to talk of these things). It satisfies the steenrod axioms when we use the case of cochain complexes of group modules and maps in its homotopy category. If you want to interpet low degree group homology then you get the kind of results you were talking about. Calculating any of these things is very difficult, even if it can be done at all. Cyclic groups can have their cohomology calculated. There are ways of calculating group cohomolgy using topological spaces too. 7. Apr 19, 2004 ### lethe too many things i don't know. it sounds to me like i simply don't have enough training in algebra to learn chomology of groups. ho hum 8. Apr 19, 2004 ### matt grime I don't know about that. It's just that these are things no one knows enough about to be able to teach courses on them at a basic level. We just can't calculate these things. There is an alternative method using simplicial complexes (or do I mean singular) where the complexes/cells are treated as free modules for the group ring. Do you know what an exact sequence is? There is another interpretation of the n'th cohomology group ext(M,N) as the set of isomorphism classes of exact sequences with n+2 entries. 9. Apr 19, 2004 Staff Emeritus In simplicial cohomology the basic units are elementary simplices of a triangulation of the manifold. Singular cohomolgy is based on continuous maps of spheres into the manifold. Where all these diffenernt cohomologies can be defined together, they coincide. Evaluation of expressions is indeed based on exact sequences, and also on known cohomology operations like the Steenrod operations. Outside of the classic manifolds (spheres, tori) it's like pulling teeth and every little advance is greeted with joy. Nash and Sen's fairly straightforward book or Nakahara's harder and comprehensive one might help you. It would be equivalent to a semester course. Nash & Sen, Topology and Geometry for Physicists, Academic Press. Nakahara, Geometry, Topology, and Physics, Institute of Physics Publishing, Graduate Student Series in Physics. 10. Apr 19, 2004 ### lethe simplicial complexes can be used to make homology groups. i am familiar with this. i did not know that you could make cohomology groups this way. as i said before, the only cohomology groups that i know are the de Rham cohomology groups. these are the groups of closed forms mod exact forms on a smooth manifold. well, the only homology groups i know are the simplicial homology groups. the group of boundaryless chains mod boundary chains. i wanted to learn more about projective reps and group cohomology and where that cocycle condition fits in. I have been looking at a book on homological algebra by Rotman, but the book is probably meant for algebraists and for me, the learning curve for that book is very steep, and so i was hoping to get something more dumbed down on the forum. but your explanations are also hard for me. i know what an exact sequence is. i don't know what ext(M,N) means though. is it a notation for exact sequences? 11. Apr 19, 2004 ### lethe i am pretty sure you mean simplicial homology are you addressing this comment to me? this seems rather strange. how would those books help me learn group cohomology? I have read both of those books, and i can assure you, they don't breath a word on the subject. they both have a nonrigorous colloquial and incomplete introduction to de Rham cohomology, simplicial homology, and the homotopy groups Map(S^n,X). 12. Apr 19, 2004 ### matt grime homology an cohomology are essentially the same thing, the only difference is that in homology the d sends degree n to n-1 and in cohomology it sends degree n to n+1. If the complexes are unbounded in both directions this is the same, otherwise it is different. See Ext versus Tor. Rotman is fairly basic for a topologist. It doesn't treat group cohomology very well at all. But then no book does really, not unless you're at least a 3rd year postgraduate. It is not an easy topic, there is not an easy way to deal with it. Sorry. It takes at least two years to even be happy with what projective means in my opinion, and then to use it. I would suggest readin Alperin's Local Representation Theory as background (omitting the last two chapters) before reading Benson's books on group cohomology. For a general overview of homological algebra then try Wiebel's book on it. Remember that any chain complex is turned into a cochain complex by applying a contravariant functor. Last edited: Apr 19, 2004 13. Apr 20, 2004 ### lethe yeah, and they are dual to one another. or at least, simplicial homology (or is it singular? did we decide what the difference is between those two?) is dual to de Rham cohomology. well, i saw Rotman cited as a reference in the bibliography of a physics book for exactly this topic. what does projective mean, anyway? here is what i think about a projective representation: it is a representation of a central extension of a group. there is another usage of projective that i am familiar with, in the context of the projective spaces RP^n, CP^n and HP^n. these are, respectively, the space of rays in R^n, C^n, and H^n. more generally, you can consider the projective space of rays through any vector space. the two notions are related by the fact that a projective representation of a group on a vector space is the same as a representation on a projective vector space. also, with regards to projective linear groups, like PGL_n. i understand PGL_n as the space of lines through the origin in the vector space M_n (nxn matrices) intersect GL_n a colleague of mine in the math department said that a definition of the projective group associated to any group G is G/Z(G). i have never seen this definition in the books, but it agrees with my definition of PGL_n, and others. do you know this definition for a projective group? but i don't know what it means for a module to be projective. thanks for the references. i will take a look when i get a chance what is the point of this statement? to show me how a homology group is related to a cohomolgy group? so what? 14. Apr 20, 2004 ### lethe i have found a nice book: "Lie groups, Lie algebras, cohomolgoy and some applications in physics" by Azcárraga and Izquierdo it is a mathematical physics book, i think. that is, it is a book on mathematical topics but mostly motivated by physical applications. so, closer to my needs than a pure math book, probably. they define group cohomology groups in chapter 5 given a group G, an abelian group A, and a realization $\sigma$ of G on the group of (outer) automorphisms of A (i.e. a homomorphism from G --> Aut(A)) we have n-cochains $C^n(G,A)$ the set of mappings $\alpha_n\colon G\times\dotsb\times G\rightarrow A$ $$\alpha_n\colon (g_1,\dotsc,g_n)\mapsto \alpha_n(g_1,\dotsc,g_n)\inA$$ and the coboundary operator $\delta_n\colon C^n\rightarrow C^{n+1}$ $$\begin{multline} (\delta_n\alpha_n)(g_1,\dotsc,g_n,g_{n+1})=\sigma(g_1)\alpha_n(g_2,\dotsc,g_n,g_{n+1}) \\+\sum_{i=1}^n(-1)^i\alpha_n(g_1,\dotsc,g_{i-1},g_ig_{i+1},g_{i+2},\dotsc,g_{n+1}) \\+(-1)^{n+1}\alpha_n(g_1,\dotsc,g_n) \end{multline}$$ this operator satisfies the usual $\delta_n\circ\delta_{n-1}=0$ and so we can define the cohomology groups $$H^n_\sigma(G,A)=\operatorname{Ker}\delta_n/\operatorname{Im}\delta_{n-1}$$ then the zeroth cohomology group is the subgroup of A of fixed invariant points under the action of G. the first cohomology group is the group of crossed homomorphisms from G to A mod by principle homomorphisms (what the **** is a crossed homomorphism?) the second cohomology group is the one we use in physics. it characterizes the way that we can centrally extend G by A. in other words, it tells us the possible projective representations of G the third cohomology group tells us the possible nonassociative representations. this seems to be the sort of stuff that i was looking for. this book is a little difficult, but more or less aimed at a guy at my level. i don't understand everything, but so far, i am learning something from this book. Last edited: Apr 20, 2004 15. Apr 20, 2004 ### lethe one interesting thing about these cohomology groups i notice is that they take their coefficients in A (not in the real line). matt grime said before that the fact that de Rham cohomology groups are over the reals makes them easier. now, how about the cup product? i didn't see any mention of this topic in the book. my understanding is that any cohomology group should have a cup product. is there one here? Last edited: Apr 20, 2004 16. Apr 20, 2004 ### lethe one of the upshots of the chapter on group cohomology is the result that nonrelativistic quantum mechanics of a neutral spin-0 particle has to use a complex Hilbert space, whereas relativistic quantum mechanics of a neutral spin-0 particle can use a real valued Hilbert space. 17. Apr 20, 2004 ### matt grime 1) Projective means a summand of a free module, or that Hom(P,?) is an exact functor. Over a field of char zero everything thing is projective, so the theory there is trivial. 2) cannot for the life of me remember why i made that parting comment, seems like one of my better non sequiturs. 3) The resolution in the book is called the free resolution. It uses G's (well, kG's tensored really) and it it semi-useful because you can easily say what it is, but the resolution grows exponentially so it's impractical. 4) There is a cup product. You can interpret the cohomology as 'n fold extensions of k by k' where k means the trivial module of the group and the n fold extenions means an exact sequence of modules starting and ending with k and with n terms in between. Thus H^1(G,k) is the set of short exact sequences starting and ending in k. It also has the other interpretations you mention you've found. If you use this idea you can splice together these exact sequences into longer ones. This is the Yoneda splice. You may also realize it as the total tensor product of the two exact sequences. 18. Apr 20, 2004 ### lethe hmm... that definition seems opaque to me. can you make contact between this usage of the word "projective" and any of the definitions i gave above? 19. Apr 20, 2004 ### matt grime A module P is projective if any short exact sequence 0-->X-->Y--->P--->0 is split exact. Ie there are no non-trivial extensions, or H^1 is zero. it is a quite opaque definition, but they are the cohomologically trivial objects, since the projective resolution is just non zero in one degree only. they do take a while to get used to. The main thing is that things which are true for free modules tend to be true fo projective modules as they are summands of free modules. They have important lifting properties that ensure eveything you want to be true about cohomology being independent of a choice of resolution is true: if P* is a complex of projectives with homology in degree zero only where it is X say, and Q* is the same, then P* and Q* are homotopic. 20. Apr 20, 2004 ### lethe I am still chewing on things. a few things, firstly, why doesn't the cohomology group that i showed above have a name? is its name simply the group cohomology group? that sounds stupid. also in what sense does a cohomology group over a char zero become trivial? i mean, de Rham cohomology groups contain nontrivial topological information about the smooth manifolds they are defined on.	2018-08-16 20:01:19	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8089531660079956, "perplexity": 547.3025020575179}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1534221211167.1/warc/CC-MAIN-20180816191550-20180816211550-00514.warc.gz"}
http://git.kamailio.org/gitlist/index.php/kamailio/logpatch/288e2739da28251e12086b52358c3a0d18e91fa5/atomic_ops_init.h	#### core, lib, modules: restructured source code tree - new folder src/ to hold the source code for main project applications - main.c is in src/ - all core files are subfolder are in src/core/ - modules are in src/modules/ - libs are in src/lib/ - application Makefiles are in src/ - application binary is built in src/ (src/kamailio) Daniel-Constantin Mierla authored on 07/12/2016 11:03:51 Showing 1 changed files 1 1 deleted file mode 100644 ... ... @@ -1,42 +0,0 @@ 1 -/* 2 - * Copyright (C) 2006 iptelorg GmbH 3 - * 4 - * This file is part of Kamailio, a free SIP server. 5 - * 6 - * Kamailio is free software; you can redistribute it and/or modify 7 - * it under the terms of the GNU General Public License as published by 8 - * the Free Software Foundation; either version 2 of the License, or 9 - * (at your option) any later version 10 - * 11 - * Kamailio is distributed in the hope that it will be useful, 12 - * but WITHOUT ANY WARRANTY; without even the implied warranty of 13 - * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 14 - * GNU General Public License for more details. 15 - * 16 - * You should have received a copy of the GNU General Public License 17 - * along with this program; if not, write to the Free Software 18 - * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA 19 - / 20 - 21 -/! 22 - * \file 23 - * \brief Kamailio core :: atomic_ops init functions 24 - * 25 - * Copyright (C) 2006 iptelorg GmbH 26 - * 27 - * \ingroup core 28 - * Module: \ref core 29 - * 30 - * Needed for lock intializing if no native asm locks are available 31 - * for the current arch./compiler combination, see \ref atomic_ops.c 32 - / 33 - 34 -#ifndef __atomic_ops_init_h 35 -#define __atomic_ops_init_h 36 - 37 -/! \brief init atomic ops / 38 -int init_atomic_ops(void); 39 -/! \brief destroy atomic ops (e.g. frees the locks, if locks are used) / 40 -void destroy_atomic_ops(void); 41 - 42 -#endif #### core : Update include files - delete IDs, update doxygen, delete history Olle E. Johansson authored on 03/01/2015 10:55:48 Showing 1 changed files ... ... @@ -1,16 +1,14 @@ 1 1 / 2 - * $Id$ 3 - * 4 2 * Copyright (C) 2006 iptelorg GmbH 5 3 * 6 - * This file is part of ser, a free SIP server. 4 + * This file is part of Kamailio, a free SIP server. 7 5 * 8 - * ser is free software; you can redistribute it and/or modify 6 + * Kamailio is free software; you can redistribute it and/or modify 9 7 * it under the terms of the GNU General Public License as published by 10 8 * the Free Software Foundation; either version 2 of the License, or 11 9 * (at your option) any later version 12 10 * 13 - * ser is distributed in the hope that it will be useful, 11 + * Kamailio is distributed in the hope that it will be useful, 14 12 * but WITHOUT ANY WARRANTY; without even the implied warranty of 15 13 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 16 14 * GNU General Public License for more details. ... ... @@ -22,7 +20,9 @@ 22 20 23 21 /! 24 22 \file 25 - * \brief SIP-router core :: atomic_ops init functions 23 + * \brief Kamailio core :: atomic_ops init functions 24 + * 25 + * Copyright (C) 2006 iptelorg GmbH 26 26 * 27 27 * \ingroup core 28 28 * Module: \ref core ... ... @@ -30,11 +30,6 @@ 30 30 * Needed for lock intializing if no native asm locks are available 31 31 * for the current arch./compiler combination, see \ref atomic_ops.c 32 32 / 33 -/ 34 - * History: 35 - * -------- 36 - * 2006-03-30 created by andrei 37 - / 38 33 39 34 #ifndef __atomic_ops_init_h 40 35 #define __atomic_ops_init_h #### all: updated FSF address in GPL text Anthony Messina authored on 04/07/2014 09:36:37 • Daniel-Constantin Mierla committed on 04/07/2014 09:37:36 Showing 1 changed files ... ... @@ -17,7 +17,7 @@ 17 17 18 18 * You should have received a copy of the GNU General Public License 19 19 * along with this program; if not, write to the Free Software 20 - * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA 20 + * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA 21 21 / 22 22 23 23 /! #### core: Added void to declarations of functions with empty argument list In C language, a declaration in the form int f(); is equivalent to int f(...);, thus being able to accept an indefinit number of parameters. With the -Wstrict-prototypes GCC options, these declarations are reported as "function declaration isn’t a prototype". On some cases, this may trick the compiler into generating unoptimized code (like preparing to handle variadic argument list). In all cases having a declaration int f() and a definition inf f(int) is missleading, even if standard compliant. This is still Work in Progress. (maybe adding the -Wstrict-prototypes option to default is desireable) Marius Zbihlei authored on 02/04/2012 14:19:17 Showing 1 changed files ... ... @@ -40,8 +40,8 @@ 40 40 #define __atomic_ops_init_h 41 41 42 42 /! \brief init atomic ops / 43 -int init_atomic_ops(); 43 +int init_atomic_ops(void); 44 44 /! \brief destroy atomic ops (e.g. frees the locks, if locks are used) / 45 -void destroy_atomic_ops(); 45 +void destroy_atomic_ops(void); 46 46 47 47 #endif oej authored on 25/10/2009 19:11:28 Showing 1 changed files ... ... @@ -10,11 +10,6 @@ 10 10 * the Free Software Foundation; either version 2 of the License, or 11 11 * (at your option) any later version 12 12 * 13 - * For a license to use the ser software under conditions 14 - * other than those described here, or to purchase support for this 15 - * software, please contact iptel.org by e-mail at the following addresses: 16 - * info@iptel.org 17 - * 18 13 * ser is distributed in the hope that it will be useful, 19 14 * but WITHOUT ANY WARRANTY; without even the implied warranty of 20 15 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ... ... @@ -24,6 +19,7 @@ 24 19 * along with this program; if not, write to the Free Software 25 20 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA 26 21 / 22 + 27 23 /! 28 24 * \file 29 25 * \brief SIP-router core :: atomic_ops init functions #### - Doxygen updates on core files - Add project name to doxygen in Makefile oej authored on 19/10/2009 20:35:43 Showing 1 changed files ... ... @@ -24,10 +24,15 @@ 24 24 * along with this program; if not, write to the Free Software 25 25 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA 26 26 / 27 -/ 28 - * atomic_ops init functions 29 - * (needed for lock intializing if no native asm locks are available 30 - * for the current arch./compiler combination, see atomic_ops.c) 27 +/! 28 + \file 29 + * \brief SIP-router core :: atomic_ops init functions 30 + * 31 + * \ingroup core 32 + * Module: \ref core 33 + * 34 + * Needed for lock intializing if no native asm locks are available 35 + * for the current arch./compiler combination, see \ref atomic_ops.c 31 36 / 32 37 / 33 38 * History: ... ... @@ -38,9 +43,9 @@ 38 43 #ifndef __atomic_ops_init_h 39 44 #define __atomic_ops_init_h 40 45 41 -/* init atomic ops / 46 +/! \brief init atomic ops / 42 47 int init_atomic_ops(); 43 -/ destroy atomic ops (e.g. frees the locks, if locks are used) / 48 +/! \brief destroy atomic ops (e.g. frees the locks, if locks are used) / 44 49 void destroy_atomic_ops(); 45 50 46 51 #endif #### - makefile: - compile in 64bit mode by default on sparc64 - sparc <= v8 support - -CC_GCC_LIKE_ASM is defined when the compiler supports gcc style inline asm (gcc and icc) - atomic operations and memory barriers support for: - x86 - x86_64 - mips (only in NOSMP mode and if it supports ll and sc) - mips2 (mips32, isa >= 2) - mips64 - powerpc - powerpc64 - sparc <= v8 (only memory barriers, the atomic operations are implemented using locks because there is no hardware support for them) - sparc64 - both 32 (v8plus) and 64 bit mode If there is no support for the compiler/arch. combination, it falls back to locks. The code is tested (only basic tests: it runs and the results are ok, but no parallel tests) on x86, x86_64, mips2, powerpc, sparc64 (both modes). The sparc version runs ok on sparc64 (so it's most likely ok). powerpc64 and mips64 not tested due to no access to the corresponding hardware, but they do compile ok. For more details see the comments at the beginning of atomic_ops.h. Andrei Pelinescu-Onciul authored on 30/03/2006 19:56:06 Showing 1 changed files 1 1 new file mode 100644 ... ... @@ -0,0 +1,46 @@ 1 +/ 2 + * $Id$ 3 + * 4 + * Copyright (C) 2006 iptelorg GmbH 5 + * 6 + * This file is part of ser, a free SIP server. 7 + * 8 + * ser is free software; you can redistribute it and/or modify 9 + * it under the terms of the GNU General Public License as published by 10 + * the Free Software Foundation; either version 2 of the License, or 11 + * (at your option) any later version 12 + * 13 + * For a license to use the ser software under conditions 14 + * other than those described here, or to purchase support for this 15 + * software, please contact iptel.org by e-mail at the following addresses: 16 + * info@iptel.org 17 + * 18 + * ser is distributed in the hope that it will be useful, 19 + * but WITHOUT ANY WARRANTY; without even the implied warranty of 20 + * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 21 + * GNU General Public License for more details. 22 + * 23 + * You should have received a copy of the GNU General Public License 24 + * along with this program; if not, write to the Free Software 25 + * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA 26 + / 27 +/ 28 + * atomic_ops init functions 29 + * (needed for lock intializing if no native asm locks are available 30 + * for the current arch./compiler combination, see atomic_ops.c) 31 + / 32 +/ 33 + * History: 34 + * -------- 35 + * 2006-03-30 created by andrei 36 + / 37 + 38 +#ifndef __atomic_ops_init_h 39 +#define __atomic_ops_init_h 40 + 41 +/ init atomic ops / 42 +int init_atomic_ops(); 43 +/ destroy atomic ops (e.g. frees the locks, if locks are used) */ 44 +void destroy_atomic_ops(); 45 + 46 +#endif	2020-09-22 08:12:50	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.6991891860961914, "perplexity": 6567.470337476786}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1600400204410.37/warc/CC-MAIN-20200922063158-20200922093158-00435.warc.gz"}
https://projecteuclid.org/euclid.aop/1022855870	## The Annals of Probability ### Lyapounov exponents and quenched large deviations for multidimensional random walk in random environment Martin P. W. Zerner #### Abstract Assign to the lattice sizes $z \epsilon \mathbb{Z}^d$ i.i.d. random 2 $d$-dimensional vectors $(\omega(z, z + e))_{\|e\|=1}$ whose entries take values in the open unit interval and add up to one. Given a realization $\omega$ of this environment, let $(X_n)_{n \geq o}$ be a Markov chain on $\mathbb{Z}^d$ which, when at $z$, moves one step to its neighbor $z + e$ with transition probability $\omega(z, z + e)$. We derive a large deviation principle for $X_n/n$ by means of a result similar to the shape theorem of first-passage percolation and related models. This result produces certain constants that are the analogue of the Lyapounov exponents known from Brownian motion in Poissonian potential or random walk in random potential. We follow a strategy similar to Sznitman. #### Article information Source Ann. Probab., Volume 26, Number 4 (1998), 1446-1476. Dates First available in Project Euclid: 31 May 2002 Permanent link to this document https://projecteuclid.org/euclid.aop/1022855870 Digital Object Identifier doi:10.1214/aop/1022855870 Mathematical Reviews number (MathSciNet) MR1675027 Zentralblatt MATH identifier 0937.60095 #### Citation Zerner, Martin P. W. Lyapounov exponents and quenched large deviations for multidimensional random walk in random environment. Ann. Probab. 26 (1998), no. 4, 1446--1476. doi:10.1214/aop/1022855870. https://projecteuclid.org/euclid.aop/1022855870 #### References • 1 DEMBO, A., PERES, Y. and ZEITOUNI, O. 1996. Tail estimates for one-dimensional random walk in random environment. Comm. Math. Phys. 181 667 683. • 2 DEMBO, A. and ZEITOUNI, O. 1993. Large Deviations Techniques. Jones and Bartlett, Boston. • 3 FREIDLIN, M. 1985. Functional Integration and Partial Differential Equations. Princeton Univ. Press. • 4 GANTERT, N. and ZEITOUNI, O. 1998. Quenched sub-exponential tail estimates for one-dimensional random walk in random environment. Comm. Math. Phys. 194 177 190. • 5 GREVEN, A. and DEN HOLLANDER, F. 1994. Large deviations for a random walk in random environment. Ann. Probab. 22 1381 1428. • 6 HUGHES, B. D. 1996. Random Walks and Random Environments 2. Clarendon Press, Oxford. • 7 KALIKOW, S. A. 1981. Generalized random walk in a random environment. Ann. Probab. 9 753 768. • 8 KESTEN, H. 1986. Aspects of first passage percolation. Ecole d'ete de Probabilites de St. ´ ´ ´ Flour. Lecture Notes in Math. 1180 125 264. Springer, Berlin. • 9 KESTEN, H. 1993. On the speed of convergence in first-passage percolation. Ann. Appl. Probab. 3 296 338. • 10 LEE, T.-Y. and TORCASO, F. 1998. Wave propagation in a lattice KPP equation in random media. Ann. Probab. 26 1179 1197. • 11 LIGGETT, T. 1985. Interacting Particle Systems. Springer, New York. • 12 LORENTZEN, L. and WAADELAND, H. 1992. Continued Fractions with Applications. NorthHolland, Amsterdam. • 13 PETROV, V. V. 1995. Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Clarendon Press, Oxford. • 14 PISZTORA, A. and POVEL, T. 1997. Large deviation principle for random walk in a quenched random environment in the low speed regime. Unpublished manuscript. • 15 PISZTORA, A., POVEL, T. and ZEITOUNI, O. 1997. Precise large deviation estimates for one-dimensional random walk in random environment. Probab. Theory Related Fields. To appear. • 16 REVESZ, P. 1990. Random Walk in Random and Non-Random Environments. World ´ ´ Scientific, Singapore. • 17 SOLOMON, F. 1975. Random walks in random environment. Ann. Probab. 3 1 31. • 18 SZNITMAN, A. S. 1994. Shape theorem, Lyapounov exponents, and large deviations for Brownian motion in a Poissonian potential. Comm. Pure Appl. Math. 47 1655 1688. • 19 SZNITMAN, A. S. 1996. Distance fluctuations and Lyapounov exponents. Ann. Probab. 24 1507 1530. • 20 SZNITMAN, A. S. 1998. Brownian Motion, Obstacles and Random Media. Springer, Berlin. • 21 ZERNER, M. P. W. 1998. Directional decay of the Green's function for a random nonnegative potential on d. Ann. Appl. Probab. 8 246 280.	2019-11-13 05:01:19	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8287382125854492, "perplexity": 2264.2780818469946}, "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/1573496665985.40/warc/CC-MAIN-20191113035916-20191113063916-00211.warc.gz"}
https://briantliao.com/data-science-notes/lectures/lec14_models_loss.html	Lecture 14: Review Models and Loss • Response var you want to estimate • Model, summarizes with parameter w • w is an estimator Loss • $$L(w)=\frac{1}{n}$$ # sum n i • sum of each $$y_i$$ • We compare the red value vs the purple value. The green value is the best w, it minimizes loss. Minimizing Loss • L(w) ? • What is your 1. data, 2. model, 3. parameters, 4. loss, (also optimization method) • From 1-d (best avg) to 2-d, best func • $$w$$ is w that minimizes L(w). The best estimator is $$\hat{y}(w^*)$$ • Can generalize our optimization to 3d! (3 weight params) • Can't plot our loss in 3d (cause it's 4d with 3 dim and loss dim) • One option is calculus set deriv of loss to 0 • Can actually do brute force. do np.linspace and try all values. O(N^2)?	2020-09-25 23:52:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.3267315626144409, "perplexity": 10518.64045807982}, "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-2020-40/segments/1600400228998.45/warc/CC-MAIN-20200925213517-20200926003517-00710.warc.gz"}
http://leancrew.com/all-this/	# Wobbly words in Tweetbot If you’re using Tweetbot 4 and have the font set to San Francisco, you may have noticed that text in the Compose screen sometimes reformats itself in ways that don’t make sense. Before last night, I thought I was imagining this, but it happened so often as I was tweeting during the Cubs wildcard win over the Pirates1 I knew it was real. We’re all used to certain types of reformatting as we type. When we get near the end of a line and start typing a long word, that word will jump to the next line when it gets long enough to hit the right edge of the text field and force a word wrap. What I’m talking about in Tweetbot is seeing words jump back and forth from line to line as I add text after them. Let me show you a couple of examples. Here’s me typing a reply to noted Yankee fan John Gruber: The last word, the, starts out on the third line of the reply, but when I type the C it suddenly jumps back to the second line. Later on, a similar thing happened: When I typed the s onto the end of Cardinals, the the jumped back down to the third line again. This all seemed very weird to me, so I asked (on Twitter, of course) if anyone knew why. Paul Haddad of Tapbots—a pretty authoritative source, I’d say—answered: @drdrang the counter is using San Francisco and it defaults to proportional numbers. Paul Haddad (@tapbot_paul) Oct 7 2015 10:29 PM The right margin of the composition field is defined by the width of the countdown field in the upper right corner. When the counter gets narrower, so does the margin, and more text can fit on every line of the tweet. But in both of my examples, the count is two digits long, which suggests an unchanging width. That’s the point of Paul’s tweet. While most fonts—even fonts that are otherwise proportional—use monospaced characters for the numerals, San Francisco is different. It has both monospaced and proportional numerals and it’s the proportional ones that happen to be used by default. So the margin width changes a little bit every time the countdown value changes, and that’s what’s causing the text to jump around. It probably doesn’t help that I use a relatively large font in Tweetbot. That magnifies the small differences in the widths of the numerals and means only a few words can fit on every line of the composition field. These two things will tend to cause more text jumps than if I were using a smaller font. But I need to see what I’m typing, so reducing the font size isn’t in the cards. According to Paul, the next revision to Tweetbot will use monospaced numerals in the countdown, which will eliminate the jumpiness except when the number of digits changes. This’ll be much better, although personally I’d prefer either a right margin with a fixed width large enough to accommodate a three-digit count or to have the counter moved from the right margin to the otherwise unused space under the user’s avatar. As tweets about this situation went back and forth between me, Paul, and the Egg McMuffinless Casey Liss, I got a tweet from Ian Bjorhovde directing me to this portion of the WWDC session on San Francisco, which talks about how to use the proportional and monospaced numerals.2 It’s an interesting subtopic and lasts only about three minutes. Well worth your time. David Loehr mentioned that he uses Avenir, the other font that Tweetbot allows. It doesn’t have the proportional numeral problem, but I find it a little too thin and “gray” for comfortable reading. I’ll stick with San Francisco and hope the Tweetbot revision makes it through Apple’s approval process quickly. The revision is also going to fix the chart labeling bug I talked about a few days ago. 1. I mention this as a way of preserving a record of this rare Cub playoff appearance and victory for future skeptical generations. 2. I always pay attention when Ian tweets at me. He’s a bright guy in his own right, and he’s also the son of a pretty famous structural engineer whose publications I’ve used countless times. # Better charts Occasionally, I write a post about making charts. Sometimes these posts are rants about poor practice or my thoughts on good practice, but usually they’re more descriptive than prescriptive. I write about how I make charts with the expectation that my beliefs and tastes will come through and that I might have some small influence in stemming the tide of bad charts. So far, I’ve had about as much effect as King Canute. The rising popularity of data journalism has brought with it some truly dumb charts. Reporters with no training or experience in communicating graphically are being given a crash course in some JavaScript plotting library and told to have at it. The result is a bunch of charts that seem OK at a casual glance but not when you look again. Here’s a political example from Morning Consult: A layout that might work with a few candidates is a mess with fifteen. The legend overwhelms the chart, and there’s no rhyme or reason to the order of the names. The colors are too close to one another. The markers, which could be used to distinguish the candidates, are the same for each.1 The labels for the horizontal axis are stupidly formatted over two lines. Worst of all, the polls are equally spaced horizontally even though the times between them vary from 5 to 14 days. You might say this is nitpicking and that the important thing is that the chart communicates who’s winning and who’s moving up or down in the polls. You could also argue that there’s no reason to wrote an article with correct tenses or to gets the verbs to agree with the subjects. Them things isn’t important to communication, is they?2 Into this mess steps Kieran Healy, associate professor of sociology at a basketball university down in North Carolina (no, the other one). Kieran is perhaps best known on the internet for a data visualization post that has, unfortunately, become something of an evergreen. His charts are always tasteful and informative because he’s a smart guy and he’s thought a lot about how to communicate through plots. This semester—half-semester, actually—Kieran’s going to impart his wisdom to grad students in his department through a special topics course. He’s starting out right, by demonstrating the evils of Excel’s overly cute 3D column charts: Image from Kieran Healy. A handful of Duke sociology students won’t fix the world’s data visualization problems, but Kieran is making his class notes available on GitHub, so there’s hope that others will find them and learn. 1. Distinguishing the data series is somewhat easier in the actual chart (as opposed to this screenshot) because you can click or tap the names in the legend and see the corresponding series light up. Of course, you have to know or guess that this is possible, otherwise you’d never try it. 2. Given my penchant for leaving typos and editing artifacts in my posts, this is a very dangerous paragraph. # Unresponsive This post from last week by Kirk McElhearn and this followup today by Michael Tsai reminded me that Safari 9 has a new feature in the menu: Responsive Design Mode. Unfortunately, it’s not as smart as I’d hoped it would be. Or maybe I’m not. The idea behind RDM is to let web designers see what a site looks like on smaller (Apple) screens without continually resizing windows or reloading pages on other devices. You can see how it looks by just clicking a button associated with the device of interest. And, unlike pinned sites, this feature is available on Safari 9 even if you’re still running Yosemite. I’m not a web designer, but I am responsible for this site, and I do occasionally fiddle with its layout. I (finally!) made a mobile layout back in June, and it would have been much easier if I’d been able to see the results of my CSS changes immediately on my Mac as I made them. But I soon found that RDM doesn’t really emulate smaller devices. Here’s what my site looks like on my iPhone 5s: And here’s what it looks like in Safari RDM: Not exactly a faithful representation. And it’s no better in landscape. I assume this is at least partially because I don’t really know what I’m doing. I have two CSS files for the site: the original style.css that’s meant for wider screens and mobile.css that’s meant for narrower screens. The file used is controlled by these three lines in the <head>: <link rel="stylesheet" type="text/css" media="all and (max-device-width:480px) and (orientation:portrait)" href="resources/mobile.css" /> <link rel="stylesheet" type="text/css" media="all and (max-device-width:480px) and (orientation:landscape)" href="resources/style.css" /> <link rel="stylesheet" type="text/css" media="all and (min-device-width:480px)" href="resources/style.css" /> So I’m really choosing the style file on the basis of the device’s width, not the view’s width. This is a simple solution, and it works, even though it isn’t fully responsive. If you’re reading this on a phone and rotate it, the layout will change; but if you’re reading this on a notebook or desktop computer and make the browser window narrow, the layout won’t change. I don’t really want to mess with the site’s layout again, and I certainly wouldn’t do so just to get it to work in Safari’s RDM, but there are two things that’ll probably force me into it: iOS 9’s Slide Over and Split View on iPads. I’m pretty sure those views don’t change the device-width, and if I want the site to display its narrow layout when it’s in those modes, I’ll have to make it truly responsive. Eventually. I’m not particularly responsive, either. # Charts in Tweetbot 4 I’ve been using Tweetbot 4 for a few hours now, and I really like it. This is not a review. There are several other places you can go for that. What I want to talk about are the little charts it provides in the Stats view. They’re fun without being obsessive the way Twitter’s own analytics charts are. At the top is an activity timeline for the past seven days. By going back only seven days, the chart is clean, uncluttered, and easy to read at a glance. You’ll notice that it’s a Bezoan chart, with no scale for the vertical axis. I think that’s OK in this context. The idea is to give you a quick sense of what’s been going on for the past week. Again, if you want to pore over the details, go to Twitter’s analytics site to see how many of your followers are self-employed weight conscious Verizon users. One thing about this chart that I’m pretty sure is a bug is the horizontal labeling. Based on how the final point starts low and rises through the day, I believe it represents the current day’s activity, even though it’s labeled as yesterday. In fact, all of the day labels appear to be one day off. Below this chart is a list of your tweets, each with bar charts showing how often it’s been favorited and retweeted. The two bars obviously aren’t drawn to the same scale, but what scales are they drawn to? Tweetbot’s Paul Haddad gave the answer in a tweet a couple of days ago: @chase_mccoy all relative to your most popular tweet of the last week. Paul Haddad (@tapbot_paul) Oct 1 2015 6:01 PM I think that’s a good choice and is in keeping with the seven-day scale of the chart at the top. In this case, the absolute numbers are given, but the bar charts are relative to the past week. If you keep scrolling down, you’ll start seeing tweets that are more than seven days old, and it’s likely you’ll run into some that have more favorites or retweets than the seven-day maximum. Those bars are pegged to the right end and don’t give an accurate representation of their value, but their numbers are still correct. You just have to keep in mind that the charts on this screen are all based on the past week. Overall, I like these new Tweetbot charts because they’re fun and casual. Social media professionals will probably not find them acceptable, but who cares what they think?	2015-10-09 15:54:41	{"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": 0.4062669277191162, "perplexity": 1485.428565392919}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443737933027.70/warc/CC-MAIN-20151001221853-00129-ip-10-137-6-227.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/fluid-mechanics-question.329104/	# Fluid mechanics question Suppose you're driving along a straight, smooth, level stretch of highway with a constant velocity $v$. There is a cup of some fluid in your cup holder. How would you go about finding the angle $\theta$ that the surface of the fluid makes with the bottom of the cup as a function of $v$? And what if you started out with a velocity $v_0$ at time $t=0$ and then accelerated at a constant rate $a$? Can we find $\theta(t)$ without much sweat? $$\theta = \tan^{-1} \left (\frac{a}{g}\right)$$ where $$a$$ is your acceleration and $$g$$ is gravity.	2021-11-30 05:34:33	{"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": 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": 0.5675606727600098, "perplexity": 166.26333307443804}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358953.29/warc/CC-MAIN-20211130050047-20211130080047-00349.warc.gz"}
https://en.wikibooks.org/wiki/Ordinary_Differential_Equations/Trajectories	Ordinary Differential Equations/Trajectories Orthogonal Trajectory Let A be a family of curves. Then B is an orthogonal trajectory of A if every member of B(also a family of curves) cuts every member of A at right angle.It is important to note that we are not insisting that B should intersect every member of A but if they intersect, the angle between their tangents, at every point of intersection, is ${\pi}/2$ Example Every straight line passing through origin is a normal to every circle having origin as the center. Hence they are orthogonal trajectories of each other. Steps to find orthogonal trajectory 1. let f(x,y,c)=0 be the equation of the family of curves, where c is an arbitrary constant. 2. Differentiate the given equation with respect to x and then eliminate c. 3. replace $dy/dx$ by $-dx/dy$ 4. Solve the obtained differential equation. You will get the required orthogonal trajectory.	2015-07-01 20:03:28	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 3, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.823606014251709, "perplexity": 352.60238329914523}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375095184.64/warc/CC-MAIN-20150627031815-00179-ip-10-179-60-89.ec2.internal.warc.gz"}
https://crypto.ethereum.org/blog/pq-ssle	• home • blog • research • bounties • team • events # Towards practical post quantum Single Secret Leader Election (SSLE) - Part 1 ## Introduction Single Secret Leader Election (SSLE from now on) is an important research problem the cryptographic community has been researching on. The SSLE protocols allow a set of users to elect a leader ensuring that the identity of the winner remains secret until he decides to reveal himself. Whisk is a block proposer election protocol tailored to the Ethereum beacon chain that protects the privacy of proposers. It relies on discrete logarithm assumptions and uses a shuffling approach and NIZK proof of shuffle to prove correctness. This year NIST announced its choice for Post-Quantum-Cryptography algorithms that are going to replace the existing public key infrastructure (Zhenfei Zhang covered this in a previous blog post). In this blog post we are going to analyze a possible Post Quantum analogue of Whisk based on Commutative Supersingular isogenies (CSIDH). N.B. If you wonder if this solution is affected by the new devastating attack on SIDH the answer is NO. The Castryck-Decru Key Recovery Attack crucially relies on torsion point information that are not present in CSIDH based solutions. ## Whisk's recap As mentioned above Whisk is a proposal to fully implement SSLE from DDHand shuffles scheme (see also section 6 from Boneh et al paper). The idea behind this solution is pretty straightforward and neat. Let's list below the key ingredients of the commitment scheme in Whisk (at the net of the shuffles): 1. Alice commits to a random long-term secret k using a tuple $(rG,krG)$ (called tracker). 2. Bob randomizes Alice’s tracker with a random secret $z$ by multiplying both elements of the tuple: $(zrG,zkrG)$. 3. Alice proves ownership of her randomized tracker (i.e. open it) by providing a proof of knowledge of a discrete log (DLOG NIZK) that proves knowledge of a k such that $k(zrG)==zkrG$ . 4. Identity binding is achieved by having Alice provide a deterministic commitment $com(k)=kG$ when she registers her tracker. 5. We also use it at registration and when opening the trackers to check that both the tracker and $com(k)$ use the same $k$ using a discrete log equivalence proof (DLEQ NIZK). Whisk can be implemented in any group where the Decisional Diffie Hellman problem (DDH) is hard. Currently Whisk is instantiated via a commitment scheme in BLS12-381. ## Commutative Supersingular isogenies (CSIDH). This section (and the remainder of the blog post) will require some knowledge about elliptic curves and isogeny based cryptography. The general reference on elliptic curves is Silverman for a thorough explanation of isogenies we refer to De Feo. CSIDH is an isogeny based post quantum key exchange presented at Asiacrypt 2018 based on an efficient commutative group action. The idea of using group actions based on isogenies finds its origins in the now well known 1997 paper by Couveignes. Almost 10 years later Rostovtsev and Stolbunov rediscovered Couveignes's ideas . Couveignes in his seminal work introduced the concept of Very Hard Homogeneous Spaces (VHHS). A VHHS is a generalization of cyclic groups for which the computational and decisional Diffie-Hellman problem are hard. The exponentiation in the group (or the scalar multiplication if we use additive notation) is replaced by a group action on a set. The main hardness assumption underlying group actions based on isogenies, is that it is hard to invert the group action: Group Action Inverse Problem (GAIP)). Given a curve $E$, with $End(E) = O$, find an ideal a ⊂ O such that $E = [a]E_0$. The GAIP (also known as vectorization) might resemble a bit the discrete logarithm problem and in this blog post we exploit this analogy to translate the commitment scheme in Whisk to the CSIDH setting. ## CSIDH Whisk In this section we will show that a 1:1 translation is indeed (almost) easily achievable. Indeed the translation from the DLOG setting to VHHS presents a caveat: in this blog post we will focus our attention on the fraud proof version of shuffle based SSLE. This is also described in the original SSLE paper (see Removing NIZKs paragraph). The reason behind this is because currently there isn't a way to have NIZK proof of shuffle based on isogenies. Apart from this, let's see how it is indeed possible to translate all the other ingredients. ### Whisk commitment scheme The hardness of the GAIP problem gives a natural translation of the Whish commitment scheme. Alice commits to a random long-term secret $[k]$ using a tuple $([r]E_0,[k][r]E_0)$, where $E_0:y^2 = x^3 + x$ over $F_p$ is the base curve (the equivalent of the generator $G$ in the elliptic curve based solution). Also the randomization phase is trivial: Bob randomizes Alice’s tracker with a random secret $[z]$ by multiplying both elements of the tuple: $([z][r]E_0,[z][k][r]E_0)$. ### DDH and CSIDH The next thing to address is ensuring DDH is a hard problem in CSIDH. Group-Action DDH the Group-Action DDH assumption holds if the two distributions $([a]E_0, [b]E_0, [a][b]E_0)$ and $([a]E_0, [b]E_0, [c]E_0)$ are computationally indistinguishable. Castryck et al showed that the DDH problem is easy in ideal-class-group actions when the class number is even. Such groups are therefore unsuited for the above construction. As a countermeasure to their attack, they suggest working with supersingular elliptic curves over Fp for $p ≡ 3 (mod 4)$, which is already the case for CSIDH. In that setting, the Group-Action DDH problem is conjectured to be hard. ### DLOG NIZK in CSIDH A sigma protocol proving knowledge of a solution of a GAIP instance in zero knowledge has been described in original Couveignes's paper and further analyzed in Stolbunov'sPhD thesis. Two incarnations of these ideas in the CSIDH setting are SeaSign and CSI-FiSh. The first paper (SeaSign) uses rejection sampling (a technique successfully employed in lattice based cryptography) to prevent signatures from leaking the private key (a problem that occurs if a sigma protocol is performed naively). The same is achieved in the latter paper (CSI-FiSh) computing the class group of the imaginary quadratic field used in the CSIDH-512 cryptosystem. ### DLEQ NIZK in CSIDH A way to solve discrete log equivalence proof (DLEQ NIZK) in the CSIDH is provided in Beullens et al. section 2.4. ## Conclusion In this blog post we briefly analyzed a possible replacement of Whisk in the Post Quantum setting. We achieved this employing the commutative supersingular isogeny (CSIDH) setting. We have seen that a direct translation from DLOG to VHHS is indeed possible with some limitations. The derived Post Quantum Whisk Protocol is restricted to the fraud proof version due the lack of NIZK proof of shuffle in the isogeny setting. The current zero-knowledge proving system is an adaptation of the Bayer-Groth shuffle argument but is currently out of reach for isogeny based cryptography. We hope this blog post stimulates researchers to look into this open problem. ## Acknowledgement We would like to thank Ward Beullens, Dan Boneh, Luca De Feo and George Kadianakis for for fruitful discussions and comments. cryptography@ethereum.org	2022-09-29 00:33:05	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 20, "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": 0.6717911958694458, "perplexity": 1781.7816919513873}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1664030335303.67/warc/CC-MAIN-20220929003121-20220929033121-00445.warc.gz"}
https://zapgeeks.com/think-before-installing-a-new-fireplace-insert-for-the-winter-burn-season	Life Style # Think Before Installing a New Fireplace Insert for the Winter Burn Season A fireplace insert is the best solution. This heater will help your fireplace get a makeover and decrease heating and cooling costs without the high price and inconvenience of a fireplace renovation ### What is the fireplace insert? A fireplace insert is a fireplace specially designed to fit inside an existing fireplace opening. Fireplace inserts are available in a wide variety of styles, from modern and stylish designs that can make your old fireplace more contemporary to rustic designs. They have in common the self-cleaning glass doors which create a nice viewing window of the fire and increase the efficiency of the fireplace. Renovating or rebuilding an existing home is incredibly expensive and impractical. It can cost up to $10,000 to rebuild a brick fireplace.This means that masonry days in your home create dust and disrupt your normal life. Conversely, most fireplace inserts cost less than$ 1,000 and can be installed in a day. Built-in fireplaces have five main advantages that make them popular: affordability, energy efficiency, low environmental impact, alternative fuel options, and convenient features. ### Less environmental impact The high efficiency of the fireplace inserts also makes them more environmentally friendly. PA certified wood-burning fireplace inserts produce low emissions. They are designed to meet and in some cases exceed clean air standards. Whether you choose a gas, wood, or pellet stove, you can rest easy knowing that your fireplace has no negative impact on the quality of the outdoor air. ### Efficiency of Energy The average masonry fireplace has less than 10% efficiency. This means that more than 90% of the heat generated by your chimney goes over the chimney stack as opposed to heating your home. Losing heat from your chimney isn’t the main issue. In any event, throughout the spring and summer, when your chimney isn’t being used, it can make your home less energy productive by permitting blistering air to descend the chimney stack and cooled air to get outside. Accordingly, having a conventional chimney in your home can expand your energy costs all year. You can solve both problems with a fireplace insert. The improved venting system is connected to the fireplace insert. Its glass doors prevent drafts from outside. They also prevent hot and cold air from escaping from the chimney. Fireplace inserts are also planned to be much more efficient. Some have up to a 70% efficiency rating which means most of the heat that they produce is radiated into your home instead of escaping outside. ### Substitute Fuel Options Installing a fireplace insert is an easy way to convert your fireplace to burn other fuels. When you select a fireplace insert, you can choose between: • Natural gas / propane • Electric • Pilates • Wood ### Suitable Features As well as being available in a wide array of different styles and fuel types, fireplace inserts also offer practical features that traditional large electric fireplaces have. Thermostat control and remote operation are two of the most popular features available with electric fireplace, gas, and pellet inserts. With a gas fireplace insert, you will have the ability to customize the display of the fire. Everything from setting the backlight to selecting the fake log set or on-screen fire glass. ### How do you select the perfect fireplace insert? The most important thing to keep in mind when choosing a fireplace insert is that it should fit the size of your existing fireplace hearth. We recommend that you consult a professional fireplace expert who has experience installing fireplace inserts in your area. They will measure the size of the fireplace and tell you the exact dimensions you need to look for. They can also tell you if there are types of inserts that are not compatible with your fireplace. How do you set up a fireplace insert? Installing a fireplace insert is not a job you should try to DIY unless you are a fireplace expert. We are not the only ones to recommend the use of an expert. The Hearth, Patio and Barbecue (HPBA) association also recommends that fireplace inserts should be installed by a certified professional. The most important reason for bringing in a professional for the installation of your fireplace insert is that they will make sure that your ventilation system is set up correctly. If your fireplace insert is improperly installed, it can present a serious risk to you and your home, especially if it does not vent properly. Check Also Close	2022-07-06 10:40:11	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.17753273248672485, "perplexity": 2366.828253536546}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1656104669950.91/warc/CC-MAIN-20220706090857-20220706120857-00479.warc.gz"}
http://openstudy.com/updates/5047590be4b0c3bb09860279	## anonymous 3 years ago When tightening cylinder-head bolts in a car, the crucial quantity is the torque applied to the bolts. why is this more important than the actual force applied to the wrench handle? pls help. tnx 1. anonymous The motion of bodies which rotate around an axis is governed by torque versus mass. The long handle of the wrench INCREASES the force appled at the end by the leverage principle 2. anonymous Torque ON the bolt is$\it F_{on-bolt}\times Radius_{bolt} = Torque$ 3. anonymous So the actual FORCE-ON-BOLT is crucial 4. anonymous Thanks for the med. PLease close the question. 5. anonymous Thank you.. :) 6. anonymous Youuu eiii-lcUmmm !	2016-07-23 11:13:50	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5073155760765076, "perplexity": 4874.307695452067}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257822172.7/warc/CC-MAIN-20160723071022-00115-ip-10-185-27-174.ec2.internal.warc.gz"}
https://www.vedantu.com/maths/odds-and-probability	Courses Courses for Kids Free study material Free LIVE classes More # Odds and Probability ## Odds in Favour Meaning Last updated date: 18th Mar 2023 Total views: 25.5k Views today: 0.10k When we do something, in mathematics we call it an event. So when an event occurred then there is some outcome or result of that event, to study that outcome or get an idea of an event we use three-term which are odds, chances, and probability Odds and Probability in Mathematics ## Definition of Odds Odds:- It is a measure of the likelihood of a particular outcome and it is generally calculated by the ratio of the number of favorable outcomes to the number of unfavorable outcomes i.e ${\rm{Odds = }}\dfrac{{{\rm{Number \; of \; favourable\; outcomes}}}}{{{\rm{Number \; of \; total \; outcomes}}}}$ ## Definition of Probability and Chances Probability:- It is the chance that something will happen, or how likely it is that an event will occur. It is calculated by the ratio of the number of favorable outcomes to the number of total outcomes i.e ${\rm{probability = }}\dfrac{{{\rm{Number\; of \; favourable\; outcomes}}}}{{{\rm{Number \; of \; total \; outcomes}}}}$ Calculating the Probability of an event ${\rm{Probability \; of \; event = }}\dfrac{{{\rm{No}}{\rm{. of \; favourable\; outcomes}}}}{{{\rm{No}}{\rm{. of \; favourable\; outcomes + No}}{\rm{. of\; unfavourable \; outcomes}}}}$ Chances:- It is, simply, the possibility of something happening, which is not planned or controlled. Its value is the same as probability. ## Odds in Favour Odds in favour of a particular event are given by the Number of favorable outcomes to the Number of unfavorable outcomes. i.e Odds Formula ${\rm{odds\; in\; favour = }}\dfrac{{{\rm{No}}{\rm{. of favourable\; outcomes}}}}{{{\rm{No}}{\rm{. \;of \;unfavourable\; outcomes}}}}$ ## Odds in Against Odds against are given by Number of unfavorable outcomes to the number of favorable outcomes. i.e ${\rm{Odds\; in\; Against = }}\dfrac{{{\rm{No}}{\rm{. of \;unfavourable \;outcomes}}}}{{{\rm{No}}{\rm{. of\; favourable\; outcomes}}}}$ ## Difference between Odds and Probability The difference between odds and probability are: Odds of an event are the ratio of success to failure. ${\rm{Odds = }}\dfrac{{{\rm{success}}}}{{{\rm{failure}}}}$ The probability of an event is the ratio of success to the sum of success and failure. ${\rm{Probability = }}\dfrac{{{\rm{success}}}}{{{\rm{success + failure}}}}$ ## Solved Examples 1. Find the odds in favor of throwing a die to get “3 dots”. Solution: Total number of outcomes in throwing a die = 6 (1,2,3,4,5,6) Number of favorable outcomes = 1 (3) Number of unfavorable outcomes = (6 - 1) = 5 Therefore, odds in favor of throwing a die to get “3 dots” is 1 : 5 or $\dfrac{1}{5}$ 2. Find the odds in favor of throwing a coin to get a “tail”. Solution: Total number of outcomes in throwing a coin = 2 (“head”,”tail”) Number of favorable outcomes = 1 (“tail”) Number of unfavorable outcomes = (2- 1) = 2 Therefore, odds in favor of throwing a coin to get a “tail” is 1 : 1 or $\dfrac{1}{1}$ 3. Find the odds against throwing a die to get “3 dots”. Solution: Total number of outcomes in throwing a die = 6 Number of favorable outcomes = 1 Number of unfavorable outcomes = (6 - 1) = 5 Therefore, odds in against of throwing a die to get “3 dots” is 5 : 1 or $\dfrac{5}{1}$ 4. Find the odds against throwing a die to get “2 dots”. Solution: Total number of outcomes in throwing a die = 6(1,2,3,4,5,6) Number of favorable outcomes = 1 Number of unfavorable outcomes = (6 - 1) = 5 Therefore, odds in against of throwing a die to get “3 dots” is 5 : 1 or $\dfrac{5}{1}$ 5.Find the probability of getting “2 dots” in throwing a die. Solution: Total number of outcomes in throwing a die = 6 Number of favorable outcomes = 1 Number of unfavorable outcomes = (6 - 1) = 5 Therefore, probability of getting “2 dots” in throwing a die. is 1: (1+5) or $\dfrac{1}{6}$ 6.If odds in favor of X solving a problem are 4 to 3 and odds against Y solving the same problem are 2 to 6. Find probability for: (i) X solving the problem (ii) Y solving the problem Solution: Given odds in favor of X solving a problem are 4 to 3. Number of favorable outcomes = 4 Number of unfavorable outcomes = 3 (i) X solving the problem P(X) = P(solving the problem) = 4/(4 + 3) = $\dfrac{4}{7}$ Given odds against Y solving the problem are 2 to 6 Number of favorable outcomes = 6 Number of unfavorable outcomes = 2 (ii) Y solving the problem P(Y) = P(solving the problem) = 6/(2 + 6) = $\dfrac{6}{8}$ = $\dfrac{3}{4}$ In this article we learned about Odds and probability and how to calculate odds and also learnt the physical meaning of both ## FAQs on Odds and Probability 1. What is the range of probability? The range of probability is between 0 and 1 2. What is meant by chance in Probability? In real life, there are numerous instances where we must take a chance or risk. The likelihood of an event occurring based on specific circumstances may be easily anticipated. Simply put, the likelihood that a specific event will occur 3. List any two uses of Probability? It helps in lending debt. • Helps in commute times. 4. What is the 0 value in terms of negative or positive? 0 is neither positive nor negative. Both the terms start after or before as negative numbers or positive numbers.	2023-03-22 06:19:49	{"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": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7767788767814636, "perplexity": 984.6805184998102}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1679296943750.71/warc/CC-MAIN-20230322051607-20230322081607-00602.warc.gz"}
https://tajassus.com/significant-figures-definition/	# Significant figures definition Significant figures The specified number of digits which are accepted to denote the approximate value of the number which is expressed as an integer and a decimal number. For example a distance of 984325 km. It can be taken as 984300 km which is correct to four significant digits, the last two zeros are not significant as they denote the order of magnitude only. Similarly, the number of significant digits in 0.00678 is 3, as the zeros, in the beginning, are used to denote the value of the place of the digits. Thus the zeros at the end of a whole number or decimal may or may not be significant according to the degrees of accuracy required but zero on the left is never the first significant figure. Similar figures Geometrical figures which have the same shapes but different sizes. More precisely, two or more figures are said to be similar if a mapping can be defined between them such that the ratio of the distance between pairs of points of one figure remains the same for the corresponding point of the other figure. Similar fraction A fraction having the same denominators. For example 4/9,5/9 and respectively 0.444444 and 0.555555 Similar terms Like terms, terms are expressions that differ in their coefficient only. Similarly, terms can always be combined into a single term. For example 8xy^2z,+6xy^2z are like terms and hence combining to give a single term 14xy^2z	2022-06-25 08:49:32	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8872840404510498, "perplexity": 339.5677276811623}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103034877.9/warc/CC-MAIN-20220625065404-20220625095404-00794.warc.gz"}
https://email.esm.psu.edu/pipermail/macosx-tex/2012-May/049082.html	# [OS X TeX] Using iwona in a beamer presentation J. McKenzie Alexander jalex at lse.ac.uk Sat May 26 13:38:33 EDT 2012 Hello, I am attempting to change the font in a beamer presentation to iwona. In my preamble, I have inserted \usepackage[math]{iwona} which does change the font used for text to iwona, but I noticed that some of the math fonts are not changed. For example, if I type $\frac{1}{2\pi i}f(X)$ it looks like the numerals and letters are in the original beamer math font, but the \pi is typeset using iwona. Any recommendations on how to change this would be greatly appreciated! Cheers, Jason -- Dr J. McKenzie Alexander Department of Philosophy, Logic and Scientific Method London School of Economics and Political Science Houghton Street, London WC2A 2AE Please access the attached hyperlink for an important electronic communications disclaimer: http://lse.ac.uk/emailDisclaimer	2018-11-21 11:43:59	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9141086339950562, "perplexity": 5782.848466666757}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039748315.98/warc/CC-MAIN-20181121112832-20181121134832-00530.warc.gz"}
https://brilliant.org/problems/sufficiently-divisible/	# Sufficiently divisible $1 \times 2\times 3\times \cdots \times n$ I've calculated the number above, but I seem to have forgotten the value of $n$. I recall that the above number is divisible 17, 18 and 19. If my recollection is correct, is this number also divisible by 20? ×	2021-07-30 15:23:11	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 2, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9857786297798157, "perplexity": 411.67322356656655}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046153966.60/warc/CC-MAIN-20210730122926-20210730152926-00091.warc.gz"}
https://myblog.maheshyadav.com.np/2016_09_10_archive.html	# Install LaTeX on Ubuntu or Debian 12:55:00 PM Mahesh Kumar Yadav 0 Comments LaTeX is a document markup language and a text preparation system to create documents. LaTeX is recommended to create technical or scientific articles, papers, reports, books and other documents like PhDs. A terminal is a Command Line Interface (CLI) where you type commands to tell the computer what to do. Make sure you've opened the terminal, if so, continues in the next step. 2. INSTALL TEX LIVE TeX Live is a TeX distribution to get up and running with the TeX document production system. To install it, once you're in the terminal, enter the following command: sudo apt-get install texlive-full Then, type your 'sudo' password and you'll have installed Tex Live. This operation may take a long time. 3. INSTALL TEXMAKER Now you need a text editor. I recommend using a specific editor for LaTeX. There are many text editors for LaTeX on the Internet as Kile, TeXworks, JLatexEditor, Gedit LaTeX Plugin, etc. My favorite text editor for Latex is Texmaker. Texmaker is a cross-platform open source LaTeX editor. To install it, go to the Ubuntu or Debian terminal and enter the following command: sudo apt-get install texmaker In a few minutes you'll have installed Texmaker. To check that everything is working properly, create a LaTeX blank document. Open Texmaker and click on File, New. Then write the following code: \documentclass{article} \begin{document} Hello, world! \end{document} Now save the document as a 'tex' file going to File, Save. Finally, compile the document clicking on Tools, PDFLaTeX. Make sure the 'pdf' file has been created and it's working. And that's it! You've created your first LaTeX document!	2020-09-20 11:21:49	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.8377758264541626, "perplexity": 6464.073147195024}, "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-2020-40/segments/1600400197946.27/warc/CC-MAIN-20200920094130-20200920124130-00240.warc.gz"}
https://www.mathplanet.com/education/algebra-2/conic-sections/distance-between-two-points-and-the-midpoint	# Distance between two points and the midpoint The distance formula is an algebraic expression used to determine the distance between two points with the coordinates (x1, y1) and (x2, y2). $D=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$ Example Find the distance between (-1, 1) and (3, 4). This problem is solved simply by plugging our x- and y-values into the distance formula: $D=\sqrt{(3-(-1))^{2}+(4-1)^{2}}=$ $=\sqrt{16+9}=\sqrt{25}=5$ Sometimes you need to find the point that is exactly between two other points. This middle point is called the "midpoint". By definition, a midpoint of a line segment is the point on that line segment that divides the segment in two congruent segments. If the end points of a line segment is (x1, y1) and (x2, y2) then the midpoint of the line segment has the coordinates: $(\frac{x_{1}+x_{2}}{2},\: \frac{y_{1}+y_{2}}{2})$ ## Video lesson Find the midpoints between the coordinates (-4, -1) and (6,7)	2018-07-22 03:10:16	{"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": 0.6846089363098145, "perplexity": 242.10057728207929}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1531676593004.92/warc/CC-MAIN-20180722022235-20180722042235-00632.warc.gz"}
https://math.stackexchange.com/questions/2835326/how-can-we-convert-double-sums-to-single-sums	# How can we convert double sums to single sums? Let's say that $n\to A(n),B(n)$ is a bijection from $\mathbb{N} \to \mathbb{N^2}$. This would be the inverse of a pairing function. The canonical example would be $B(n)= n-\frac{1}{2}\lfloor \frac{\sqrt{8n+1}-1}{2}\rfloor \lfloor \frac{\sqrt{8n+1}+1}{2} \rfloor$ $A(n) = \lfloor \frac{\sqrt{8n+1}-1}{2}\rfloor-B\left(n\right)$ Then is it the case that $$\sum_{b=0}^\infty \sum_{a=0}^\infty{f(a,b)}=\sum_{n=0}^\infty f(A(n),B(n))?$$ I assume that if I am given that LHS is absolutely convergent the equality holds for any pairing function. I would guess that this is overkill however. If $\sum_{b=0}^\infty \sum_{a=0}^\infty{f(a,b)}$ is conditionally convergent we still may be able to find a suitable $A(n),B(n)$. Questions 1) What conditions are required for this equality to hold? 2) I don't know a whole lot about double sums. Is this a standard technique? Where can I learn more? 3) Where can I find more pairing functions? I can't imagine that the one above is the most convenient to work with... Example Let's say $f(a,b)=\frac{1}{(a+1)^2(b+1)^2}$ Then it's easy enough (Given we know the solution of the Basel problem) to find the left hand side is $\pi^4/36$. But to find what the righthand side looks like is going to be tricky. We have a new series of rationals that approaches $\pi^4/36$. It's $\frac{1}{4}+\frac{1}{4}+\frac{1}{9}+\frac{1}{25}+\dots$ it doesn't have such a nice pattern to it because of our selection of the functions $A,B$. The fact that you're initially presenting this as a double sum is irrelevant. In the end, this is just about rearrangements of series. Different bijections of $\mathbb N$ to $\mathbb N^2$ correspond to different rearrangements of the series. The Riemann series theorem says that a conditionally convergent series can be rearranged to converge to any real number, or diverge to $+\infty$ or $-\infty$.	2019-08-26 09:13:43	{"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": 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": 0.9424268007278442, "perplexity": 135.3612133608712}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027331485.43/warc/CC-MAIN-20190826085356-20190826111356-00042.warc.gz"}
https://mathematica.stackexchange.com/questions/188915/how-to-implement-an-asynchronous-pipeline-with-queue	# How to implement an asynchronous pipeline with queue Assume I have three tasks that must be performed in order: Do[ The three tasks above involve usage of different resources. For example, TaskA imports an image file from disk. TaskB performs some data augmentation via image processing. TaskC sends augmented data to another node via WSTP on TCP. I would like to improve performance by making a asynchronous pipeline. That is, starting three threads for each task. Each task puts/retrieves intermediate results to/from a queue with fixed size. (hypothetical code) queueAB = MakeQueue[16]; queueBC = MakeQueue[16]; Do[ EnQueue[queueAB, $$TaskEnded], {taskId, 10000}] ]; taskB = MakeThread[ Module[{productA}, While[True, productA = DeQueue[queueAB]; If[productA ===$$TaskEnded, EnQueue[queueBC, $$TaskEnded]; Break[], EnQueue[queueBC, TaskB[productA]] ] ] ] ]; taskC = MakeThread[ Module[{productB}, While[True, productB = DeQueue[queueBC]; If[productB ===$$TaskEnded,	2022-08-08 13:28:41	{"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": 2, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3415842354297638, "perplexity": 5685.984613785055}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882570827.41/warc/CC-MAIN-20220808122331-20220808152331-00583.warc.gz"}
https://blender.stackexchange.com/questions/12204/how-to-set-import-units-when-importing-a-collada-file-with-python-script	# How to set Import Units when importing a collada file with python script? Blender shows the python command in the tooltip, for example, hover the mouse over File > Import > Collada (Default) (.dae), it shows: Load a Collada file Python: bpy.ops.wm.collada_import() Then I can find the bpy.ops.wm.collada_import() command in the script console. I want to set the Import Units to True before I import the collada file, hover the mouse over the ticker it says Python: WM_OT_collada_import.import_units But I can't find any clue on how to use this piece of code. WM_OT_collada_import is the bl_idname of the Collada Import Operator in the C-code notation (in Python it's wm.collada_import, but the importer is a C operator). import_units is an operator property. If you want to set this property to True in an operator call, give it a keyword-argument: bpy.ops.wm.collada_import(filepath=..., import_units=True)	2019-10-19 04:11:28	{"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": 0.3655535876750946, "perplexity": 4777.104491421776}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1570986688826.38/warc/CC-MAIN-20191019040458-20191019063958-00351.warc.gz"}
https://www.math.ias.edu/seminars/abstract?event=52304	# The communication complexity of distributed subgraph detection Computer Science/Discrete Mathematics Seminar I Topic: The communication complexity of distributed subgraph detection Speaker: Rotem Oshman Affiliation: Tel Aviv University Date: Monday, October 6 Time/Room: 11:15am - 12:15pm/S-101 Video Link: http://video.ias.edu/csdm/2014/1006-RotemOshman In distributed systems, communication between the participants in the computation is usually the most expensive part of the computation. Theoretical models of distributed systems usually reflect this by neglecting the cost of local computation, and charging only for messages sent between the participants; in particular, we usually assume that the computation proceeds in rounds, and in each round each participant can send only a limited number of bits. We are interested in characterizing the number of rounds required to perform various tasks. In this talk we discuss the complexity of distributed subgraph detection: there are $n$ servers, each representing a node in an undirected graph, and each server receives as input its adjacent edges in the graph. The goal of the computation is to determine whether the global input graph contains some fixed subgraph. In the talk I will describe upper and lower bounds for several classes of subgraphs, through a connection to Turan numbers. The general case remains open. We also point out a connection between this problem and number-on-forehead communication complexity, through which we are able to obtain a tight lower bound on deterministic triangle detection.	2017-12-15 14:09:32	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.605072557926178, "perplexity": 477.59658689326307}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948572676.65/warc/CC-MAIN-20171215133912-20171215155912-00700.warc.gz"}
https://en.wikipedia.org/wiki/Loop_tiling	# Loop nest optimization (Redirected from Loop tiling) In computer science and particularly in compiler design, loop nest optimization (LNO) is an optimization technique that applies a set of loop transformations for the purpose of locality optimization or parallelization or other loop overhead reduction of the loop nests. One classical usage is to reduce memory access latency or the cache bandwidth necessary due to cache reuse for some common linear algebra algorithms. The technique used to produce this optimization is called loop tiling,[1] also known as loop blocking[2] or strip mine and interchange. ## Overview Loop tiling partitions a loop's iteration space into smaller chunks or blocks, so as to help ensure data used in a loop stays in the cache until it is reused. The partitioning of loop iteration space leads to partitioning of large array into smaller blocks, thus fitting accessed array elements into cache size, enhancing cache reuse and eliminating cache size requirements. An ordinary loop ```for(i=0; i<N; ++i){ ... } ``` can be blocked with a block size B by replacing it with ```for(j=0; j<N; j+=B){ for(i=j; i<min(N, j+B); ++i){ .... } } ``` where min() is a function returning the minimum of its arguments. ## Example: matrix-vector multiplication The following is an example of matrix vector multiplication. There are three arrays, each with 100 elements. The code does not partition the arrays into smaller sizes. ``` int i, j, a[100][100], b[100], c[100]; int n = 100; for (i = 0; i < n; i++) { c[i] = 0; for (j = 0; j < n; j++) { c[i] = c[i] + a[i][j] * b[j]; } } ``` After we apply loop tiling using 2 * 2 blocks, our code looks like: ``` int i, j, x, y, a[100][100], b[100], c[100]; int n = 100; for (i = 0; i < n; i += 2) { c[i] = 0; c[i + 1] = 0; for (j = 0; j < n; j += 2) { for (x = i; x < min(i + 2, n); x++) { for (y = j; y < min(j + 2, n); y++) { c[x] = c[x] + a[x][y] * b[y]; } } } } ``` The original loop iteration space is n by n. The accessed chunk of array a[i, j] is also n by n. When n is too large and the cache size of the machine is too small, the accessed array elements in one loop iteration (for example, `i = 1`, `j = 1 to n`) may cross cache lines, causing cache misses. ## Tiling size It is not always easy to decide what value of tiling size is optimal for one loop because it demands an accurate estimate of accessed array regions in the loop and the cache size of the target machine. The order of loop nests (loop interchange) also plays an important role in achieving better cache performance. Explicit blocking requires choosing a tile size based on these factors. By contrast, cache-oblivious algorithms are designed to make efficient use of cache without explicit blocking. ## Example: matrix multiplication Many large mathematical operations on computers end up spending much of their time doing matrix multiplication. The operation is: C = A×B where A, B, and C are N×N arrays. Subscripts, for the following description, are in the form `C[row][column]`. The basic loop is: ```int i, j, k; for (i = 0; i < N; ++i) { for (j = 0; j < N; ++j) { C[i][j] = 0; for (k = 0; k < N; ++k) C[i][j] += A[i][k] * B[k][j]; } } ``` There are three problems to solve: • Floating point additions take some number of cycles to complete. In order to keep an adder with multiple cycle latency busy, the code must update multiple accumulators in parallel. • Machines can typically do just one memory operation per multiply–add, so values loaded must be reused at least twice. • Typical PC memory systems can only sustain one 8-byte doubleword per 10–30 double-precision multiply–adds, so values loaded into the cache must be reused many times. The original loop calculates the result for one entry in the result matrix at a time. By calculating a small block of entries simultaneously, the following loop reuses each loaded value twice, so that the inner loop has four loads and four multiply–adds, thus solving problem #2. By carrying four accumulators simultaneously, this code can keep a single floating point adder with a latency of 4 busy nearly all the time (problem #1). However, the code does not address the third problem. (Nor does it address the cleanup work necessary when N is odd. Such details will be left out of the following discussion.) ```for (i = 0; i < N; i += 2) { for (j = 0; j < N; j += 2) { acc00 = acc01 = acc10 = acc11 = 0; for (k = 0; k < N; k++) { acc00 += B[k][j + 0] * A[i + 0][k]; acc01 += B[k][j + 1] * A[i + 0][k]; acc10 += B[k][j + 0] * A[i + 1][k]; acc11 += B[k][j + 1] * A[i + 1][k]; } C[i + 0][j + 0] = acc00; C[i + 0][j + 1] = acc01; C[i + 1][j + 0] = acc10; C[i + 1][j + 1] = acc11; } } ``` This code has had both the `i` and `j` iterations blocked by a factor of two, and had both the resulting two-iteration inner loops completely unrolled. This code would run quite acceptably on a Cray Y-MP (built in the early 1980s), which can sustain 0.8 multiply–adds per memory operation to main memory. A machine like a 2.8 GHz Pentium 4, built in 2003, has slightly less memory bandwidth and vastly better floating point, so that it can sustain 16.5 multiply–adds per memory operation. As a result, the code above will run slower on the 2.8 GHz Pentium 4 than on the 166 MHz Y-MP! A machine with a longer floating-point add latency or with multiple adders would require more accumulators to run in parallel. It is easy to change the loop above to compute a 3x3 block instead of a 2x2 block, but the resulting code is not always faster. The loop requires registers to hold both the accumulators and the loaded and reused A and B values. A 2x2 block requires 7 registers. A 3x3 block requires 13, which will not work on a machine with just 8 floating point registers in the ISA. If the CPU does not have enough registers, the compiler will schedule extra loads and stores to spill the registers into stack slots, which will make the loop run slower than a smaller blocked loop. Matrix multiplication is like many other codes in that it can be limited by memory bandwidth, and that more registers can help the compiler and programmer reduce the need for memory bandwidth. This register pressure is why vendors of RISC CPUs, who intended to build machines more parallel than the general purpose x86 and 68000 CPUs, adopted 32-entry floating-point register files. The code above does not use the cache very well. During the calculation of a horizontal stripe of C results, one horizontal stripe of A is loaded, and the entire matrix B is loaded. For the entire calculation, C is stored once (that's good), A is loaded into the cache once (assuming a stripe of A fits in the cache with a stripe of B), but B is loaded N/ib times, where ib is the size of the strip in the C matrix, for a total of N3/ib doubleword loads from main memory. In the code above, ib is 2. The next step to reducing the memory traffic is to make ib as large as possible. It needs to be larger than the "balance" number reported by streams. In the case of one particular 2.8 GHz Pentium 4 system used for this example, the balance number is 16.5. The second code example above cannot be extended directly, since that would require many more accumulator registers. Instead, the loop is blocked over i. (Technically, this is actually the second time i is blocked, as the first time was the factor of 2.) ```for (ii = 0; ii < N; ii += ib) { for (j = 0; j < N; j += 2) { for (i = ii; i < ii + ib; i += 2) { acc00 = acc01 = acc10 = acc11 = 0; for (k = 0; k < N; k++) { acc00 += B[k][j + 0] * A[i + 0][k]; acc01 += B[k][j + 1] * A[i + 0][k]; acc10 += B[k][j + 0] * A[i + 1][k]; acc11 += B[k][j + 1] * A[i + 1][k]; } C[i + 0][j + 0] = acc00; C[i + 0][j + 1] = acc01; C[i + 1][j + 0] = acc10; C[i + 1][j + 1] = acc11; } } } ``` With this code, we can set ib to be anything we like, and the number of loads of the B matrix will be reduced by that factor. This freedom has a cost: we are now keeping a N×ib slices of the A matrix in the cache. As long as that fits, this code will not be limited by the memory system. So what size matrix fits? Our example system, a 2.8 GHz Pentium 4, has a 16KB primary data cache. With ib=20, the slice of the A matrix in this code will be larger than the primary cache when N > 100. For problems larger than that, we'll need another trick. That trick is reducing the size of the stripe of the B matrix by blocking the k loop, so that the stripe is of size ib × kb. Blocking the k loop means that the C array will be loaded and stored N/kb times, for a total of ${\displaystyle 2N^{3}/kb}$ memory transfers. B is still transferred N/ib times, for ${\displaystyle N^{3}/ib}$ transfers. So long as 2N/kb + N/ib < N/balance the machine's memory system will keep up with the floating point unit and the code will run at maximum performance. The 16KB cache of the Pentium 4 is not quite big enough: we might choose ib=24 and kb=64, thus using 12KB of the cache—we don't want to completely fill it, since the C and B arrays have to have some room to flow through. These numbers comes within 20% of the peak floating-point speed of the processor. Here is the code with loop `k` blocked. ```for (ii = 0; ii < N; ii += ib) { for (kk = 0; kk < N; kk += kb) { for (j=0; j < N; j += 2) { for(i = ii; i < ii + ib; i += 2 ) { if (kk == 0) acc00 = acc01 = acc10 = acc11 = 0; else { acc00 = C[i + 0][j + 0]; acc01 = C[i + 0][j + 1]; acc10 = C[i + 1][j + 0]; acc11 = C[i + 1][j + 1]; } for (k = kk; k < kk + kb; k++) { acc00 += B[k][j + 0] * A[i + 0][k]; acc01 += B[k][j + 1] * A[i + 0][k]; acc10 += B[k][j + 0] * A[i + 1][k]; acc11 += B[k][j + 1] * A[i + 1][k]; } C[i + 0][j + 0] = acc00; C[i + 0][j + 1] = acc01; C[i + 1][j + 0] = acc10; C[i + 1][j + 1] = acc11; } } } } ``` The above code examples do not show the details of dealing with values of N which are not multiples of the blocking factors. Compilers which do loop nest optimization emit code to clean up the edges of the computation. For example, most LNO compilers would probably split the kk == 0 iteration off from the rest of the `kk` iterations, in order to remove the if statement from the `i` loop. This is one of the values of such a compiler: while it is straightforward to code the simple cases of this optimization, keeping all the details correct as the code is replicated and transformed is an error-prone process. The above loop will only achieve 80% of peak flops on the example system when blocked for the 16KB L1 cache size. It will do worse on systems with even more unbalanced memory systems. Fortunately, the Pentium 4 has 256KB (or more, depending on the model) high-bandwidth level-2 cache as well as the level-1 cache. We are presented with a choice: • We can adjust the block sizes for the level-2 cache. This will stress the processor's ability to keep many instructions in flight simultaneously, and there is a good chance it will be unable to achieve full bandwidth from the level-2 cache. • We can block the loops again, again for the level-2 cache sizes. With a total of three levels of blocking (for the register file, for the L1 cache, and for the L2 cache), the code will minimize the required bandwidth at each level of the memory hierarchy. Unfortunately, the extra levels of blocking will incur still more loop overhead, which for some problem sizes on some hardware may be more time consuming than any shortcomings in the hardware's ability to stream data from the L2 cache. Rather than specifically tune for one particular cache size, as in the first example, a cache-oblivious algorithm is designed to take advantage of any available cache, no matter what its size is. This automatically takes advantage of two or more levels of memory hierarchy, if available. Cache-oblivious algorithms for matrix multiplication are known.	2019-09-18 03:36:31	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 2, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.42645859718322754, "perplexity": 1237.6155937922174}, "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-2019-39/segments/1568514573176.51/warc/CC-MAIN-20190918024332-20190918050332-00190.warc.gz"}
https://studydaddy.com/question/what-is-the-valence-electron-configuration-for-potassium	Waiting for answer This question has not been answered yet. You can hire a professional tutor to get the answer. QUESTION # What is the valence electron configuration for potassium? K = 1s^2 2s^2 2p^6 3s^2 3p^6 4s^1. This simplifies to a noble gas notation of:K = [Ar] 4s^1 Potassium (K) is an Alkali metal found in the first column (Group I) and in the fourth row (period 4) of . All alkali metals end in s^1 for their . The total electron configuration for potassium is K = 1s^2 2s^2 2p^6 3s^2 3p^6 4s^1 This simplifies to a noble gas notation of K = [Ar] 4s^1 We use the symbol for argon (Ar) because it is the last noble gas in the period (row) above potassium on the periodic table. The are the highest energy level s and p orbital electrons of the atom. This would mean potassium has a valence shell of 4s^1 I hope this was helpful. SMARTERTEACHER	2019-04-21 20:57:58	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6255771517753601, "perplexity": 3099.571624557513}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1555578532882.36/warc/CC-MAIN-20190421195929-20190421220805-00022.warc.gz"}
https://clay6.com/qa/sat/chemistry/chemical-equilibrium	# Recent questions and answers in chemical equilibrium ### What is the best description of the change that occurs when $Na_2O(s)$ is dissolved in water? To see more, click for all the questions in this category.	2020-09-21 00:14:07	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.538528561592102, "perplexity": 551.0490947212734}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1600400198868.29/warc/CC-MAIN-20200920223634-20200921013634-00245.warc.gz"}
https://socratic.org/questions/how-do-you-find-the-first-and-second-derivatives-of-y-6x-2-4-2x-2-using-the-quot	× # How do you find the first and second derivatives of y= (6x^2+4)/ (2x+2) using the quotient rule? Jun 17, 2018 $y ' = \frac{2 {x}^{2} + 6 x - 2}{x + 1} ^ 2$ $y ' ' = \frac{10}{x + 1} ^ 3$ #### Explanation: Using the Quotient rule we get $y ' = \frac{6 x \left(x + 1\right) - \left(3 {x}^{2} + 2\right)}{x + 1} ^ 2$ which simplifyes to $y ' = \frac{3 {x}^{2} + 6 x - 2}{x + 1} ^ 2$ Again by the Quotient rule we get $y ' ' = \frac{\left(6 x + 6\right) {\left(x + 1\right)}^{2} - \left(3 {x}^{2} + 6 x - 2\right) \left(2 \left(x + 1\right)\right)}{x + 1} ^ 3$ Simplifying we obtain $y ' ' = \frac{10}{x + 1} ^ 3$	2018-09-25 02:50:44	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 6, "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": 0.9733036756515503, "perplexity": 2168.4426675842547}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267160923.61/warc/CC-MAIN-20180925024239-20180925044639-00228.warc.gz"}
https://academy.vertabelo.com/course/postgresql-insert-update-delete-commands/advanced-features/values-from-another-column/update-using-value-from-another-column	Deals Of The Week - hours only!Up to 80% off on all courses and bundles.-Close Introduction Inserting and updating NULLs Conditions in UPDATE and DELETE Updating the list of columns in one query Using values from another column 13. Update using a value from another column Return rows in INSERT, UPDATE, DELETE Summary ## Instruction Good job! So far, we've always inserted fixed values when updating data. However, when we update, we can also refer to values from another column. Suppose someone has found mistakes made in the scoring of written and oral English exams. Say that each written English exam score needs four points added to it and each oral English exam score is actually six points less than the written exam score. Here's the statement to correct this mistake: UPDATE exam SET written_exam_score = written_exam_score + 4, oral_exam_score = written_exam_score - 6 WHERE subject = 'English' With the SET command, we assigned to the written_exam_score column its current value plus 4 points. Similarly, we updated the oral_exam_score column, only we subtracted six points from the written exam score (we took the value from written_exam_score). In the WHERE clause, we limited this UPDATE operation to exams for English. ## Exercise The oral exams with no date assigned take place four days after the written_exam_date. Update the data. ### Stuck? Here's a hint! To add four days to written_exam_date, use: written_exam_date + 4	2023-03-25 23:54:06	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.2759171724319458, "perplexity": 5392.143983280208}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945376.29/warc/CC-MAIN-20230325222822-20230326012822-00568.warc.gz"}
https://courses.ansys.com/index.php/courses/structures-in-consumer-goods/lessons/analyzing-the-auxetic-design-of-a-shoe-base-lesson-2/	# Analyzing the Auxetic Design of a Shoe Base — Lesson 2 An auxetic structure expands in the transverse direction while being stretched in the longitudinal direction. Although the material could have a positive Poisson’s ratio, the overall geometry exhibits a negative Poisson’s ratio effect. Due to this unique property, the auxetic structure is widely applied to sports. This lesson illustrates a simulation of the auxetic structure of a shoe base.	2022-01-28 08:36:58	{"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": 0.8158724904060364, "perplexity": 1972.5173829355278}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320305423.58/warc/CC-MAIN-20220128074016-20220128104016-00168.warc.gz"}
https://zbmath.org/0764.33009	# zbMATH — the first resource for mathematics Functional inequalities for hypergeometric functions and complete elliptic integrals. (English) Zbl 0764.33009 The authors obtain a number of inequalities for the classical $$_ 2F_ 1$$ hypergeometric functions and for two of its special cases, the complete elliptic integrals of the first and second kind. A typical one is $$-g(x)>{_ 2F_ 1}(a,b;a-1b:X)>-g(x)/B(a,b)$$ for $$0<a,b,x<1$$, where $$g(x)=x^{-1}\log(1-x)$$. The lower estimate is sharp at $$x=1$$ and the upper estimate is sharp at $$x=0$$. For $$a,b>1$$, $$0<x<1$$, these inequalities swap and for $$a=b=1$$ there is equality. 33E05 Elliptic functions and integrals 33C05 Classical hypergeometric functions, $${}_2F_1$$	2021-06-24 00:55:14	{"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": 0.8685802817344666, "perplexity": 297.5949161010631}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488544264.91/warc/CC-MAIN-20210623225535-20210624015535-00617.warc.gz"}
http://rcd.ics.org.ru/authors/detail/627-nataliya_stankevich	0 2013 Impact Factor # Nataliya Stankevich 410054, Saratov, Politehnicheskaya str., 77 Yuri Gagarin State Technical University of Saratov ## Publications: Stankevich N. V., Dvorak A., Astakhov V. V., Jaros P., Kapitaniak M., Perlikowski P., Kapitaniak T. Chaos and Hyperchaos in Coupled Antiphase Driven Toda Oscillators 2018, vol. 23, no. 1, pp. 120-126 Abstract The dynamics of two coupled antiphase driven Toda oscillators is studied. We demonstrate three different routes of transition to chaotic dynamics associated with different bifurcations of periodic and quasi-periodic regimes. As a result of these, two types of chaotic dynamics with one and two positive Lyapunov exponents are observed. We argue that the results obtained are robust as they can exist in a wide range of the system parameters. Keywords: chaos, hyperchaos, Toda oscillator Citation: Stankevich N. V., Dvorak A., Astakhov V. V., Jaros P., Kapitaniak M., Perlikowski P., Kapitaniak T., Chaos and Hyperchaos in Coupled Antiphase Driven Toda Oscillators, Regular and Chaotic Dynamics, 2018, vol. 23, no. 1, pp. 120-126 DOI:10.1134/S1560354718010094 Kuptsov P. V., Kuznetsov S. P., Stankevich N. V. A Family of Models with Blue Sky Catastrophes of Different Classes 2017, vol. 22, no. 5, pp. 551-565 Abstract A generalized model with bifurcations associated with blue sky catastrophes is introduced. Depending on an integer index $m$, different kinds of attractors arise, including those associated with quasi-periodic oscillations and with hyperbolic chaos. Verification of the hyperbolicity is provided based on statistical analysis of intersection angles of stable and unstable manifolds. Keywords: dynamical system, blue sky catastrophe, quasi-periodic oscillations, hyperbolic chaos, Smale–Williams solenoid Citation: Kuptsov P. V., Kuznetsov S. P., Stankevich N. V., A Family of Models with Blue Sky Catastrophes of Different Classes, Regular and Chaotic Dynamics, 2017, vol. 22, no. 5, pp. 551-565 DOI:10.1134/S1560354717050069	2018-12-13 21:27:52	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.36766085028648376, "perplexity": 2325.45618680635}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1544376825098.68/warc/CC-MAIN-20181213193633-20181213215133-00498.warc.gz"}
https://web2.0calc.com/questions/what-is-the-largest-n-such-that-a-2-306-cdot-3-340	+0 # What is the largest $n$ such that $a = 2^{306} \cdot 3^{340}$ is a perfect $n$th power? 0 123 3 What is the largest $n$ such that $a = 2^{306} \cdot 3^{340}$ is a perfect $n$th power? Jun 11, 2022 #1 +2448 0 Hint: it will be $$\sqrt{2^{306}} \times \sqrt {3^{340}}$$, that way $$n^2 = {2^{306}} \times {3^{340}}$$ Jun 11, 2022 #2 0 No we are asked to find the $n$ for which is the square of that. We need the largest n th root. Jun 12, 2022 #3 0 n= Greatest Common Divisor of 306 and 340 306=2×3^2×17 340=2^2×5×17 n=GCD(306,340)=2×17=34 a=2^306×3^340=2^(9×34)×3^(10×34)=(2^9)^34×(3^10)^34 a=(2^9×3^10)^34=30,233,088^34 n=34 Jun 12, 2022	2022-12-03 05:09:32	{"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": 0.9619250297546387, "perplexity": 975.6535469733412}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1669446710924.83/warc/CC-MAIN-20221203043643-20221203073643-00776.warc.gz"}
https://brightside.live/e1s6ja/axial-flow-compressor-formula-db7876	As with any heat engine, fuel efficiency is strongly related to the compression ratio, so there is very strong financial need to improve the compressor stages beyond these sorts of ratios. The rotating airfoils, also known as blades or rotors, accelerate the fluid. It is found that for the system investigated there is an important nondimensional parameter on which this response depends. P in kW = Here. As Griffith had originally noted in 1929, the large frontal size of the centrifugal compressor caused it to have higher drag than the narrower axial-flow type. α Calculate actual velocity to standard volume flow, and reversely In a multi-stage compressor, at the high pressure stages, axial velocity is very small. Current turbofan engines have fans that operate at Mach 1.7 or more, and require significant containment and noise suppression structures to reduce blade loss damage and noise. It is a rotating, airfoil-based compressor in which the gas or working fluid principally flows parallel to the axis of rotation. The only obvious effort was a test-bed compressor built by Hayne Constant, Griffith's colleague at the Royal Aircraft Establishment. V r Additionally the axial-flow design could improve its compression ratio simply by adding additional stages and making the engine slightly longer. (Eds.) Non-uniformity of air flow in the rotor blades may disturb local air flow in the compressor without upsetting it. Positive Stalling- Flow separation occur on the suction side of the blade. for positive values of J, slope of the curve is negative and vice versa. α{\displaystyle \alpha } is the guide vane angle and β{\displaystyle \beta } is the blade angle. The energy level of the fluid increases as it flows through the compressor due to the a… If the downstream pressure is increased beyond the maximum possible the compressor will stall and become unstable. As with any heat engine, fuel efficiency is strongly related to the compression ratio, so there is very strong financial need to improve the compressor stages beyond these sorts of ratios. Axial compressors consist of rotating and stationary components. centrifugal compressors). As an aircraft changes speed or altitude, the pressure of the air at the inlet to the compressor will vary. Whether this parameter is above or below a critical value determines which mode of compressor instability, rotating stall or surge, will be encountered at the stall line. The General Electric J79 was the first major example of a variable stator design, and today it is a common feature of most military engines. Other early jet efforts, notably those of Frank Whittle and Hans von Ohain, were based on the more robust and better understood centrifugal compressor which was widely used in superchargers. w Once in flight the inlet pressure drops, but the inlet speed increases (due to the forward motion of the aircraft) to recover some of this pressure, and the compressor tends to run at a single speed for long periods of time. The airfoil profiles are optimized and matched for specific velocities and turning. {{#invoke:Citation/CS1\|citation 12. Inorder to provide a better understanding of 'Mechanics and Thermodynamics of Propulsion,' 2nd edn, Prentice Hall, 1991. By the 1950s every major engine development had moved on to the axial-flow type. Contours of efficiency are drawn as well as performance lines for operation at particular rotational speeds. {\displaystyle \psi \,} The stationary airfoils, also known as vanes or stators, convert the increased kinetic energy into static pressure through diffusion and redirect the flow direction of the fluid to prepare it for the rotor blades of the next stage. It shows the mass flow along the horizontal axis, typically as a percentage of the design mass flow rate, or in actual units. Though valve position is set for lower flow rate say point G but compressor will work according to normal stable operation point say E, so path E-F-P-G-E will be followed leading to breakdown of flow, hence pressure in the compressor falls further to point H(PH{\displaystyle P_{H}\,}). {\displaystyle r_{2}\,} A shaft drives a central drum which is retained by bearings inside of a stationary tubular casing. In England, Hayne Constant reached an agreement with the steam turbine company Metropolitan-Vickers (Metrovick) in 1937, starting their turboprop effort based on the Griffith design in 1938. This phenomenon depending upon the blade-profile leads to reduced compression and drop in engine power. Compressors are typically driven by an electric motor or a steam or a gas turbine.[1]. Early engines were designed for simplicity, and used a single large compressor spinning at a single speed. S. 'Turbo-Machinery Dynamics: Design and Operation,' New York: McGraw-Hill: 2005. Fixed geometry compressors, like those used on early jet engines, are limited to a design pressure ratio of about 4 or 5:1. Hence axial flow compressors have many more stages than axial flow turbines. Axial compressors are integral to the design of large gas turbines such as jet engines, high speed ship engines, and small scale power stations. Modern jet engines use a series of compressors, running at different speeds; to supply air at around 40:1 pressure ratio for combustion with sufficient flexibility for all flight conditions. In the centrifugal-flow design the compressor itself had to be larger in diameter, which was much more difficult to "fit" properly on the aircraft. (2007). Known and measured properties are: a. The pressure rise is indicated on the vertical axis as a ratio between inlet and exit stagnation pressures. It had a critical value which predicted either rotating stall or surge where the slope of pressure ratio against flow changed from negative to positive. Bleed systems were already commonly used anyway, to provide airflow into the turbine stage where it was used to cool the turbine blades, as well as provide pressurized air for the air conditioning systems inside the aircraft. Due to this back flow, pressure in pipe will decrease because this unequal pressure condition cannot stay for a long period of time. Bleed systems were already commonly used anyway, to provide airflow into the turbine stage where it was used to cool the turbine blades, as well as provide pressurized air for the air conditioning systems inside the aircraft. [3] The cross-sectional area between rotor drum and casing is reduced in the flow direction to maintain an optimum Mach number using variable geometry as the fluid is compressed. }} It is a rotating, airfoil-based compressor in which the gas or working fluid principally flows parallel to the axis of rotation, or axially. tan For a given geometry the temperature rise depends on the square of the tangential Mach number of the rotor row. Standard flow calculation with temperature and pressure compensation. Thus, rotating stall deceases the effectiveness of the compressor. {{#invoke:Category handler\|main}}{{#invoke:Category handler\|main}}[citation needed] Thus, rotating stall decreases the effectiveness of the compressor. Typically the instability will be at the Helmholtz frequency of the system, taking the downstream plenum into account. 20. Pump input power calculation formula or pump shaft power calculation formula. Operating efficiency is highest close to the stall line. Schematic sketch showing location of compressor in typical jet propulsion engine. Stages losses in compressor are mainly due to blade friction, flow separation, unsteady flow and vane-blade spacing. Designing the rotor passage with a diffusing capability can produce a pressure rise in addition to its normal functioning. They grow larger at very low flow rate and affect the entire blade height. H Losses are due to blade friction, flow separation, unsteady flow and vane-blade spacing. The law of moment of momentum states that the sum of the moments of external forces acting on a fluid which is temporarily occupying the control volume is equal to the net change of angular momentum flux through the control volume. 'Mechanics and Thermodynamics of Propulsion,' 2nd edn, Prentice Hall, 1991. Higher stage pressure ratios are also possible if the relative velocity between fluid and rotors is supersonic, but this is achieved at the expense of efficiency and operability. Between the drum and the casing are rows of airfoils, each row connected to either the drum or the casing in an alternating manner. By incorporating variable stators in the first five stages, General Electric Aircraft Engines has developed a ten-stage axial compressor capable of operating at a 23:1 design pressure ratio. Therefore the flow range of the meter covers this application. I. N. T. A., D. Carlos Sánchez Tarifa, a su extremada Axial Flow and Radial Flow Gas Turbines 7.1 INTRODUCTION TO AXIAL FLOW TURBINES The axial ﬂow gas turbine is used in almost all applications of gas turbine power plant. 'The Design of High-Efficiency Turbomachinery and Turbines,' 2nd edn, Prentice Hall, 1998. Axial flow compressors produce a continuous flow of compressed gas, and have the benefits of high efficiency and large mass flow rate, particularly in relation to their size and cross-section. This produces greater pressure rise per stage which constitutes a stator and a rotor together. This differs from other rotating compressors such as centrifugal compressor, axi-centrifugal compressors and mixed-flow compressors where the fluid flow will include a "radial component" through the compressor. This page was last edited on 8 January 2015, at 10:49. Additionally the compressor may stall if the inlet conditions change abruptly, a common problem on early engines. Perry, R.H. and Green, D.W. crossing the surge line, is caused by the compressor trying to deliver air, still running at the same speed, to a higher exit pressure. For startup they are rotated to "closed", reducing compression, and then are rotated back into the airflow as the external conditions require. Hence the rear stage develops a significantly lower pressure ratio than the first stage. Bibliography Includes bibliographical references (p. [262]-267). His rotor and stator blading described in one of his patents[7] had little or no camber although in some cases the blade design was based on propeller theory. 1 [4] 1 A nonlinear model is developed to predict the transient response of a compression system subsequent to a perturbation from steady operating conditions. This is achieved normally through the use of adjustable stators or with valves that can bleed fluid from the main flow between stages (inter-stage bleed). This two-spool design, pioneered on the Bristol Olympus, resulted in increased efficiency. The fluid velocity is increased through the rotor, and the stator converts kinetic energy to pressure energy. Thus the left blade will receive the flow at higher incidence and the blade to its right with decreased incidence. The left blade will experience more stall while the blade to its right will experience lesser stall. In the previous research, it is very difficult to accurately model the fouled axial flow compressor. 3. Because early axial compressors were not efficient enough a number of papers in the early 1920s claimed that a practical jet engine would be impossible to construct. This was also used to help start the engine, allowing it to be spun up without compressing much air by bleeding off as much as possible. Axial flow compressor computer program for calculating off design performance /Program 4/ Computer program for calculating off-design aerodynamic performance of axial flow compressor. {\displaystyle \alpha _{1}=\alpha _{3}\,} The performance of a compressor is defined according to its design. Negative stall is negligible compared to the positive stall because flow separation is least likely to occur on the pressure side of the blade. Contents. used in the parametric study of axial compressors: – Flow coefficient, – Stage loading, – Degree of reaction, R. x – Diffusion factor, D * φ=C. α ⁡ Steam-turbine designer Charles Algernon Parsons, for example, recognized that a turbine which produced work by virtue of a fluid's static pressure (i.e. Perry's Chemical Engineers' Handbook (8th ed.). for positive values of J, slope of the curve is negative and vice-versa. 'The Design of High-Efficiency Turbomachinery and Turbines,' 2nd edn, Prentice Hall, 1998. J=tan⁡β2+tan⁡α3){\displaystyle J=\tan \beta _{2}+\tan \alpha _{3})\,} is constant, ψ′=1−J⁡(ϕ′){\displaystyle \psi ^{'}=1-J(\phi ^{'})\,}, J=1−ψ′ϕ′{\displaystyle J={\frac {1-\psi ^{'}}{\phi ^{'}}}\,}....................(3), ψ=1−J⁡(ϕ){\displaystyle \psi =1-J(\phi )\,}, ψ=1−ϕ⁡(1−ψ′ϕ′){\displaystyle \psi =1-\phi ({\frac {1-\psi ^{'}}{\phi ^{'}}})\,} .. [from (3)]. Cascade Data for the Blade Design of High-Performance Axial Flow Com-pressors", por S. M. BOGDONOFF. Template:Rellink, An axial compressor is a machine that can continuously pressurise gases. Axial compressors consist of rotating and stationary components. The relative motion of the blades to the fluid adds velocity or pressure or both to the fluid as it passes through the rotor. Surge cycle Fouling is the most important performance degradation factor, so it is necessary to accurately predict the effect of fouling on engine performance. For all of these reasons, axial compressors on modern jet engines are considerably more complex than those on earlier designs. The blade profile and variation of pressure and velocity in the stage are shown in the Fig. This differs from other rotating compressors such as centrifugal compressor, axi-centrifugal compressors and mixed-flow compressors where the fluid flow will include a "radial component" through the compressor. = The value of (tan⁡β2+tan⁡α1){\displaystyle (\tan \beta _{2}+\tan \alpha _{1})\,} doesn't change for a wide range of operating points till stalling. A map shows the performance of a compressor and allows determination of optimal operating conditions. If the downstream pressure is increased beyond the maximum possible the compressor will stall and become unstable. centrifugal compressors). a / U h /U C. w / U. That is why left portion of the curve from the surge point is called unstable region and may cause damage to the machine. }}. As a general rule-of-thumb we can assume that each stage in a given compressor has the same temperature rise (Delta T). [5], we can determine performance of axial compressor. In 1940, after the successful run of Whittle's centrifugal-flow design, their effort was re-designed as a pure jet, the Metrovick F.2. If 50% of the pressure rise in a stage is obtained at the rotor section, it is said to have a 50% reaction. He showed that the use of airfoils instead of the flat blades would increase efficiency to the point where a practical jet engine was a real possibility. The stage efficiency drops with higher losses. 2 For a given geometry the temperature rise depends on the square of the tangential Mach number of the rotor row. What happens, i.e. Let some blades receives flow at higher incidence, this blade will stop positively. In a multi-stage compressor, at the high pressure stages, axial velocity is very small. An axial compressor is a gas compressor that can continuously pressurize gases. The change in pressure energy is calculated through degree of reaction. per minute at pressures up to 45 p.s.i.[9]. Though valve position is set for lower flow rate say point G but compressor will work according to normal stable operation point say E, so path E-F-P-G-E will be followed leading to breakdown of flow, hence pressure in the compressor falls further to point H( ). The rotor reduces the relative kinetic head of the fluid and adds it to the absolute kinetic head of the fluid i.e., the impact of the rotor on the fluid particles increases its velocity (absolute) and thereby reduces the relative velocity between the fluid and the rotor. 3 The Euler Turbine Equation . !y ivvesti- gated and gradually minimized. The stationary blades slow the fluid, converting the circumferential component of flow into pressure. The rotor reduces the relative kinetic head of the fluid and adds it to the absolute kinetic head of the fluid i.e., the impact of the rotor on the fluid particles increases their velocity (absolute) and thereby reduces the relative velocity between the fluid and the rotor. ISBN 0-13-312000-7. In some cases, if the stall occurs near the front of the engine, all of the stages from that point on will stop compressing the air. If 50% of the pressure rise in a stage is obtained at the rotor section, it is said to have a 50% reaction. Due to high performance, high reliability and flexible operation during the flight envelope, they are also used in aerospace engines.[2]. Delivery pressure significantly drops with large stalling which can lead to flow reversal. {{#invoke: Navbox \| navbox }}, {{#invoke:citation/CS1\|citation The pressure rise results in a stagnation temperature rise. Other early jet efforts, notably those of Frank Whittle and Hans von Ohain, were based on the more robust and better understood centrifugal compressor which was widely used in superchargers. The axial flow compressors can be of impulse type or reaction type. . 4 Multistage Axial Compressors. ISBN 0-07-145369-5. Once in flight the inlet pressure drops, but the inlet speed increases (due to the forward motion of the aircraft) to recover some of this pressure, and the compressor tends to run at a single speed for long periods of time. Also From an energy exchange point of view axial compressors are reversed turbines. Development of the axial ﬂow gas turbine was hindered by the need to obtain both a high-enough ﬂow rate and compression ratio from a compressor … doesn't change for a wide range of operating points till stalling. But in actual practice, the operating point of the compressor deviates from the design- point which is known as off-design operation. The stationary airfoils, also known as stators or vanes, convert the increased rotational kinetic energy into static pressure through diffusion and redirect the flow direction of the fluid, preparing it for the rotor blades of the next stage. Westinghouse also entered the race in 1942, their project proving to be the only successful one of the US efforts, later becoming the J30. }} the whole engine dramatically. Northrop also started their own project to develop a turboprop, which the US Navy eventually contracted in 1943. In the centrifugal-flow design the compressor itself had to be larger in diameter, which was much more difficult to "fit" properly on the aircraft. When the compressor is operating as part of a complete gas turbine engine, as opposed to on a test rig, a higher delivery pressure at a particular speed can be caused momentarily by burning too-great a step-jump in fuel which causes a momentary blockage until the compressor increases to the speed which goes with the new fuel flow and the surging stops. Thus, a practical limit on the number of stages, and the overall pressure ratio, comes from the interaction of the different stages when required to work away from the design conditions. There is simply no "perfect" compressor for this wide range of operating conditions. In the United States, both Lockheed and General Electric were awarded contracts in 1941 to develop axial-flow engines, the former a pure jet, the latter a turboprop. 2. ψ=∆ 0 =∆ In the jet engine application, the compressor faces a wide variety of operating conditions. 3 They do, however, require several rows of airfoils to achieve a large pressure rise, making them complex and expensive relative to other designs (e.g. Treager, Irwin E. 'Aircraft Gas Turbine Engine Technology' 3rd edn, McGraw-Hill Book Company, 1995. A surge or stall line identifies the boundary to the left of which the compressor performance rapidly degrades and identifies the maximum pressure ratio that can be achieved for a given mass flow. In a rotor with blades moving say towards right. 19690001542 . A pair of one row of rotating airfoils and the next row of stationary airfoils is called a stage. {\displaystyle V_{w2}\,} The increase in velocity of the fluid is primarily in the tangential direction (swirl) and the stator removes this angular momentum. In an upcoming blog, Flow Coefficient and Work Coefficient Application, we’ll see how they can be used to properly choose the right class of machine for a given task. McGraw Hill. So the recommended operation range is on the right side of the surge line. P There is simply no "perfect" compressor for this wide range of operating conditions. These “off-design” conditions can be mitigated to a certain extent by providing some flexibility in the compressor. The diffusing action in the stator converts the absolute kinetic head of the fluid into a rise in pressure. Also α1=α3{\displaystyle \alpha _{1}=\alpha _{3}\,} because of minor change in air angle at rotor ans stator, where α3{\displaystyle \alpha _{3}\,} is diffuser blade angle. off-design, of the compressor from ground idle to its highest corrected rotor speed, which for a civil engine may occur at top-of-climb, or, for a military combat engine, at take-off on a cold day. The pressure difference between the entry and exit of the rotor blade is called reaction pressure.The change in pressure energy is calculated through Degree of Reaction. [6] It is a situation of separation of air flow at the aero-foil blades of the compressor. In the jet engine application, the compressor faces a wide variety of operating conditions. 2000= (V∗π (〖100/2)〗^2∗3600)/1000,000. The only obvious effort was a test-bed compressor built by Hayne Constant, Griffith's colleague at the Royal Aircraft Establishment. The stage efficiency drops with higher losses. del Departamento de Motopropulsión del. 143 IB-ATE-nts10.01.doc Difference between compressors and turbines (page 95): Compressor Turbine Work transfer work input work output Pressure change pressure rise pressure drop D The energy level of the fluid increases as it flows through the compressor due to the action of the rotor blades which exert a torque on the fluid. In short, the rotor increases the absolute velocity of the fluid and the stator converts this into pressure rise. The compressor continues to work normally but with reduced compression. An axial compressor is typically made up of many alternating rows of rotating and stationary blades called rotors and stators, respectively, as shown in Figures 12.3 and 12.4.The first stationary row (which comes in front of the rotor) is … Later designs added a second turbine and divided the compressor into "low-pressure" and "high-pressure" sections, the latter spinning faster. Movement of the rotating stall can be observed depending upon the chosen reference frame. In some cases, if the stall occurs near the front of the engine, all of the stages from that point on will stop compressing the air. Real work on axial-flow engines started in the late 1930s, in several efforts that all started at about the same time. , with tangential velocity, 1 Fig. ψ=ϕ⁢(tan⁡α2−tan⁡α1){\displaystyle \psi =\phi (\tan \alpha _{2}-\tan \alpha _{1})\,}......... (1), tan⁡α2=1ϕ−tan⁡β2{\displaystyle \tan \alpha _{2}={\frac {1}{\phi }}-\tan \beta _{2}\,}................. (2), ψ=1−ϕ⁢(tan⁡β2+tan⁡α1){\displaystyle \psi =1-\phi (\tan \beta _{2}+\tan \alpha _{1})\,}. Models were developed for the blade row, the control perturbations launched by the moving blades, and the compressor. It creates obstruction in the passage between the blade to its left and itself. 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Altitude, the difference in the compressor calculate the compressor continues to work but!	2021-04-23 12:34:23	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.596645176410675, "perplexity": 2071.2595378502488}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039617701.99/warc/CC-MAIN-20210423101141-20210423131141-00113.warc.gz"}
http://susam.in/blog/apache-nutch-10-released/	## Apache Nutch 1.0 released Today, we received an announcement from the Nutch committer, Sami Siren that Apache Nutch 1.0 has been released. An extract from the announcement: Apache Nutch, a subproject of Apache Lucene, is open source web-search software. It builds on Lucene Java, adding web-specifics, such as a crawler, a link-graph database, parsers for HTML and other document formats. Apache Nutch 1.0 contains a number of bug fixes and improvements such as Solr Integration, new indexing framework and new scoring framework just to mention a few. Details can be found in the changes file: I have been waiting for this release for a long time as I made some contributions to this project and I wanted them to be available in official release so that I didn't have to maintain a separate set of patches for myself. These contributions were also my first contributions to an open source project. Let me list my contributions from the CHANGES.txt file. 62. NUTCH-559 - NTLM, Basic and Digest Authentication schemes for web/proxy server. (Susam Pal via dogacan) 77. NUTCH-44 - Too many search results, limits max results returned from a single search. (Emilijan Mirceski and Susam Pal via kubes) 80. NUTCH-612 - URL filtering was disabled in Generator when invoked from Crawl (Susam Pal via ab) 81. NUTCH-601 - Recrawling on existing crawl directory (Susam Pal via ab) In 2007, while playing with the search engine, I found that there was no way for Nutch to authenticate itself to intranet sites requiring HTTP authentication. So, I modified the module that deals with the HTTP protocol so that it could authenticate itself with configured credentials when challenged with authentication. With this change, Nutch now supports NTLM, Basic and Digest authentication schemes. More details on this can be found in NUTCH-559 (JIRA) and the Nutch wiki entry on HTTP authentication schemes. NUTCH-44 and NUTCH-612 were bug fixes. NUTCH-601 involved the removal of a minor irritant. In the days of Nutch 0.9, the crawler complained if a directory with the name 'crawl' already existed in the current directory. As a result, before beginning a re-crawl using the bin/nutch crawl command, we had to move the existing crawl directory to another location. After a discussion in the community, we agreed that it was better to avoid shuffling the crawl directories by allowing re-crawls on the same directory. The change was made and committed. Nutch users' mailing list has often received mails from users who wanted to know how they can enable support for authentication schemes in Nutch 0.9 by applying the patch in NUTCH-559. Patching Nutch 0.9 was a little cumbersome as the patch was generated against the trunk. With this release, the users can simply download Nutch 1.0 and configure the authentication schemes. #### Life Mysterious said: Congrats and thanx! I'm sure it would have been a long wait for a lot of users for configuring their authentication schemes. #### Paritosh said: Hey Dude, Finally caught up with your blog. Nice posts, quite informative. Have started following your blog. Cheers! Paritosh #### Utkarshraj Atmaram said: Congrats! Good to see your work being useful to the open source community.	2017-09-26 18:04:45	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.24350564181804657, "perplexity": 6239.852393968158}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1505818696677.93/warc/CC-MAIN-20170926175208-20170926195208-00697.warc.gz"}
http://physics.stackexchange.com/tags/voltage/new	Tag Info 1 How do voltage and voltage drops over a circuit relate to work done? The Volt unit is energy normalized to unit charge; Joule per Coulomb. Since the Amp unit is Coulomb per second, the product of the voltage across and current through a circuit element is the power associated with the circuit element. For a DC circuit, voltage and current are constant ... 1 But then that means that the electron in the 5 ohm circuit would have done 5x the amount of work (or work done on it) of the 1 ohm circuit over 5x the duration. You're confusing work with power here. Work has nothing to do with duration. If an electron crosses a potential difference of $V$ with any resistance in between, the work done is the same, ... 0 Typically the field strength is proportional to the voltage, so to get a higher field strength you need to increase the voltage. To see why this is you start from the basic formula for the field strength: $$B = k N I$$ where $B$ is the field strength, $N$ is the number of turns and $I$ is the current in the coil. $k$ is a constant that we'll ignore for ... 1 A few preliminary ideas which might help: It doesn't really matter what the speed of the electrons is - a current of 1 C/s (=1 A) just means that a coulomb worth of charge (equal to $6.2 \times 10^{18}$ electrons) passes each point in the circuit each second. Perhaps there is one electron travelling so fast that it does $6.2 \times 10^{18}$ laps of the ... 5 Voltage is similar to height. It plays the same role for electric charge as heightgravity does for a ball on a hill. So high voltage means high potential energy the same way a ball being high up on a hill means high potential energy. Voltage is not potential energy, the same way height is not energy. However, if you have a certain amount of charge $q$, you ... 0 Why does a capacitor charge only upto the voltage of the source? Step by step: (1) When the capacitor voltage equals the source voltage, the voltage across the resistor in the series RC circuit is zero (2) By Ohm's Law, the current through the resistor must be zero too. (3) Because it is a series circuit, if there is zero current through the ... 0 Here's how I would convince myself of the correct answer. Draw a circuit diagram showing the voltage source, resistor, and capacitor. (I assume it's in a simple series circuit?) Next, write out Kirchhoff's loop rule. You should find something like $V_\text{source}-V_\text{resistor}-V_\text{cap}=0$. Note that this equation is true at any time, not only for ... 0 What kills you is the current not the voltage, as you read on your books. Of course that you to have a voltage difference so the current can flow, but it does not determine how strong the current will be. I do not know if I would die if I touch something with 1 kV. That's because the current will depend on the sum of the resistance between me and the ground ... 0 I always thought this rule of thumb was a bit silly - current kills because it was driven by a voltage, otherwise there would have been no current. The rule arises because of the variability of skin resistance. Little voltage applied internally across your heart will kill you, but the skin's variability means that it is impossible to say what external ... 1 The current you are going to get through your body depends on the voltage and on the resistance. You can touch a 110 V exposed cable using a piece of metal or a piece of plastic - in both cases the voltage is the same, but the resulting current - and hence the danger - is greater in the first (metal) case. -1 Wires are in fact resistors, but with VERY VERY tiny amounts of energy being thermally dissipated by the current due to EXTREMELY low drops in voltage over large lenths of the wire. Thus, current DOES flow through the neutral wire, but the drop in potential along a length is literally far too small for your voltmeter to detect. Review Kirchoff's Voltage Law ... 2 steps: emf= change in magnetic flux/ time Therefore, emf=(BA)/time Magnetic field strength B is constant. So, we just have to find change in area by time The length of AC will increase by 0.6m per second. As six second passed, AC=0.66. The BC can be calculated, BC= ACtan19 At the beginning, area is zero. After 6 seconds, area=0.5AC*BC. Thus, change is ... 3 Draw the circuit using ideal circuit elements: Now, the series current is: $$I = \dfrac{\mathcal{E}}{R_{internal}+ R_{load}}$$ The voltage across the internal resistance is: $$V = \mathcal{E} \dfrac{R_{internal}}{R_{internal}+ R_{load}}$$ The power dissipated by the internal resistance can by found three equivalent ways: P = VI = ... 2 The analogy is wrong. A voltage source can only shock us if it is able to pass a considerable amount of current through our body ( ~ 250 mA or so, I dont know the exact value but you can Google it ). The circuit that you are trying to discuss, does indeed have 36 Amps of current flowing through it, but once you connect yourself to the circuit, you are in ... 0 well basically your missing some additional equations. The current coming out of the $10 \Omega$ resistor is the same as the one going into it. Hence $I_3$ in your equation would be just $0.5A$. Similar thing for $I_1$: The current going through the $20 \Omega$ resistor is $0.2A$. Note that the power supply also doesn't alter the current only the voltage. ... Top 50 recent answers are included	2013-12-21 03:54:11	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6481686234474182, "perplexity": 324.81531397763143}, "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-2013-48/segments/1387345774929/warc/CC-MAIN-20131218054934-00002-ip-10-33-133-15.ec2.internal.warc.gz"}
https://projecteuclid.org/ebooks/geometry-integrability-and-quantization/Proceedings-of-the-Second-International-Conference-on-Geometry-Integrability-and/chapter/Symplectic-Leaves-of-W-Algebras-From-the-Reduced-KacMoody-Point/10.7546/giq-2-2001-99-109	Translator Disclaimer VOL. 2 \| 2001 Symplectic Leaves of W-Algebras From the Reduced Kac–Moody Point of View Z. Bajnok, D. Nógrádi Editor(s) Ivaïlo M. Mladenov, Gregory L. Naber ## Abstract The symplectic leaves of W-algebras are the intersections of the symplectic leaves of the Kac-Moody algebras and the hypersurface of the second class constraints, which define the $W-$algebra. This viewpoint enables us to classify the symplectic leaves and also to give a representative for each of them. The case of the $W_2$ (Virasoro) algebra is investigated in detail, where the positivity of the energy functional is also analyzed. ## Information Published: 1 January 2001 First available in Project Euclid: 5 June 2015 zbMATH: 1062.81524 MathSciNet: MR1815634 Digital Object Identifier: 10.7546/giq-2-2001-99-109	2021-11-28 03:43:08	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4590570032596588, "perplexity": 2184.5467756403473}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358443.87/warc/CC-MAIN-20211128013650-20211128043650-00113.warc.gz"}
https://blender.stackexchange.com/questions/79783/what-do-the-tools-in-options-panel-like-tag-bevel-in-edit-mode-do	What do the tools in options panel( like Tag Bevel )in edit mode do? I was trying to get to know everything in Blender and I came across the options tab in Edit Mode and there's whole bunch of options like Tag Seam, Tag Bevel etc. I know how to use all the suffix functions (i.e every tool without "tag" ).But I am wondering if this has any specific function/usability. So, if anyone has any idea can you please explain all those options(Tag Seam, Tag Sharp, Tag Crease, Tag Bevel)? Thanks. • As far as I know those sound like the options in the W menu for use with the corresponding modifiers like Bevel, or Subsurf) – Duarte Farrajota Ramos May 17 '17 at 12:41 • Thanks for the response man.But could you please clarify,what purpose the tools/options in W menu serve when used with corresponding modifier? Lets take SubSurf modifier as example,the suitable options I see for them is Shade smooth and flat.But these only save the time and/or trouble of going back to object mode.Or is there some other use that I seem to not know about? – KnowledgeSeeker May 17 '17 at 13:46 • I've been testing here I could not figure out what those options are really for. I know tagging edges from the W menu is used in parallel with modifiers to specify different transformation values for different parts of the model. Say the Bevel Weight can make different edges beveled a different distance without requiring multiple Bevel modifiers. Different Crease values will change how smoothed an edge is with a Subsurf modifier – Duarte Farrajota Ramos May 17 '17 at 18:02 The options in the Options Panel for tagging edges are the same as the edges menu with only a slight difference that I can see. Most notably in Bevel and Crease, in that the weight added is always at 1. • Crease marks edges for use with subsurf, with a weight of 1 • Bevel marks edges for use with Bevel modifier, with a weight of 1	2020-01-20 20:59:13	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2871171832084656, "perplexity": 2590.9741501842977}, "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-2020-05/segments/1579250599789.45/warc/CC-MAIN-20200120195035-20200120224035-00389.warc.gz"}
http://www.pcc.edu/ccog/default.cfm?fa=ccog&subject=ABE&course=0744	# Portland Community College Course Number: ABE 0744 Course Title: Credit Hours: 0 Lecture Hours: 0 Lecture/Lab Hours: 0 Lab Hours: 0 Special Fee: #### Course Description Provides instruction for adults who wish to improve skills in reading, math, and writing, or who wish to prepare for the State GED Examination. #### Intended Outcomes for the course Upon completion of the course students will be able to: • Use writing as a communication tool. • Comprehend and interpret a variety of reading materials • Apply math systems to daily life situations. • Plan for employment and life skills development. • Use the GED Certificate to go to college, get a job, or improve skills. • Use the computer for basic word processing and internet activities. • Compute math problems using a calculator. #### Aspirational Goals • Love of learning • Ability to apply scientific reasoning in daily life • Appreciation of learning mathematics • Use math in a powerful way to achieve goals • Ability to apply scientific reasoning in daily life • Exhibit persistence, self-motivation, self-advocacy, and personal responsibility • Reflect upon, assess, identify, and celebrate ones own learning gains • Explore, develop, and monitor appropriate academic and professional goals • Advance knowledge and skills to make independent choices as a citizen, family member, worker, and life-long learner #### Course Activities and Design 1. Interpret and apply a few common patterns, functions, and relationships, using technology strategically. 1. Recognize and develop repeating patterns and generalize the relationship with a table, rule, graph, or one step formula (e.g., I make $7 an hour. If I work 30 hours a week, I can figure out how much I make in N weeks by multiplying N x 7 x 30, or Total wages = N(7 x 30).) 2. Identify, describe, and use common properties of operation (e.g., associative and distributive property) 2. Read and interpret common symbolic information. 1. Show repeated multiplication for simple whole numbers using exponents (e.g., 34 = 81) 2. Use variables to explain real life situations (e.g., If there are 8 items in each box, then I can figure out that the total number of items, N, is 8 times the number of boxes, or N = 8b.) 3. Apply order of operations to evaluate expressions 4. Write statements of equality and inequality (e.g., 3 > 4 3) 5. Solve simple one step equations by using number sense, properties of operations, and the idea of maintaining equality on both sides of an equation 3. Read and interpret common data and statistical information. 1. Extract discrete information from lists, tables, bar graphs, pictographs, or line plots 2. Describe how the scale in a bar or line graph can distort interpretations of data 3. Make statements and numerical comparisons about relative values on a bar graph (e.g., One category is 3 times greater than another. or This bar extends more than halfway between 25 and 50.) 4. Identify the range, median, mean and mode of small data sets (e.g., the ages of the students in the class) 4. Pose questions that can be answered with common data and collect, organize, and represent the relevant data to answer them. 1. Design simple data investigations to address a question and collect categorical data 2. Organize categorical data and represent them in a line graph or stem and leaf plots 3. Verify that data represented are the actual data collected 4. Make simple, straightforward inferences based on the data 5. Interpret and apply basic probability concepts. 1. Predict and then conduct simple probability experiments with outcomes limited to between one and four choices (e.g., four color spinner) 2. Connect a percent (0%, 25%, 50%, 75%, or 100%) and their fraction equivalents to the expected probability (e.g., of a four color spinner) 3. Compare the likelihood of two uncertain outcomes using simple language (e.g., one spinner with half red and half yellow and another spinner with one fourth red and three fourths yellow, asking students, Do you think you will have the same chance at landing on red in each of these spinners? Why?) 6. Read, write, interpret, and apply common types of information related to measurement and geometry, using technology strategically. 1. Recognize and use commonly used standard units of measure to the nearest eighths, tenths, and thirds 2. Use measurement units to describe the environment (e.g., Do you measure wire thickness in inches or mm? Is your height measured in cm or m?) 3. Recognize and describe two dimensional shapes, including basic angle descriptions (such as acute, right and obtuse) and properties of lines (e.g., perpendicular; parallel) 4. Measure and compare radius, diameter, and circumference of a circle and informally develop an equation for determining the diameter or circumference (e.g., C is about 3d, so pi is about 3) 5. Make conjectures about the formulas for simple two dimensional shapes (e.g., Since I can cut a rectangle into two equal triangles, I think that I can find the area of a triangle if I can create the rectangle it came from.) 6. Demonstrate an informal understanding of the coordinate graph system (e.g., find locations on a map using a grid system) 7. Select and apply mathematical procedures, using technology strategically. 1. Identify and use appropriate tools to measure to the nearest benchmark fractional unit (both decimal and fraction), including metric units 2. Make simple conversions within the same measurement system (e.g., inches to feet; cm to m) 3. Use direction, distance, labels, simple scales, and symbols to read and use maps and plans 4. Determine whether two dimensional shapes have similar attributes and properties (e.g., Are they congruent?) 5. Determine the area and perimeter of common two dimensional shapes and explain what happens to the area and perimeter when a dimension is changed 6. Measure size of angles and use benchmark angles (e.g., 90° and 45°) to estimate size of angles 7. Use ratio and proportion to solve problems involving scale drawings or similar figures 8. Apply common types of mathematical information and concepts to real life and theoretical problems involving whole numbers/integers, using technology strategically. 1. Use the knowledge that multiplication and division are inversely related to develop efficient and accurate strategies for multiplying and dividing three digit numbers by onedigit numbers 2. Multiply and divide to solve a variety of problems, including those related to geometry, measurement, and data 3. Estimate to predict answer when an exact answer is not needed or to determine reasonableness of computation 4. Recognize and apply negative integers in real contexts (e.g., The temperature was 20 degrees but went down to5 overnight. How much did the temperature drop?) 5. Identify prime and composite numbers and describe the difference between them 6. Use divisibility rules for 2, 3, 5, 10 and explain why they work 9. Apply common types of mathematical information and concepts to reallife and theoretical problems involving rational numbers, using technology strategically. 1. Use the commutative, associative, and distributive properties to create equivalent representations of numbers up to 10,000 (e.g., 8,900 = 9(1000) 100) and to the nearest hundredth (e.g.,$28.98 = 2(1.000) + 9.00 .02 ) 2. Extend benchmark fractions to equivalent decimals and percents (1/8, 1/6, 1/10, 1/100, etc.) and explain how these relate on a number line 3. Explain ratios as equivalent forms of benchmark fractions (e.g., 2/4 = 1/2) 4. Demonstrate that multiplying by a fraction is the same as dividing by the whole number in the denominator (e.g., 10 x 12 is the same as 10 Ã· 2.) 5. Use benchmark fractions, decimals, and percents (e.g., 34 and 1/10) to estimate relative sizes (e.g., 11/16 is close to 34 because 12/16 is the same as 34.) 6. Apply proportional reasoning to simple, one-step problems (e.g., If 5 pounds of potatoes cost \$4, how much would 10 pounds cost?) 10. Apply common types of mathematical information and concepts to real life and theoretical problems involving exponents, using technology strategically. 1. Use the exponent 2 to express the area of two dimensional figures #### Outcome Assessment Strategies • Take and pass a CASAS reading post-test and improve one level or pass the GED math, science test, reasoning through language arts and social studies exam • Take and pass a CASAS math post-test and improve one level or pass the GED math, science test, reasoning through language arts, and social studies • Develop a transition plan for short and long range life planning • Take COMPASS test after completion of the GED Exam if college bound • Complete computer-based assignments • Complete a post-test of math word problems using a GED Specific calculator • Create writing portfolios, including reflections, drafts that show evidence of editing and revising • Write paragraphs, essays, letters, poems, resumes, journal entries, notes, writing in response in response to text, and annotations • Develop projects, presentations, and debates • Complete Reading with Understanding Diary • Assess comprehension with quizzes, multiple choice questions, written response, and discussion questions • Complete a computer-based assignment #### Course Content (Themes, Concepts, Issues and Skills) Themes • Life (e.g. family and citizen) and employability ( i.e. worker) planning • Transition from ABE to post-secondary or ABE to work • Lifelong learning • Goal setting • Critical Thinking • Metacognition • Confidence Building Concepts • Time Management (attendance and completing tasks) • Social skills (communication, and diversity) • Collaboration • Student Success Issues • Barriers to student success • Learning Style Differences • Life Instability • Communication • Employability Skills: Communication Skills • Awareness of writing as a process (planning, developing, organizing, revising, editing) • Clarify purpose of the writer(s) and reader(s) in a specific situation • Draw on prior experience, research, new knowledge, and ones own questions, interests and observations to generate ideas • Choose among a variety of strategies appropriate to planning and organizing texts types • Develop and organize ideas and information in varied genres, including the presentation of an argument • Summarize and paraphrase ideas in a text while avoiding plagiarism • Introduce claims, acknowledge alternate or opposing claims and organize the reasons logically • Support claim(s) with logical reasoning and relevant evidence, using accurate, credible source • Use words, phrases and clauses to create cohesion and clarify the relationships among claims, reasons, and evidence • Provide a concluding statement or section that follows from and supports the argument presented • Use basic and complex grammar to construct coherent text with sentences that vary in style, length and complexity • Draw from a broad vocabulary that includes words needed for specialized and/or academic purposes • Express ones own thoughts and ideas in a way that is clear and compelling • Consider and apply feedback from self and others to enhance the impact of the writing and better address the writing purpose • Use writing conventions appropriate for complex text types in multiple genres, including academic or occupational texts • Draw from a variety of technologies and media appropriate for the writing purposes. • Carry out writing tasks that involve presentations of information, or require the synthesis, analysis and/or evaluation of ideas • Select the writing development strategy appropriate to writing purposes and needs • Read regularly for own purposes • Identify, clarify, and/or prepare for complex reading purpose • Pronounce on sight words, abbreviations, and acronyms found in everyday texts and a range of terms related to areas of interest or study • Recognize on sight syllable patterns/types, root words, and affixes in multi-syllabic words • Acquire and apply meanings of most words and phrases found in everyday and academic texts, including terms related to specialized topics • Accurately read text composed of dense or long, complex sentences and paragraphs with appropriate pacing, phrasing, and expression • Evaluate and/or apply prior knowledge of the content and situation, including cultural understanding, to support comprehension • Use strategies easily and in combination to pronounce and/or discern the meanings of unfamiliar words found in a complex text • Choose from a range of strategies, including some sophisticated ones, and integrate them to monitor and/or enhance text comprehension (e.g. scan/skim, make inferences, mark text and/or make notes, organize notes and/or make graphic organizers and text maps, write a summary to check understanding, discuss with others) • Use text format and features (e.g. search engines, drop down menus, headings) to enhance comprehension • Locate, analyze, and critique stated and unstated information, ideas/arguments, and/ or themes in a complex functional, informational, or persuasive text • Determine, analyze and summarize the authors central idea and major points over multiple paragraphs/pages • Evaluate the reliability, accuracy, and sufficiency of information, claims, or arguments • Draw conclusions related to the structural elements of a complex literary work, using literary terms • Analyze and evaluate an authors style, attending to the use of language and literary techniques and to influences on the writing • Integrate the people/characters, events, information, ideas/arguments, themes, or writing styles in lengthy or multiple complex tests with each other and/or with knowledge of the world to address a complex reading purpose • Agree or disagree with an idea/argument/claim or theme, and explain reasoning • Follow complex, multi step directions, integrating written and graphic information (e.g., science experiment) • Compare and Contrast people/characters, events and ideas in different texts • Combine, compare, contrast and/or critique ideas/arguments/claims or themes in different texts Job Skills • Use career search resources • Develop job search strategies and skills (interviewing, resume, and cover letter) • Use technology effectively • Communicate effectively at work Math Skills • Basic Arithmetic Facts • Solve numerical and application problems with GED specific calculator • Perform order of operations accurately using whole numbers • Develop skills in estimation and number sense • Master fraction and decimal vocabulary • Solve numerical and application problems with fractions and decimals • Round a given number to a specified place • Arrange numbers in numerical order • Perform order of operations accurately using fractions and decimals • Determine whether a given whole number is prime or composite • Evaluate expressions containing exponents and square roots • Perform operations accurately using fractions, decimals, and percents • Solve application problems with fractions, decimals, and percents • Read and interpret data from bar, pictorial, line, circle graphs, tables, charts and various graphs • Find statistical measures such as median, mode, mean and apply to scientific problems • 1.15 Apply scientific reasoning to problem-solving activities • 1.16 Scientific Notation • Ratios and Proportions • Interest • Mean, Median, Mode and Range and apply to scientific problems • Probability • Graphs, Tables, Charts and Graphic Methods of Data Reporting • Coordinate Plane • Slope Formula • Distance between two points (Coordinates) • Pythagorean Theorem • Absolute Value • Official GED Calculator • Algebra one step and multistep Equations • Algebraic Factoring • Geometric Measurement (Area, Perimeter, Area and Surface Area) • Apply scientific reasoning to problem-solving activities Problem Solving Skills • Apply study skills, develop test-taking strategies • Assess and re-assess short and long-term goals • Develop critical thinking, problem solving and decision-making skills • Data analysis Personal/Interpersonal Skills • Build confidence and self-esteem • Interact with others	2017-07-20 22:53:32	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.23217307031154633, "perplexity": 11925.11187064}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": 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https://www.nature.com/articles/s41467-018-07030-2?error=cookies_not_supported&code=afab7027-d139-4ee8-8527-2854aeaef007	Article \| Open \| Published: Three-dimensional topological acoustic crystals with pseudospin-valley coupled saddle surface states Abstract Topological valley states at the domain wall between two artificial crystals with opposite valley Chern numbers offer a feasible way to realize robust wave transport since only broken spatial symmetry is required. In addition to the valley, spin and crystal dimension are two other important degrees of freedom, particularly in realizing spin-related topological phenomena. Here we experimentally demonstrate that it is possible to construct two-dimensional acoustic topological pseudospin-valley coupled saddle surface states, designed from glide symmetry in a three-dimensional system. By taking advantage of such two-dimensional surface states, a full set of acoustic pseudospins can be realized, exhibiting pseudospin-valley dependent transport. Furthermore, due to the hyperbolic character of the dispersion of saddle surface states, multi-directional anisotropic controllable robust sound transport with little backscattering is observed. Our findings may open research frontiers for acoustic pseudospins and provide a satisfactory platform for exploring unique acoustic topological properties in three-dimensional structures. Introduction The discovery of topological phases of matter has renewed our understanding of condensed matter physics over the past few decades1,2 and has inspired studies of classical bosonic systems such as photonics3,4,5,6,7,8,9,10,11,12 and phononics13,14,15,16,17,18,19,20. Without considering the difference in spin between fermions (half-integer spin) and bosons (integer spin), their wavefunctions share a similar form associated with a similar topology. This condition gives rise to the search for photonic/phononic analogues of quantum Hall effect with broken time-reversal (TR) symmetry3,4,14, topological valley states with a broken mirror or inversion symmetry10,11,12,18,19,20, Floquet topological states due to temporal (or spatial) modulation21,22,23 and Weyl semimetals with chiral structures24,25. However, regarding spin-related topological phenomena, degenerate polarizations or Bloch states must be introduced to construct pseudospins (with pseudo-TR squares to −1)26. Therefore, counterparts of the two-dimensional (2D) quantum spin Hall effect8,15 and of three-dimensional (3D) topological states27,28 for photons/phonons can be designed in principle as electrons in electronic systems by using pseudo-TR instead of natural TR. For airborne sound such as a spinless wave, an additional degree of freedom (DOF) such as crystal symmetry needs to be considered to construct acoustic pseudospins29. Here, we resort to 3D artificial acoustic systems, which provide more flexible platforms to search for this kind of pseudospin among all 230 types of space groups and provide a pre-designable artificial unit structure. Typically, the topological nature is manifested in its character in lower dimension, e.g. 2D systems can possess topological protected one-dimensional (1D) edge or zero-dimensional (0D) corner states30, therefore, the 3D models can provide rich topologically protected 2D surface states beyond 1D and 0D lowering from 2D systems. In addition, a typical artificial structure has no more than 104 artificial atoms as a whole, which enable us to accurately manipulate every atom and to deliberately introduce defects, take measurements without limitation of the Fermi level31 and to create arbitrary interfaces32. In particular, due to the lack of strong spin–orbit coupling and efficiently TR-breaking method for airborne sound, the valley DOF can provide a convenient way to realize acoustic topological states since only broken mirror (or inversion) spatial symmetry is required. The degenerate point of band structures, such as Dirac degeneracy in 2D case, can be lifted to form the K (K′) valley in the momentum space associated with non-trivial Berry curvature. Although the summation of Berry curvatures over the whole Brillouin zone (BZ) is trivial in TR symmetric cases, robust edge transports still exist along some particular directions. In this article, we focus on 3D valley acoustic crystals with pseudospin-related topological phenomena. By elaborately designing the symmetries of 3D lattices, the four-fold degenerate point of bulk band structures is lifted to two-fold degenerate valleys, hosting a pair of acoustic pseudospins. The 2D acoustic topological pseudospin-valley coupled saddle surface states and the corresponding anisotropic robust sound transport with little backscattering can be observed in our experiments. Results Crystal structure and the bulk band structure Our 3D periodical crystal structure is composed of stacked double-layer honeycomb lattices along the z axis, containing two kinds of acoustic atoms in each layer (Fig. 1a). Two adjacent layers show glide symmetry that is the combination of reflection symmetry (xz plane) and a translation by half a lattice constant h (along the z axis). This crystal structure belongs to the non-symmorphic space group No. 194 (P63/mmc). The acoustic atom is a triangular prism cavity with five tubes connecting the nearest neighbours (Fig. 1b). For simplicity, different sidelengths (lg and lr) of the triangular prisms represent different kinds of acoustic atoms with the other parameters fixed (lattice constant a = 8.7 mm, height h = 0.54a, height of the prism hp = a/3, radius of the tube r = 0.13a). One-half of the first BZ is shown in Fig. 1c. To show the evolution of the band structure, we start from what is likely the simplest 3D structure: a stacked monolayer graphene structure (graphite). The primary unit cell (inset) with all its identical atoms and band structures is shown in Fig. 1d. The side length of the prism is lg = lr = 0.7a. The bulk band projected onto the kxy and kyz momentum planes at the frequency near the centre of the K–H direction is shown in the lower panel. Due to the D6h symmetry, the kxy projected bulk band is a Dirac cone, where the Dirac point is a two-fold nodal line along the K–H direction. However, just two-fold degenerate states are not enough to construct acoustic pseudospins. The key point is to increase the DOF to form four-fold degeneracy26. An efficient method is the BZ folding approach. As shown in Fig. 1e, we choose a double-sized unit cell (associated with one-half of the BZ). Then, the first two bands are folded into four bands with four-fold degenerate states at the H point. The kxy projected bulk bands become doubly Dirac cones. Due to the unbroken TR symmetry, the states at the H′ point are also four-fold degenerate. Then, we introduce a sublattice structure by choosing two different acoustic atoms (lg = 0.6a and lr = 0.8a) with glide symmetry (Fig. 1f). In this case, a complete band gap is created. More importantly, the four-fold degeneracy is split into two two-fold degeneracies, which can be used to form acoustic pseudospins. The glide symmetry here can be described as G:(x, y, z) → (x,y, z+h). Then, G2:(x, y, z) → (x, y, z+2h). Due to the Bloch theorem, the Bloch phase function under lattice translation can be described as $$e^{ - i{\boldsymbol{k}} \cdot {\boldsymbol{r}}}$$. At the H point (kz = π/2 h), G = $$e^{ - i\pi /2}$$ (G2 = $$e^{ - i\pi }$$). The pseudo-TR written as GK, squares to −1 (the complex conjugation K represents the TR of sound)26,27, ensuring that the completely two-fold degenerate Bloch states on the kz = π/2h plane form acoustic pseudospins (lower panel of Fig. 1f). The pseudospin-valley Chern numbers Our model is based on a four-band model, which can be treated as a kind of acoustic topological pseudospin-valley states. As shown in Fig. 1e, the four-fold degenerate states at the H (H′) point are doubly Dirac cones projecting onto the kxy plane. After introducing the sublattice structure (glide), the mirror symmetry respective to the xz-plane is broken, with the four-fold degeneracy splitting into two two-fold degenerate acoustic pseudospin± associated with nonzero Berry curvatures. The valley Chern numbers at the H point for the lower two bands (acoustic pseudospin±) (Fig. 1f) are half-integer with opposite signs $${\mathrm{C}}_{\mathrm{H}}^ \pm =\pm 1/2$$, while $${\mathrm{C}}_{{\mathrm{H}}^\prime }^ \pm = \mp 1/2$$ at the H′ point. Thus, the acoustic pseudospin-valley Chern numbers33 can be described as $${\mathrm{C}}_{\mathrm{v}}^{\mathrm{s}} = ({\mathrm{C}}_{\mathrm{H}}^ + - {\mathrm{C}}_{{\mathrm{H}}^\prime }^ + - {\mathrm{C}}_{\mathrm{H}}^ - + {\mathrm{C}}_{{\mathrm{H}}^\prime }^ - )/2$$ = +1 (see Supplementary Note 1). It should be noticed that unlike recently realized topological spin-valley-locked states in 2D photonic systems10 [where the spins are two intrinsic electromagnetic polarizations with the same signs for the valley Chern numbers at the H(H′) point (associated with $${\mathrm{C}}_{\mathrm{v}}^{\mathrm{s}} = 0$$)], our acoustic pseudospins are artificially constructed via crystal symmetry with nontrivial $${\mathrm{C}}_{\mathrm{v}}^{\mathrm{s}}$$. Acoustic pseudospins for topological saddle surface states To study the 2D topological pseudospin-valley surface states, we introduce a domain wall (zigzag) between two opposite domains on the xz-plane. Figure 2a, b shows one-half of the surface BZ projected onto the kxz plane and a schematic for the interface. Notably, there are two pairs of topological surface states with surface nodal lines, constructed by two opposite saddle surfaces accidentally touching at their saddle points (Fig. 2c, d). Here, we choose four points (marked in Fig. 2c) near the centre of the BZ to investigate the acoustic pseudospins. The acoustic fields are constructed by a symmetric (S) real part and anti-symmetric (A) imaginary part of the acoustic fields, forming S ± iA states (Fig. 2e). Then, acoustic pseudospin± can be defined as $$\nabla \times$$ (S ± iA). More importantly, the A state has two independent components: anti-symmetric respective to the yz-plane (noted as Ax) and anti-symmetric respective to the xz-plane (noted as Az). This result can be attributed to the existence of two mirror symmetric planes perpendicular to the domain wall. Consequently, a full set of acoustic pseudospins on the Bloch sphere (linear, circular or elliptical) can be constructed as shown in Fig. 2f, while in 2D systems, the pseudospins are limited in a 1D space7,15. It should be noticed that such a zigzag domain wall has a pseudospin-valley Chern number difference: $$\Delta {\mathrm{C}}_{\mathrm{v}}^{\mathrm{s}} = 2$$ corresponding to two pairs of topological surface states (Supplementary Figure 3). We can also design a $$\Delta {\mathrm{C}}_{\mathrm{v}}^{\mathrm{s}} = 1$$ domain wall with only a pair of acoustic pseudospins (see Supplementary Figure 6). It is also worth noting that these acoustic surface states are gapped at the kxz (armchair) or kxy interfaces, because the H and H′ points are projected onto the same point of the surface BZ; thus, the valley Chern numbers with opposite signs will be cancelled out (see Supplementary Figure 4). Anisotropic robust sound transport In our experiment, we choose two orthogonal directions to show multi-directional robust acoustic pseudospin-valley transport (Supplementary Figure 1). Figure 3a shows the measured transmission spectra (10 periods) for surface states along the $${\tilde{\mathrm \Gamma }}$$$${\tilde{\mathrm X}}$$ direction with both straight (red line) and z-shape (blue line) waveguides. In contrast, the measured transmission spectra for bulk states are shown by the black line, with a relative band gap width of over 15%. Following the incidence of the pseudospin+ acoustic wave, the transmissions maintain a very high transmission value in the bulk band gap frequency region for both straight and z-shape waveguides, indicating a strongly suppressed backscattering property (Supplementary Figure 2). The overall transmission of the saddle surface states is 20 dB larger than that of the bulk within the bulk band gap, representing the surface’s gapless behaviour, except for a transmission dip near 19.678 kHz according to the quadratic saddle point. Near the saddle point, the extremely flat dispersion results in the rapidly enhanced state intensity of sound; thus, the transmission is intensely attenuated even at a very low loss. The simulation results for the pressure field at frequencies of 19 kHz (in bulk band gap) and 19.678 kHz (near the saddle point) are shown in Fig. 3b. In simulations, the loss is introduced by adding an imaginary part (10−3) for the sound speed. On the other hand, even without the nonlinear effect of loss, the transmission near the saddle point is still sharply decreased because of the diminishing group velocity down to zero. A similar robust transmission along the $${\tilde{\mathrm \Gamma }}$$$${\tilde{\mathrm Z}}$$ direction is experimentally observed (Fig. 3c), matching well with the simulation results (Fig. 3d). The lossless condition at the saddle point is also provided for comparison (right panel). To further verify such saddle surface states, we increase the height (h) between two layers from 0.54a to 0.67a (Fig. 4a). The interaction between the two layers is weaker. Thus, the whole band structures show a slight redshift. Interestingly, two opposite saddles are separated to form an eye-shape (Fig. 4b, c). The experimental transmission spectra along the $${\tilde{\mathrm \Gamma }}$$$${\tilde{\mathrm X}}$$ and $${\tilde{\mathrm \Gamma }}$$$${\tilde{\mathrm Z}}$$ directions are shown in Fig. 4d, e. There is no obviously decreased transmission along the $${\tilde{\mathrm \Gamma }}$$$${\tilde{\mathrm X}}$$ direction. However, due to the eye opening, a wider transmission dip along the $${\tilde{\mathrm \Gamma }}$$$${\tilde{\mathrm Z}}$$ direction can be found than that in Fig. 3c. The dashed box represents the frequency region of the eye. Based on such strong hyperbolic behaviour for the topological saddle surface states, multi-directional anisotropic controllable robust sound transport with little backscattering can be obtained (see Supplementary Figures 710). Discussion In summary, we experimentally demonstrate 2D acoustic pseudospin-valley coupled saddle surface states in 3D topological acoustic crystals generated due to glide-symmetry design29. Compared to the electronic topological crystalline insulator with saddle dispersion in condensed matter physics34,35, this acoustic model exhibits surface nodal lines for the surface states, which can hardly shrink to a Dirac cone because of the lack of intrinsic acoustic spins (Supplementary Figure 5). Our acoustic pseudospins (satisfying pseudo-TR) are constructed by crystal symmetry which cannot be kept intact on the domain wall. Unlike the intrinsic spins of electrons, these acoustic pseudospins are gradually changed along the surface, e.g. becoming linear pseudospins at the saddle point. However, this 3D acoustic topological model still shows strongly suppressed backscattering behaviour on the whole 2D surface, which resembles the 2D topological cases with a tiny gap in the middle of the 1D edge states7,15,36 due to spatial symmetry breaking on the boundary. The results we revealed here may pave the way towards acoustic pseudospins and valleytronics in 3D structures37. The saddle surface states could be applied to realize hyperbolic pseudospin filters37. The robust pseudospin-valley propagation within a large topological band gap and the extremely flat dispersion near the saddle point may give rise to an ultraslow sound, ultrahigh-Q acoustic resonator. Methods Experiments Our samples are fabricated by 3D printing with commercial low-viscosity liquid photopolymer materials (Somos Imagine 8000). The tolerance of the fabrication is ±0.1 mm, which is less than 5% compared to the smallest feature size of 2.2 mm in our model. Due to the fabrication tolerance of different samples, the measured transmission spectra for the saddle points in Fig. 3a, c show a slight blue or redshift. A B&K-4939-2670 microphone acts as a detector, which is placed 1 cm from the boundary with its response acquired and analysed in B&K-3560-C. The frequencies are swept from 14 to 24 kHz with an increment of 0.02 kHz. The experimental transmission spectra plotted in this article are normalized to the acoustic wave transmission through the same distance in air. The slight deviation recorded in experiments is due primarily to the frequency dependent coupling into and out efficiency. Simulations Numerical investigations used to calculate band structures (Figs. 1d–f, 2c, d and 4b) and field distributions (Figs. 2e and 3b, d) are conducted by using an acoustic model in commercial FEM software (COMSOL MULTIPHYSICS). Due to the large acoustic impedance mismatch between air and photopolymer material (modulus 2765 MPa, density 1.3 g cm−3), the models in the numerical calculation are constructed using only acoustic cavities with hard boundaries, without considering the polymer background. The density and velocity of sound are chosen to be 1.25 kg m−3 and 343 m s−1, respectively. The numerical results agree well with experimental results. Data availability The data that support the plots within this paper and other findings of this study are available from the corresponding author upon request. Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. References 1. 1. Hasan, M. Z. & Kane, C. L. 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Acknowledgements The work was jointly supported by the National Key R&D Program of China (Grant Nos. 2017YFA0303702 and 2017YFA0305100) and the National Natural Science Foundation of China (Grant Nos. 11874196, 11625418, 11474158, 51732006, 51721001, 51702152 and 51472114). We also acknowledge the support of the Fundamental Research Funds for the Central Universities (Grant No. 14380097) and the Natural Science Foundation of Jiangsu Province (BK20140019). Author information Author notes 1. These authors contributed equally: Cheng He, Si-Yuan Yu, Hao Ge. Affiliations 1. National Laboratory of Solid State Microstructures & Department of Materials Science and Engineering, Nanjing University, Nanjing, 210093, China • Cheng He • , Si-Yuan Yu • , Hao Ge • , Yuan Tian • , Xiao-Chen Sun • , Jian Zhou • , Ming-Hui Lu • & Yan-Feng Chen 2. Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing, 210093, China • Cheng He • , Si-Yuan Yu • , Haijun Zhang • , Ming-Hui Lu • & Yan-Feng Chen 3. National Laboratory of Solid State Microstructures & School of Physics, Nanjing University, Nanjing, 210093, China • Huaiqiang Wang • , Haijun Zhang • & Y. B. Chen Contributions C.H. conceived the idea and performed the numerical simulations. C.H., S.-Y.Y., H.G. and Y.T. carried out the experimental measurements. H.W. and C.H. carried out the theoretical analysis. H.Z., X.-C.S., Y.B.C. and J.Z. contributed to the discussion of the results. C.H., M.-H.L. and Y.-F.C. supervised all the aspects of this work and managed this project. Competing interests The authors declare no competing interests. Corresponding authors Correspondence to Cheng He or Ming-Hui Lu or Yan-Feng Chen.	2018-11-15 15:29:34	{"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": 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": 0.7572539448738098, "perplexity": 2727.6028545552313}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039742779.14/warc/CC-MAIN-20181115141220-20181115163220-00490.warc.gz"}
https://proofwiki.org/wiki/Infinite_Ramsey%27s_Theorem_implies_Finite_Ramsey%27s_Theorem	# Infinite Ramsey's Theorem implies Finite Ramsey's Theorem ## Theorem $\forall l, n, r \in \N: \exists m \in \N: m \to \left({l}\right)_r^n$ where $\alpha \to \left({\beta}\right)^n_r$ means that: for any assignment of $r$-colors to the $n$-subsets of $\alpha$ there is a particular color $\gamma$ and a subset $X$ of $\alpha$ of size $\beta$ such that all $n$-subsets of $X$ are $\gamma$. ## Proof Aiming for a contradiction, assume that there is a $l$ such that: $\forall m \in \N: m \nrightarrow \left({l}\right)_r^n$ Let $\hat{K_i}$ denote a hypergraph on $i$ vertices where all possible $n$-subsets of the vertices are the hyperedges. Let $G$ be a hypergraph with vertices $V = \left\{ {v_i: i \in \N}\right\}$. Let the hyperedges of $G <$ be enumerated by: $E = \left\{ {E_i: E_i \subset \N, \left\vert{E_i}\right\vert = n}\right\}$ We construct a (rooted) tree $T$ as follows: $(1): \quad$ First introduce a root node $r t$. $(2): \quad$ Each node is allowed to have at most $r<$ children which correspond to the $r$-colors, subject to it satisfying the criteria: A child is always labeled by one among the $r$-colors. Let the colors be denoted $c_1, c_2, \ldots, c_r$. $(3): \quad$ A child $c_i$ is permitted if and only if its introduction creates a path of some finite length $k$ starting from the root. So, if the hyperedges $E_1, E_2, \ldots, E_k$ are colored by the colors used in the path in the same order, then the corresponding subgraph of $G$ does not contain a monochromatic $\hat{K_l}$. For example: if the introduction of a child $c_i$ creates the $k$ length path $r t, c_a, c_b, \ldots, c_i$ and the hyperedges $E_1, E_2, \ldots, E_k$ when colored $c_a, c_b, \ldots, c_i$ do not contain a monochromatic $\hat{K_l}$, the child $c_i$ is permitted to be added to $T$. Note that for all $m$, there always exists a coloring of $\hat{K_m}$ such that no monochromatic $\hat{K_l}$ exists within. Thus the situation that a child cannot be added to any vertex at a given level $k$ cannot arise. For we can always take a coloring of $\hat{K_{k+n}}$ containing no monochromatic $\hat{K_l}$. Since any $k$ hyperedges in it would yield a sequence of colors already existing in $T$, we know which vertex to add the child to. We give the child the color corresponding to any other edge. Hence we can forever keep adding children and so $T$ is infinite. It is also obvious that each level $k$ of $T$ has at most $r^k<$ vertices. So each level is finite. By König's Tree Lemma there will be an infinite path $P$ in $T$. $P$ provides a $r$-coloring of $G$ that contains no monochromatic $\hat{K_i}$. Hence $P$ contains no monochromatic infinite hypergraph. This contradicts the Infinite Ramsey's Theorem. The result follows by Proof by Contradiction. $\blacksquare$	2020-01-25 02:45:15	{"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": 0.8077398538589478, "perplexity": 204.10203899997512}, "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-2020-05/segments/1579250628549.43/warc/CC-MAIN-20200125011232-20200125040232-00056.warc.gz"}
https://www.projectrhea.org/rhea/index.php/Z_Transform_table	Table of (double-sided) Z Transform Pairs and Properties (Used in ECE301, ECE438, ECE538) (double-sided) Z Transform and its Inverse (Double-side) Z Transform $X(z)=\mathcal{Z}(x[n])=\sum_{n=-\infty}^{\infty}x[n]z^{-n} \$ (info) Inverse Z Transform $x[n]=\mathcal{Z}^{-1}(X(z))=\frac{1}{2\pi j}\oint_{c}X(z)z^{n-1}dz \$ (info) (double-sided) Z Transform Pairs Signal Transform Region of convergence (ROC) Unit impulse signal $\delta[n]\$ $1\$ All complex $z\$ including $\infty$ Unit step signal $u[n]\$ $\frac{1}{1-z^{-1}} \$ $\|z\| > 1\$ (computation) $-u[-n-1]\$ $\frac{1}{1-z^{-1}}$ $\|z\| < 1\$ Shifted unit impulse signal $\delta[n-m]\$ $z^{-m}\$ $All\ z,\ except\$ $0\ (if\ m>0)\ or\$ $\infty \ (if\ m<0)\$ $\alpha^{n}u[n]\$ $\frac{1}{1-\alpha z^{-1}}$ $\|z\| > \| \alpha \|\$ $-\alpha^{n}u[-n-1]\$ $\frac{1}{1-\alpha z^{-1}}$ $\|z\| < \| \alpha \|\$ $n\alpha^{n}u[n]\$ $\frac{\alpha z^{-1}}{(1-\alpha z^{-1})^{2}}$ $\|z\| > \| \alpha \|\$ $-n\alpha^{n}u[-n-1]\$ $\frac{\alpha z^{-1}}{(1-\alpha z^{-1})^{2}}$ $\|z\| < \| \alpha \|\$ Single-side cosine signal $[\cos{\omega_{0}n}]u[n]\$ $\frac{1-[\cos{\omega_{0}}]z^{-1}}{1-[2\cos{\omega_{0}}]z^{-1}+z^{-2}}$ $\|z\| > 1\$ Single-side sine signal $[\sin{\omega_{0}n}]u[n]\$ $\frac{1-[\sin{\omega_{0}}]z^{-1}}{1-[2\cos{\omega_{0}}]z^{-1}+z^{-2}}$ $\|z\| > 1\$ $[r^{n}\cos{\omega_{0}n}]u[n]\$ $\frac{1-[r\cos{\omega_{0}}]z^{-1}}{1-[2r\cos{\omega_{0}}]z^{-1}+r^{2}z^{-2}}$ $\|z\| > r\$ $[r^{n}\sin{\omega_{0}n}]u[n]\$ $\frac{1-[r\sin{\omega_{0}}]z^{-1}}{1-[2r\cos{\omega_{0}}]z^{-1}+r^{2}z^{-2}}$ $\|z\| > r\$ (double-sided) Z Transform Properties Below $x[n]$, $x_1[n]$ and $x_2[n]$ are DT signals with z-transforms $X(z)$, $X_1(Z)$, $X_2(z)$, and region of convergence (ROC) $R$, $R_1$, $R_2$, respectively. Signal Z-Transform ROC Linearity $ax_{1}[n]+bx_{2}[n]\$ $aX_{1}(z)+bX_{2}[z]\$ $At\ least\ the\ intersection\ of\ R_{1}\ and\ R_{2}\$ Time shifting $x[n-n_{0}]\$ $z^{-n_{0}}X(z)\$ $R,\ except\ for\ the\ possible\ addition\$ $or\ deletion\ of\ the\ origin\$ Scaling in the z-domain $e^{j\omega_{0}n}x[n]\$ $X(e^{j\omega_{0}}z)\$ $R\$ Modulation (proof) $a^{n}x[n]\$ $X(a^{-1}z)\$ $\|a_0\| R$ (Scaled version of) $R\$ $(i.e.,\ \|a\|R=\ the\ set\ of\ points\ {\|a\|z}\ for\ z\ in\ R)\$ Time reversal $x[-n]\$ $X(z^{-1})\$ $R^{1/k}\ (i.e.,\ the\ set\ of\ points\ z^{1/k},\$ $where\ z\ is\ in\ R)\$ Time expansion $x^{(k)}= \begin{cases} x[r], &n=rk \\ 0, &n\neq rk \end{cases}$ $X(z^{k})\$ $R^{1/k}\ (i.e.,\ the\ set\ of\ points\ z^{1/k},\$ $where\ z\ is\ in\ R)\$ Conjugation $x^{}[n]\$ $X^{}(z^{})\$ $R\$ Convolution $x_{1}[n]x_{2}[n]\$ $X_{1}(z)X_{2}(z)\$ $At\ least\ the\ intersection\ of\ R_{1}\ and\ R_{2}\$ First difference $x[n]-x[n-1]\$ $(1-z^{-1})X(z)\$ $At\ least\ the\ intersection\ of\ R\ and\ \|z\|>0\$ Accumulation $\sum_{k=-\infty}^{n}x[k]\$ $\frac{1}{(1-z^{-1})}X(z)\$ $At\ least\ the\ intersection\ of\ R\ and\ \|z\|>1\$ Differentiation in the z-domain $nx[n]\$ $-z\frac{dX(z)}{dz}\$ $R\$ Other Z Transform Properties Initial Value Theorem $If\ x[n]=0\ for\ n<0,\ then\ x[0]=\lim_{z\rightarrow \infty}X(z)\$	2019-04-20 03:22:53	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8870634436607361, "perplexity": 5029.327143305201}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1555578528481.47/warc/CC-MAIN-20190420020937-20190420042033-00011.warc.gz"}
https://byjus.com/jee/rank-of-a-matrix-and-special-matrices/	# Rank of a Matrix and Some Special Matrices The maximum number of its linearly independent columns (or rows ) of a matrix is called the rank of a matrix. The rank of a matrix cannot exceed the number of its rows or columns. If we consider a square matrix, the columns (rows) are linearly independent only if the matrix is nonsingular. In other words, the rank of any nonsingular matrix of order m is m. Rank of a matrix A is denoted by ρ(A). The rank of a null matrix is zero. A null matrix has no non-zero rows or columns. So, there are no independent rows or columns. Hence the rank of a null matrix is zero. ## How to find the Rank of a Matrix? To find the rank of a matrix, we will transform that matrix into its echelon form. Then determine the rank by the number of non zero rows. Consider the following matrix. A = $\begin{bmatrix} 2 & 4 &6 \\ 4& 8& 12 \end{bmatrix}$ While observing the rows, we can see that the second row is two times the first row. Here we have two rows. But it does not count. The rank is considered as 1. Consider the unit matrix A = $\begin{bmatrix} 1 &0 &0 \\ 0& 1 & 0\\ 0 & 0 &1 \end{bmatrix}$ We can see that the rows are independent. Hence the rank of this matrix is 3. The rank of a unit matrix of order m is m. If A matrix is of order m×n, then ρ(A ) ≤ min{m, n } = minimum of m, n. If A is of order n×n and \|A\| ≠ 0, then the rank of A = n. If A is of order n×n and \|A\| = 0, then the rank of A will be less than n. ### Rank of a Matrix by Row- Echelon Form We can transform a given non-zero matrix to a simplified form called a Row-echelon form, using the row elementary operations . In this form, we may have rows all of whose entries are zero. Such rows are called zero rows. A non-zero row is one in which at least one of the elements is not zero. For example, consider the following matrix. A = $\begin{bmatrix} 1 &0 &2 \\ 0& 0 & 1\\ 0 & 0 & 0 \end{bmatrix}$ Here R1 and R2 are non zero rows. R3 is a zero row. A non-zero matrix A is said to be in a row-echelon form if: (i) All zero rows of A occur below every non-zero row of A. (ii) The first non-zero element in any row i of A occurs in the jth column of A, then all other elements in the jth column of A below the first non-zero element of row i are zeros. (iii) The first non-zero entry in the ith row of A lies to the left of the first non-zero entry in ( i + 1)th row of A. Note: A non-zero matrix is said to be in a row-echelon form, if all zero rows occur as bottom rows of the matrix and if the first non-zero element in any lower row occurs to the right of the first non-zero entry in the higher row. If a matrix is in row-echelon form, then all elements below the leading diagonal are zeros. Consider the following matrix. A = $\begin{bmatrix} 0 &0 &1 \\ 0& 0 & 5\\ 0 & 0 & 0 \end{bmatrix}$ Check the rows from the last row of the matrix. The third row is a zero row. The first non-zero element in the second row occurs in the third column and it lies to the right of the first non-zero element in the first row which occurs in the second column. Hence the matrix A is in row echelon form. ### Rank of a Matrix Solved Examples Example 1: Find the rank of matrix A by using the row echelon form. A = $\begin{bmatrix} 1 &2 &3 \\ 2& 1 & 4\\ 3 & 0 & 5 \end{bmatrix}$ Solution: Given A = $\begin{bmatrix} 1 &2 &3 \\ 2& 1 & 4\\ 3 & 0 & 5 \end{bmatrix}$ Now we apply elementary transformations. R2 → R2 – 2R1 R3 → R3 – 3R1 We get $\begin{bmatrix} 1 &2 &3 \\ 0& -3 & -2\\ 0 & -6 & -4 \end{bmatrix}$ R3 → R3 – 2R2 $\begin{bmatrix} 1 &2 &3 \\ 0& -3 & -2\\ 0 & 0 & 0 \end{bmatrix}$ The above matrix is in row echelon form. Number of non-zero rows = 2 Hence the rank of matrix A = 2 Example 2: Find the rank of the matrix A = $\begin{bmatrix} 1 &2 &3 \\ 2& 3 &4\\ 3 & 5 & 7 \end{bmatrix}$ Solution: Given A = $\begin{bmatrix} 1 &2 &3 \\ 2& 3 &4\\ 3 & 5 & 7 \end{bmatrix}$ Now we transform the matrix A to echelon form by using elementary transformation. R2 → R2 – 2R1 R3 → R3 – 3R1 $\begin{bmatrix} 1 &2 &3 \\ 0& -1 &-2\\ 0 & -1 & -2 \end{bmatrix}$ R3 → R3 – R2 $\begin{bmatrix} 1 &2 &3 \\ 0& -1 &-2\\ 0 & 0 & 0 \end{bmatrix}$ Number of non-zero rows = 2 Hence the rank of matrix A = 2 Example 3: Find the rank of the matrix. $\begin{bmatrix} 1 &1 &1 \\ 1& 1 &1\\ 1 & 1 & 1 \end{bmatrix}$ Solution: Given $\begin{bmatrix} 1 &1 &1 \\ 1& 1 &1\\ 1 & 1 & 1 \end{bmatrix}$ R2 → R2 – R1 R3 → R3 – R1 We get $\begin{bmatrix} 1 &1 &1 \\ 0& 0 &0\\ 0 & 0 & 0 \end{bmatrix}$ Here number of non zero rows = 1 Hence the rank of the matrix = 1 Example 4: Find the rank of the 2×2 matrix B = $\begin{bmatrix} 5 & 6\\ 7& 8 \end{bmatrix}$ Solution: Given B = $\begin{bmatrix} 5 & 6\\ 7& 8 \end{bmatrix}$ Order of B = 2×2 \|B\| = 40 – 42 = -2 ≠ 0 So the rank of B = 2 Example 5: Given A = $\begin{bmatrix} 4& 7\\ 8& 14 \end{bmatrix}$ Find the rank of matrix A. Solution: Given A = $\begin{bmatrix} 4& 7\\ 8& 14 \end{bmatrix}$ By observing the rows, we can see that elements of the second row are twice the elements of the first row. R1→ 2R1 – R2 $\begin{bmatrix} 0& 0\\ 8& 14 \end{bmatrix}$ Number of non zero rows = 1 Rank of matrix A = 1. Example 6: The rank of the matrix M is M = $\begin{bmatrix} 0 & 1 & 1\\ 1& 0 &1 \\ 1& 1& 0 \end{bmatrix}$ a) 1 b) 2 c) 3 d) 0 Solution: M = $\begin{bmatrix} 0 & 1 & 1\\ 1& 0 &1 \\ 1& 1& 0 \end{bmatrix}$ Apply row transformations to make the matrix into echelon form. Interchange R2 and R1. $\begin{bmatrix} 1 & 0 & 1\\ 0& 1 &1 \\ 1& 1& 0 \end{bmatrix}$ R3 → R3 – R1 $\begin{bmatrix} 1 & 0 & 1\\ 0& 1 &1 \\ 0& 1& -1 \end{bmatrix}$ R3 → R3 – R2 $\begin{bmatrix} 1 & 0 & 1\\ 0& 1 &1 \\ 0& 0& -2 \end{bmatrix}$ Divide R3 by -2 $\begin{bmatrix} 1 & 0 & 1\\ 0& 1 &1 \\ 0& 0& 1 \end{bmatrix}$ Since there are three non zero rows, rank = 3 Hence option (c) is the answer. Example 7: The rank of the following matrix is $\begin{bmatrix} 1 & 1 & 0& -2\\ 2& 0& 2 & 2\\ 4& 1 & 3 & 1 \end{bmatrix}$ a) 1 b) 2 c) 3 4) 4 Solution: Given $\begin{bmatrix} 1 & 1 & 0& -2\\ 2& 0& 2 & 2\\ 4& 1 & 3 & 1 \end{bmatrix}$ We transform the matrix using elementary row operations. R2 → R2 – 2R1 $\begin{bmatrix} 1 & 1 & 0& -2\\ 0& -2& 2 & 6\\ 0& -3 & 3 & 9 \end{bmatrix}$ R2 → R2/-2 $\begin{bmatrix} 1 & 1 & 0& -2\\ 0& 1& -1 &-3\\ 0& -3 & 3 & 9 \end{bmatrix}$ R3 → R3 + 3R2 $\begin{bmatrix} 1 & 1 & 0& -2\\ 0& 1& -1 &-3\\ 0& 0 & 0 & 0 \end{bmatrix}$ Since the number of non zero rows is 2, rank = 2 Hence option (b) is the answer. Example 8: Let P = $\begin{bmatrix} 1 & 1 & -1\\ 2 & -3& 4\\ 3 & -2 & 3 \end{bmatrix}$ And Q = $\begin{bmatrix} -1 & -2 & -1\\ 6& 12& 6\\ 5 & 10 & 5 \end{bmatrix}$ be two matrices. Then the rank of P + Q = a) 1 b) 0 c) 2 d) 3 Solution: Given P = $\begin{bmatrix} 1 & 1 & -1\\ 2 & -3& 4\\ 3 & -2 & 3 \end{bmatrix}$ Q = $\begin{bmatrix} -1 & -2 & -1\\ 6& 12& 6\\ 5 & 10 & 5 \end{bmatrix}$ P + Q = $\begin{bmatrix} 0 & -1 & -2\\ 8& 9& 10\\ 8& 8 & 8 \end{bmatrix}$ Interchange C1 and C2 $\begin{bmatrix} -1 & 0 & -2\\ 9& 8& 10\\ 8& 8 & 8 \end{bmatrix}$ R2 → R2 + 9R1 R3 → R3 + 8R1 $\begin{bmatrix} -1 & 0 & -2\\ 0& 8& -8\\ 0& 8 & -8 \end{bmatrix}$ R3 → R3 – R2 $\begin{bmatrix} -1 & 0 & -2\\ 0& 8& -8\\ 0& 0 & 0 \end{bmatrix}$ R2 → R2/8 $\begin{bmatrix} -1 & 0 & -2\\ 0& 1& -1\\ 0& 0 & 0 \end{bmatrix}$ Number of non zero rows = 2 So the rank = 2 Hence option (c) is the answer.	2021-06-16 10:25:39	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 40, "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": 0.7673259377479553, "perplexity": 492.031514577451}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487623596.16/warc/CC-MAIN-20210616093937-20210616123937-00320.warc.gz"}
https://www.physicsforums.com/threads/electric-field-vector-surface-integral.530159/	# Electric field vector & surface integral 1. Sep 14, 2011 ### jegues 1. The problem statement, all variables and given/known data See figure attached. 2. Relevant equations 3. The attempt at a solution See figure attached. The solution shows that, $\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}$ How did they obtain this? #### Attached Files: File size: 17.9 KB Views: 75 • ###### AT1.5A.jpg File size: 41.5 KB Views: 74 2. Sep 14, 2011 ### DiracRules First of all, you did not write correctly the expression of the electric field in Cartesian coordinates. (Remember that \|r\| is not -usually- equals to 1) Then, how do you usually evaluate the surface integral (the flux) over a certain surface? 3. Sep 14, 2011 ### jegues It's not a vector, it's a unit vector, that's what the hat signifies. For the flux since it's a closed surface I can apply divergence theorem, $\oint_{S} \vec{F} \cdot \hat{n}dS = \int\int\int_{V}\vec{\nabla} \cdot \vec{F}dV$ I'm still confused as how to get $\vec{r}$. 4. Sep 15, 2011 ### jegues Bump, still looking for some help on this one! 5. Sep 15, 2011 ### DiracRules Sorry for misunderstanding r, but everyone has his own conventions and symbols :D In Cartesian coordinates, how do you write the position occupied by an object? By giving the three coordinates. You can write both $\vec{r}_P=\left[\begin{array}{c}x\\y\\z\end{array}\right]$ and $\vec{r}_P=x\hat{i}+y\hat{j}+z\hat{k}$ since$\left[\begin{array}{c}x\\y\\z\end{array}\right]=x\left[\begin{array}{c}1\\0\\0\end{array}\right]+y\left[\begin{array}{c}0\\1\\0\end{array}\right]+z\left[\begin{array}{c}0\\0\\1\end{array}\right]$ Now, I think that for the first part of the question you cannot use the flux theorem because the problem asks to calculate the flux through one face of the cube, not through the whole surface: I fear you need to calculate explicitly the flux by evaluating the surface integral or something like that (it shouldn't be too difficult, however). You'd better express the electric field in term of its cartesian components and then try to integrate. 6. Sep 15, 2011 ### jegues So for the first part of the question, "Express the electric field vector in its rectangular coordinate components" does this have any significance with the cube at all? The electric field vector is going to generate a radial vector field from the point charge outward everywhere in space. The radius depends on which point in space you are observing (i.e. any point (x,y,z)). Thus, $\vec{E} = \frac{q \vec{r}}{4\pi \epsilon_{o}r^{3}}$ Where, $$\vec{r} = x\hat{i} + y\hat{j} + z\hat{k} \quad \text{Giving,} \quad r = \sqrt{x^{2} + y^{2} + z^{2}}$$ I'm moving onto the cube portion now, I'll post my results. 7. Sep 15, 2011 ### jegues Here's what I've got so far, can't remember how to evaluate such an integral, #### Attached Files: • ###### AT1.5a.JPG File size: 48.7 KB Views: 75 8. Sep 16, 2011 ### DiracRules I think it is right. To evaluate the integral, you can use http://en.wikipedia.org/wiki/List_of_integrals_of_irrational_functions" [Broken] > List of integrals involving $R=\sqrt{ax^2+bx+c}$ > $\int \frac{dx}{R^3}$ Last edited by a moderator: May 5, 2017 9. Sep 16, 2011 ### jegues How would I do that? I don't have $ax^{2} + bx + c$, we would be missing the term with the b cofficient. Last edited by a moderator: May 5, 2017 10. Sep 16, 2011 put b=0 :D	2017-11-23 10:16:29	{"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": 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": 0.823095977306366, "perplexity": 1505.1302478808102}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1510934806768.19/warc/CC-MAIN-20171123085338-20171123105338-00577.warc.gz"}
https://indico.physik.uni-muenchen.de/event/5/contributions/161/	# Laser-Plasma Accelerator Workshop 2019 5-10 May 2019 MedILS Europe/Berlin timezone ## Multi-Species dynamics in the radiation pressure acceleration of ions from ultra-thin foils Not scheduled 20m MedILS #### MedILS Meštrovićevo šetalište 45 HR – 21000 Split Republic of Croatia Poster Contribution ### Speaker Aodhan McIlvenny (Queen's University Belfast) ### Description As the community prepares for the next generation of laser facilities coming online in the near future, attention will shift towards advanced mechanisms such as the radiation pressure acceleration (RPA) which has been predicted to be the dominant ion acceleration mechanism at intensities >10$^{22}$W/cm2 [1]. Recent studies have shown that current facilities can also enter this regime by irradiating ultra-thin foils with circularly polarised (CP) pulses at intensities [2], I ~6x10$^{20}$W/cm2 on target with the use of double plasma mirrors for contrast enhancement [2]. The use of CP light helps to reduce electron heating thus mitigating relativistic transparency and allowing the target to remain opaque and efficiently accelerated by RPA in the Light Sail mode. The work presented here will focus on a recent campaign on the GEMINI laser system at the Rutherford Appleton laboratory which has advanced the results reported in [2], by improving the efficiency of the bulk species (Carbon) acceleration and demonstrating the existence of an optimal thickness for Light Sail acceleration. Additionally, the data highlight the importance of multispecies dynamics during the acceleration with clear evidence for a different acceleration mechanism for Carbon ions and protons ions. Ultra-thin (2-100nm) amorphous carbon foils were irradiated at normal incidence with an f/2 parabola by a high contrast 40fs laser pulse with ~6J on target, producing an intensity of ~5x10$^{20}$W/cm2. The data shows a clear difference between the effects of linearly polarized and CP light on the ion energies with CP generating significantly higher carbon energies for thinner targets, with an optimum thickness of 15nm. For this type of target, experimental data shows the acceleration of C6+ up to 33MeV/n (400MeV) while the corresponding proton energies are less than 18MeV. 2D PIC simulations (carried out with the EPOCH code) suggest that this may be associated to a non-negligible laser pedestal on the sub 6ps timescale (within the reflection window of the plasma mirror). Protons, with the higher q/m ratio, will expand much faster than C6+ beyond the short Rayleigh range associated with the f/2 parabola before the peak of the pulse arrives. The remaining plasma will remain an over-dense, sub-wavelength scale, carbon-electron plasma that can still be efficiently accelerated by RPA. [1] A. Macchi, M. Borghesi, and M. Passoni, “Ion acceleration by superintense laser-plasma interaction,” Rev. Mod. Phys., vol. 85, no. 2, pp. 751–793, 2013. [2] C. Scullion et al., “Polarization Dependence of Bulk Ion Acceleration from Ultrathin Foils Irradiated by High-Intensity Ultrashort Laser Pulses,” Phys. Rev. Lett., vol. 119, no. 5, pp. 1–6, 2017. Working group Laser-driven ion acceleration ### Primary author Aodhan McIlvenny (Queen's University Belfast) ### Co-authors Dr Domencio Doria (Extreme Light Infrastructure – Nuclear Physics (ELI-NP), ) Dr Lorenzo Romagnani (LULI) Dr Hamad Ahmed (QUB) Mr Philip Martin (QUB ) Mr Samuel Williamson (SUPA) Ms Emma Ditter (The John Adam’s Institute) Dr Oliver Ettlinger (The John Adam’s Institute) Dr George Hicks (The John Adam’s Institute) Dr Andrea Macchi (Universita di Pisa, Italy) Prof. Paul McKenna (SUPA) Prof. Zulfikar Najmudin (The John Adam’s Institute,) Prof. David Neely (Science and Technology Facilities Council, UNITED KINGDOM) Prof. Satya Kar (QUB) Prof. Marco Borghesi (QUB) ### Presentation Materials There are no materials yet.	2021-10-18 16:26:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5545517206192017, "perplexity": 10244.010872542727}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1634323585204.68/warc/CC-MAIN-20211018155442-20211018185442-00588.warc.gz"}
https://www.zbmath.org/authors/?q=ai%3Asenateur.henri	# zbMATH — the first resource for mathematics ## Senateur, Henri Compute Distance To: Author ID: senateur.henri Published as: Senateur, Henri; Sénateur, Henri Documents Indexed: 6 Publications since 1984 #### Co-Authors 0 single-authored 6 Dunau, Jean-Louis #### Serials 1 Journal of Multivariate Analysis 1 Probability Theory and Related Fields 1 Journal of Theoretical Probability 1 Annales de l’Institut Henri Poincaré. Probabilités et Statistiques #### Fields 5 Probability theory and stochastic processes (60-XX) 3 Statistics (62-XX) all top 5 #### Cited by 8 Authors 2 Dunau, Jean-Louis 2 Senateur, Henri 1 Abdesselam, Abdelmalek 1 Bernadac, Évelyne 1 Hassenforder, Claudie 1 Kato, Shogo 1 Letac, Gérard G. 1 McCullagh, Peter #### Cited in 5 Serials 3 Journal of Theoretical Probability 1 Journal of Functional Analysis 1 Probability Theory and Related Fields 1 Bernoulli 1 $$p$$-Adic Numbers, Ultrametric Analysis, and Applications #### Cited in 4 Fields 5 Probability theory and stochastic processes (60-XX) 2 Statistics (62-XX) 1 Number theory (11-XX) 1 Quantum theory (81-XX)	2021-05-15 08:09:46	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.35026538372039795, "perplexity": 14625.804818505381}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1620243991378.48/warc/CC-MAIN-20210515070344-20210515100344-00534.warc.gz"}
https://wikimili.com/en/Antiprism	# Antiprism Last updated Set of uniform n-gonal antiprisms Uniform hexagonal antiprism (n = 6) Type uniform in the sense of semiregular polyhedron Faces 2 regular n-gons 2n equilateral triangles Edges 4n Vertices 2n Vertex configuration 3.3.3.n Schläfli symbol { }⊗{n} [1] s{2,2n} sr{2,n} Conway notation An Coxeter diagram Symmetry group Dnd, [2+,2n], (2*n), order 4n Rotation group Dn, [2,n]+, (22n), order 2n Dual polyhedron convex dual-uniform n-gonal trapezohedron Properties convex, vertex-transitive, regular polygon faces, congruent & coaxial bases Net Net of uniform enneagonal antiprism (n = 9) In geometry, an n-gonal antiprism or n-antiprism is a polyhedron composed of two parallel direct copies (not mirror images) of an n-sided polygon, connected by an alternating band of 2n triangles. They are represented by the Conway notation An. ## Contents Antiprisms are a subclass of prismatoids, and are a (degenerate) type of snub polyhedron. Antiprisms are similar to prisms, except that the bases are twisted relatively to each other, and that the side faces (connecting the bases) are 2n triangles, rather than n quadrilaterals. The dual polyhedron of an n-gonal antiprism is an n-gonal trapezohedron. ## History At the intersection of modern-day graph theory and coding theory, the triangulation of a set of points have interested mathematicians since Isaac Newton, who fruitlessly sought a mathematical proof of the kissing number problem in 1694. [2] The existence of antiprisms was discussed, and their name was coined by Johannes Kepler, though it is possible that they were previously known to Archimedes, as they satisfy the same conditions on faces and on vertices as the Archimedean solids.[ citation needed ] According to Ericson and Zinoviev, Harold Scott MacDonald Coxeter wrote at length on the topic, [2] and was among the first to apply the mathematics of Victor Schlegel to this field. Knowledge in this field is "quite incomplete" and "was obtained fairly recently", i.e. in the 20th century[ citation needed ]. For example, as of 2001 it had been proven for only a limited number of non-trivial cases that the n-gonal antiprism is the mathematically optimal arrangement of 2n points in the sense of maximizing the minimum Euclidean distance between any two points on the set: in 1943 by László Fejes Tóth for 4 and 6 points (digonal and trigonal antiprisms, which are Platonic solids); in 1951 by Kurt Schütte and Bartel Leendert van der Waerden for 8 points (tetragonal antiprism, which is not a cube). [2] The chemical structure of binary compounds has been remarked to be in the family of antiprisms; [3] especially those of the family of boron hydrides (in 1975) and carboranes because they are isoelectronic. This is a mathematically real conclusion reached by studies of X-ray diffraction patterns, [4] and stems from the 1971 work of Kenneth Wade, [5] the nominative source for Wade's rules of polyhedral skeletal electron pair theory. Rare-earth metals such as the lanthanides form antiprismatic compounds with some of the halides or some of the iodides. The study of crystallography is useful here. [6] Some lanthanides, when arranged in peculiar antiprismatic structures with chlorine and water, can form molecule-based magnets. [7] ## Right antiprism For an antiprism with regular n-gon bases, one usually considers the case where these two copies are twisted by an angle of 180/n degrees. The axis of a regular polygon is the line perpendicular to the polygon plane and lying in the polygon centre. For an antiprism with congruent regularn-gon bases, twisted by an angle of 180/n degrees, more regularity is obtained if the bases have the same axis: are coaxial ; i.e. (for non-coplanar bases): if the line connecting the base centers is perpendicular to the base planes. Then the antiprism is called a right antiprism, and its 2n side faces are isosceles triangles. ## Uniform antiprism A uniform n-antiprism has two congruent regular n-gons as base faces, and 2n equilateral triangles as side faces. Uniform antiprisms form an infinite class of vertex-transitive polyhedra, as do uniform prisms. For n = 2, we have the regular tetrahedron as a digonal antiprism (degenerate antiprism); for n = 3, the regular octahedron as a triangular antiprism (non-degenerate antiprism). Family of uniform n-gonal antiprisms Antiprism name Digonal antiprism (Trigonal) Triangular antiprism (Tetragonal) Square antiprism Pentagonal antiprism Hexagonal antiprism Heptagonal antiprism Octagonal antiprism Enneagonal antiprism Decagonal antiprism Hendecagonal antiprismDodecagonal antiprism... Apeirogonal antiprism Polyhedron image ... Spherical tiling image Plane tiling image Vertex config. 2.3.3.33.3.3.34.3.3.35.3.3.36.3.3.37.3.3.38.3.3.39.3.3.310.3.3.311.3.3.312.3.3.3...∞.3.3.3 ### Schlegel diagrams A3 A4 A5 A6 A7 A8 ## Cartesian coordinates Cartesian coordinates for the vertices of a right n-antiprism (i.e. with regular n-gon bases and 2n isosceles triangle side faces) are: ${\displaystyle \left(\cos {\frac {k\pi }{n}},\sin {\frac {k\pi }{n}},(-1)^{k}h\right)}$ where 0 ≤ k ≤ 2n – 1; if the n-antiprism is uniform (i.e. if the triangles are equilateral), then: ${\displaystyle 2h^{2}=\cos {\frac {\pi }{n}}-\cos {\frac {2\pi }{n}}.}$ ## Volume and surface area Let a be the edge-length of a uniform n-gonal antiprism; then the volume is: ${\displaystyle V={\frac {n~{\sqrt {4\cos ^{2}{\frac {\pi }{2n}}-1}}\sin {\frac {3\pi }{2n}}}{12\sin ^{2}{\frac {\pi }{n}}}}~a^{3},}$ and the surface area is: ${\displaystyle A={\frac {n}{2}}\left(\cot {\frac {\pi }{n}}+{\sqrt {3}}\right)a^{2}.}$ There are an infinite set of truncated antiprisms, including a lower-symmetry form of the truncated octahedron (truncated triangular antiprism). These can be alternated to create snub antiprisms, two of which are Johnson solids, and the snub triangular antiprism is a lower symmetry form of the regular icosahedron. Antiprisms ... s{2,4} s{2,6} s{2,8} s{2,10} s{2,2n} Truncated antiprisms ... ts{2,4} ts{2,6} ts{2,8}ts{2,10}ts{2,2n} Snub antiprisms J84IcosahedronJ85Irregular faces... ... ss{2,4} ss{2,6} ss{2,8} ss{2,10} ss{2,2n} ## Symmetry The symmetry group of a right n-antiprism (i.e. with regular bases and isosceles side faces) is Dnd = Dnv of order 4n, except in the cases of: • n = 2: the regular tetrahedron, which has the larger symmetry group Td of order 24 = 3×(4×2), which has three versions of D2d as subgroups; • n = 3: the regular octahedron, which has the larger symmetry group Oh of order 48 = 4×(4×3), which has four versions of D3d as subgroups. The symmetry group contains inversion if and only if n is odd. The rotation group is Dn of order 2n, except in the cases of: • n = 2: the regular tetrahedron, which has the larger rotation group T of order 12 = 3×(2×2), which has three versions of D2 as subgroups; • n = 3: the regular octahedron, which has the larger rotation group O of order 24 = 4×(2×3), which has four versions of D3 as subgroups. Note: The right n-antiprisms have congruent regular n-gon bases and congruent isosceles triangle side faces, thus have the same (dihedral) symmetry group as the uniform n-antiprism, for n ≥ 4. ## Star antiprism 5/2-antiprism 5/3-antiprism 9/2-antiprism 9/4-antiprism 9/5-antiprism Uniform star antiprisms are named by their star polygon bases, {p/q}, and exist in prograde and in retrograde (crossed) solutions. Crossed forms have intersecting vertex figures, and are denoted by "inverted" fractions: p/(p q) instead of p/q; example: 5/3 instead of 5/2. A right star antiprism has two congruent coaxial regular convex or star polygon base faces, and 2n isosceles triangle side faces. Any star antiprism with regular convex or star polygon bases can be made a right star antiprism (by translating and/or twisting one of its bases, if necessary). In the retrograde forms but not in the prograde forms, the triangles joining the convex or star bases intersect the axis of rotational symmetry. Thus: • Retrograde star antiprisms with regular convex polygon bases cannot have all equal edge lengths, so cannot be uniform. "Exception": a retrograde star antiprism with equilateral triangle bases (vertex configuration: 3.3/2.3.3) can be uniform; but then, it has the appearance of an equilateral triangle: it is a degenerate star polyhedron. • Similarly, some retrograde star antiprisms with regular star polygon bases cannot have all equal edge lengths, so cannot be uniform. Example: a retrograde star antiprism with regular star 7/5-gon bases (vertex configuration: 3.3.3.7/5) cannot be uniform. Also, star antiprism compounds with regular star p/q-gon bases can be constructed if p and q have common factors. Example: a star 10/4-antiprism is the compound of two star 5/2-antiprisms. ## Related Research Articles A (symmetric) n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base. An n-gonal bipyramid has 2n triangle faces, 3n edges, and 2 + n vertices. In geometry, an octahedron is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent regular polygons, and the same number of faces meet at each vertex. There are only five such polyhedra: In geometry, a prism is a polyhedron comprising an n-sided polygon base, a second base which is a translated copy of the first, and n other faces, necessarily all parallelograms, joining corresponding sides of the two bases. All cross-sections parallel to the bases are translations of the bases. Prisms are named after their bases, e.g. a prism with a pentagonal base is called a pentagonal prism. Prisms are a subclass of prismatoids. In geometry, a decagon is a ten-sided polygon or 10-gon. The total sum of the interior angles of a simple decagon is 1440°. In Euclidean geometry, a regular polygon is a polygon that is direct equiangular and equilateral. Regular polygons may be either convex, star or skew. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon, if the edge length is fixed. In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon. In geometry, a dodecagon or 12-gon is any twelve-sided polygon. In geometry, the triangular bipyramid is a type of hexahedron, being the first in the infinite set of face-transitive bipyramids. It is the dual of the triangular prism with 6 isosceles triangle faces. In geometry, a cupola is a solid formed by joining two polygons, one with twice as many edges as the other, by an alternating band of isosceles triangles and rectangles. If the triangles are equilateral and the rectangles are squares, while the base and its opposite face are regular polygons, the triangular, square, and pentagonal cupolae all count among the Johnson solids, and can be formed by taking sections of the cuboctahedron, rhombicuboctahedron, and rhombicosidodecahedron, respectively. In geometry, an n-gonaltrapezohedron, n-trapezohedron, n-antidipyramid, n-antibipyramid, or n-deltohedron is the dual polyhedron of an n-gonal antiprism. The 2n faces of an n-trapezohedron are congruent and symmetrically staggered; they are called twisted kites. With a higher symmetry, its 2n faces are kites. The triaugmented triangular prism, in geometry, is a convex polyhedron with 14 equilateral triangles as its faces. It can be constructed from a triangular prism by attaching equilateral square pyramids to each of its three square faces. The same shape is also called the tetrakis triangular prism, tricapped trigonal prism, tetracaidecadeltahedron, or tetrakaidecadeltahedron; these last names mean a polyhedron with 14 triangular faces. It is an example of a deltahedron and of a Johnson solid. In geometry, the gyrobifastigium is the 26th Johnson solid. It can be constructed by joining two face-regular triangular prisms along corresponding square faces, giving a quarter-turn to one prism. It is the only Johnson solid that can tile three-dimensional space. In geometry, a uniform 4-polytope is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons. In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent. In geometry, the square antiprism is the second in an infinite family of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It is also known as an anticube. In geometry, the hexagonal antiprism is the 4th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face. It is a conic solid with polygonal base. A pyramid with an n-sided base has n + 1 vertices, n + 1 faces, and 2n edges. All pyramids are self-dual. In geometry, a prismatic compound of antiprism is a category of uniform polyhedron compound. Each member of this infinite family of uniform polyhedron compounds is a symmetric arrangement of antiprisms sharing a common axis of rotational symmetry. In geometry, an icosahedron is a polyhedron with 20 faces. The name comes from Ancient Greek εἴκοσι (eíkosi) 'twenty' and from Ancient Greek ἕδρα (hédra) ' seat'. The plural can be either "icosahedra" or "icosahedrons". ## References 1. N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite symmetry groups, 11.3 Pyramids, Prisms, and Antiprisms, Figure 11.3c 2. Ericson, Thomas; Zinoviev, Victor (2001). "Codes in dimension n = 3". Codes on Euclidean Spheres. North-Holland Mathematical Library. Vol. 63. pp. 67–106. doi:10.1016/S0924-6509(01)80048-9. ISBN 9780444503299. 3. Beall, Herbert; Gaines, Donald F. (2003). "Boron Hydrides". Encyclopedia of Physical Science and Technology. pp. 301–316. doi:10.1016/B0-12-227410-5/00073-9. ISBN 9780122274107. 4. “Boron Hydride Chemistry” (E. L. Muetterties, ed.), Academic Press, New York 5. Wade, K. (1971). "The structural significance of the number of skeletal bonding electron-pairs in carboranes, the higher boranes and borane anions, and various transition-metal carbonyl cluster compounds". J. Chem. Soc. D. 1971 (15): 792–793. doi:10.1039/C29710000792. 6. Meyer, Gerd (2014). [10.1016/B978-0-444-63256-2.00264-3 "Symbiosis of Intermetallic and Salt"]. Including Actinides. Handbook on the Physics and Chemistry of Rare Earths. Vol. 45. pp. 111–178. doi:10.1016/B978-0-444-63256-2.00264-3. ISBN 9780444632562.{{cite book}}: Check \|chapter-url= value (help) 7. Bartolomé, Elena; Arauzo, Ana; Luzón, Javier; Bartolomé, Juan; Bartolomé, Fernando (2017). Magnetic Relaxation of Lanthanide-Based Molecular Magnets. Handbook of Magnetic Materials. Vol. 26. pp. 1–289. doi:10.1016/bs.hmm.2017.09.002. ISBN 9780444639271. ### Bibliography • Anthony Pugh (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0-520-03056-7. Chapter 2: Archimedean polyhedra, prisms and antiprisms	2023-03-26 06:23:14	{"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": 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": 0.7153171896934509, "perplexity": 2591.2868523692205}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1679296945433.92/warc/CC-MAIN-20230326044821-20230326074821-00604.warc.gz"}
http://codeforces.com/problemset/problem/1055/A	A. Metro time limit per test 1 second memory limit per test 256 megabytes input standard input output standard output Alice has a birthday today, so she invited home her best friend Bob. Now Bob needs to find a way to commute to the Alice's home. In the city in which Alice and Bob live, the first metro line is being built. This metro line contains $n$ stations numbered from $1$ to $n$. Bob lives near the station with number $1$, while Alice lives near the station with number $s$. The metro line has two tracks. Trains on the first track go from the station $1$ to the station $n$ and trains on the second track go in reverse direction. Just after the train arrives to the end of its track, it goes to the depot immediately, so it is impossible to travel on it after that. Some stations are not yet open at all and some are only partially open — for each station and for each track it is known whether the station is closed for that track or not. If a station is closed for some track, all trains going in this track's direction pass the station without stopping on it. When the Bob got the information on opened and closed stations, he found that traveling by metro may be unexpectedly complicated. Help Bob determine whether he can travel to the Alice's home by metro or he should search for some other transport. Input The first line contains two integers $n$ and $s$ ($2 \le s \le n \le 1000$) — the number of stations in the metro and the number of the station where Alice's home is located. Bob lives at station $1$. Next lines describe information about closed and open stations. The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($a_i = 0$ or $a_i = 1$). If $a_i = 1$, then the $i$-th station is open on the first track (that is, in the direction of increasing station numbers). Otherwise the station is closed on the first track. The third line contains $n$ integers $b_1, b_2, \ldots, b_n$ ($b_i = 0$ or $b_i = 1$). If $b_i = 1$, then the $i$-th station is open on the second track (that is, in the direction of decreasing station numbers). Otherwise the station is closed on the second track. Output Print "YES" (quotes for clarity) if Bob will be able to commute to the Alice's home by metro and "NO" (quotes for clarity) otherwise. You can print each letter in any case (upper or lower). Examples Input 5 31 1 1 1 11 1 1 1 1 Output YES Input 5 41 0 0 0 10 1 1 1 1 Output YES Input 5 20 1 1 1 11 1 1 1 1 Output NO Note In the first example, all stations are opened, so Bob can simply travel to the station with number $3$. In the second example, Bob should travel to the station $5$ first, switch to the second track and travel to the station $4$ then. In the third example, Bob simply can't enter the train going in the direction of Alice's home.	2020-01-20 13:17:30	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.581950843334198, "perplexity": 853.6704672048616}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250598726.39/warc/CC-MAIN-20200120110422-20200120134422-00231.warc.gz"}
https://dsp.stackexchange.com/questions/10438/can-you-quarter-the-processing-time-for-real-symmetric-ffts	# Can you quarter the processing time for real, symmetric FFTs? The DFT of a real signal is Hermite-symmetric, so you can roughly halve the computation time/memory by not bothering to calculate half the values of the spectrum (and complex conjugating the existing values and copying them to the second half if needed). So the rfft operation takes N samples and outputs N/2 spectrum bins in half the time, for instance. Signals which are even-symmetrical and real have even-symmetrical and real spectra (and real spectra take half the memory to store as complex spectra), so for symmetrical input, can the calculation be halved again by only using the first half of the signal (N/2) to generate the first half of the spectrum (N/2)? How? Is there a way to take the regular FFT of the N/2 half-signal and manipulate the output to produce the N/2 half-spectrum? (real/odd ⇔ imaginary/odd would work, too, but real/even ⇔ real/even case is simpler to follow.) • I'll defer to someone with more knowledge, but one thing worth noting is that the discrete cosine transform enforces even symmetry. The "fast" DCT algorithms employ FFT structures that are optimized for these input constraints. I would start your search there. Other than that, I know you can pack a length N sequency into N/2 complex samples and for not much added complexity you can recover a length N FFT. So you can sort of attack it from the other angle of doing a half-length complex FFT. Aug 22, 2013 at 17:57 • @Bryan: Wikipedia says "DCTs are equivalent to DFTs of roughly twice the length, operating on real data with even symmetry (since the Fourier transform of a real and even function is real and even)" Aug 23, 2013 at 16:07 • As I said, I'll defer to someone with more knowledge :) However, to your point, the DCT in general assumes the input is a length N sequence with no symmetry constraints, so it implicity enforces even-symmetry by "mirroring" the sequence about the origin. The method of mirroring is what distinguishes the DCT variants. If your length 'N' sequence is already even-symmetric, you could equivalently perform the DCT of half the sequence. Aug 23, 2013 at 16:42 • Done. I'm not resistent to posting answers, but rather formally answering in domains I'm not comfortable in. Despite reading papers on DCTs, I've never implemented one myself. I've actually implemented the complex packing scheme in hardware before, but that didn't seem like it would exploit all of your input constraints at first glance. Glad I could help. Aug 23, 2013 at 19:08 There are many possible reductions in the number of computations for the DFT for constrained inputs. Looking closer at what well known DFT libraries do (FFTW for example) should be a good resource for how one can exploit these constraints. For your case, I would look at the discrete cosine transform (DCT). The discrete cosine transform enforces even symmetry. The "fast" DCT algorithms employ FFT structures that are optimized for these input constraints. The types of implicit even-symmetry mirroring that the DCT perform are what differentiate the various DCT types (Type I, Type II, ...). If your sequence is truly even-symmetric, then you could use a fast DCT algorithm (one based on FFT structures) with half of your data knowing the algorithm is assuming the other half is equivalent (albeit time-reversed). This should result in a reduction in the number of overall operations, as it would require a DFT of twice this length if you went the normal route. You can also pack a length N sequence into N/2 complex samples and for not much added complexity you can recover a length N FFT. So you can sort of attack it from the other angle of doing a half-length complex FFT. While this seems less efficient, for a typical FFT structure you are complex after the first stage anyways, and so for large DFT sizes this becomes very efficient. • Seems the Type I DCT is exactly what I was looking for. I've tried to learn about the DCT in the past but could never get it. Today I finally do. :) SciPy's dct(a[:N/2+1], 1) produces the same output as rfft(a), but ~3 times as fast, or ~5 times as fast as fft(a). I made an example here Aug 23, 2013 at 21:26 I'm not sure about quartering the process time but this is what I can think of for now. N point DFT of x. $X_k = \sum_{n=0}^{N-1}x_nW_N^{kn}$ By the even symmetry assumption $X_k = \sum_{n=0}^{N/2-1}x_n(W_N^{kn}+W_N^{k(N-1-n)})$ Then we simplify $X_k = W_N^{k(N-1)/2}\sum_{n=0}^{N/2-1}x_n(W_N^{kn}W_N^{-k(N-1)/2}+W_N^{-kn}W_N^{k(N-1)/2})$ $X_k = 2W_N^{k(N-1)/2}\sum_{n=0}^{N/2-1}x_n\cos(2\pi k\frac{n-(N-1)/2}{N})$ There are probably some errors but if simplifications are valid the 'FFT' size is reduced by a factor of 2 and the twiddle factors are real (this maybe corresponds to your quartering). As in normal FFT the symmetry and periodicity can be exploited to decompose the 'symmetric FFT' into radix-2 steps for instnace.	2022-08-13 00:19:15	{"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": 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": 0.6831204295158386, "perplexity": 1129.5421760556433}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571847.45/warc/CC-MAIN-20220812230927-20220813020927-00045.warc.gz"}
http://archive.numdam.org/item/AIHPC_1993__10_2_131_0/	Blow-up behaviour of one-dimensional semilinear parabolic equations Annales de l'I.H.P. Analyse non linéaire, Tome 10 (1993) no. 2, pp. 131-189. @article{AIHPC_1993__10_2_131_0, author = {Herrero, Miguel A. and Velazquez, Juan J. L.}, title = {Blow-up behaviour of one-dimensional semilinear parabolic equations}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {131--189}, publisher = {Gauthier-Villars}, volume = {10}, number = {2}, year = {1993}, zbl = {0813.35007}, mrnumber = {1220032}, language = {en}, url = {http://archive.numdam.org/item/AIHPC_1993__10_2_131_0/} } Herrero, M. A.; Velázquez, J. J. L. Blow-up behaviour of one-dimensional semilinear parabolic equations. Annales de l'I.H.P. Analyse non linéaire, Tome 10 (1993) no. 2, pp. 131-189. http://archive.numdam.org/item/AIHPC_1993__10_2_131_0/ [A] S. 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Widder, The Heat Equation, Academic Press, New York, 1975. \| MR 466967 \| Zbl 0322.35041	2021-04-23 05:41:53	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.28335994482040405, "perplexity": 2200.5203836180026}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1618039601956.95/warc/CC-MAIN-20210423041014-20210423071014-00484.warc.gz"}
https://www.physicsforums.com/threads/why-the-sum-of-cosines-between-v-and-any-vector-1.903658/	# Why the sum of cosines between "v" and any vector =1? 1. The problem statement, all variables and given/known data Given that matrix, A can be decomposed using SVD (Singular Value Decomposition) into $A=USV^T$, why does always the sum of the square of cosines between v vectors and any other column vector q representation of arbitrarily column vector Q vector sum up to 1? 2. Relevant equations $A=USV^T$. $Q=USq$ 3. The attempt at a solution I tried a simple 2x2 matrix but even with this simple matrix, the calculation goes missy. In addition, I seek a rigorous proof. Related Calculus and Beyond Homework News on Phys.org #### Stephen Tashi the sum of the square of cosines between v vectors and any other column vector q representation of arbitrarily column vector Q vector sum up to 1? It isn't clear (to me) what those words mean. $1 = \sum_{k =1}^N \frac{ \overrightarrow{v_k} \cdot \overrightarrow{q_j}}{ \|\overrightarrow{v_k}\|\|\overrightarrow{q_j}\|}$ (?) Exactly, $1 = \sum_{k =1}^N (\frac{ \overrightarrow{v_k} \cdot \overrightarrow{q_j}}{ \|\overrightarrow{v_k}\|\|\overrightarrow{q_j}\|})^2$ I discovered this fact by coincidence but it turns out that it may have a nice link to the quantum mechanics. For example, if the cosine of the angle represents the inner products (the eigen function which is also the inner product between the system state and the eigen state in, say position representation), then the sum of the cosine square is equal to 1. In other words, if the quantum system is complete, then all information of the system is encoded in the wave-function(s). Consequently, it is natural to think about the square of the wave function (or the square of the cosine) as a probability amplitude and because the system is complete, all probabilities should sum up to 1. #### marcusl Gold Member You are projecting an arbitrary unit vector onto a complete orthonormal basis set, and then have defined (or "discovered") the standard Euclidean vector norm. "Why the sum of cosines between "v`" and any vector =1?" ### 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	2019-05-24 02:46:36	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.8773897886276245, "perplexity": 808.085023280668}, "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-2019-22/segments/1558232257497.4/warc/CC-MAIN-20190524024253-20190524050253-00548.warc.gz"}
https://theaisummer.com/text-to-speech/	📖 You can now grab a copy of our new Deep Learning in Production Book 📖 # Speech synthesis: A review of the best text to speech architectures with Deep Learning Sergios Karagiannakoson2021-05-13·14 mins Audio Speech synthesis is the task of generating speech from some other modality like text, lip movements, etc. In most applications, text is chosen as the preliminary form because of the rapid advance of natural language systems. A Text To Speech (TTS) system aims to convert natural language into speech. Over the years there have been many different approaches, with the most prominent being concatenation synthesis and parametric synthesis. ## Concatenation synthesis Concatenation synthesis, as the name suggests, is based on the concatenation of pre-recorded speech segments. The segments can be full sentences, words, syllables, diphones, or even individual phones. They are usually stored in the form of waveforms or spectrograms. We acquire the segments with the help of a speech recognition system and we then label them based on their acoustic properties (e.g. their fundamental frequency). At run time, the desired sequence is created by determining the best chain of candidate units from the database (unit selection). ## Statistical Parametric Synthesis Parametric synthesis utilizes recorded human voices as well. The difference is that we use a function and a set of parameters to modify the voice. Let’s break that down: In statistical parametric synthesis, we generally have two parts. The training and the synthesis. During training, we extract a set of parameters that characterize the audio sample such as the frequency spectrum (vocal tract), fundamental frequency (voice source), and duration (prosody) of speech. We then try to estimate those parameters using a statistical model. The one that has been proven to provide the best results historically is the Hidden Markov Model (HMM). During synthesis, HMMs generate a set of parameters from our target text sequence. The parameters are used to synthesize the final speech waveforms. • No need to store audio sample in a database • Language independence • Flexibility in voice characteristics However, in most cases, the quality of the synthesized speech is not ideal. This is where Deep Learning based methods come into play. But before that, I would like to open a small parenthesis and discuss how we evaluate speech synthesis models. ## Speech synthesis evaluation Mean Opinion Score (MOS) is the most frequently used method to evaluate the quality of the generated speech. MOS has a range from 0 to 5 where real human speech is between 4.5 to 4.8 MOS comes from the telecommunications field and is defined as the arithmetic mean over single ratings performed by human subjects for a given stimulus in a subjective quality evaluation test. This historically means that a group of people sits in a quiet room, listens to the generated sample, and gives it a score. MOS is nothing more than the average of all “people’s opinion” Today’s benchmarks are performed over different speech synthesis datasets in English, Chinese, and other popular languages. You can find such benchmarks in paperswithcode.com ## Speech synthesis with Deep Learning Before we start analyzing the various architectures, let’s explore how we can mathematically formulate TTS. Given an input text sequence $\mathbf{Y}$ , the target speech $\mathbf{X}$ can be derived by: $X=\arg \max P(X\|Y,\theta )$ where $\theta$ is the model’s parameters. In most models, we first pass the input text to an acoustic feature generator, which produces a set of acoustic features such as the fundamental frequency or spectrogram. To generate the final speech segment, a Neural vocoder is typically used. A traditional vocoder is a category of voice codec which encrypts and compresses the audio signal and vice versa. This was traditionally accomplished through digital signal processing techniques. A neural vocoder achieves the encoding/decoding using a neural network. ## WaveNet WaveNet was the first model that successfully modeled the raw waveform of the audio signal instead of the acoustic features. It is able to generate new speech-like waveforms at 16,000 samples per second. WaveNet in its core is an autoregressive model where each sample depends on the previous ones. Mathematically, this can be expressed as : $p_{\theta }(\mathbf {x} )=\prod _{t=1}^{T}p(x_{t}\|x_{1},...,x_{t-1})$ In essence, we factorize the joining probability of the waveform as a product of conditional probabilities of the previous time steps. To build such autoregressive models, the authored used a fully convolutional neural network with dilated convolutions. WaveNet was inspired by PixelCNN and PixelRNN, which are able to generate very complex natural images. As we can see in the image above, each convolutional layer has a dilation factor. They used real waveforms recorded from human speakers during training. After training, the final waveform is produced by sampling from the network. How the sampling is performed? The autoregressive model computes the probability distribution $p_{\theta }(\mathbf {x} )$. At each timestep: • We sample a value from the distribution • We feed the value back to the input, and the model generates the new prediction • We continue this procedure one step at a time to generate the entire speech waveform. This is the main shortcoming of WaveNet. Because we need to perform this for every simple sample, inferences can become very slow and computationally expensive The first version of WaveNet managed to has a MOS of 4.21 in the English language where for previous state of art models, MOS was between 3.67 and 3.86. ### Fast WaveNet Fast WaveNet managed to reduce the complexity of the original WaveNet from $O( 2^L)$ to $O(L)$ where $L$ is the number of layers in the network. This was achieved by introducing a caching system that stored previous calculations. That way no redundant convolutions were ever be calculated. The caching scheme of Fast WaveNet. Source: Fast WaveNet Generation Algorithm ## Deep Voice Deep Voice by Baidu laid the foundation for the later advancements on end-to-end speech synthesis. It consists of 4 different neural networks that together form an end-to-pipeline. 1. A segmentation model that locates boundaries between phonemes. It is a hybrid CNN and RNN network that is trained to predict the alignment between vocal sounds and the target phonemes using the CTC loss. 2. A model that converts graphemes to phonemes. A multi-layer encoder-decoder model with GRU cells was chosen for this task. 3. A model to predict phonemes duration and the fundamental frequencies. Two fully connected layers followed by two unidirectional GRU layers and another fully connected layer, were trained to learn both tasks simultaneously 4. A model to synthesize the final audio. Here the authors implemented a modified WaveNet. The WaveNet consists of a conditioning network that upsamples linguistic features to the desired frequency, and an autoregressive network, which generates a probability distribution P over discretized audio samples System diagram depicting (a) training procedure and (b) inference procedure of DeepVoice. Source: Deep Voice: Real-time Neural Text-to-Speech They also managed to achieve real-time inference by constructing highly optimized CPU and GPU kernels to speed up the inference. It received a MOS of 2.67 in US English. ## Tacotron Tacotron was released by Google in 2017 as an end-to-end system. It is basically a sequence to sequence model that follows the familiar encoder-decoder architecture. An attention mechanism was also utilized. Let’s break down the above diagram. The model takes as input characters and outputs the raw spectrogram of the final speech, which is then converted to waveform. ### The CBHG module You might wonder what is this CBHG. CBHG stands for: 1-D convolution bank + highway network + bidirectional GRU. The CBHG module is used to extract representations from sequences, and it was originally developed for neural machine translation. The below diagram will give you a better understanding: The CBHG module. Source: Tacotron: Towards End-to-End Speech Synthesis Back to Tacotron. The encoder’s goal is to extract robust sequential representations of text. It receives a character sequence represented as one-hot encoding and through a stack of PreNets and CHBG modules, it outputs the final representation. PreNet is used to describe the non-linear transformations applied to each embedding. Content-based attention is used to pass the representation to the decoder, where a recurrent layer produces the attention query at each time step. The query is concatenated with the context vector and passed to a stack of GRU cells with residual connections. The output of the decoder is converted to the end waveform with a separate post-processing network, containing a CBHG module. Tacotron achieved a MOS of 3.82 on an US English evaluation set. ## Deep Voice 2 Deep Voice 2 came as an improvement of the original Deep Voice architecture. While the main pipeline was quite similar, each model was created from scratch to enhance its performance. Another big enhancement was the addition of multi-speaker support. Key points of the architecture: • Separation of the phoneme duration and fundamental frequency models • Speaker embeddings were introduced on each model to achieve multiple-speaker capabilities. The speaker embeddings hold the unique information per speaker and are used to produce recurrent neural network (RNN) initial states, nonlinearity biases, and multiplicative gating factors, used throughout the networks. • Batch normalization and residual connections were applied to the basic models Segmentation, duration and frequency models of DeepVoice 2. Source: Deep Voice 2: Multi-Speaker Neural Text-to-Speech A surprising fact is that the authors showed, in the same paper, that we can also enhance Tacotron to support multi-speakers using similar techniques. Moreover, they replace Tacotron’s spectrogram-to-waveform Model with their own WaveNet-based neural vocoder and the results were very promising DeepVoice 2 with an 80-layer WaveNet, as the sound synthesizer model, achieved a MOS of 3.53 ## Deep Voice 3 Deep Voice 3 is a complete redesign of the previous versions. Here we have a single model instead of four different ones. More specifically, the authors proposed a fully-convolutional character-to-spectrogram architecture which is ideal for parallel computation. As opposed to RNN-based models. They were also experimenting with different waveform synthesis methods with the WaveNet achieving the best results once again. As you can see, Deep Voice 3 is an encoder-decoder architecture and is able to produce a variety of textual features(character, phonemes, etc.) to a variety of vocoder parameters. The encoder is a fully-convolutional neural network that transforms textual features into a compact representation. The decoder is another fully-convolutional network that converts the learned representation into a low-dimensional audio representation. This is achieved using a multi-hop convolutional attention mechanism. The convolution block comprises 1-D convolutions followed by a GRU cell and a residual connection. The convolution block of Deep Voice 3. Source: Deep Voice 3: Scaling Text-to-Speech with Convolutional Sequence Learning The attention mechanism uses a query vector (the hidden states of the decoder) and the per-timestep key vectors from the encoder to compute attention weights. It then outputs a context vector as the weighted average of the value vectors. Attention block of Deep Voice 3. Source: Deep Voice 3: Scaling Text-to-Speech with Convolutional Sequence Learning Deep Voice 3 with WaveNet achieved a MOS of 3.78 at the time of publishing ## Parallel WaveNet Parallel WaveNet aims to solve the complexity and performance issues of the original WaveNet, which relies on sequential generation of the audio, one sample at a time. They introduced a concept called Probability Density Distillation that tries to marry Inverse autoregressive flows with efficient WaveNet training methods. Inverse autoregressive flows (IAFs) represent a kind of dual formulation of deep autoregressive modelling, in which sampling can be performed in parallel. IAFs are stochastic generative models whose latent variables are arranged so that all elements of a high dimensional observable sample can be generated in parallel Let’s break that down and explain it in simple terms: Because each sample depends on the previous ones, we can’t simple parallelized this process and compute them in parallel. Instead, we start out from simple white noise and apply changes over time until it morphs to the desired output waveform. These changes are applied to the entire signal in a parallel fashion. How? We use a teacher-student relationship. The teacher is the original Network that holds the ground truth but it is quite slow. The student is the new network that tries to mimic the teacher but in a more efficient way. According to the authors: “To stress the fact that we are dealing with normalized density models, we refer to this process as Probability Density Distillation (in contrast to Probability Density Estimation). The basic idea is for the student to attempt to match the probability of its own samples under the distribution learned by the teacher” Overview of Probability Density Distillation. Source: Parallel WaveNet: Fast High-Fidelity Speech Synthesis Parallel WaveNet is 1000 times faster than the original networks and can produce 20 seconds of audio in 1 second. Also, note that similar techniques with IAFs to parallelize wave generation have also been used by other architectures such as ClariNet ## Tacotron 2 Tacotron 2 improves and simplifies the original architecture. While there are no major differences, let’s see its key points: • The encoder now consists of 3 convolutional layers and a bidirectional LSTM replacing PreNets and CHBG modules • Location sensitive attention improved the original additive attention mechanism • The decoder is now an autoregressive RNN formed by a Pre-Net, 2 uni-directional LSTMs, and a 5-layer Convolutional Post-Net • A modified WaveNet is used as the Vocoder that follows PixelCNN++ and Parallel WaveNet • Mel spectrograms are generated and passed to the Vocoder as opposed to Linear-scale spectrograms • WaveNet replaced the Griffin-Lin algorithm used in Tacotron 1 Tacotron 2. Source: Natural TTS Synthesis by Conditioning WaveNet on Mel Spectrogram Predictions Tacotron 2 received an impressive MOS of 4.53. ## Global Style Tokens (GST) Global Style Tokens is a new idea to augment Tacotron-based architectures . The authors proposed a bank of embeddings that can be trained jointly with Tacotron in an unsupervised manner (also referred as GST-Tacotron). The embeddings represent the acoustic expressiveness of different speakers and are trained with no explicit labels. In other words, they aim to model different speaking styles. During training, a reference encoder is used to extract a fixed-length vector which encodes the information about the speaking style (also known as prosody). This is then passed to the “style token layer”, an attention layer that calculates the contribution of each token to the resulting style embedding. During inference, a reference audio sequence can be used to produce a style embedding or we can manually control the speech style. ## TTS with Transformers Transformers are dominating the Natural Language field for a while now, so it was inevitable that they will gradually enter the TTS field. Transformers-based models aim to tackle two problems of previous TTS methods such as Tacotron2: • Low efficiency during training and inference • Difficulty in modeling long dependencies using RNNs The first transformers-based architecture introduced in 2018 and replaced RNNs with multi-head attention mechanisms that can be trained in parallel. As you can see above, the proposed architecture resembles the Transformer proposed in the famous “Attention is all you need” paper. In more details we have: • A Text-to-Phoneme Converter: converts text to phonemes • Scaled positional encoding: they use a sinusoidal form that captures information about the position of phonemes • An Encoder Pre-Net: a 3-layer CNN similar to Tacotron 2, which learns the phonemes embeddings • A Decoder Pre-Net: consumes a mel spectogram and projects it into the same subspace as phoneme embeddings • The Encoder: The bi-directional RNN is replaced with a Transformer Encoder with multi-head attention • The Decoder: The 2-layer RNN with location-sensitive attention is replaced by a Transformer decoder with multi-head self-attention • Mel Liner and Stop Linear: Two different linear projections are used to predict the mel spectrogram and the stop token respectively The Transformer-based system achieved a MOS of 4.39. ### FastSpeech A similar approach with Transformers is followed by FastSpeech. FastSpeech managed to speed up the aforementioned architecture by 38x. In short, this was accomplished by the following 3 things: • Parallel mel-specogram generation • Hard alignment between phonemes and their mel-spectograms in contrast to soft attention alignments in the previous model • A length regulator that can easily adjust voice speed by lengthening or shortening the phoneme duration to determine the length of the generated mel spectrograms, In the same direction, Fast Speech 2 and FastPitch came later and improved upon the original idea. ## Flow-based TTS Before we examine flow-based TTS, let’s explain what flow-based models are. Contrary with GANs and VAEs which approximate the probability density function of our data $p(x)$, Flow-based models do exactly that with the help of normalizing flows. Normalizing Flows are a method for constructing complex distributions by transforming a probability density through a series of invertible mappings. By repeatedly applying a predefined rule for change of variables, the initial density ‘flows’ through the sequence of invertible mappings. At the end of this sequence, we obtain a valid probability distribution and hence this type of flow is referred to as a normalizing flow. For more details, check out the original paper A lot of models have been proposed based om that idea with the most popular being RealNVP, NICE and Glow. You can have a look at this excellent article by Lillian Weng to get a more complete understanding. So as you may have guessed, Flow-based TTS models take advantage of this idea and apply it on speech synthesis. ### WaveGlow WaveGlow by Nvidia is one of the most popular flow-based TTS models. It essentially tries to combine insights from Glow and WaveNet in order to achieve fast and efficient audio synthesis without utilizing auto-regression. Note that WaveGlow is used strictly to generated speech from mel spectograms replacing WaveNets. They are not end-to-end TTS systems. Waveglow. Source: WaveGlow: A Flow-based Generative Network for Speech Synthesis The model is trained by minimizing the negative log-likelihood function of the data. To achieve that, we need to use Invertible Neural Networks because otherwise, the function is intractable. I won’t go into many details because we would need a separate article to explain everything but here are a few things to remember: • Invertible neural networks are usually constructed using coupling layers. In this case, the authors used affine coupling layers • They also used 1x1 invertible convolutions following the Glow paradigm Once the model is trained, the inference is simply a matter of randomly sampling values and run them through the network Similar models include Glow-TTS and Flow-TTS. Flowtron, on the other hand, uses an Autoregressive Flow-based Generative Network to generate speech. So we can see that there are research works in both areas of flow-based models. ## GAN-based TTS and EATS Finally, I’d like to close with one of the most recent and impactful works. End-to-End Adversarial Text-to-Speech by Deepmind. EATS falls into the category of GAN-based TTS and is inspired by a previous work called GAN-TTS EATS takes advantage of the adversarial training paradigm used in Generative Adversarial Networks. It operates on pure text or raw phoneme sequences and produces raw waveforms as outputs. EATS consists of two basic submodules: the aligner and the decoder. The aligner receives the raw input sequence and produces low-frequency aligned features in an abstract feature space. The aligner’s job is to map the unaligned input sequence to a representation that is aligned with the output. The decoder takes the features and upsamples them using 1D convolutions to produce audio waveforms. The whole system is trained as a whole entity in an adversarial manner A few key things worth mentioning are: • The generator is a feed-forward neural network that uses a differentiable alignment scheme based on token length prediction • To allow the model to capture temporal variation in the generated audio, soft dynamic time warping is also employed. EATS achieved a MOS of 4.083 You can also find a great explanation of this architecture by Yannic Kilcher, on his Youtube channel. ## Conclusion Text to speech is an area of research with a lot of novel ideas. It is evident that the field has come a long way over the past few years. Take a look at smart devices such as Google assistant, Amazon’s Alexa and Microsoft’s Cortana. If you want to experiment with some of the above models, all you have to do is go into Pytorch’s or TensorFlow model hub, find your model and play around with it. Another great resource is the following repo by Mozilla: TTS: Text-to-Speech for all. If you also want us to explore a different architecture, feel free to ping us and we can include it here as well. ## Deep Learning in Production Book 📖 #### Learn how to build, train, deploy, scale and maintain deep learning models. Understand ML infrastructure and MLOps using hands-on examples. * Disclosure: Please note that some of the links above might be affiliate links, and at no additional cost to you, we will earn a commission if you decide to make a purchase after clicking through.	2022-01-24 17:12:33	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 10, "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": 0.6005449295043945, "perplexity": 1824.2960791140065}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1642320304572.73/warc/CC-MAIN-20220124155118-20220124185118-00471.warc.gz"}
https://mathleaks.com/study/solving_one-step_equations/grade-1-3/solution	{{ item.displayTitle }} navigate_next No history yet! Progress & Statistics equalizer Progress expand_more Student navigate_next Teacher navigate_next {{ filterOption.label }} {{ item.displayTitle }} {{ item.subject.displayTitle }} arrow_forward {{ searchError }} search {{ courseTrack.displayTitle }} {{ printedBook.courseTrack.name }} {{ printedBook.name }} # Solving One-Step Equations ## Solving One-Step Equations 1.3 - Solution a To solve the equation, we need to isolate $x$ on one side of the equation. Subtracting $2$ from both sides will "cancel" it from the right-hand side. $6=x+2$ $6-2=x+2-2$ $4=x$ $x=4$ b To solve the equation we have to isolate $x$ on one side. By adding $11$ to both sides, we can cancel out $\text{-} 11$ and isolate $q.$ $q-11=\text{-}9$ $q-11+11=\text{-}9+11$ $q=2$ c To solve the equation we have to isolate $y$ on one side of the equation. By dividing both sides of the equation by $\text{-}7,$ we can remove the coefficient on the left-hand side. $\text{-}7y=28$ $\dfrac{\text{-}7y}{\text{-}7}=\dfrac{28}{\text{-}7}$ $y=\dfrac{28}{\text{-}7}$ $y=\text{-}\dfrac{28}{7}$ $y=\text{-}4$ d By multiplying both sides of the equation by $2,$ we can eliminate the denominator and isolate $z.$ $14=\dfrac{z}{2}$ $14\cdot 2=\dfrac{z}{2}\cdot 2$ $14\cdot 2=z$ $28=z$ $z=28$	2021-02-26 16:00:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 32, "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": 0.8269956111907959, "perplexity": 864.033439325845}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178357929.4/warc/CC-MAIN-20210226145416-20210226175416-00526.warc.gz"}
https://brilliant.org/discussions/thread/why-phasor-multiplication-dont-agree-with-sinusoid/	× # Why phasor multiplication don't agree with sinusoid multiplication? let us we have two sinusoids sin(wt+30) and sin(wt+45). if we multiply them, when t=0s, we get sin30xsin45=.35355339 Now if we transform these two sinusoids into phasor, we get (co30+j sin30) and (cos45+j sin45) respectively. Multiplying these two phasors we get a new phasor (cos75+j sin75). This new phasor should represent the result sinusoid of above multiplication of two sinusoid, as in phasor addition. But at t=0s, this phasor shows sin75. But clearly sin30xsin45 is not equal to sin75. So why do we use this in AC circuit analysis? I know I am doing something wrong, but where is it? Note by Fahim Khan 1 year, 9 months ago	2016-10-26 07:43:18	{"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": 0.9382340908050537, "perplexity": 4094.798229104048}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1476988720760.76/warc/CC-MAIN-20161020183840-00399-ip-10-171-6-4.ec2.internal.warc.gz"}
https://codereview.stackexchange.com/questions/205458/sourcing-content-from-a-webpage-in-an-organized-manner	# Sourcing content from a webpage in an organized manner I've written a script in python to grab different title and address from different pages of a website. Firstly the script will collect all the property links from the landing page and then go one layer deep to collect the title and address. When I run my script, I get the results accordingly. Should it not be a better approach If I call a single function and the rest of the functions work like a chain to produce the same results? If so, what is the right way to do so? This is the link to that site Here is the working script: import requests from urllib.parse import urljoin from bs4 import BeautifulSoup headers = {"User-Agent":"Mozilla/5.0 (Windows NT 6.1) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/69.0.3497.100 Safari/537.36"} def get_token(): #getting csrf_token r = requests.get(url) soup = BeautifulSoup(r.text,"lxml") item = soup.select_one("[name='bhc_csrf_token']")['value'] 'bhc_csrf_token': item, 'logout': 1 } sauce = BeautifulSoup(res.text,"lxml") soupobj = BeautifulSoup(response.text,"lxml") name = soupobj.select_one("h2#hp_hotel_name").get_text(strip=True) Perhaps I misunderstand you. But it looks as though you've already written the functions to call one another: get_info() calls make_post() internally, and make_post() then delegates getting the token to get_token() by calling it internally to itself. So in your final line of code, you should be able to replace the three calls with a single call to get_info().	2021-08-01 02:15:26	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.24250662326812744, "perplexity": 2622.4445163533414}, "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-2021-31/segments/1627046154127.53/warc/CC-MAIN-20210731234924-20210801024924-00712.warc.gz"}
https://socratic.org/questions/how-many-unpaired-electrons-are-in-a-boron-atom	# How many unpaired electrons are in a boron atom? The configuration of boron is $1 {s}^{2} 2 {s}^{2}$$2 {p}^{1}$ which means only the $p$ electron is unpaired. However, when bonding in a compound, one of the $2 s$ electrons is promoted into the $2 p$ subshell, with the result that $s {p}^{2}$ hybrid orbitals are created, and three bonding orbitals exist.	2020-06-04 15:51:15	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 6, "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": 0.2511052191257477, "perplexity": 9896.96876087474}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347441088.63/warc/CC-MAIN-20200604125947-20200604155947-00516.warc.gz"}
http://askubuntu.com/questions/177650/shutdown-button-not-working	Shutdown button not working [closed] I have 12.04 installed on my laptop. It was working perfectly until yesterday. Since then, no matter whether I choose restart shutdown or suspend option from GUI, my computer suspends. Please give suggestions. - closed as off topic by Eliah Kagan, jokerdino♦, belacq, Mitch♦, fossfreedom♦Aug 20 '12 at 10:20 Questions on Ask Ubuntu are expected to relate to Ubuntu within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here.If this question can be reworded to fit the rules in the help center, please edit the question. This should be reported as a bug. (We might close this as off-topic.) Also, to get help with this or effectively report it as a bug, it will be important for you to specify the exact brand and model of your laptop, whether you're using the 32-bit or 64-bit version of Ubuntu, if running sudo shutdown -P now still successfully shuts the machine down, and if any updates were installed yesterday. – Eliah Kagan Aug 19 '12 at 4:25 (/var/log/dpkg.log, /var/log/apt/history.log, and /var/log/apt/term.log contain detailed information about that--so you can check them, and even attach them to the bug). If the above shutdown command also suspends, run ubuntu-bug linux to begin reporting the bug. Otherwise, probably run ubuntu-bug \$(pidof indicator-session-service) (and manually attach /var/log/dmesg and /var/log/kern.log). – Eliah Kagan Aug 19 '12 at 4:26 the laptop worked fine yesterday without me trying anything. But its again giving problems today. – Shagun Aug 20 '12 at 6:55	2015-01-27 14:49:01	{"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": 0.20729483664035797, "perplexity": 5902.458539609802}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422121981339.16/warc/CC-MAIN-20150124175301-00238-ip-10-180-212-252.ec2.internal.warc.gz"}
https://academ.us/list/math/	### Essential holonomy of Cantor actions A group action has essential holonomy if the set of points with non-trivial holonomy has positive measure. If such an action is topologically free, then having essential holonomy is equivalent to the action not being essentially free, that is, the set of points with non-trivial stabilizer has positive measure. In the paper, we investigate the relation between the property of having essential holonomy and structure of the acting group for minimal equicontinuous actions on Cantor sets. We show that if such a group action is locally quasi-analytic and has essential holonomy, then every commutator subgroup in the group lower central series has elements with positive measure set of points with non-trivial holonomy. We deduce that all minimal equicontinuous Cantor actions by nilpotent groups have no essential holonomy. We also introduce a local version of the Farber criterion, which allows to determine when a locally quasi-analytic action has no essential holonomy. ### Poles of degenerate Eisenstein series and Siegel-Weil identities for exceptional split groups Let $G$ be a linear split algebraic group. The degenerate Eisenstein series associated to a maximal parabolic subgroup $E_{P}(f^{0},s,g)$ with the spherical section $f^{0}$ is studied in the first part of the thesis. In this part, we study the poles of $E_{P}(f^{0},s,g)$ in the region $\operatorname{Re} s >0$. We determine when the leading term in the Laurent expansion of $E_{P}(f^{0},s,g)$ around $s=s_0$ is square integrable. The second part is devoted to finding identities between the leading terms of various Eisenstein series at different points. We present an algorithm to find this data and implement it on \textit{SAGE}. While both parts can be applied to a general algebraic group, we restrict ourself to the case where $G$ is split exceptional group of type $F_4,E_6,E_7$, and obtain new results. ### A remark on the Castelnuovo-Mumford regularity of powers of ideal sheaves We show that a bound of the Castelnuovo-Mumford regularity of any power of the ideal sheaf of a smooth projective complex variety $X\subseteq\mathbb{P}^r$ is sharp exactly for complete intersections, provided the variety $X$ is cut out scheme-theoretically by several hypersurfaces in $\mathbb{P}^r$. This generalizes a result of Bertram-Ein-Lazarsfeld. ### [4] 2205.06298 The Dedekind zeta function of a quadratic number field factors as a product of the Riemann zeta function and the $L$-function of a quadratic Dirichlet character. We categorify this formula using objective linear algebra in the abstract incidence algebra of the division poset. ### An Equivalence Principle for the Spectrum of Random Inner-Product Kernel Matrices We consider random matrices whose entries are obtained by applying a (nonlinear) kernel function to the pairwise inner products between $n$ independent data vectors drawn uniformly from the unit sphere in $\mathbb{R}^d$. Our study of this model is motivated by problems in machine learning, statistics, and signal processing, where such inner-product kernel random matrices and their spectral properties play important roles. Under mild conditions on the kernel function, we establish the weak-limit of the empirical spectral distribution of these matrices when $d, n \to \infty$ such that $n / d^\ell \to \kappa \in (0, \infty)$, for some fixed $\ell \in \mathbb{N}$ and $\kappa \in \mathbb{R}$. This generalizes an earlier result of Cheng and Singer (2013), who studied the same model in the linear scaling regime (with $\ell = 1$ and $n/d \to \kappa$). The main insight of our work is a general equivalence principle: the spectrum of the random kernel matrix is asymptotically equivalent to that of a simpler matrix model, constructed as the linear combination of a (shifted) Wishart matrix and an independent matrix drawn from the Gaussian orthogonal ensemble. The aspect ratio of the Wishart matrix and the coefficients of the linear combination are determined by $\ell$ and by the expansion of the kernel function in the orthogonal Hermite polynomial basis. Consequently, the limiting spectrum of the random kernel matrix can be characterized as the free additive convolution between a Marchenko-Pastur law and a semicircle law. ### Continuity in right semitopological groups Groups with a topology that is in one way or another consistent with the algebraic structure are considered. Classical groups with topology -- topological, paratopological, semitopological, quasitopological groups. We also study other ways of matching topology and algebraic structure. The minimum requirement in this paper is that the group is a right semitopological group (such groups are often called right topological groups). Are studied when a group with topology is a topological group, research in this direction began with the work of Deane Montgomery and Robert Ellis. The (invariant) semi-neighborhoods of the diagonal are used as a means of study. ### Frucht's theorem in Borel setting In this paper, we show that Frucht's theorem holds in Borel setting. More specifically, we prove that any standard Borel group can be realized as the Borel automorphism group of a Borel graph. A slight modification of our construction also yields the following result in topological setting: Any Polish group can be realized as the homeomorphic automorphism group of a $\mathbf{\Delta^0_2}$-graph on a Polish space. ### Reversing monoid actions and domination in graphs Given a graph $G=(V,E)$, a set of vertices $D\subseteq V$ is called a dominating set if every vertex in $V\backslash D$ is adjacent to a vertex in $D$, and a subset $B\subseteq V$ is called a non-blocking set if $V\backslash B$ is a dominating set. In this paper, we introduce a graph dynamical system and establish a one to one correspondence between the periodic points of this system and subsets of $V$ that are simultaneously dominating and non-blocking sets. Besides, by using the action of this graph dynamical system we obtain actions of free monoid on two letters, for which fixed points of the action and more specialized dominating sets, such as, independent dominating sets and perfect dominating sets, coincide. ### Average capacity of quantum entanglement As an alternative to entanglement entropies, the capacity of entanglement becomes a promising candidate to probe and estimate the degree of entanglement of quantum bipartite systems. In this work, we study the typical behavior of entanglement capacity over major models of random states. In particular, the exact and asymptotic formulas of average capacity have been derived under the Hilbert-Schmidt and Bures-Hall ensembles. The obtained formulas generalize some partial results of average capacity computed recently in the literature. As a key ingredient in deriving the results, we make use of recent advances in random matrix theory pertaining to the underlying orthogonal polynomials and special functions. Numerical study has been performed to illustrate the usefulness of average capacity as an entanglement indicator. ### Solvability of some Stefan type problems In this paper, we interest on some class of Stefan type problems. We prove the existence and uniqueness of renormalized solution in anisotropic Sobolev spaces with data belongs to $L^1- data,$ based on the properties of the renormalized trunctions and the generalized monotonicity method in the functional spaces. ### Fractional-Step Runge--Kutta Methods: Representation and Linear Stability Analysis Fractional-step methods are a popular and powerful divide-and-conquer approach for the numerical solution of differential equations. When the integrators of the fractional steps are Runge--Kutta methods, such methods can be written as generalized additive Runge--Kutta (GARK) methods, and thus the representation and analysis of such methods can be done through the GARK framework. We show how the general Butcher tableau representation and linear stability of such methods are related to the coefficients of the splitting method, the individual sub-integrators, and the order in which they are applied. We use this framework to explain some observations in the literature about fractional-step methods such as the choice of sub-integrators, the order in which they are applied, and the role played by negative splitting coefficients in the stability of the method. ### Continuous Interior Penalty stabilization for divergence-free finite element methods In this paper we propose, analyze, and test numerically a pressure-robust stabilized finite element for a linearized case of the Navier-Stokes' equation in the high Reynolds number regime, also known as Oseen's problem. Stabilization terms are defined by jumps of different combinations of derivatives for the convective term over the element faces of the triangulation of the domain. With the help of these stabilizing terms, and the fact the finite element space is assumed to provide a point-wise divergence-free velocity, an $\mathcal O\big(h^{k+\frac12}\big)$ error estimate in the $L^2$-norm is proved for the method (in the convection-dominated regime), and optimal order estimates in the remaining norms of the error. Numerical results supporting the theoretical findings are provided. ### A description of the Zeta map on Dyck paths area sequences We give a description of the well known Zeta map on Dyck paths which sends the dinv,area statistics to the area,bounce statistics. Our description uses Dyck paths area sequences and can be implemented easily. ### Regular theory in complex braid groups In his seminal paper on complex reflection arrangements, Bessis introduces a Garside structure for the braid group of a well-generated irreducible complex reflection group. Using this Garside structure, he establishes a strong connection between regular elements in the reflection group, and roots of the "full twist" element of the pure braid group. He then suggests that it would be possible to extend the conclusion of this theorem to centralizers of regular elements in well-generated groups. In this paper we give a positive answer to this question and we show moreover that these results hold for an arbitrary reflection group. ### Degree Based Topological Indices of a General Random Chain Let G=(V(G),E(G)) a graph, many important topological indices can be defined as TI(G)= \sum_{vu\in E(G)} h(d_{v},d_{u}). In this paper, we look at one type of general random chains and an alternative approach to study these kinds of topological indices is proposed. In which their explicit analytical expressions of the expected values and variances are obtained. Moreover, the asymptotic normality of topological indices of a random chain is established through the Martingale Central Limit Theorem. ### Determining the collision kernel in the Boltzmann equation near the equilibrium We consider an inverse problem for the nonlinear Boltzmann equation near the equilibrium. Our goal is to determine the collision kernel in the Boltzmann equation from the knowledge of the Albedo operator. Our approach relies on a linearization technique as well as the injectivity of the Gauss-Weierstrass transform. ### Virtually Full-duplex Cell-Free Massive MIMO with Access Point Mode Assignment We consider a cell-free massive multiple-input multiple-output (MIMO) network utilizing a virtually full-duplex (vFD) mode, where access points (APs) with a downlink (DL) mode and those with an uplink (UL) mode simultaneously serve DL and UL users (UEs). In order to maximize the sum spectral efficiency (SE) of the DL and UL transmissions, we formulate a mixed-integer optimization problem to jointly design the AP mode assignment and power control. This problem is subject to minimum per-UE SE requirements, per-AP power control, and per-UL UE power constraints. By employing the successive convex approximation technique, we propose an algorithm to obtain a stationary solution of the formulated problem. Numerical results show that the proposed vFD approach can provide a sum SE that is $2.5$ and $1.5$ times larger than the traditional half-duplex and heuristic baseline schemes, respectively, in terms of $95\%$-likely sum SE. ### Optimal Order of Encoding for Gaussian MIMO Multi-Receiver Wiretap Channel The Gaussian multiple-input multiple-output (MIMO) multi-receiver wiretap channel is studied in this paper. The base station broadcasts confidential messages to K intended users while keeping the messages secret from an eavesdropper. The capacity of this channel has already been characterized by applying dirty-paper coding and stochastic encoding. However, K factorial encoding orders may need to be enumerated for that, which makes the problem intractable. We prove that there exists one optimal encoding order and reduced the K factorial times to a one-time encoding. The optimal encoding order is proved by forming a secrecy weighted sum rate (WSR) maximization problem. The optimal order is the same as that for the MIMO broadcast channel without secrecy constraint, that is, the weight of users' rate in the WSR maximization problem determines the optimal encoding order. Numerical results verify the optimal encoding order. ### Representations of Quantum Coordinate Algebras at Generic $q$ and Wiring Diagrams This paper is devoted to the representation theory of quantum coordinate algebra $\mathbb{C}_q[G]$, for a semisimple Lie group $G$ and a generic parameter $q$. By inspecting the actions of normal elements on tensor modules, we generalize a result of Levendorski\u{\i} and Soibelman for highest weight modules. For a double Bruhat cell $G^{w_1,w_2}$, we describe the primitive spectra $\mathrm{prim}\,\mathbb{C}_q[G]_{w_1,w_2}$ in a new fashion, and construct a bundle of $(w_1,w_2)$ type simple modules onto $\mathrm{prim}\,\mathbb{C}_q[G]_{w_1,w_2}$, provided $\mathrm{Supp}(w_1)\cap\mathrm{Supp}(w_2)=\varnothing$ or enough pivot elements. The fibers of the bundle are shown to be products of the spectrums of simple modules of 2-dimensional quantum torus $L_q(2)$. As an application of our theory, we deduce an equivalent condition for the tensor module to be simple, and construct some simple modules for each primitive ideal when $G=SL_3(\mathbb{C})$. This completes Dixmier's program for $\mathbb{C}_q[SL_3(\mathbb{C})]$. ### Ballisticity of Random walks in Random Environments on $\mathbb{Z}$ with Bounded Jumps We characterize ballistic behavior for general i.i.d. random walks in random environments on $\mathbb{Z}$ with bounded jumps. The two characterizations we provide do not use uniform ellipticity conditions. They are natural in the sense that they both relate to formulas for the limiting speed in the nearest-neighbor case. Note: This paper duplicates results from some versions of the preprint "Random walks in Dirichlet random environments on $\mathbb{Z}$ with bounded jumps." (arxiv: 2104.14950). The present paper is being split off for reasons of length, and the plan is to remove these results from a future version of the previous paper and replace them with a citation of the present preprint. ### Scaling limit of a generalized contact process We derive macroscopic equations for a generalized contact process that is inspired by a neuronal integrate and fire model on the lattice $\mathbb{Z}^d$. The states at each lattice site can take values in $0,\ldots,k$. These can be interpreted as neuronal membrane potential, with the state $k$ corresponding to a firing threshold. In the terminology of the contact processes, which we shall use in this paper, the state $k$ corresponds to the individual being infectious (all other states are noninfectious). In order to reach the firing threshold, or to become infectious, the site must progress sequentially from $0$ to $k$. The rate at which it climbs is determined by other neurons at state $k$, coupled to it through a Kac-type potential, of range $\gamma^{-1}$. The hydrodynamic equations are obtained in the limit $\gamma\rightarrow 0$. Extensions of the microscopic model to include excitatory and inhibitory neuron types, as well as other biophysical mechanisms, are also considered. ### On multilevel Monte Carlo methods for deterministic and uncertain hyperbolic systems In this paper, we evaluate the performance of the multilevel Monte Carlo method (MLMC) for deterministic and uncertain hyperbolic systems, where randomness is introduced either in the modeling parameters or in the approximation algorithms. MLMC is a well known variance reduction method widely used to accelerate Monte Carlo (MC) sampling. However, we demonstrate in this paper that for hyperbolic systems, whether MLMC can achieve a real boost turns out to be delicate. The computational costs of MLMC and MC depend on the interplay among the accuracy (bias) and the computational cost of the numerical method for a single sample, as well as the variances of the sampled MLMC corrections or MC solutions. We characterize three regimes for the MLMC and MC performances using those parameters, and show that MLMC may not accelerate MC and can even have a higher cost when the variances of MC solutions and MLMC corrections are of the same order. Our studies are carried out by a few prototype hyperbolic systems: a linear scalar equation, the Euler and shallow water equations, and a linear relaxation model, the above statements are proved analytically in some cases, and demonstrated numerically for the cases of the stochastic hyperbolic equations driven by white noise parameters and Glimm's random choice method for deterministic hyperbolic equations. ### A quantitative Khintchine-Groshev theorem for S-arithmetic Diophantine approximation In his 1960 paper, Schmidt studied a quantitative type of Khintchine-Groshev theorem. Recently, a new proof of the theorem was found, which made it possible to relax the dimensional constraint and more generally, to add on the congruence condition by M. Alam, A. Ghosh, and S. Yu. In this paper, we generalize this new approach to S-arithmetic spaces and obtain a quantitative version of an S-arithmetic Khintchine-Groshev theorem. In fact, we consider a new S-arithmetic analog of Diophantine approximation, which is different from the one formerly established (see the 2007 paper of D. Kleinbock and G. Tomanov). Hence for the sake of completeness, we also deal with the convergence case of the Khintchine-Groshev theorem, based on this new generalization. ### Double crossed biproducts and related structures Let $H$ be a bialgebra. Let $\sigma: H\otimes H\to A$ be a linear map, where $A$ is a left $H$-comodule coalgebra, and an algebra with a left $H$-weak action $\triangleright$. Let $\tau: H\otimes H\to B$ be a linear map, where $B$ is a right $H$-comodule coalgebra, and an algebra with a right $H$-weak action $\triangleleft$. In this paper, we improve the necessary conditions for the two-sided crossed product algebra $A\#^{\sigma} H~{^{\tau}\#} B$ and the two-sided smash coproduct coalgebra $A\times H\times B$ to form a bialgebra (called double crossed biproduct) such that the condition $b_{[1]}\triangleright a_0\otimes b_{[0]}\triangleleft a_{-1}=a\otimes b$ in Majid's double biproduct (or double-bosonization) is one of the necessary conditions. On the other hand, we provide a more general two-sided crossed product algebra structure via Brzez\'nski's crossed product and give some applications. ### Heat kernel estimate in a conical singular space Let $(X,g)$ be a product cone with the metric $g=dr^2+r^2h$, where $X=C(Y)=(0,\infty)_r\times Y$ and the cross section $Y$ is a $(n-1)$-dimensional closed Riemannian manifold $(Y,h)$. We study the upper boundedness of heat kernel associated with the operator $L_V=-\Delta_g+V_0 r^{-2}$, where $-\Delta_g$ is the positive Friedrichs extension Laplacian on $X$ and $V=V_0(y) r^{-2}$ and $V_0\in\mathcal{C}^\infty(Y)$ is a real function such that the operator $-\Delta_h+V_0+(n-2)^2/4$ is a strictly positive operator on $L^2(Y)$.The new ingredient of the proof is the Hadamard parametrix and finite propagation speed of wave operator on $Y$. ### Rigid and Separable Algebras in Compact Semisimple Monoidal 2-Categories Given a monoidal 2-category that has right and left adjoints, we prove that the 2-categories of modules and of bimodules over a rigid algebra have right and left adjoints. Given a compact semisimple monoidal 2-category, we use this result to prove that the 2-categories of modules and of bimodules over a separable algebra are compact semisimple. Finally, we define the dimension of a connected rigid algebra in a compact semisimple monoidal 2-category, and prove that such an algebra is separable if and only if its dimension is non-zero. ### Blind Deconvolution with Non-smooth Regularization via Bregman Proximal DCAs Blind deconvolution is a technique to recover an original signal without knowing a convolving filter. It is naturally formulated as a minimization of a quartic objective function under some assumption. Because its differentiable part does not have a Lipschitz continuous gradient, existing first-order methods are not theoretically supported. In this letter, we employ the Bregman-based proximal methods, whose convergence is theoretically guaranteed under the $L$-smad property. We first reformulate the objective function as a difference of convex (DC) functions and apply the Bregman proximal DC algorithm (BPDCA). This DC decomposition satisfies the $L$-smad property. The method is extended to the BPDCA with extrapolation (BPDCAe) for faster convergence. When our regularizer has a sufficiently simple structure, each iteration is solved in a closed-form expression, and thus our algorithms solve large-scale problems efficiently. We also provide the stability analysis of the equilibriums and demonstrate the proposed methods through numerical experiments on image deblurring. The results show that BPDCAe successfully recovered the original image and outperformed other existing algorithms. ### Disjoint Total Dominating Sets in Near-Triangulations We show that every simple planar near-triangulation with minimum degree at least three contains two disjoint total dominating sets. The class includes all simple planar triangulations other than the triangle. This affirms a conjecture of Goddard and Henning [Thoroughly dispersed colorings, J. Graph Theory, 88 (2018) 174-191]. ### Strongly First Order, Domain Independent Dependencies: the Union-Closed Case Team Semantics generalizes Tarski's Semantics by defining satisfaction with respect to sets of assignments rather than with respect to single assignments. Because of this, it is possible to use Team Semantics to extend First Order Logic via new kinds of connectives or atoms - most importantly, via dependency atoms that express dependencies between different assignments. Some of these extensions are more expressive than First Order Logic proper, while others are reducible to it. In this work, I provide necessary and sufficient conditions for a dependency atom that is domain independent (in the sense that its truth or falsity in a relation does not depend on the existence in the model of elements that do not occur in the relation) and union closed (in the sense that whenever it is satisfied by all members of a family of relations it is also satisfied by their union) to be strongly first order, in the sense that the logic obtained by adding them to First Order Logic is no more expressive than First Order Logic itself. ### Extinction of multiple shocks in the modular Burgers equation A traveling viscous shock was previously studied in the Burgers equation with the modular advection term. It was shown that small, smooth, and exponentially decaying in space perturbations to the viscous shock decay in time. The present work addresses multiple shocks of the same model. We first prove that no traveling viscous shocks with multiple interfaces exist. We then suggest with the help of a priori energy estimates and numerical simulations that the evolution of viscous shocks with multiple interfaces leads to the finite-time extinction of compact regions between two consequent interfaces. We specify a precise scaling law of the finite-time extinction supported by the interface equations and by numerical simulations. ### A class of few-Lee weight $\mathbb{Z}_2[u]$-linear codes using simplicial complexes and minimal codes via Gray map Recently some mixed alphabet rings are involved in constructing few-Lee weight codes with optimal or minimal Gray images using suitable defining sets or down-sets. Inspired by these works, we choose the mixed alphabet ring $\mathbb{Z}_2\mathbb{Z}_2[u]$ to construct a special class of linear code $C_L$ over $\mathbb{Z}_2[u]$ with $u^2=0$ by employing simplicial complexes generated by a single maximal element. We show that $C_L$ has few-Lee weights by determining the Lee weight distribution of $C_L$. Also we have an infinite family of minimal codes over $\mathbb{Z}_2$ via Gray map, which can be used to secret sharing schemes. ### Data-Driven Upper Bounds on Channel Capacity We consider the problem of estimating an upper bound on the capacity of a memoryless channel with unknown channel law and continuous output alphabet. A novel data-driven algorithm is proposed that exploits the dual representation of capacity where the maximization over the input distribution is replaced with a minimization over a reference distribution on the channel output. To efficiently compute the required divergence maximization between the conditional channel and the reference distribution, we use a modified mutual information neural estimator that takes the channel input as an additional parameter. We evaluate our approach on different memoryless channels and show that the estimated upper bounds closely converge either to the channel capacity or to best-known lower bounds. ### Monomial reduction of knot polynomials For all natural numbers $N$ and prime numbers $p$, we find a knot $K$ whose skein polynomial $P_K(a,z)$ evaluated at $z=N$ has trivial reduction modulo $p$. An interesting consequence of our construction is that all polynomials $P_K(a,N)$ (mod~$p$) with bounded $a$-span are realised by knots with bounded braid index. As an application, we classify all polynomials of the form $P_K(a,1)$ (mod $2$) with $a$-span $\leq 10$. ### Existence and weak-strong uniqueness for Maxwell-Stefan-Cahn-Hilliard systems A Maxwell-Stefan system for fluid mixtures with driving forces depending on Cahn-Hilliard-type chemical potentials is analyzed. The corresponding parabolic cross-diffusion equations contain fourth-order derivatives and are considered in a bounded domain with no-flux boundary conditions. The main difficulty of the analysis is the degeneracy of the diffusion matrix, which is overcome by proving the positive definiteness of the matrix on a subspace and using the Bott--Duffin matrix inverse. The global existence of weak solutions and a weak-strong uniqueness property are shown by a careful combination of (relative) energy and entropy estimates, yielding $H^2(\Omega)$ bounds for the densities, which cannot be obtained from the energy or entropy inequalities alone. ### Frame set for Gabor systems with Haar window We show the full structure of the frame set for the Gabor system $\mathcal{G}(g;\alpha,\beta):=\{e^{-2\pi i m\beta\cdot}g(\cdot-n\alpha):m,n\in\Bbb Z\}$ with the window being the Haar function $g=-\chi_{[-1/2,0)}+\chi_{[0,1/2)}$. The strategy of this paper is to introduce the piecewise linear transformation $\mathcal{M}$ on the unit circle, and to provide a complete characterization of structures for its (symmetric) maximal invariant sets. This transformation is related to the famous three gap theorem of Steinhaus which may be of independent interest. Furthermore, a classical criterion on Gabor frames is improved, which allows us to establish {a} necessary and sufficient condition for the Gabor system $\mathcal{G}(g;\alpha,\beta)$ to be a frame, i.e., the symmetric invariant set of the transformation $\mathcal{M}$ is empty. Compared with the previous studies, the present paper provides a self-contained environment to study Gabor frames by a new perspective, which includes that the techniques developed here are new and all the proofs could be understood thoroughly by the readers without reference to the known results in the previous literature. ### Opportunistic Routing aided Cooperative Communication Network with Energy-Harvesting Nodes In this paper, a cooperative communication network based on two energy-harvesting (EH) decode-and-forward (DF) relays which harvest energy from the ambience using buffers with harvest-store-use (HSU) architecture is considered. For improving the data delivery in this network, an opportunistic routing (OR) algorithm considering channel status information (CSI), location and energy buffer status of relays is proposed. For the sake of deriving the theoretical expressions for limiting distribution of energy stored in buffers with discrete-time continuous-state space Markov chain (DCSMC) model, the probabilities that the packet to be forwarded exists in one and more transmitting nodes are obtained based on the state transition matrix (STM). Moreover, the closed-form expressions for outage probability and throughput of the network based on the CSI and the limiting distributions of energy stored in buffers are presented. Numerous experiments have been performed to verify the derived theoretical expressions. ### Asymptotics for connected graphs and irreducible tournaments We compute the whole asymptotic expansion of the probability that a large uniform labeled graph is connected, and of the probability that a large uniform labeled tournament is irreducible. In both cases, we provide a combinatorial interpretation of the involved coefficients. ### Ordering sequence for link diagrams with respect to Ridemeister moves I and III We prove that there exist infinitely many pairs of RI-III related (see Definition 2.1 in this paper) trivial knot diagrams that are not transformed into each other by a sequence of Reidemeister moves I, followed by a sequence of Reidemeister moves III, followed by a sequence of Reidemeister moves I. To create a simple sequence for RI-III related link diagrams instead of the ordinary ordering sequence, we prove that RI-III related link diagrams are always transformed into each other by applying an I-generalized ordering sequence. ### Regularization Theory of the Analytic Deep Prior Approach The analytic deep prior (ADP) approach was recently introduced for the theoretical analysis of deep image prior (DIP) methods with special network architectures. In this paper, we prove that ADP is in fact equivalent to classical variational Ivanov methods for solving ill-posed inverse problems. Besides, we propose a new variant which incorporates the strategy of early stopping into the ADP model. For both variants, we show how classical regularization properties (existence, stability, convergence) can be obtained under common assumptions. ### Coded Caching at the Edge of Satellite Networks Caching multimedia contents at the network edge is a key solution to decongest the amount of traffic in the backhaul link. In this paper, we extend and analyze the coded caching technique [1] in an unexplored scenario, i.e. at the edge of two-tier heterogeneous networks with an arbitrary number of users. We characterize the performance of such scheme by deriving a closed-form expression of the average backhaul load and reveal a significant gain compared to other benchmark caching schemes proposed in the literature. ### On Fekete Points for a Real Simplex We survey what is known about Fekete points/optimal designs for a simplex in $\R^d.$ Several new results are included. The notion of Fej\'er exponenet for a set of interpolation points is introduced. ### The Prime Number Theorem and Pair Correlation of Zeros of the Riemann Zeta-Function We prove that the error in the prime number theorem can be quantitatively improved beyond the Riemann Hypothesis bound by using versions of Montgomery's conjecture for the pair correlation of zeros of the Riemann zeta-function which are uniform in long ranges and with suitable error terms. ### Characters and projective characters of alternating and symmetric groups determined by values on $l'$-classes This paper identifies all pairs of ordinary irreducible characters of the alternating group which agree on conjugacy classes of elements of order not divisible by a fixed integer $l$, for $l \neq 3$. We do the same for the double covers of the symmetric and alternating groups. The only such characters are the conjugate or associate pairs labelled by partitions with a certain parameter divisible by $l$. When $l$ is prime, this implies that the rows of the $l$-modular decomposition matrix are distinct except for the rows labelled by these pairs. When $l=3$ we exhibit many additional examples of such pairs of characters. ### When all Permutations are Combinatorial Similarities Let $(X, d)$ be a semimetric space. A permutation $\Phi$ of the set $X$ is a combinatorial self similarity of $(X, d)$ if there is a bijective function $f \colon d(X^2) \to d(X^2)$ such that $$d(x, y) = f(d(\Phi(x), \Phi(y)))$$ for all $x$, $y \in X$. We describe the set of all semimetrics $\rho$ on an arbitrary nonempty set $Y$ for which every permutation of $Y$ is a combinatorial self similarity of $(Y, \rho)$. ### An affine Birkhoff--Kellogg type result in cones with applications to functional differential equations In this short note we prove, by means of classical fixed point index, an affine version of a Birkhoff--Kellogg type theorem in cones. We apply our result to discuss the solvability of a class of boundary value problems for functional differential equations subject to functional boundary conditions. We illustrate our theoretical results in an example. ### Representations of the Kottwitz gerbes Let $F$ be a local or global field and let $G$ be a linear algebraic group over $F$. We study Tannakian categories of representations of the Kottwitz gerbes $\text{Rep}(\text{Kt}_{F})$ and the functor $G\mapsto B(F, G)$ defined by Kottwitz in [28]. In particular, we show that if $F$ is a function field of a curve over $\mathbb{F}_q$, then $\text{Rep}(\text{Kt}_F)$ is equivalent to the category of Drinfeld isoshtukas. In the case of number fields, we establish the existence of various fiber functors on $\text{Rep}(\text{Kt}_{\mathbb{Q}})$ and its subcategories and show that Scholze's conjecture [41, Conjecture 9.5] follows from the full Tate conjecture over finite fields [47]. ### An Information-theoretic Method for Collaborative Distributed Learning with Limited Communication In this paper, we study the information transmission problem under the distributed learning framework, where each worker node is merely permitted to transmit a $m$-dimensional statistic to improve learning results of the target node. Specifically, we evaluate the corresponding expected population risk (EPR) under the regime of large sample sizes. We prove that the performance can be enhanced since the transmitted statistics contribute to estimating the underlying distribution under the mean square error measured by the EPR norm matrix. Accordingly, the transmitted statistics correspond to the eigenvectors of this matrix, and the desired transmission allocates these eigenvectors among the statistics such that the EPR is minimal. Moreover, we provide the analytical solution of the desired statistics for single-node and two-node transmission, where a geometrical interpretation is given to explain the eigenvector selection. For the general case, an efficient algorithm that can output the allocation solution is developed based on the node partitions. ### Transmission operators for the non-overlapping Schwarz method for solving Helmholtz problems in rectangular cavities In this paper we discuss different transmission operators for the non-overlapping Schwarz method which are suited for solving the time-harmonic Helmholtz equation in cavities (i.e. closed domains which do not feature an outgoing wave condition). Such problems are heavily impacted by back-propagating waves which are often neglected when devising optimized transmission operators for the Schwarz method. This work explores new operators taking into account those back-propagating waves and compares them with well-established operators neglecting these contributions. Notably, this paper focuses on the case of rectangular cavities, as the optimal (non-local) transmission operator can be easily determined. Nonetheless, deviations from this ideal geometry are considered as well. In particular, computations of the acoustic noise in a three-dimensional model of the helium vessel of a beamline cryostat with optimized Schwarz schemes are discussed. Those computations show a reduction of 46% in the iteration count, when comparing an operator optimized for cavities with those optimized for unbounded problems. ### Simplex Closing Probabilities in Directed Graphs Recent work in mathematical neuroscience has calculated the directed graph homology of the directed simplicial complex given by the brains sparse adjacency graph, the so called connectome. These biological connectomes show an abundance of both high-dimensional directed simplices and Betti-numbers in all viable dimensions - in contrast to Erd\H{o}s-R\'enyi-graphs of comparable size and density. An analysis of synthetically trained connectomes reveals similar findings, raising questions about the graphs comparability and the nature of origin of the simplices. We present a new method capable of delivering insight into the emergence of simplices and thus simplicial abundance. Our approach allows to easily distinguish simplex-rich connectomes of different origin. The method relies on the novel concept of an almost-d-simplex, that is, a simplex missing exactly one edge, and consequently the almost-d-simplex closing probability by dimension. We also describe a fast algorithm to identify almost-d-simplices in a given graph. Applying this method to biological and artificial data allows us to identify a mechanism responsible for simplex emergence, and suggests this mechanism is responsible for the simplex signature of the excitatory subnetwork of a statistical reconstruction of the mouse primary visual cortex. Our highly optimised code for this new method is publicly available. ### Deterministic Identification over Channels without CSI Identification capacities of randomized and deterministic identification were proved to exceed channel capacity for Gaussian channels \emph{with} channel side information (CSI). In this work, we extend deterministic identification to the block fading channels without CSI by applying identification codes for both channel estimation and user identification. We prove that identification capacity is asymptotically higher than transmission capacity even in the absence of CSI. And we also analyze the finite-length performance theoretically and numerically. The simulation results verify the feasibility of the proposed blind deterministic identification in finite blocklength regime. ### Valuatuions and orderings on the real Weyl algebra ### Characterization of Lipschitz Functions via the Commutators of Maximal Function on Stratified Lie Groups In this paper, the main aim is to consider the boundedness of the Hardy-Littlewood maximal commutator $M_{b}$ and the nonlinear commutator $[b, M]$ on the Lebesgue spaces and Morrey spaces over some stratified Lie group $\mathbb{G}$ when $b$ belongs to the Lipschitz space, by which some new characterizations of the Lipschitz spaces on Lie group are given. ### Reduced modelling and optimal control of epidemiological individual-based models with contact heterogeneity Modelling epidemics via classical population-based models suffers from shortcomings that so-called individual-based models are able to overcome, as they are able to take heterogeneity features into account, such as super-spreaders, and describe the dynamics involved in small clusters. In return, such models often involve large graphs which are expensive to simulate and difficult to optimize, both in theory and in practice. By combining the reinforcement learning philosophy with reduced models, we propose a numerical approach to determine optimal health policies for a stochastic epidemiological graph-model taking into account super-spreaders. More precisely, we introduce a deterministic reduced population-based model involving a neural network, and use it to derive optimal health policies through an optimal control approach. It is meant to faithfully mimic the local dynamics of the original, more complex, graph-model. Roughly speaking, this is achieved by sequentially training the network until an optimal control strategy for the corresponding reduced model manages to equally well contain the epidemic when simulated on the graph-model. After describing the practical implementation of this approach, we will discuss the range of applicability of the reduced model and to what extent the estimated control strategies could provide useful qualitative information to health authorities. ### Phase-field approximation of a vectorial, geometrically nonlinear cohesive fracture energy We consider a family of vectorial models for cohesive fracture, which may incorporate $\mathrm{SO}(n)$-invariance. The deformation belongs to the space of generalized functions of bounded variation and the energy contains an (elastic) volume energy, an opening-dependent jump energy concentrated on the fractured surface, and a Cantor part representing diffuse damage. We show that this type of functional can be naturally obtained as $\Gamma$-limit of an appropriate phase-field model. The energy densities entering the limiting functional can be expressed, in a partially implicit way, in terms of those appearing in the phase-field approximation. ### Extension Operators for Trimmed Spline Spaces We develop a discrete extension operator for trimmed spline spaces consisting of piecewise polynomial functions of degree $p$ with $k$ continuous derivatives. The construction is based on polynomial extension from neighboring elements together with projection back into the spline space. We prove stability and approximation results for the extension operator. Finally, we illustrate how we can use the extension operator to construct a stable cut isogeometric method for an elliptic model problem. ### On sets with sum and difference structure For nonempty sets $A,B$ of nonnegative integers and an integer $n$, let $r_{A,B}(n)$ be the number of representations of $n$ as $a+b$ and $d_{A,B}(n)$ be the number of representations of $n$ as $a-b$, where $a\in A, b\in B$. In this paper, we determine the sets $A,B$ such that $r_{A,B}(n)=1$ for every nonnegative integer $n$. We also consider the \emph{difference} structure and prove that: there exist sets $A$ and $B$ of nonnegative integers such that $r_{A,B}(n)\ge 1$ for all large $n$, $A(x)B(x)=(1+o(1))x$ and for any given nonnegative integer $c$, we have $d_{A,B}(n)=c$ for infinitely many positive integers $n$. Other related results are also contained. ### From octonions to composition superalgebras via tensor categories The nontrivial unital composition superalgebras, of dimension 3 and 6, which exist only in characteristic 3, are obtained from the split Cayley algebra and its order 3 automorphisms, by means of the process of semisimplification of the symmetric tensor category of representations of the cyclic group of order 3. Connections with the extended Freudenthal Magic Square in characteristic 3, that contains some exceptional Lie superalgebras specific of this characteristic are discussed too. In the process, precise recipes to go from (nonassociative) algebras in this tensor category to the corresponding superalgebras are given. ### Random cluster model on regular graphs For a graph $G=(V,E)$ with $v(G)$ vertices the partition function of the random cluster model is defined by $$Z_G(q,w)=\sum_{A\subseteq E(G)}q^{k(A)}w^{\|A\|},$$ where $k(A)$ denotes the number of connected components of the graph $(V,A)$. In this paper we show that if $(G_n)_n$ is a sequence of $d$-regular graphs such that the girth $g(G_n)\to \infty$, the length of the shortest cycle, then the limit $$\lim_{n\to \infty} \frac{1}{v(G_n)}\ln Z_{G_n}(q,w)=\ln \Phi_{d,q,w}$$ exists if $q\geq 2$ and $w\geq 0$, and is equal to $$\Phi_{d,q,w}:=\max_{t\in [-\pi,\pi]}\Phi_{d,q,w}(t),$$ where $$\Phi_{d,q,w}(t):=\left(\sqrt{1+\frac{w}{q}}\cos(t)+\sqrt{\frac{(q-1)w}{q}}\sin(t)\right)^{d}+(q-1)\left(\sqrt{1+\frac{w}{q}}\cos(t)-\sqrt{\frac{w}{q(q-1)}}\sin(t)\right)^{d}.$$ The same conclusion holds true for a sequence of random $d$-regular graphs with probability $1$. We extend the work of Dembo, Montanari, Sly and Sun for the Potts model (integer $q$) and we prove a conjecture of Helmuth, Jenssen and Perkins about the phase transition of the random cluster model with fixed $q$. ### Comparison bounds for perturbed Schrödinger operators with single-well potentials We prove bounds on the sum of the differences between the eigenvalues of a Schr\"odinger operator and its perturbation. Our results hold for operators in one dimension with single-well potentials. We rely on a variation of the well-known factorisation method. In the P\"oschl-Teller and Coulomb cases we are able to use the explicit factorisations to establish improved bounds. ### A dynamic approach to heterogeneous elastic wires We consider closed planar curves with fixed length whose elastic energy depends on an additional density variable and a spontaneous curvature. Working with the inclination angle, the associated $L^2$-gradient flow is a nonlocal quasilinear coupled parabolic system of second order. We show local well-posedness and global existence of solutions for initial data in a weak regularity class and with arbitrary winding number. ### Energy bounds of sign-changing solutions to Yamabe equations on manifolds with boundary We study the Yamabe equation in the Euclidean half-space. We prove that any sign-changing solution has at least twice the energy of a standard bubble. Moreover, a sharper energy lower bound of the sign-changing solution set is also established via the method of moving planes. This bound increases the energy range for which Palais-Smale sequences of related variational problem has a non-trivial weak limit. ### Discrete density comonads and graph parameters Game comonads have brought forth a new approach to studying finite model theory categorically. By representing model comparison games semantically as comonads, they allow important logical and combinatorial properties to be exressed in terms of their Eilenberg-Moore coalgebras. As a result, a number of results from finite model theory, such as preservation theorems and homomorphism counting theorems, have been formalised and parameterised by comonads, giving rise to new results simply by varying the comonad. In this paper we study the limits of the comonadic approach in the combinatorial and homomorphism-counting aspect of the theory, regardless of whether any model comparison games are involved. We show that any standard graph parameter has a corresponding comonad, classifying the same class. This comonad is constructed via a simple Kan extension formula, making it the initial solution to this problem and, furthermore, automatically admitting a homomorphism-counting theorem. ### Urysohn and Hammerstein operators on H"older spaces We present an application-oriented approach to Urysohn and Hammerstein integral operators acting between spaces of H"older continuous functions over compact metric spaces. These nonlinear mappings are formulated by means of an abstract measure theoretical integral involving a finite measure. This flexible setting creates a common framework to tackle both such operators based on the Lebesgue integral like frequently met in applications, as well as e.g.\ their spatial discretization using stable quadrature/cubature rules (Nystr"om methods). Under suitable Carath{\'e}odory conditions on the kernel functions, properties like well-definedness, boundedness, (complete) continuity and continuous differentiability are established. Furthermore, the special case of Hammerstein operators is understood as composition of Fredholm and Nemytskii operators. While our differentiability results for Urysohn operators appear to be new, the section on Nemytskii operators has a survey character. Finally, an appendix provides a rather comprehensive account summarizing the required preliminaries for H\"older continuous functions defined on metric spaces. ### On varieties with Ulrich twisted normal bundles We characterize smooth irreducible varieties with Ulrich twisted normal bundle. ### $\times a$ and $\times b$ empirical measures, the irregular set and entropy We consider the $\times a$ and $\times b$ maps: $T_a$ and $T_b$ on $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ for integers $a$ and $b\geq 2$. It is known that, if $a$ and $b$ are multiplicatively independent, then the only $T_a,T_b$-invariant and ergodic measure with positive entropy of $T_a$ or $T_b$ is the Lebesgue measure. However, whether there exists a nontrivial $T_a,T_b$-invariant and ergodic measure is not known. In this paper, we study the empirical measures of $x\in\mathbb{T}$ with respect to the $T_a,T_b$-action and show that the set of $x$ such that the empirical measures of $x$ do not converge to any measure has Hausdorff dimension $1$ and the set of $x$ such that the empirical measures can approach a nontrivial $T_a,T_b$-invariant measure has Hausdorff dimension $0$. Furthermore, we obtain some equidistribution result about the $T_a,T_b$-orbit of $x$ in the complement of a set of Hausdorff dimension $0$. ### Smooth $l$-Fano weighted complete intersections In this paper we prove that for $n$-dimensional smooth $l$-Fano well formed weighted complete intersections, which is not isomorphic to a usual projective space, the upper bound for $l$ is equal to $\lceil \log_2(n+2) \rceil-1 .$ We also prove that the only $l$-Fano of dimension $n$ among such manifolds with inequalities $\lceil \log_3(n+2) \rceil \leqslant l \leqslant \lceil \log_2(n+2) \rceil -1$ is a complete intersection of quadrics in a usual projective space. ### Fine Selmer groups of modular forms We compare the Iwasawa invariants of fine Selmer groups of $p$-adic Galois representations over admissible $p$-adic Lie extensions of a number field $K$ to the Iwasawa invariants of ideal class groups along these Lie extensions. More precisely, let $K$ be a number field, let $V$ be a $p$-adic representation of the absolute Galois group $G_K$ of $K$, and choose a $G_K$-invariant lattice ${T \subseteq V}$. We study the fine Selmer groups of ${A = V/T}$ over suitable $p$-adic Lie extensions $K_\infty/K$, comparing their corank and $\mu$-invariant to the corank and the $\mu$-invariant of the Iwasawa module of ideal class groups in $K_\infty/K$. In the second part of the article, we compare the Iwasawa $\mu$- and $l_0$-invariants of the fine Selmer groups of CM modular forms on the one hand and the Iwasawa invariants of ideal class groups on the other hand over trivialising multiple $\mathbb{Z}_p$-extensions of $K$. ### Existence of real algebraic hypersurfaces with many prescribed components Given a real algebraic variety $X$ of dimension $n$, a very ample divisor $D$ on $X$ and a smooth closed hypersurface $\Sigma$ of $\mathbf{R}^n$, we construct real algebraic hypersurfaces in the linear system $\|mD\|$ whose real locus contains many connected components diffeomorphic to $\Sigma$. As a consequence, we show the existence of real algebraic hypersurfaces in the linear system $\|mD\|$ whose Betti numbers grow by the maximal order, as $m$ goes to infinity. As another application, we recover a result by D. Gayet on the existence of many disjoint lagrangians with prescribed topology in any smooth complex hypersurface of $\mathbf{C}\mathbf{P}^n$. The results in the paper are proved more generally for complete intersections. The proof of our main result uses probabilistic tools. ### The probability of random trees being isomorphic We show that the probability that two randomly chosen trees are isomorphic decays exponentially for rooted labelled trees as well as Galton--Watson trees with bounded degrees. In the former case a full asymptotic expansion is derived. We also show that, in general, we cannot obtain exponential decay for Galton--Watson trees. Lastly, we prove joint convergence to a multivariate normal distribution for vertices of given degrees in pairs of labelled trees conditioned on being isomorphic. ### The disguised toric locus and affine equivalence of reaction networks Under the assumption of mass-action kinetics, a dynamical system may be induced by several different reaction networks and/or parameters. It is therefore possible for a mass-action system to exhibit complex-balancing dynamics without being weakly reversible or satisfying toric constraints on the rate constants; such systems are called disguised toric dynamical systems. We show that the parameters that give rise to such systems are preserved under invertible affine transformations of the network. We also consider the dynamics of arbitrary mass-action systems under affine transformations, and show that there is a canonical bijection between their sets of positive steady states, although their qualitative dynamics can differ substantially. ### Controlled continuous $\ast$-$g$-Frames in Hilbert $C^{\ast}$-Modules The frame theory is dynamic and exciting with various pure and applied mathematics applications. In this paper, we introduce and study the concept of Controlled Continuous $\ast$-$g$-Frames in Hilbert $C^{\ast}$-Modules, which is a generalization of discrete controlled $\ast$-$g$-Frames in Hilbert $C^{\ast}$-Modules. Also, we give some properties. ### A Polar Subcode Approach to Belief Propagation List Decoding Permutation decoding gained recent interest as it can exploit the symmetries of a code in a parallel fashion. Moreover, it has been shown that by viewing permuted polar codes as polar subcodes, the set of usable permutations in permutation decoding can be increased. We extend this idea to pre-transformed polar codes, such as cyclic redundancy check (CRC)-aided polar codes, which previously could not be decoded using permutations due to their lack of automorphisms. Using belief propagation (BP)-based subdecoders, we showcase a performance close to CRC-aided SCL (CA-SCL) decoding. The proposed algorithm outperforms the previously best performing iterative CRC-aided belief propagation list (CA-BPL) decoder both in error-rate performance and decoding latency. ### Scattered linear sets in a finite projective line and translation planes Lunardon and Polverino construct a translation plane starting from a scattered linear set of pseudoregulus type in $\mathrm{PG}(1,q^t)$. In this paper a similar construction of a translation plane $\mathcal A_f$ obtained from any scattered linearized polynomial $f(x)$ in $\mathbb F_{q^t}[x]$ is described and investigated. A class of quasifields giving rise to such planes is defined. Denote by $U_f$ the $\mathbb F_q$-subspace of $\mathbb F_{q^t}^2$ associated with $f(x)$. If $f(x)$ and $f'(x)$ are scattered, then $\mathcal A_f$ and $\mathcal A_{f'}$ are isomorphic if and only if $U_f$ and $U_{f'}$ belong to the same orbit under the action of $\Gamma\mathrm L(2,q^t)$. This gives rise to as many distinct translation planes as there are inequivalent scattered linearized polynomials. In particular, for any scattered linear set $L$ of maximum rank in $\mathrm{PG}(1,q^t)$ there are $c_\Gamma(L)$ pairwise non-isomorphic translation planes, where $c_\Gamma(L)$ denotes the $\Gamma\mathrm L$-class of $L$, as defined by Csajb\'ok, Marino and Polverino. A result by Jha and Johnson allows to describe the automorphism groups of the planes obtained from the linear sets not of pseudoregulus type defined by Lunardon and Polverino. ### Robust Fundamental Lemma for Data-driven Control The fundamental lemma by Willems and coauthors facilitates a parameterization of all trajectories of a linear time-invariant system in terms of a single, measured one. This result plays an important role in data-driven simulation and control. Under the hood, the fundamental lemma works by applying a persistently exciting input to the system. This ensures that the Hankel matrix of resulting input/output data has the "right" rank, meaning that its columns span the entire subspace of trajectories. However, such binary rank conditions are known to be fragile in the sense that a small additive noise could already cause the Hankel matrix to have full rank. Therefore, in this extended abstract we present a robust version of the fundamental lemma. The idea behind the approach is to guarantee certain lower bounds on the singular values of the data Hankel matrix, rather than mere rank conditions. This is achieved by designing the inputs of the experiment such that the minimum singular value of a deeper input Hankel matrix is sufficiently large. This inspires a new quantitative and robust notion of persistency of excitation. The relevance of the result for data-driven control will also be highlighted through comparing the predictive control performance for varying degrees of persistently exciting data. ### A scalable space-time domain decomposition approach for solving large-scale nonlinear regularized inverse ill-posed problems in 4D variational data assimilation We develop innovative algorithms for solving the strong-constraint formulation of four-dimensional variational data assimilation in large-scale applications. We present a space-time decomposition approach that employs domain decomposition along both the spatial and temporal directions in the overlapping case and involves partitioning of both the solution and the operators. Starting from the global functional defined on the entire domain, we obtain a type of regularized local functionals on the set of subdomains providing the order reduction of both the predictive and the data assimilation models. We analyze the algorithm convergence and its performance in terms of reduction of time complexity and algorithmic scalability. The numerical experiments are carried out on the shallow water equation on the sphere according to the setup available at the Ocean Synthesis/Reanalysis Directory provided by Hamburg University. ### Turning grain maps into diagrams The present paper studies mathematical models for representing, imaging, and analyzing polycrystalline materials. We introduce various techniques for converting grain maps into diagram or tessellation representations that rely on constrained clustering. In particular, we show how to significantly accelerate the generalized balanced power diagram method from [1] and how to extend it to allow for optimization over all relevant parameters. A comparison of the accuracies of the proposed approaches is given based on a 3D real-world data set of $339\times 339 \times 599$ voxels. ### Univalent typoids A typoid is a type equipped with an equivalence relation, such that the terms of equivalence between the terms of the type satisfy certain conditions, with respect to a given equivalence relation between them, that generalise the properties of the equality terms. The resulting weak 2-groupoid structure can be extended to every finite level. The introduced notions of typoid and typoid function generalise the notions of setoid and setoid function. A univalent typoid is a typoid satisfying a general version of the univalence axiom. We prove some fundamental facts on univalent typoids, their product and exponential. As a corollary, we get an interpretation of propositional truncation within the theory of typoids. The couple typoid and univalent typoid is a weak groupoid-analogue to the couple precategory and category in homotopy type theory. ### Pullback and forward attractors of contractive difference equations The construction of attractors of a dissipative difference equation is usually based on compactness assumptions. In this paper, we replace them with contractivity assumptions under which the pullback and forward attractors are identical. As a consequence, attractors degenerate to unique bounded entire solutions. As an application, we investigate attractors of integrodifference equations which are popular models in theoretical ecology. ### Discrete m-functions with Doubly Palindromic Continued Fraction Coefficients We demonstrate that discrete m-functions with eventually periodic continued fraction coefficients have an algebraic relationship to their second solution if and only if the periodic part of the sequence of continued fraction coefficients is doubly palindromic. In this setting, doubly palindromic means that each sequence is a repeated concatenation of two palindromes and a compatibility condition between the lengths of these palindromes is satisfied. ### Complete monotonicity of time-changed Lévy processes at first passage We consider the class of (possibly killed) spectrally positive L\'evy process that have been time-changed by the inverse of an integral functional. Within this class we characterize the family of those processes which observe the following property: as functions of point of issue, the Laplace transforms of their first-passage times downwards are completely monotone. A wide (dense, in a sense) subfamily of this family admits closed form expressions for said Laplace transforms. ### Dynamical boundary conditions for time-dependent fractional operators on extension domains We consider a parabolic semilinear non-autonomous problem $(\tilde P)$ for a fractional time-dependent operator $\mathcal{B}^{s,t}_\Omega$ with Venttsel'-type boundary conditions in an extension domain $\Omega\subset\mathbb{R}^N$ having as boundary a $d$-set. We prove existence and uniqueness of the mild solution of the associated semilinear abstract Cauchy problem via an evolution family $U(t,\tau)$. We then prove that the mild solution of the abstract problem actually solves problem $(\tilde P)$ via a generalized fractional Green formula. ### A weighted first-order formulation for solving anisotropic diffusion equations with deep neural networks In this paper, a new weighted first-order formulation is proposed for solving the anisotropic diffusion equations with deep neural networks. For many numerical schemes, the accurate approximation of anisotropic heat flux is crucial for the overall accuracy. In this work, the heat flux is firstly decomposed into two components along the two eigenvectors of the diffusion tensor, thus the anisotropic heat flux approximation is converted into the approximation of two isotropic components. Moreover, to handle the possible jump of the diffusion tensor across the interface, the weighted first-order formulation is obtained by multiplying this first-order formulation by a weighted function. By the decaying property of the weighted function, the weighted first-order formulation is always well-defined in the pointwise way. Finally, the weighted first-order formulation is solved with deep neural network approximation. Compared to the neural network approximation with the original second-order elliptic formulation, the proposed method can significantly improve the accuracy, especially for the discontinuous anisotropic diffusion problems. ### Long term analysis of splitting methods for charged-particle dynamics In this paper, we rigorously analyze the energy, momentum and magnetic moment behaviours of two splitting methods for solving charged-particle dynamics. The near-conservations of these invariants are given for the system under constant magnetic field or quadratic electric potential. By the approach named as backward error analysis, we derive the modified equations and modified invariants of the splitting methods and based on which, the near-conservations over long times are proved. Some numerical experiments are presented to demonstrate these long time behaviours. ### Injectivity for algebras and categories with quantum symmetry We study completely positive maps and injectivity for Yetter-Drinfeld algebras over compact quantum groups, and module categories over rigid C-tensor categories. This gives a generalization of Hamana's theory of injective envelope to the framework of dynamical systems over quantum groups. As a byproduct we also establish a duality between the Yetter-Drinfeld algebras and certain bimodule categories with central generators. ### Higher Reciprocity Laws and Ternary Linear Recurrence Sequences We describe the set of prime numbers splitting completely in the non-abelian splitting field of certain monic irreducible polynomials of degree three. As an application we establish some divisibility properties of the associated ternary recurrence sequence by primes $p$, thus greatly extending recent work of Evink and Helminck and of Faisant. We also prove some new results on the number of solutions of the characteristic equation of the recurrence sequence modulo $p,$ extending and simplifying earlier work of Zhi-Hong Sun (2003). ### Pebble trees A pebble tree is an ordered tree where each node receives some colored pebbles, in such a way that each unary node receives at least one pebble, and each subtree has either one more or as many leaves as pebbles of each color. We show that the contraction poset on pebble trees is isomorphic to the face poset of a convex polytope called pebble tree polytope. Beside providing intriguing generalizations of the classical permutahedra and associahedra, our motivation is that the faces of the pebble tree polytopes provide realizations as convex polytopes of all assocoipahedra constructed by K. Poirier and T. Tradler only as polytopal complexes. ### Co-spectral radius, equivalence relations and the growth of unimodular random rooted trees We define the co-spectral radius of inclusions $\mathcal{S}\leq \mathcal{R}$ of discrete, probability measure-preserving equivalence relations, as the sampling exponent of a generating random walk on the ambient relation. The co-spectral radius is analogous to the spectral radius for random walks on $G/H$ for inclusion $H\leq G$ of groups. The almost sure existence of the sampling exponent is already new for i.i.d. percolation clusters on countable groups. For the proof, we develop a general method called 2-3-method that is based on the mass-transport principle. As a byproduct, we show that the growth of a unimodular random rooted tree of bounded degree always exists, assuming its upper growth passes a critical threshold. This complements Timar's work who showed the possible nonexistence of growth below this threshold. We also show that the walk growth exists for an arbitrary unimodular random rooted graph of bounded degree. We also investigate how the co-spectral radius behaves for Property (T) and hyperfinite relations. ### On the use of a local R-hat to improve MCMC convergence diagnostic Diagnosing convergence of Markov chain Monte Carlo is crucial and remains an essentially unsolved problem. Among the most popular methods, the potential scale reduction factor, commonly named $\hat{R}$, is an indicator that monitors the convergence of output chains to a target distribution, based on a comparison of the between- and within-variances. Several improvements have been suggested since its introduction in the 90s. Here, we aim at better understanding the $\hat{R}$ behavior by proposing a localized version that focuses on quantiles of the target distribution. This new version relies on key theoretical properties of the associated population value. It naturally leads to proposing a new indicator $\hat{R}_\infty$, which is shown to allow both for localizing the Markov chain Monte Carlo convergence in different quantiles of the target distribution, and at the same time for handling some convergence issues not detected by other $\hat{R}$ versions. ### Metric lines in Jet Space The space $J^k$ of $k$-jets of a real function of one real variable $x$ admits the structure of a Carnot group. $J^k$ has a natural \sR submersion onto the Euclidean plane and curves in the Euclidean plane can be horizontal lifted to $J^k$. The horizontal lifts of Euclidean lines are thus metric lines in $J^k$. Are other metric lines in $J^k$ besides the horizontal lifts lines? This work is the first of a sequence of papers, where we attempt to make a complete classification of the metric lines in $J^k$. ### Hyper-parameter tuning of physics-informed neural networks: Application to Helmholtz problems We consider physics-informed neural networks [Raissi et al., J. Comput. Phys. 278 (2019) 686-707] for forward physical problems. In order to find optimal PINNs configuration, we introduce a hyper-parameter tuning procedure via Gaussian processes-based Bayesian optimization. We apply the procedure to Helmholtz problems for bounded domains and conduct a thorough study, focusing on: (i) performance, (ii) the collocation points density $r$ and (iii) the frequency $\kappa$, confirming the applicability and necessity of the method. Numerical experiments are performed in two and three dimensions, including comparison to finite element methods. ### The Capacity of Causal Adversarial Channels We characterize the capacity for the discrete-time arbitrarily varying channel with discrete inputs, outputs, and states when (a) the encoder and decoder do not share common randomness, (b) the input and state are subject to cost constraints, (c) the transition matrix of the channel is deterministic given the state, and (d) at each time step the adversary can only observe the current and past channel inputs when choosing the state at that time. The achievable strategy involves stochastic encoding together with list decoding and a disambiguation step. The converse uses a two-phase "babble-and-push" strategy where the adversary chooses the state randomly in the first phase, list decodes the output, and then chooses state inputs to symmetrize the channel in the second phase. These results generalize prior work on specific channels models (additive, erasure) to general discrete alphabets and models. ### Linesearch Newton-CG methods for convex optimization with noise This paper studies the numerical solution of strictly convex unconstrained optimization problems by linesearch Newton-CG methods. We focus on methods employing inexact evaluations of the objective function and inexact and possibly random gradient and Hessian estimates. The derivative estimates are not required to satisfy suitable accuracy requirements at each iteration but with sufficiently high probability. Concerning the evaluation of the objective function we first assume that the noise in the objective function evaluations is bounded in absolute value. Then, we analyze the case where the error satisfies prescribed dynamic accuracy requirements. We provide for both cases a complexity analysis and derive expected iteration complexity bounds. We finally focus on the specific case of finite-sum minimization which is typical of machine learning applications. ### The height of binomial ideals and toric $K$-algebras with isolated singularity We give an upper bound for the height of an arbitrary binomial ideal $I$ in terms of the dimension of a vector space spanned by integer vectors corresponding to a set of binomial generators of $I$. When $I$ is an unmixed binomial ideal, this dimension is precisely the height of $I$. Applying this result to the ideal of inner $2$-minors $I_\Pc$ of a finite set of cells $\Pc$, one gets a nice interpretation for $\height I_\Pc$ for an unmixed ideal $I_\Pc$, in terms of the number of cells of $\Pc$. Moreover, we study some families of toric $K$-algebras such as Hibi rings and toric rings defined by inner $2$-minors to determine when they have isolated singularity. ### The existence of $m$-tree-connected $(g,f+f'-m)$-factors using $(g,f)$-factors and $m$-tree-connected $(m,f')$-factors Let $G$ be a graph and let $g$, $f$, and $f'$ be three positive integer-valued functions on $V(G)$ with $g\le f$. Tokuda, Xu, and Wang (2003) showed that if $G$ contains a $(g,f)$-factor and a spanning $f'$-tree, then $G$ also contains a connected $(g,f+f'-1)$-factor. In this note, we develop their result to a tree-connected version by proving that if $G$ contains a $(g,f)$-factor and an $m$-tree-connected $(m,f')$-factor, then $G$ also contains an $m$-tree-connected $(g,f+f'-m)$-factor, provided that $f\ge m$. In addition, we show that $g$ allows to be nonnegative. ### Equivalent Boundary Conditions for an Elasto-Acoustic Problem set in a Domain with a Thin Layer We present equivalent conditions and asymptotic models for the diffraction problem of elastic and acoustic waves in a solid medium surrounded by a thin layer of fluid medium. Due to the thinness of the layer with respect to the wavelength, this problem is well suited for the notion of equivalent conditions and the effect of the fluid medium on the solid is as a first approximation local. We derive and validate equivalent conditions up to the fourth order for the elastic displacement. These conditions approximate the acoustic waves which propagate in the fluid region. This approach leads to solve only elastic equations. The construction of equivalent conditions is based on a multiscale expansion in power series of the thickness of the layer for the solution of the transmission problem. ### Proofs For Progressively Generalized Fibonacci Identities Using Maximal Independent Sets of Tree Graphs This paper generalizes a graph theoretic proof technique for a Fibonacci identity proposed by Lee Knisley Sanders, and explores characteristics of these generalized theorems ad infinitum. ### Multi-Marginal Gromov-Wasserstein Transport and Barycenters Gromov-Wasserstein (GW) distances are generalizations of Gromov-Haussdorff and Wasserstein distances. Due to their invariance under certain distance-preserving transformations they are well suited for many practical applications. In this paper, we introduce a concept of multi-marginal GW transport as well as its regularized and unbalanced versions. Then we generalize a bi-convex relaxation of the GW transport to our multi-marginal setting which is tight if the cost function is conditionally negative definite in a certain sense. The minimization of this relaxed model can be done by an alternating algorithm, where each step can be performed by a Sinkhorn scheme for a multi-marginal transport problem. We show a relation of our multi-marginal GW problem for a tree-structured cost function to an (unbalanced) GW barycenter problem and present different proof-of-concept numerical results. ### Hybridized Discontinuous Galerkin Methods for a Multiple Network Poroelasticity Model with Medical Applications The quasi-static multiple network poroelastic theory (MPET) model, first introduced in the context of geomechanics, has recently found new applications in medicine. In practice, the parameters in the MPET equations can vary over several orders of magnitude which makes their stable discretization and fast solution a challenging task. Here, a new efficient parameter-robust hybridized discontinuous Galerkin method, which also features fluid mass conservation, is proposed for the MPET model. Its stability analysis which is crucial for the well-posedness of the discrete problem is performed and cost-efficient fast parameter-robust preconditioners are derived. We present a series of numerical computations for a 4-network MPET model of a human brain which support the performance of the new algorithms. ### Dynamical maps and symmetroids Starting from the canonical symmetroid $\mathcal{S}(G)$ associated with a groupoid $G$, the issue of describing dynamical maps in the groupoidal approach to Quantum Mechanics is addressed. After inducing a Haar measure on the canonical symmetroid $\mathcal{S}(G)$, the associated von-Neumann groupoid algebra is constructed. It is shown that the left-regular representation allows to define linear maps on the groupoid-algebra of the groupoid $G$ and given subsets of functions are associated with completely positive maps. Some simple examples are also presented. ### A model of invariant control system using mean curvature drift from Brownian motion under submersions Given a submersion $\phi: M \to N$, where $M$ is Riemannian, we construct a stochastic process $X$ on $M$ such that the image $Y:=\phi(X)$ is a (reversed, scaled) mean curvature flow of the fibers of the submersion. The model example is the mapping $\pi: GL(n) \to GL(n)/O(n)$, whose image is equivalent to the space of $n$-by-$n$ positive definite matrices, $\pdef$, and the said MCF has deterministic image. We are able to compute explicitly the mean curvature (and hence the drift term) of the fibers w.r.t. this map, (i) under diagonalization and (ii) in matrix entries, writing mean curvature as the gradient of log volume of orbits. As a consequence, we are able to write down Brownian motions explicitly on several common homogeneous spaces, such as Poincar\'e's upper half plane and the Bures-Wasserstein geometry on $\pdef$, on which we can see the eigenvalue processes of Brownian motion reminiscent of Dyson's Brownian motion. By choosing the background metric via natural $GL(n)$ action, we arrive at an invariant control system on the $GL(n)$-homogenous space $GL(n)/O(n)$. We investigate feasibility of developing stochastic algorithms using the mean curvature flow. KEY WORDS: mean curvature flow, gradient flow, Brownian motion, Riemannian submersion, random matrix, eigenvalue processes, geometry of positive definite matrices, stochastic algorithm, control theory on homogeneous space ### Corner asymptotics of the magnetic potential in the eddy-current model In this paper, we describe the magnetic potential in the vicinity of a corner of a conducting body embedded in a dielectric medium in a bidimensional setting. We make explicit the corner asymptotic expansion for this potential as the distance to the corner goes to zero. This expansion involves singular functions and singular coefficients. We introduce a method for the calculation of the singular functions near the corner and we provide two methods to compute the singular coefficients: the method of moments and the method of quasi-dual singular functions. Estimates for the convergence of both approximate methods are proven. We eventually illustrate the theoretical results with finite element computations. The specific non-standard feature of this problem lies in the structure of its singular functions: They have the form of series whose first terms are harmonic polynomials and further terms are genuine non-smooth functions generated by the piecewise constant zeroth order term of the operator. ### A uniqueness criterion and a counterexample to regularity in an incompressible variational problem In this paper we consider the problem of minimizing functionals of the form $E(u)=\int_B f(x,\nabla u) \,dx$ in a suitably prepared class of incompressible, planar maps $u: B \rightarrow \mathbb{R}^2$. Here, $B$ is the unit disk and $f(x,\xi)$ is quadratic and convex in $\xi$. It is shown that if $u$ is a stationary point of $E$ in a sense that is made clear in the paper, then $u$ is a unique global minimizer of $E(u)$ provided the gradient of the corresponding pressure satisfies a suitable smallness condition. We apply this result to construct a non-autonomous, uniformly convex functional $f(x,\xi)$, depending smoothly on $\xi$ but discontinuously on $x$, whose unique global minimizer is the so-called $N-$covering map, which is Lipschitz but not $C^1$. ### On the size and local equations of fibres of general projections For a general birational projection of a smooth nondegenerate projective $n$-fold from $\mathbb P^{n+c}$ to $\mathbb P^m$, $n<m\leq(n+c)/2$, all fibres have total length asymptotically bounded by $2^{\sqrt{n}+1}$ and the fibres are locally defined by linear and quadratic equations. ### Twisted Gan-Gross-Prasad conjecture for certain tempered L-packets In this paper, we investigate the twisted GGP conjecture for certain tempered representations using the theta correspondence and establish some special cases, namely when the L-parameter of the unitary group is the sum of conjugate-dual characters of the appropriate sign. ### Codes for Preventing Zeros at Partially Defect Memory Positions This work deals with error correction for non-volatile memories that are partially defect-at some levels. Such memory cells can only store incomplete information since some of their levels cannot be utilized entirely due to, e.g. wearout. On top of that, this paper corrects random errors $t\geq 1$ that could happen among $u$ partially defective cells while preserving their constraints. First, we show that the probability of violating the partially defective cells' restriction due to random errors is not trivial. Next, we update the models in [1] such that the coefficients of the output encoded vector plus the error vector at the partially defect positions are nonzero. Lastly, we state a simple theorem for masking the partial defects using a code with a minimum distance $d$ such that $d\geq (u+t)+1$. "Masking" means selecting a word whose entries correspond to writable levels in the (partially) defect positions. A comparison shows that, for a certain BCH code, masking $u$ cells by this theorem is as good as using the complicated coding scheme proven in [1, Theorem 1]. ### Symplectic geometry and Toeplitz operators on Cartan domains of type IV Let us consider, for $n \geq 3$, the Cartan domain $\mathrm{D}_n^{\mathrm{IV}}$ of type IV. On the weighted Bergman spaces $\mathcal{A}^2_\lambda(\mathrm{D}_n^{\mathrm{IV}})$ we study the problem of the existence of commutative $C^$-algebras generated by Toeplitz operators with special symbols. We focus on the subgroup $\mathrm{SO}(n) \times \mathrm{SO}(2)$ of biholomorphisms of $\mathrm{D}_n^{\mathrm{IV}}$ that fix the origin. The $\mathrm{SO}(n) \times \mathrm{SO}(2)$-invariant symbols yield Toeplitz operators that generate commutative $C^$-algebras, but commutativity is lost when we consider symbols invariant under a maximal torus or under $\mathrm{SO}(2)$. We compute the moment map $\mu^{\mathrm{SO}(2)}$-action for the $\mathrm{SO}(2)$-action on $\mathrm{D}_n^{\mathrm{IV}}$ considered as a symplectic manifold for the Bergman metric. We prove that the space of symbols of the form $a = f \circ \mu^{\mathrm{SO}(2)}$, denoted by $L^\infty(\mathrm{D}_n^{\mathrm{IV}})^{\mu^{\mathrm{SO}(2)}}$, yield Toeplitz operators that generate commutative $C^$-algebras. Spectral integral formulas for these Toeplitz operators are also obtained. ### Encodings of trajectories and invariant measures We consider a discrete dynamical system on a compact manifold M generated by a homeomorphism f. Let C = {M(i)} be a finite covering of M by closed cells. The symbolic image of a dynamical system is a directed graph G with vertices corresponding to cells in which vertices i and j are joined by an arc i to j if the image f(M(i)) intersects M(j). We show that the set of paths of the symbolic image converges to the set of trajectories of the system in the Tychonoff topology as the diameter of the covering tends to zero. For a cycle on G going through different vertices, a simple flow is by definition a uniform distribution on arcs of this cycle. We show that simple flows converge to ergodic measures in the weak topology as the diameter of the covering tends to zero. ### Partitioning through projections: strong SDP bounds for large graph partition problems The graph partition problem (GPP) aims at clustering the vertex set of a graph into a fixed number of disjoint subsets of given sizes such that the sum of weights of edges joining different sets is minimized. This paper investigates the quality of doubly nonnegative (DNN) relaxations, i.e., relaxations having matrix variables that are both positive semidefinite and nonnegative, strengthened by polyhedral cuts for two variations of the GPP: the $k$-equipartition and the graph bisection problem. After reducing the size of the relaxations by facial reduction, we solve them by a cutting-plane algorithm that combines an augmented Lagrangian method with Dykstra's projection algorithm. Since many components of our algorithm are general, the algorithm is suitable for solving various DNN relaxations with a large number of cutting planes. We are the first to show the power of DNN relaxations with additional cutting planes for the GPP on large benchmark instances up to 1,024 vertices. Computational results show impressive improvements in strengthened DNN bounds. ### Schur-Sato theory for quasi-elliptic rings The notion of quasi-elliptic rings appeared as a result of an attempt to classify a wide class of commutative rings of operators found in the theory of integrable systems, such as rings of commuting differential, difference, differential-difference, etc. operators. They are contained in a certain non-commutative "universal" ring - a purely algebraic analogue of the ring of pseudodifferential operators on a manifold, and admit (under certain mild restrictions) a convenient algebraic-geometric description. An important algebraic part of this description is the Schur-Sato theory - a generalisation of the well known theory for ordinary differential operators. Some parts of this theory were developed earlier in a series of papers, mostly for dimension two. In this paper we present this theory in arbitrary dimension. We apply this theory to prove two classification theorems of quasi-elliptic rings in terms of certain pairs of subspaces (Schur pairs). They are necessary for the algebraic-geometric description of quasi-elliptic rings mentioned above. The theory is effective and has several other applications, among them is a new proof of the Abhyankar inversion formula. ### Bandwidth Cost of Code Conversions in the Split Regime Distributed storage systems must store large amounts of data over long periods of time. To avoid data loss due to device failures, an $[n,k]$ erasure code is used to encode $k$ data symbols into a codeword of $n$ symbols that are stored across different devices. However, device failure rates change throughout the life of the data, and tuning $n$ and $k$ according to these changes has been shown to save significant storage space. Code conversion is the process of converting multiple codewords of an initial $[n^I,k^I]$ code into codewords of a final $[n^F,k^F]$ code that decode to the same set of data symbols. In this paper, we study conversion bandwidth, defined as the total amount of data transferred between nodes during conversion. In particular, we consider the case where the initial and final codes are MDS and a single initial codeword is split into several final codewords ($k^I=\lambda^F k^F$ for integer $\lambda^F \geq 2$), called the split regime. We derive lower bounds on the conversion bandwidth in the split regime and propose constructions that significantly reduce conversion bandwidth and are optimal for certain parameters. ### On degenerate blow-up profiles for the subcritical semilinear heat equation We consider the semilinear heat equation with a superlinear power nonlinearity in the Sobolev subcritical range. We construct a solution which blows up in finite time only at the origin, with a completely new blow-up profile, which is cross-shaped. Our method is general and extends to the construction of other solutions blowing up only at the origin, with a large variety of blow-up profiles, degenerate or not. ### A note on the involutive concordance invariants for certain (1,1)-knots We compute the involutive concordance invariants for the 10- and 11-crossing (1,1)-knots. ### Global Convergence of Hessenberg Shifted QR III: Approximate Ritz Values via Shifted Inverse Iteration We give a self-contained randomized algorithm based on shifted inverse iteration which provably computes the eigenvalues of an arbitrary matrix $M\in\mathbb{C}^{n\times n}$ up to backward error $\delta\\|M\\|$ in $O(n^4+n^3\log^2(n/\delta)+\log(n/\delta)^2\log\log(n/\delta))$ floating point operations using $O(\log^2(n/\delta))$ bits of precision. While the $O(n^4)$ complexity is prohibitive for large matrices, the algorithm is simple and may be useful for provably computing the eigenvalues of small matrices using controlled precision, in particular for computing Ritz values in shifted QR algorithms as in (Banks, Garza-Vargas, Srivastava, 2022). ### Global Convergence of Hessenberg Shifted QR II: Numerical Stability We develop a framework for proving rapid convergence of shifted QR algorithms which use Ritz values as shifts, in finite arithmetic. Our key contribution is a dichotomy result which addresses the known forward-instability issues surrounding the shifted QR iteration [Parlett and Le 1993]: we give a procedure which provably either computes a set of approximate Ritz values of a Hessenberg matrix with good forward stability properties, or leads to early decoupling of the matrix via a small number of QR steps. Using this framework, we show that the shifting strategy introduced in Part I of this series [Banks, Garza-Vargas, and Srivastava 2021] converges rapidly in finite arithmetic with a polylogarithmic bound on the number of bits of precision required, when invoked on matrices of controlled eigenvector condition number and minimum eigenvalue gap. ### Expectation of the Maximum of a Pair of Random Variables with Zero-Truncated Bivariate Normal Distribution Calculating the expectation of the maximum of normally distributed variables arises in many applications. We derive a closed-form solution for the expectation of the maximum of a zero-truncated bivariate normal distribution, and conduct a simulation study to numerically confirm the solution by comparing it with a Monte-Carlo method. ### $α$-GAN: Convergence and Estimation Guarantees We prove a two-way correspondence between the min-max optimization of general CPE loss function GANs and the minimization of associated $f$-divergences. We then focus on $\alpha$-GAN, defined via the $\alpha$-loss, which interpolates several GANs (Hellinger, vanilla, Total Variation) and corresponds to the minimization of the Arimoto divergence. We show that the Arimoto divergences induced by $\alpha$-GAN equivalently converge, for all $\alpha\in \mathbb{R}_{>0}\cup\{\infty\}$. However, under restricted learning models and finite samples, we provide estimation bounds which indicate diverse GAN behavior as a function of $\alpha$. Finally, we present empirical results on a toy dataset that highlight the practical utility of tuning the $\alpha$ hyperparameter. ### Learning Based User Scheduling in Reconfigurable Intelligent Surface Assisted Multiuser Downlink Reconfigurable intelligent surface (RIS) is capable of intelligently manipulating the phases of the incident electromagnetic wave to improve the wireless propagation environment between the base-station (BS) and the users. This paper addresses the joint user scheduling, RIS configuration, and BS beamforming problem in an RIS-assisted downlink network with limited pilot overhead. We show that graph neural networks (GNN) with permutation invariant and equivariant properties can be used to appropriately schedule users and to design RIS configurations to achieve high overall throughput while accounting for fairness among the users. As compared to the conventional methodology of first estimating the channels then optimizing the user schedule, RIS configuration and the beamformers, this paper shows that an optimized user schedule can be obtained directly from a very short set of pilots using a GNN, then the RIS configuration can be optimized using a second GNN, and finally the BS beamformers can be designed based on the overall effective channel. Numerical results show that the proposed approach can utilize the received pilots more efficiently than the conventional channel estimation based approach, and can generalize to systems with an arbitrary number of users. ### Counting unique molecular identifiers in sequencing using a decomposable multitype branching process with immigration Detection of extremely rare variant alleles, such as tumour DNA, within a complex mixture of DNA molecules is difficult. Barcoding of DNA template molecules early in the next-generation sequencing library construction provides a way to identify and bioinformatically remove polymerase errors. During the PCR-based barcoding procedure consisting of $t$ consecutive PCR-cycles, DNA molecules become barcoded by random nucleotide sequences. Previously, values 2 and 3 of $t$ have been used, however even larger values of $t$ might be relevant. This paper proposes using a multi-type branching process with immigration as a model describing the random outcome of imperfect PCR-barcoding procedure, with variable $t$ treated as the time parameter. For this model we focus on the expected numbers of clusters of molecules sharing the same unique molecular identifier. ### Skew-sparse matrix multiplication Based on the observation that $\mathbb{Q}^{(p-1) \times (p-1)}$ is isomorphic to a quotient skew polynomial ring, we propose a new method for $(p-1)\times (p-1)$ matrix multiplication over $\mathbb{Q}$, where $p$ is a prime number. The main feature of our method is the acceleration for matrix multiplication if the product is skew-sparse. Based on the new method, we design a deterministic algorithm with complexity $O(T^{\omega-2} p^2)$, where $T\le p-1$ is a parameter determined by the skew-sparsity of input matrices and $\omega$ is the asymptotic exponent of matrix multiplication. Moreover, by introducing randomness, we also propose a probabilistic algorithm with complexity $O^\thicksim(t^{\omega-2}p^2+p^2\log\frac{1}{\nu})$, where $t\le p-1$ is the skew-sparsity of the product and $\nu$ is the probability parameter. ### Continuous-time mean-variance portfolio selection under non-Markovian regime-switching model with random horizon In this paper, we consider a continuous-time mean-variance portfolio selection with regime-switching and random horizon. Unlike previous works, the dynamic of assets are described by non-Markovian regime-switching models in the sense that all the market parameters are predictable with respect to the filtration generated jointly by Markov chain and Brownian motion. We formulate this problem as a constrained stochastic linear-quadratic optimal control problem. The Markov chain is assumed to be independent of the Brownian motion. So the market is incomplete. We derive closed-form expressions for both the optimal portfolios and the efficient frontier. All the results are different from those in the problem with fixed time horizon. ### Exponential Integral Solutions for Fixation Time in Wright-Fisher Model With Selection In this work we derive new analytic expressions for fixation time in Wright-Fisher model with selection. The three standard cases for fixation are considered: fixation to zero, to one or both. Second order differential equations for fixation time are obtained by a simplified approach using only the law of total probability and Taylor expansions. The obtained solutions are given by a combination of exponential integral functions with elementary functions. We then state approximate formulas involving only elementary functions valid for small selection effects. The quality of our results are explored throughout an extensive simulation study. We show that our results approximate the discrete problem very accurately even for small population size (a few hundreds) and large selection coefficients. ### Convergence of Deep Neural Networks with General Activation Functions and Pooling Deep neural networks, as a powerful system to represent high dimensional complex functions, play a key role in deep learning. Convergence of deep neural networks is a fundamental issue in building the mathematical foundation for deep learning. We investigated the convergence of deep ReLU networks and deep convolutional neural networks in two recent researches (arXiv:2107.12530, 2109.13542). Only the Rectified Linear Unit (ReLU) activation was studied therein, and the important pooling strategy was not considered. In this current work, we study the convergence of deep neural networks as the depth tends to infinity for two other important activation functions: the leaky ReLU and the sigmoid function. Pooling will also be studied. As a result, we prove that the sufficient condition established in arXiv:2107.12530, 2109.13542 is still sufficient for the leaky ReLU networks. For contractive activation functions such as the sigmoid function, we establish a weaker sufficient condition for uniform convergence of deep neural networks. ### Convergence Analysis of Deep Residual Networks Various powerful deep neural network architectures have made great contribution to the exciting successes of deep learning in the past two decades. Among them, deep Residual Networks (ResNets) are of particular importance because they demonstrated great usefulness in computer vision by winning the first place in many deep learning competitions. Also, ResNets were the first class of neural networks in the development history of deep learning that are really deep. It is of mathematical interest and practical meaning to understand the convergence of deep ResNets. We aim at characterizing the convergence of deep ResNets as the depth tends to infinity in terms of the parameters of the networks. Toward this purpose, we first give a matrix-vector description of general deep neural networks with shortcut connections and formulate an explicit expression for the networks by using the notions of activation domains and activation matrices. The convergence is then reduced to the convergence of two series involving infinite products of non-square matrices. By studying the two series, we establish a sufficient condition for pointwise convergence of ResNets. Our result is able to give justification for the design of ResNets. We also conduct experiments on benchmark machine learning data to verify our results. ### Blind Image Inpainting with Sparse Directional Filter Dictionaries for Lightweight CNNs Blind inpainting algorithms based on deep learning architectures have shown a remarkable performance in recent years, typically outperforming model-based methods both in terms of image quality and run time. However, neural network strategies typically lack a theoretical explanation, which contrasts with the well-understood theory underlying model-based methods. In this work, we leverage the advantages of both approaches by integrating theoretically founded concepts from transform domain methods and sparse approximations into a CNN-based approach for blind image inpainting. To this end, we present a novel strategy to learn convolutional kernels that applies a specifically designed filter dictionary whose elements are linearly combined with trainable weights. Numerical experiments demonstrate the competitiveness of this approach. Our results show not only an improved inpainting quality compared to conventional CNNs but also significantly faster network convergence within a lightweight network design. ### Rectangular mesh contour generation algorithm for finite differences calculus In this work, a 2D contour generation algorithm is proposed for irregular regions. The contour of the physical domain is approximated by mesh segments using the known coordinates of the contour. For this purpose, the algorithm uses a repeating structure that analyzes the known irregular contour coordinates to approximate the physical domain contour by mesh segments. To this end, the algorithm calculates the slope of the line defined by the known point of the irregular contours and the neighboring vertices. In this way, the algorithm calculates the points of the line and its distance to the closest known nodes of the mesh, allowing to obtain the points of the approximate contour. This process is repeated until the approximate contour is obtained. Therefore, this approximate contour generation algorithm, from known nodes of a mesh, is suitable for describing meshes involving geometries with irregular contours and for calculating finite differences in numerical simulations. The contour is evaluated through three geometries, the difference between the areas delimited by the given contour and the approximate contour, the number of nodes and the number of internal points. It can be seen that the increase in geometry complexity implies the need for a greater number of nodes in the contour, generating more refined meshes that allow reaching differences in areas below 2%. ### Improved Upper Bound on Independent Domination Number for Hypercubes We revisit the problem of determining the independent domination number in hypercubes for which the known upper bound is still not tight for general dimensions. We present here a constructive method to build an independent dominating set $S_n$ for the $n$-dimensional hypercube $Q_n$, where $n=2p+1$, $p$ being a positive integer $\ge 1$, provided an independent dominating set $S_p$ for the $p$-dimensional hypercube $Q_p$, is known. The procedure also computes the minimum independent dominating set for all $n=2^k-1$, $k>1$. Finally, we establish that the independent domination number $\alpha_n\leq 3 \times 2^{n-k-2}$ for $7\times 2^{k-2}-1\leq n<2^{k+1}-1$, $k>1$. This is an improved upper bound for this range as compared to earlier work. ### Heavy-Tail Phenomenon in Decentralized SGD Recent theoretical studies have shown that heavy-tails can emerge in stochastic optimization due to `multiplicative noise', even under surprisingly simple settings, such as linear regression with Gaussian data. While these studies have uncovered several interesting phenomena, they consider conventional stochastic optimization problems, which exclude decentralized settings that naturally arise in modern machine learning applications. In this paper, we study the emergence of heavy-tails in decentralized stochastic gradient descent (DE-SGD), and investigate the effect of decentralization on the tail behavior. We first show that, when the loss function at each computational node is twice continuously differentiable and strongly convex outside a compact region, the law of the DE-SGD iterates converges to a distribution with polynomially decaying (heavy) tails. To have a more explicit control on the tail exponent, we then consider the case where the loss at each node is a quadratic, and show that the tail-index can be estimated as a function of the step-size, batch-size, and the topological properties of the network of the computational nodes. Then, we provide theoretical and empirical results showing that DE-SGD has heavier tails than centralized SGD. We also compare DE-SGD to disconnected SGD where nodes distribute the data but do not communicate. Our theory uncovers an interesting interplay between the tails and the network structure: we identify two regimes of parameters (stepsize and network size), where DE-SGD %addition of network structure can have lighter or heavier tails than disconnected SGD depending on the regime. Finally, to support our theoretical results, we provide numerical experiments conducted on both synthetic data and neural networks. ### Sparsity and $\ell_p$-Restricted Isometry A matrix $A$ is said to have the $\ell_p$-Restricted Isometry Property ($\ell_p$-RIP) if for all vectors $x$ of up to some sparsity $k$, $\\|Ax\\|_p$ is roughly proportional to $\\|x\\|_p$. It is known that with high probability, random dense $m\times n$ matrices (e.g., with i.i.d.\ $\pm 1$ entries) are $\ell_2$-RIP with $k \approx m/\log n$, and sparse random matrices are $\ell_p$-RIP for $p \in [1,2)$ when $k, m = \Theta(n)$. However, when $m = \Theta(n)$, sparse random matrices are known to \emph{not} be $\ell_2$-RIP with high probability. With this backdrop, we show that there are no sparse matrices with $\pm 1$ entries that are $\ell_2$-RIP. On the other hand, for $p \neq 2$, we show that any $\ell_p$-RIP matrix \emph{must} be sparse. Thus, sparsity is incompatible with $\ell_2$-RIP, but necessary for $\ell_p$-RIP for $p \neq 2$. ### Artificial Intelligence-Assisted Optimization and Multiphase Analysis of Polygon PEM Fuel Cells This article presents new PEM fuel cell models with hexagonal and pentagonal designs. After observing cell performance improvement in these models, we optimized them. Inlet pressure and temperature were used as input parameters, and consumption and output power were the target parameters of the multi-objective optimization algorithm. Then we used artificial intelligence techniques, including deep neural networks and polynomial regression, to model the data. Next, we employed the RSM (Response Surface Method) method to derive the target functions. Furthermore, we applied the NSGA-II multi-objective genetic algorithm to optimize the targets. Compared to the base model (Cubic), the optimized Pentagonal and Hexagonal models averagely increase the output current density by 21.819% and 39.931%, respectively. ### Principal-Agent Hypothesis Testing Consider the relationship between the FDA (the principal) and a pharmaceutical company (the agent). The pharmaceutical company wishes to sell a product to make a profit, and the FDA wishes to ensure that only efficacious drugs are released to the public. The efficacy of the drug is not known to the FDA, so the pharmaceutical company must run a costly trial to prove efficacy to the FDA. Critically, the statistical protocol used to establish efficacy affects the behavior of a strategic, self-interested pharmaceutical company; a lower standard of statistical evidence incentivizes the pharmaceutical company to run more trials for drugs that are less likely to be effective, since the drug may pass the trial by chance, resulting in large profits. The interaction between the statistical protocol and the incentives of the pharmaceutical company is crucial to understanding this system and designing protocols with high social utility. In this work, we discuss how the principal and agent can enter into a contract with payoffs based on statistical evidence. When there is stronger evidence for the quality of the product, the principal allows the agent to make a larger profit. We show how to design contracts that are robust to an agent's strategic actions, and derive the optimal contract in the presence of strategic behavior.	2022-05-16 18:40:04	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.7560212016105652, "perplexity": 384.00089512966514}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1652662512229.26/warc/CC-MAIN-20220516172745-20220516202745-00534.warc.gz"}
https://www.wyzant.com/resources/answers/496359/how_many_of_each_does_he_have	Taylor K. # how many of each does he have? Ronald has been saving his winning lottery tickets. He has 23 tickets that are worth a total of $175. If each ticket is worth either$5 or \$10, how many of each does he have? By:	2019-10-16 09:19:44	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5156561136245728, "perplexity": 1990.4246653931732}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1570986666959.47/warc/CC-MAIN-20191016090425-20191016113925-00252.warc.gz"}
https://homework.study.com/explanation/synthesis-reactions-complete-and-balance-1-na-plus-o-2-2-al-plus-br-2-3-cl-2-plus-mg.html	# Synthesis Reactions Complete and balance: {eq}1.\ Na + O_2 \rightarrow\\ 2.\ Al + Br_2 \rightarrow\\ 3.\ Cl_2 + Mg \rightarrow {/eq} ## Question: Synthesis Reactions Complete and balance: {eq}1.\ Na + O_2 \rightarrow\\ 2.\ Al + Br_2 \rightarrow\\ 3.\ Cl_2 + Mg \rightarrow {/eq} ## Balanced Reactions: When we use suitable coefficients for the reactants and products of a chemical reaction, such that the number of atoms of every element is the same on both sides of the equation, we call the reaction a balanced reaction. The coefficients represent the ratio by the number of moles of the reactants used in relation to each other and to the products formed.	2023-03-22 02:35:50	{"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": 0.9244801998138428, "perplexity": 2358.9026543017335}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1679296943749.68/warc/CC-MAIN-20230322020215-20230322050215-00219.warc.gz"}
http://www.acmerblog.com/POJ-1887-Testing-the-CATCHER-blog-543.html	2013 11-10 # Testing the CATCHER A military contractor for the Department of Defense has just completed a series of preliminary tests for a new defensive missile called the CATCHER which is capable of intercepting multiple incoming offensive missiles. The CATCHER is supposed to be a remarkable defensive missile. It can move forward, laterally, and downward at very fast speeds, and it can intercept an offensive missile without being damaged. But it does have one major flaw. Although it can be fired to reach any initial elevation, it has no power to move higher than the last missile that it has intercepted. The tests which the contractor completed were computer simulations of battlefield and hostile attack conditions. Since they were only preliminary, the simulations tested only the CATCHER’s vertical movement capability. In each simulation, the CATCHER was fired at a sequence of offensive missiles which were incoming at fixed time intervals. The only information available to the CATCHER for each incoming missile was its height at the point it could be intercepted and where it appeared in the sequence of missiles. Each incoming missile for a test run is represented in the sequence only once. The result of each test is reported as the sequence of incoming missiles and the total number of those missiles that are intercepted by the CATCHER in that test. The General Accounting Office wants to be sure that the simulation test results submitted by the military contractor are attainable, given the constraints of the CATCHER. You must write a program that takes input data representing the pattern of incoming missiles for several different tests and outputs the maximum numbers of missiles that the CATCHER can intercept for those tests. For any incoming missile in a test, the CATCHER is able to intercept it if and only if it satisfies one of these two conditions: The incoming missile is the first missile to be intercepted in this test. -or- The missile was fired after the last missile that was intercepted and it is not higher than the last missile which was intercepted. The input data for any test consists of a sequence of one or more non-negative integers, all of which are less than or equal to 32,767, representing the heights of the incoming missiles (the test pattern). The last number in each sequence is -1, which signifies the end of data for that particular test and is not considered to represent a missile height. The end of data for the entire input is the number -1 as the first value in a test; it is not considered to be a separate test. Output for each test consists of a test number (Test #1, Test #2, etc.) and the maximum number of incoming missiles that the CATCHER could possibly intercept for the test. That maximum number appears after an identifying message. There must be at least one blank line between output for successive data sets. Note: The number of missiles for any given test is not limited. If your solution is based on an inefficient algorithm, it may not execute in the allotted time. 389 207 155 300 299 170 158 65 -1 23 34 21 -1 -1 Test #1: maximum possible interceptions: 6 Test #2: maximum possible interceptions: 2 import java.io.BufferedReader; import java.util.*; public class Main { public static void main(String[] args) throws Exception{ int totalnum; int count = 0; while(true) { int num; ArrayList < Integer> array = new ArrayList< Integer>(); if(totalnum==-1) break; } int length = array.size(); int [] num_array = new int[length]; int [] max_array = new int[length]; for(int i=0;i< length;i++){ num_array[i] = array.get(i); max_array[i] = 1; } int max_value = 1; for(int i=1;i< length;i++){ for(int j=0;j< i;j++){ if(num_array[i]<=num_array[j]&&max_array[i]<=max_array[j]) max_array[i]++; } max_value = (max_array[i]>max_value)?max_array[i]:max_value; } if(count!=0) System.out.println(); System.out.println("Test #"+(++count)+":"); System.out.println(" maximum possible interceptions: "+max_value); } } } 1. 博主您好，这是一个内容十分优秀的博客，而且界面也非常漂亮。但是为什么博客的响应速度这么慢，虽然博客的主机在国外，但是我开启VPN还是经常响应很久，再者打开某些页面经常会出现数据库连接出错的提示	2017-01-22 14:16:22	{"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": 0.48482003808021545, "perplexity": 1568.723415290947}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560281426.63/warc/CC-MAIN-20170116095121-00439-ip-10-171-10-70.ec2.internal.warc.gz"}
http://www.computerconservationsociety.org/resurrection/res24.htm	Resurrection Home Previous issue Next issue View Original Cover Computer RESURRECTION The Bulletin of the Computer Conservation Society ISSN 0958 - 7403 Number 24 Autumn 2000 ## Contents Editorial Nicholas Enticknap News Round-up Recollections of Early Vacuum Tube Circuits Maurice Wilkes Obituary: John Grover CCS Web site information In the Footsteps of the Conqueror Nicholas Enticknap Society Activity Impact Line Printers - an ICL Perspective Tony Wix Letters to the Editor Forthcoming Events Committee of the Society Aims and Objectives ## Editorial Nicholas Enticknap This issue has taken an unusually long time to put together. We apologise for the delay, and hope you feel it is better late than never. Much has happened since the last issue. Your Society has a new chairman, Ernest Morris, and one of the first tasks of his reign will be to superintend changes necessary to the financing of our society. The AGM authorised the Committee to introduce membership subscriptions should it become necessary. This is quite likely, as the Society's finances are at a low ebb at the moment - the Treasurer reported that there was just £120 in the kitty at the end of the financial year in April. For the time being though, the Committee is still inviting voluntary donations, especially from Society members who are not also members of the BCS. Our outgoing chairman Brian Oakley recorded at the AGM our pleasure at the knighthood awarded to Maurice Wilkes. It was pleasant to see IT further recognised with the award of a knighthood to programming pioneer Tony Hoare and of a damehood to Steve Shirley. Significant computing anniversaries are now following one another thick and fast. Following the celebrations for Manchester University's 1948 breakthrough and for the start of the world's first computing service based on Edsac at Cambridge University in 1949, the highlight this year has been the golden jubilee of the National Physical Laboratory's Pilot Ace, which we celebrated with a one day seminar in May. Sir Maurice Wilkes was inspired by this event to reflect on early computer circuitry, and the results of his thinking form our first feature article in this issue. Output printing is today taken for granted, but it took many years for the pioneer designers to perfect an output device capable of matching the internal speed of the computer. In our second feature, Tony Wix describes the engineering difficulties involved from the perspective of a British designer working for ICL and its component companies. The Internet is also taken for granted now, even though it has only been part of most of our lives for around five years. The World Wide Web itself dates back only to 1989, and that is later than the Domesday Project, which forms the subject matter of our third feature article. ## News Round-Up Brian Oakley has relinquished the chairmanship of the Society after four years at the helm. The AGM elected Ernest Morris as his successor. -101010101- Simon Lavington is looking for contemporary photographs of people involved in the design and development of Pegasus for use in his new booklet on this pioneering Ferranti computer. Photographers at the time concentrated perhaps naturally on the machinery, and the sung and unsung heroes and heroines of the project were neglected in comparison. Anyone who can help can find Simon's contact details on page 32. -101010101- Doron Swade and his team at the Science Museum completed assembly of the printer for the Difference Engine in April, in time for the launch of Doron's book "The Cogwheel Brain". -101010101- In December the Society will present a talk on early calculators, including the BTM 541, 542, 550 and 555 and the Powers Samas EMP, and associated early computers including the BTM 1201 and 1202, the Powers Samas PCC and the ICT 558. Hamish Carmichael would love hear from any member who has documentation of any of these machines - manuals, brochures, plugboard layouts and the like. Anyone who would like to contribute a talk on any of these machines at the meeting should likewise get in touch with Hamish. -101010101- The Science Museum's new Wellcome Wing was formally opened by HM The Queen at the end of June. Billed as "the world's leading centre for the public presentation of contemporary science", it features over 40 terminals linked in an intranet to a database that is updated daily. -101010101- As we were going to press we heard the sad news that Derek Milledge had died. Derek was an enthusiastic supporter and a leading light in the Pegasus Working Party, knowing more than anyone of the details of its programming and operation. He had been designing software since the early 1950s, and we will greatly miss his detailed knowledge of the pioneering times. We plan to publish a fuller appreciation in our next issue. -101010101- We regret to report also that Donald Davies died in May aged 75. Donald, whose work with computers started on the Pilot Ace in 1947 and who subsequently became best known for his pioneering work with packet-switching, was a good friend of the Society who often contributed to Resurrection. We publish a last letter from him in this issue. -101010101- Another who is no longer with us is the Earl of Halsbury, who died in January aged 91. Lord Halsbury was as managing director of the National Research Development Corporation from 1949 to 1959 a highly influential figure in the development of the British computer industry. A former president of the British Computer Society, he attended some of the earlier Society functions. -101010101- Paul Rojas of the Freie Universitaet Berlin tells us that he and his colleagues "have re-implemented in Java the first chess program ever written". The original author was none other than Konrad Zuse, who wrote it during the war in a high level language called Plankalkül. No compiler was ever produced for this language until Professor Rojas and his team did so in February this year. Readers interested in the program can get at it at the Zuse Internet Archive at <www.zib.de/zuse>, using the button marked 'simulations'. -101010101- John Deane of the Australian Computer Museum Society is writing a history of Silliac, Sydney University's copy of John von Neumann's IAS computer. As part of this project John has compiled a list of all the machines inspired by and based on the IAS design, 18 in all. He is looking for information about the shutdown dates for all of these machines. Anybody who can help should contact John at <John.Deane at tip dot csiro dot au>. -101010101- Can any member give one, or even two, Philips minicomputers a good home? A P851, built around 1978, and a P854, probably a couple of years younger, are offered, complete with manuals for hardware, software and programming. For further details contact Ray W Clarke at <ray at clarke dot demon dot co dot uk> or <rwclarke at iee dot org>. -101010101- The Winston Churchill Memorial Trust has sent the Society details of an attractive offer. The Trust provides a number of Travelling Fellowships which are available to British citizens of any age and from all walks of life, irrespective of academic or professional qualifications. They enable their holders to travel overseas to undertake study projects related to their trade, profession or particular interest. Anyone interested please contact Hamish Carmichael. -101010101- The Committee is planning to make Resurrection available from our Web site in PDF format, instead of LaTeX, Word and ASCII as at present. -101010101- Readers who have general queries to put to the Society should address them to the Secretary at the address given on the inside back cover. Members who move house should notify Hamish Carmichael of their new address to ensure that they continue to receive copies of Resurrection. This is because the CCS membership is different from the BCS list. -101010101- ### North West Group contact details Chairman Professor Frank Sumner FBCS, Department of Computer Science, University of Manchester, M13 9PL. Tel: 0161 275 6196. Secretary William Gunn: Tel: 01663 764997. Email: bengunn at compuserve dot com Science & Industry Museum representative Jenny Wetton, Museum of Science & Industry, Liverpool Road, Castlefield, Manchester M3 4JP. Tel: 0161 832 2244. Email: curatorial at msim dot org dot uk ## Recollections of Early Vacuum Tube Circuits Maurice Wilkes A remark made by one of the speakers during the Ace 2000 seminar brought back to me very vividly the climate of the late 1940s when the first stored program digital computers were being designed. It was to the effect that Ted Newman did not care for the circuits for a possible Pilot Ace that Harry Huskey had designed, and preferred to design the real Pilot Ace in his own way. The early radio engineers were concerned with sine waves of various frequencies - radio, intermediate, audio - and nothing else. By the 1930s cathode ray tubes were coming into use and bringing with them new and strange wave forms, particularly time bases and strobes. Primitive analogue computing devices were also appearing. A new term, 'electronics', was coined for the new technology. Electronic techniques were much to the fore in ionosphere research and in television. They were vigorously exploited during the war for radar and other applications and, by the end of the war, knowledge of electronics had become widespread. The designers of the early digital computers felt entirely confident that electronic techniques would meet the challenge. In fact, electronics offered them an embarrassingly wide range of alternative techniques to choose from. The first thing they had to do was to decide on the best way to realise gates and flip-flops and to evolve a consistent set of principles for putting them together to make a computer. There was not time for a careful and exhaustive appraisal, and each designer made his choice largely on the basis of personal preference. Although their experience in other applications of electronics stood them in good stead, computer designers soon found they had to learn a few new tricks, such as how to handle non-repetitive wave forms. There were three main approaches to the design of trees and gating circuits. One was by means of what were called Kirchov circuits, that is, resistor networks feeding amplitude discriminators. This was essentially an analogue approach. Another was to make use of pentodes with independent inputs applied to the control grid and to the suppresser grid. Thirdly, use could be made of diodes. Obviously vacuum tubes would be used for amplifiers and this seemed straightforward enough. However, the output was at a much higher voltage than the input, and the inter-stage coupling circuits had to allow for this. The designer could either use capacitors or pulse transformers for inter-stage coupling, with diodes for zero restoration (otherwise called clamping), or he could use a resistor chain, perhaps with capacitors for frequency correction. Having made his choice, every designer was firmly convinced that his way was the best. This was only natural. I myself was no exception to the rule. I would stand up stoutly for the superior merits, as I saw them, of the Edsac design philosophy. Likewise, it was inevitable that Ted Newman, an ex-EMI man and a disciple of Blumlein, should have no time at all for Harry Huskey's Eniac-style circuits. Yet in spite of all the strong feelings, it was found, when the chips were down, that all the early computers worked with much the same degree of reliability. It was not that the doubts which had been expressed about pattern sensitivity, stability and so on were not well founded. What experience showed was that, if the engineering were carefully and competently done, most schemes could be made to work. The chart (see overleaf) is intended to illustrate the great diversity that existed in the way selected circuit functions were implemented in the first wave of computers. It was constructed partly from memory and I make no great claim for its accuracy. Not all the functions required in a computer are included in the chart; for example, there is no mention of control logic. Events moved fast in the first few years. Kirchov circuits dropped out and pentode gates became unpopular. Germanium diodes, which were not available when the Edsac design started, soon came along. At first, there were doubts about their reliability and recovery time, but confidence was soon established, and the Seac made free and elegant use of them. The merits of parallel architectures became recognised, one being that they opened the way to DC inter-stage connection. Finally, when all seemed set for a great future with vacuum tubes, transistors came along and we were all back at square one. Blumlein died early in the war and we can only speculate as to what his approach to digital design would have been. He is famous for his insistence that a circuit should be designed on paper, with the expectation that it would work first time. This used to puzzle me, until I realised that he must have been referring to Kirchov circuits. How right he was! Anyone who has worked with such circuits will have found that to proceed without working out a properly toleranced design in advance is a good way to hang oneself! Blumlein would have approved of one feature in the design of the Edsac, namely the use of cathode-coupled amplifiers. These are essentially long-tailed pairs, a special favourite of Blumlein's. If the tail is not made too long they have very good clipping properties and they do not invert the pulses. For this latter reason the Edsac contained no inverters. Eniac SSEM Edsac Pilot Ace Seac Swac IAS Interstage Coupling Kirchov * Capacitor and DC restorer * * Pulse transformer and DC restorer * DC * Trees Kirchov * * Diode * * Adder Kirchov * * Pentode * * * Triode * Diode * * Flip-Flops Static * * * * * * Dynamic * This chart shows the great diversity in the way in which circuit functions were implemented in first generation computers. The computers were: Eniac - Electronic Numerical Integrator and Calculator (University of Pennsylvania): SSEM - Small-Scale Experimental Machine (Manchester University 1948); Edsac - Electronic Delay Storage Automatic Calculator (Cambridge University): Pilot Ace - Automatic Computing Engine (National Physical Laboratory): Seac - Standards Eastern Automatic Computer (US National Bureau of Standards); Swac - Standards Western Automatic Computer (US National Bureau of Standards); IAS (called after the Princeton Institute of Advanced Study). Top Previous Next ## Obituary: John Grover Leo pioneer John Grover has died. John played a principal part in the coding of the world's first routine business application on an electronic computer. The application went live in November 1951. John joined Lyons, the catering company that built Leo, after service in the RAF. He was recruited as a management trainee, but was selected to work on Leo in the first trawl through the company's promising young people. David Caminer, who was Leo's systems and programming manager at this time, has paid this tribute to John Grover. "John played an invaluable part in out very very small team. He followed the methodology that we laid down unswervingly and made it possible to get it firmly established as newcomers were drawn in. He was a fine trainer and many of the young men and women who were recruited learned the new discipline working under him." John later joined the computer enterprise at EMI. When EMI was absorbed by ICT he was appointed to a senior sales management post in that company and subsequently within ICL, where he was re-united with several of his old Leo colleagues again. Top Previous Next ## CCS Web site information The Society has its own World Wide Web (WWW) site: it is located at http://www.cs.man.ac.uk/CCS. This is in addition to the FTP site at ftp.cs.man.ac.uk/pub/CCS-Archive (please note that these URLs are case-sensitive). Our Web site includes information about the SSEM project as well as selected papers from Resurrection. Readers can download files, including the current and all past issues of Resurrection and simulators for historic machines. Top Previous Next ## In the Footsteps of the Conqueror Nicholas Enticknap The 900th anniversary of the Domesday Book, William the Conqueror's great survey of all his domains, seemed a suitable occasion for the BBC to try to produce a modern equivalent. As things have turned out, 1986 proved to be a couple of decades too early, and the Domesday Project can be voted at best only a partial success. It did however provide the BBC with valuable experience which is today being used to create innovative historical Web sites. William the Conqueror's audacious attempt to create an inventory of everything he owned in the whole of England still excites admiration, and the result is still invaluable to scholars. It was not until the 19th century that a comparable census of the population was taken, and not until the 20th Century that subsequent administrators assembled anything like as complete a picture of the country's economic assets. As the 900th anniversary of the Domesday Book approached, it seemed to the BBC a good idea to celebrate the occasion by producing a modern variant. As a result the Domesday Project was born. The idea was to create a similar survey of the whole country visually rather than verbally, by means of a series of layered maps. Starting with a map of the whole country, you could drill down layer by layer until, in some towns at least, you could reach a level of detail showing individual buildings. Each layer was accompanied by text and other materials explaining what could be seen. Fourteen years later, the BBC hosted a presentation to the Society at the Science Museum describing the Project. The story started in the late seventies and early eighties. At that time, the BBC had embraced computer technology in a big way, producing a variety of television programmes about computing in general, and following that up by joining forces with Acorn, at the time a major player in the computer education market, to produce the BBC micro. This led to a series of 'how-to-do-it' programmes, created by the lead speaker at the Society's presentation, George Auckland, who is now Head of Digital Media Presentation for the BBC Education Department. The awareness of computer technology within the BBC generated by the BBC micro project led to the idea of the Domesday Project. Furthermore, as George Auckland said, "That machine had qualities which lent themselves to education". A most important feature was the Video Editing System chip within the machine, which had 32Kb memory of its own, a huge amount in 1983 when the Domesday Project started. The initial idea was that the eventual end product should be sold as an add-on to the BBC system at a price of around £1000. But things did not work out quite like that. Apart from using the BBC micro, "we had no idea of what technologies should be used", the main speaker, Professor Stephen Heppel, told the Society. Ultimately, the choice fell on LV-ROM video discs, the forerunner of today's CD-ROMs, with a capacity of 650Gb per side. They contained both moving sequences and programs written in BCPL. These programs were controlled via the BBC system's Video Editing chip, allowing you to jump from sequence to sequence via menu bars at the bottom of the screen. For data gathering the BBC called on the assistance of the nation's schoolchildren. According to Heppel, "Children were given a 1km square to look after. They wrote about it, and photographed it." Writing about it was a harder task than it would be today, as that was before the days when every schoolchild learnt keyboard skills as a matter of course. It was a massive logistical exercise, said Heppel, as "One million kids were involved. In places where there were no schools, we drafted in the WIs. "The photographs were either commissioned or sent in by the schools. They were then mounted on walls in a virtual gallery. There were doors in the gallery leading to the outside world. There was the idea of a 3-D space you could explore: it would be a hot Web site today. Three photos only were associated with each square, because of limitations on storage space. But it would still take seven years to see it all on a 9-to-5 day basis." Nonetheless the result, though an invaluable archive from the future historian's point of view, did not meet all its objectives. Part of the reason for that was the costs turned out to be much higher than foreseen, and so the price of the system was at £6000 rather too much for most individuals and even for many schools. Heppel admitted that the Domesday Project, judged from the short term perspective at least, was only a qualified success. "There were three fatal errors. First, it was very expensive - the price of a small family car at the time. Second it was jolly hard to find your way around - making the map layers work, for example. Third, it lacked extensibility. People were disappointed with what it said about them." Many of these limitations have been resolved with the passage of time. Storage space is much cheaper today, for example, so it would have been possible to add in many more pictures. Object-oriented programming technologies have also moved on apace, which would have made the provision of database navigation tools much easier. The BBC itself has moved on, and now has its own history Web site, at <bbc.co.uk/history>. According to the BBC Education Department's Chris Warren, this "is a top line generic site dealing with many different aspects of history. We are trying to build a history educational resource. It is aimed at everyone, primarily at adults but small children as well." Another current project is the History 2000 Web site, commissioned by George Auckland. "We are trying to create a live Web site. Our brief is broad. Others who engage in it have deep knowledge, and we want them to join with us in creating content. The Web was a perfect vehicle for getting this kind of thing going." Chris Warren described the History 2000 project. "History 2000 is a Millennium project to encourage audiences inspired by BBC programmes to find out more about history. We have 1000 partners around the UK providing content on the Web site. There is a calendar of events on the Web site. "We want to encourage people to submit photos to the Web site so that we can publish them fairly quickly. It was launched in September 1999. It will be part of the National Grid for Learning." All of these initiatives can trace their origins back to the Domesday Project. As Chris Warren said, "The Domesday Project proved you can build a useful resource in partnership with the whole country. So the BBC site is the great-grandchild of the Domesday Project." Editor's note: this is a report on the seminar hald by the Society at the Science Museum on 2 March 2000. Top Previous Next ## Society Activity Bombe Rebuild Project John Harper Manufacturing progress has been excellent over the past few months. There is not enough space here to cover everything, so I will just pick on some highlights. The Society of Model and Experimental Engineers has completed the machining of the main gearbox and clutch housings. This was a highly specialised operation requiring the use of precision machinery. The result is excellent and very impressive. Future machining is simple in comparison, so we treat this as a major step forward. On the Bletchley Park site, but not directly associated with the Trust, is a Mechanical Engineering Apprentices College. We have met with the management at various functions arranged by the Trust, and as a result the apprentices are making intricate parts. In addition, the staff has been helping us by providing the use of CNC machinery. Examples of the parts that have been made are the steel cams that drive the Bombe carry mechanism. They were machined using computer files generated on our CAD systems. Another example of the College's work is the manufacture of tapered brass wiring pins. This work has started, but with 6000 required we still have a long way to go. It might be worth mentioning here for those who have not seen it elsewhere that the Bletchley Park Trust has now signed the lease for the main part of the site. This allows the Trust to move forward at last, and to plan the future. The level of cooperation between the Trust and the Rebuild Project is very good, and is improving as BP plans develop. While manufacturing has been progressing well, we have also been drawn into a diversion we very much welcome. Many readers will have read Robert Harris's novel "Enigma". This novel is currently being turned into a major film, and the producers have gone to great lengths to have everything as accurate as possible. When it came to filming a Bombe room, we were asked to assist a film effects company to create 'prop' Bombes. The level of detail which these people are prepared to go to is amazing. They used many of our AutoCad drawings to make the external parts. In some cases our drawings were taken as a file, converted to a suitable format and fed directly into a CNC machine. From this process, many parts were made 'exactly to drawing'. Having British Bombes appear in a major WW2 film will greatly improve our public awareness. However there is a great deal more benefit to both the Bletchley Park Trust and the Rebuild Project when these 'prop' Bombes go on display at BP. They will not just be static displays: they have been made so as to allow the drums to rotate. Our project is committed to motorise one Bombe such that the rotation and carry action of the drums is made to be as original. By the time this report appears in print, the Bombes should be on display at BP, with the motorising planned to be working in September. Our requests for assistance in {\em Resurrection} issue 23 for people to help with drilling and tapping have produced an excellent response. We now have a team of retired Nortel people working away in the basement of the Conference Centre at Nortel Networks Harlow, and the work is progressing extremely well. The cableforming is also making good progress in the same area, as mentioned last time. We have also had a good response to our previous cry for help in the area of hardening and grinding. A technical college is carrying out initial experiments with the grade of steel we have used, and we are optimistic about the ongoing processing. Lathe work is perhaps our largest single manufacturing operation and literally hundreds of parts have been made. However, there are still very large numbers of parts that need turning, and further help would be greatly appreciated. We are now looking for help in refurbishing and rebuilding our stock of Hollerith relays. We also need a quantity of coils wound from scratch. For this, we have a coil-winding machine available. This work could be carried out by one person or split if required. All help would be welcome, and knowledge of Hollerith relays would be a distinct advantage. Readers who feel they would like to help our Project can find my contact details inside the back cover. Top Previous Next ## Impact Line Printers - an ICL Perspective Tony Wix Line printers were developed to produce computer output more quickly than the teleprinters and similar devices used with the early computers. Line printers had to meet four basic and different requirements: a) to print data onto preprinted or constant format stationery, such as payslips, invoices, orders and standard letters, where accurate and non-ambiguous printing was a higher priority than pure speed (often printing was concentrated into an area of a few lines over the full page width); b) to tabulate scientific data (to a maximum width of 160 columns); c) to produce standard personal letters, where print quality was more important than speed; and d) to print characters that could be read by an OCR B reader. Overall, line printers were expected to achieve quality printing over long periods with minimal operator intervention, maintenance and parts replacement, and to handle the output of high fanfold paper flow tidily. The first computer printers were generally solenoid-operated typewriters and modified tabulator printers operating at speeds of up to 100 lines per minute (lpm). A major advance was made in the late 1950s in the US when Shephard proposed and patented the principle of 'hit on the fly' line printing. He used a print drum made up of discs, each with 48 characters embossed around its periphery. They were clamped together to cover 120 horizontal print positions. Print speed was 150 lpm. Line printer mechanisms Impact printing involved the movement of mechanical parts under the control of electronics. Embossed characters were presented serially to the printing area, either vertically (on a print drum with its axis parallel to the paper) or horizontally (by a chain, belt or train moving across the paper). Print hammers, one for each column of print, were situated on one side of the paper with the ribbon and drum on the other side. Hammers impacted the paper and ribbon against the drum to transfer an inked character onto the paper. Embossed (reversed) characters moved continually at constant speed. As a result, during impact there was always some degree of smear - vertical from drum printers, sideways from train printers. This effect increased with speed, and thus was a factor in limiting speed. Designers attempted to minimise this smear effect by reducing the limb width of the embossed characters. They also minimised the period of impact by using high hammer speeds. A print drum typically had a 64 character repertoire around its periphery, repeated up to a maximum 160 times to provide multiple print columns. A subset of 48 of the most frequently used characters was grouped together, so that a line shift could occur during passage of the other 16 characters on most occasions. When this happened, the printing speed equalled the print drum speed. Nominal speeds quoted used this 'synchronous' subset. There were two principal types of dynamics for line printing: controlled penetration printing, and ballistic hammer printing. The early line printers were all controlled penetration systems. A pivoted arm had a hammer head at one end and an armature at the other, the latter being attracted by an electromagnet. The hammer was further away from the pivot than the armature. A front stop with adjustable screw was positioned between the hammer head and the pivot, and an adjustable screw backstop determined start position and flight time. Energising the electromagnet caused the arm to hit the front stop, flexing and overtravelling slightly to hit the paper, ribbon and embossed character and then quickly whip away. This fast rebound minimised smear. The front stop position determined the depth of the impact, and thus the print density. In theory, a front stop control should have given the crispest print because of the short dwell time; in practice, it was difficult to maintain the front stop position. A very small change in distance had a profound effect on ink transfer and thus on print density, and in the extreme could cause random wear on the faces of the embossed characters. On the early printers, the front stop was closer to the pivot, with large overthrow, and it was difficult to set front and back stop positions because they interacted. Adjusting flight time varied the initial air gap, energy input, hammer velocity and impact energy. The time taken to settle the arm was one limitation on printing speed. Later printers, such as the Anelex Series 5 from the USA, had the front stop much closer to the hammer and less overthrow, and so were relatively more stable. They achieved speeds up to 1250 lines per minute. In linear ballistic hammer printers each hammer was a free item, constrained only by a settling spring and guides to keep it along the flight path, which was a straight line through the centre of the drum. The hammer head was shaped to match the print drum curvature. An electromagnetic actuator was energised to move a hammer sitting in contact with the arm. The hammer travelled in guided free flight to impact the ribbon, paper and embossed character, then rebounded to settle under control of the return spring. The print density was set by adjusting the electrical energy supply to the actuator, while flight time was set by adjusting the arm rest position when fitted in the printer. Among the advantages of the ballistic approach were automatic compensation for variations in stationery weight and thickness, which was considerable when for example you changed from a single part set to a six part set with interleaved carbons. The hammer geometry automatically ensured that the curved face of the hammer exactly fitted the drum face, which avoided clipping of printed characters. RCA, English Electric Computers and ICL all used linear ballistic hammers. There were many variants on the theme: for example, CDC used rotary hammers and Data Products used flexure mounted hammers. Media and Paper Feed Ribbons were loaded with sufficient ink to allow them to transfer a clear character without smudging, and to have an acceptable life at reasonable cost. To start with, ribbons were made from silk, imported from China. Silk had mechanically stable fibres which did not distort under impact, retained ink well, and gave the best quality print. Later suppliers became concerned about the continuity of supply from China, and put much effort into developing nylon ribbons. By optimising warp, weft and strand thickness they developed ribbons of lower cost which became the workhorse, with silk only used where the highest quality printing was essential. Single pass melinex or mylar ribbons with a deposited ink coating were also developed for printing of highly defined characters. This development had to overcome tracking and spooling problems. The paper used had to withstand the acceleration forces as it passed through the printer, and the hammer impact force at the time of printing. Printer design teams usually included a media section, with a brief to maintain and improve quality by testing and providing customer service. Line printers used continuous fanfold stationery with sprocket holes punched down both sides. They engaged on the pintles of pairs of tractors, which moved print line by print line so the paper was nominally stationary during printing. The tractors were flat to have as many pintles in line contact as possible, to spread the load. Early printers used only one tractor pair, positioned after the print station. Later, faster printers used two pairs, one before and one after the print station. Each tractor slid along a plain round shaft for lateral positioning, and was locked into place before printing. A rotating drive shaft passed through each tractor gear wheel to provide the drive to the continuous chain or belt on which the pintles were set. The drive shafts were connected to an induction motor and flywheel through an electromagnetic friction clutch and brake system. In the early years friction clutch systems provided the best high torque, low inertia drive characteristic required for the fastest single line shifts and therefore the fastest print speed. Later magnetic particle clutches and printed circuit low inertia servo motors began to appear. When the rotating print drum was under impact, the drum tended to grab the paper, moving it backwards and so misplacing some printed characters. Paper was held taut between the tractors to resist such movement but, since it was only held by the pintles, it could tear if it was too tightly stretched under acceleration. When a print hammer impacted a character and the adjacent character was not to be printed, the paper and ribbon were drawn towards that character and a ghostly image of it could appear on the paper. Ghosting was minimised by the correct choice of character pitch and by guiding the ribbon and paper in the print area. Control Electronics Core memory was used on early line printers. A core store consisted of a matrix of, say, 120 print columns by 52 characters, giving a total of 6240 cores, each of which had a column wire and a character wire threaded through it. Coincidence of two currents in a core located the character to the column to cause a print out. Later, single line buffers holding a line of character codes were used. As a row of characters moved opposite the hammers, detectors on a coded disc on the drum axis defined the current character code. The store was scanned through the line for that character code, and where coincidence was found, hammers were fired at the drum. When all characters in the store had been read, a line feed was initiated and the store refreshed with the next line of print. Scanning could start with the drum in any position, so that printing all the characters on the drum took one complete revolution, which was followed by a line shift. The characters along the drum length were set in a slight spiral, to compensate for the time taken to read a character from store. The clutch brake unit generated a signal for every $\frac{1 6}$ or $\frac{1 8}$ of an inch of incremental paper movement. A counter was set to the number of lines to be moved, and decremented by three pulses to zero, to stop paper motion. ICT line printers to 1969 The first line printer developed by ICT was the model 600, which was introduced around 1962 for the 1301 computer. This was a controlled penetration printer with the front stop closer to the pivot, accentuating the overthrow. It had a 48 character set, 110 print positions, and a maximum speed of 120 lpm. The model 600 used a print barrel constructed from a number of print wheels. The wheels were produced by pressure-rolling the wheel periphery backwards and forwards along a linear master, engraved along its length with the full character set and laid out on the bed of a milling machine: this process eventually cold formed the characters on the wheel periphery. The 600 employed an interesting mixture of contemporary technologies: 3000 series Post Office relays, KT66 drivers, 2D21 thyratrons, thermionic valves, core stores and germanium transistors. Power consumption was in excess of 4.5 kilowatts. Paper line shifting used a wrapped spring clutch. Next came the model 666, designed for the 1900 series. ICT's marketing team set a performance target of 1500 lpm, but ICT had no new printing dynamics to achieve this, so the company copied the Anelex Series 5 printer hammer module in the model 666. The Anelex device operated at 1250 lpm: ICT uprated the barrel speed to 1350 lpm, which sacrificed print quality for speed, so a 660 lpm option was included as well. The 666 had a repertoire of 64 characters and printed across 132 columns. The design of the 666 concentrated more on production engineering aspects - the ability to produce printer components in-house at low cost - than on the development of printer technology. ICT had in 1965 invested heavily in factory automation, including expensive Milwaukee-matic computer-controlled milling machines and fine blanking. ICT manufactured the 666 in quantity at its Letchworth factory from around 1966, both for the 1900 series and for the OEM market. The official ICL history states that the 666 was "the apple of ICT's eye". ICT's last model was the 667, a low cost 600 lpm printer designed at Stevenage. A major feature was a new miniature front stop hammer module (MFSH) - ICT's advertising made much of the fact that it would fit inside a matchbox. It allowed a dramatic reduction in the mechanism size. A novel feature was the operator exchangeable print barrel. The 667 was not a success. The MFSH overheated when in the packed conditions of a full hammer tray, and the magnetic circuits of adjacent units were so close that they interacted, causing distorted printing. Magnetic shields proved no solution, since they reduced efficiency, so more power input was required, which produced more heating. To keep costs down, the paper feed path used only one set of tractors, set horizontally. But they were positioned after a right angle bend from the vertical, which compromised the control of paper between the print drum and the tractors, particularly on multipart sets. In short, this project demonstrated that small is not always beautiful. When problems arise, there is no room for manoeuvre. It also showed that basic development work should be proven before you commit to product design expenditure. After a large outlay on development, nobody dared stop the 667 project until some time after the formation of ICL. One ambitious plan for the 667 was to fit it to 1900 series printers at 600 lpm as a cost reduction. This development project also continued for too long. One idea we tried was intricate forms of air blowing, but this cooled the print hammers differentially making the printed line unstable. In the end, 'Echo' Organ closed down both projects. English Electric printers to 1969 English Electric Computers was formed in the early 1960s. This company eventually took over the computer businesses of Leo, Marconi and Elliott, changing its name several times in the process. The company's first commercial computer, the KDP 10, was manufactured at Kidsgrove under licence from RCA. It used an RCA Series 500 printer, which was costly to support as components wore out quickly. English Electric improved the design and renamed it the model 1035. The modifications that were made significantly improved print quality and printer life. The 1035 used ballistic hammers. The paper path was a horizontal table with the print drum above and the hammers below. The print drum, supplied by Mark Stamp Steel of the USA, was 120 columns wide, and was built from steel discs each of which was two columns wide and had 52 characters embossed by pressure rolling around the periphery. The 1035 was further enhanced to become the 1040 in time for the launch of the first English Electric-designed computer, the KDF9. The 1040 was demonstrated at the Business Efficiency Exhibition at Olympia in 1963. The 1040 featured new logic and diode gates and a range of plug-in printed circuit boards with higher packing density. English Electric used Mullard mesa transistors with low storage charge and higher speed, eliminating the need for speed-up capacitors, and employed a new high speed diode for the gating function, with faster rise times and good noise reduction. For the System 4 computer (another product manufactured under licence from RCA), the same printer development team produced the model 4560, which operated at 750 lpm. It was introduced in 1966 and manufactured in English Electric's Winsford factory. The 4560 employed a new mechanism with new hammer and actuator. The print repertoire was increased to 64 characters to include a lower case alphabet. For this product English Electric turned to a different print drum supplier, Caracteres of Neuchatel, Switzerland. Caracteres had developed a new machining process which increased life by a factor of 10. The drum was made up of two-column discs fixed in a slight skew around the shaft, to allow time to scan the core store sequentially as the shaft rotated. The top and bottom tractors were now linked by a belt. Paper throw rate was 26.6 inches per second. Testing of the 4560 showed that the mechanism was capable of meeting the requirement for higher speed printing, at 1350 lpm. But time was needed to evaluate performance at this speed and to develop a high speed paper throw feature and a dynamic paper stacker. So English Electric decided to buy time for the System 4 introduction phase by purchasing OEM mechanisms. Both ICT 666 and Anelex Series 5 mechanisms were examined in detail, and the choice fell on Anelex as that was the more established product. This was incorporated in a new printer known as the 4552: like the 4560 it was manufactured at Winsford and introduced in 1966. The Anelex Series 5 print mechanism used front stop controlled penetration hammers with an eight segment print barrel. Each segment was 20 columns wide and had 64 characters around the periphery. The rated speed was 1250 lpm at 48 characters. English Electric was ready to introduce the 1350 lpm version of the 4560 in the following year. Known as the 4554, it had a fast paper throw of 75 inches per second and a paper stacker which operated at the same speed. The 4554 had a barrel speed of 1350 lpm and was fitted with a second, higher speed clutch, which engaged after four lines of paper movement. The printer reverted to the lower speed clutch four lines from the end of throw. Paper formatting was controlled by punched paper tape. Anelex Series 5 ICT 666 EEC 4554 OEM mechanism cost (£) 4700 3780 3565 Maintenance time (hrs) 154 248 154 Scheduled parts replacement costs (£) 1879 4800 827 Total cost 6579 8580 4392 Figure 1 English Electric printer development involved a continuous pro\-gramme of learning and improving, particularly in the area of printing dynamics. Great attention was paid to metallurgical aspects of design, looking for the optimum choice of materials and protective finishes by a process of exhaustive life testing. The result was printing components with long life times, which reduced the total cost of ownership, as shown in Figure 1. This compares the cost of ownership of high speed printer mechanisms at the time of the introduction of the System 4. The figures are arrived at from product specifications and quotations, and are based on an assumption of 20,000 hours switched on with 50\% usage (the typical workload of a printer used on two shifts per day, six days per week over four years). ICL line printers In 1968 the Ministry of Technology inspired ICT and EEC to merge, forming ICL. ICL's North (Kidsgrove) and South (Stevenage) peripheral development groups were reorganised: Kidsgrove was dedicated to magnetic peripherals and Stevenage to paper and card peripherals. Following this reorganisation, ICL continued production of both 1900 series and System 4 printers. The company also continued development of the ICT 667 as a high priority, including the plan to use the miniature front stop hammer module (MFSH) in 666 mechanisms operating at 600 lpm as a cost reduction. The EEC 600 lpm low cost shuttle printer, now tested and proven but aimed at the same market, was scrapped. The subsequent failure of the 667 project meant that ICL would have no low cost printer to offer. As part of the reorganisation, core English Electric printer staff were relocated to Stevenage. But the ICT printer management already there did not want them, and a very unproductive period followed until two EEC executives, Roman Derc and myself, were appointed to manage new positions, Printer Development and Printer Products respectively. The first new printer requirement after the formation of ICL was for a train printer for "New Range", later to become the 2900 series. With this product, development work in the areas most critical to printing performance was carried out and proven before the company committed to a printer design. The Printer Development group studied how to improve the System 4 printing components to achieve higher printing rates without sacrificing print quality. This was essential, because the objective was to incorporate the enhanced components in a train printer. The print quality of a train printer will always be inferior to that of a drum device for a given hammer speed and mass, because the characters are not rigidly fixed on all axes and so there have to be greater misalignment tolerances. So the ICL team needed to make significant improvements to the print actuator, hammer and paper feed, as well as designing a print cartridge. Testing showed that the energy loss occurred in impacting a loosely mounted print slug (with 0.002" front to back clearance) was up to 60\%. So relative to a drum printer, the actuator needed to impart a higher energy to drive a lighter hammer. The development team increased the System 4 printer actuator energy level output from 3mJ to 9mJ with the same input level. They achieved it using a CAD program for optimising magnetic parameters and driving circuits, and by improving the geometry. Air gaps were now greater than on the System 4 drum printers. Shields were not needed since the mean operating conditions were in the unsaturated state and would reduce efficiency. Calculation and testing optimised the hammer mass around 0.75 gms. The hammer was made in a lightweight, high strength aluminium alloy. Two prototypes were built and performance tested: settling times were close to 8 msec, as the CAD software had predicted. ICL further developed the actuator driver circuit. A capacitor was charged via a resistor to a voltage preset by a potentiometer, and was then discharged into the actuator coil via a diode when a thyristor was triggered to conduct. To minimise the physical size of the capacitor for packaging, a high voltage (80v) was used. For paper movement, the System 4 friction clutch response was improved by laminating the main parts of the clutch magnetic circuit. After all these developments were complete, the Printer Development team modified an existing System 4 printer to take a train cartridge, and configured it with the newly developed hammers, actuators, drivers and clutches. This was used as a testbed to optimise printing performance. ICL director 'Echo' Organ was given a demonstration of this modified System 4 printer by the printer development and product managers. After receiving a guarantee from us that we could deliver a production train printer product within one year, having passed all the product assurance tests, he gave us the go ahead. The schedule just met the marketing requirements, and we did in fact deliver the new printer on time. The new product, called the TP1500, was exhibited at the Hanover Fair in April 1972. It had prime position on the ICL stand, and attracted great interest as the fastest current train printer worldwide. It was released on 1900 series mainframes in 1972-73, and was the standard line printer on the 2900 series when that came along in 1974. A 2000 lpm version of the TP1500 was used from 1974 in an ICL bureau continually printing high quality manuals - 2000 lpm was about the limit for train printing. But even while the TP1500 was being developed, the writing was on the wall for future ICL printer designs. In 1971 management under the leadership of Geoff Cross decided to abandon peripheral design and manufacture in the UK, and to purchase requirements instead from OEM suppliers in the USA, especially Control Data. In early 1975 the printer development and product teams were given 90 days notice of redundancy. We met the Minister for Industry and our local MP in a House of Commons committee room to discuss this abandonment of UK printer development, but got nowhere, because the civil service advisers to the Minister took the view that ICL could not compete successfully with USA printer manufacturers because it could not achieve their volumes. So later in 1975 peripheral development in Stevenage closed down. Towards the end of the decade band printers took over from train printers. In the 1980s laser printing was perfected, and page printers became the normal computer output device. However, impact printers are still in use today printing those confidential codes sent through the post. Editor's note: This article is based on a talk given by the author to the North West Group of the Society on 22 February 2000. Tony Wix was ICL Printer Products Manager from 1968 to 1975. Top Previous Next ## Letters to the Editor Dear Nicholas, It was fascinating to read Conway Berners-Lee's account of his visit to the Indian Statistical Institute. I was there four years earlier to advise the UN on whether the Russian Ural Computer should be supplied! In 1954 I was at MIT with a Commonwealth Fund Fellowship when I had a call to come to the United Nations Technical Assistance office in New York. They explained that the ISI had requested a computer and other things using money from USSR which they were quite keen to spend. Apparently it would offend the Russians if an American was sent to vet the proposal and my presence in the US made me an obvious choice. Would I go to India to look at the proposals and say whether the equipment should be supplied? Yes please! Calling in at the UK for Christmas, I went on to Delhi where I was told that the Institute was full of Russians and it would be diplomatic for me to stay in Delhi until they left. No problem! Then on to Calcutta, a town which was still crowded with the aftermath of partition; in the evening any walk outside meant stepping over people. By comparison the ISI was delightful and its hospitality superb. The equipment comprised a Russian computer, a colour lithographic printing press for the journal and machine tools (punches and presses) for making a hand calculator said to be of Indian origin. It was clear they had plenty of work for the computer and could make good use of the printing press but I wondered about the calculator. They let me have the prototype in my room and I took it apart. The parts were badly made, with poor tolerances and sharp edges that made its operation shaky. Also it could overflow in multiplication and there seemed to be a small missing part for which the inter-working parts were prepared and which should have prevented the overflow. I returned via a debriefing in Paris and then had an office in the UN building for two weeks in which to make my report, which was in favour of the computer and printing press but not the machine tools. But such was the motivation to spend Russian funds that all of it was approved. Later I visited a Block and Anderson showroom and quickly found the 'Indian' calculator which was a copy of an Italian model. The missing part was just where I expected to find it. Years later Prof Mahalanobis rang me. They were delighted with the lithographic press but were having some trouble getting the Russian computer working. The machine tools had not been unpacked for some time and were badly corroded in the humid atmosphere. Much more happened in my visit which involved Indian politics, metrication to replace the thousands of local units of measurement and stays in special government guest houses in Delhi. The whole thing was a great adventure and the bugs I picked up tuned my immune system ready for many more visits to India, which became my favourite overseas country. I went back to the ISI once more but without Mahalanobis it was a shadow of its former days of glory. With best regards, Donald W Davies Sunbury-on-Thames Middlesex 12 January 2000 Dear Editor, For those interested in the Imperial College Computing Engines, built in the late forties and early fifties, I have placed a short list of references to ICCE I and II, plus 10 photographs of ICCE II, at: <www.cee.hw.ac.uk/∼greg/icce/index.html> Best wishes, Greg Michaelson <greg at cee dot hw dot ac dot uk> 8 December 1999 ### Editorial contact details Readers wishing to contact the Editor may do so by fax to 020 8715 0484 or by e-mail to NEnticknap at compuserve dot com. ## Forthcoming Events Every Tuesday at 1200 and 1400 Demonstrations of the replica Small-Scale Experimental Machine at Manchester Museum of Science and Industry 14-15 October 2000, and fortnightly thereafter Guided tours and exhibition at Bletchley Park, price £3.00, or £2.00 for concessions Exhibition of wartime code-breaking equipment and procedures, including the replica Colossus, plus 90 minute tours of the wartime buildings 24 October 2000 North West Group meeting on "The Use of the Ferranti Mark I* in Aircraft Design" Speakers R Lane, H Malbon, P Morton 28 November 2000 North West Group meeting "Do Fish See in Colour?" (a talk on electronic publishing) Speaker D Griffiths 23 January 2001 North West Group meeting on "Early Design Automation" 20 February 2001 North West Group meeting on "Weather Forecasting" Speaker F Bushby The North West Group meetings will take place in the Conference room at the Manchester Museum of Science and Industry, Liverpool Road, Manchester, starting at 1730; tea is served from 1700. Queries about London meetings should be addressed to George Davis on 020 8681 7784, and about Manchester meetings to William Gunn on 01663 764997 or at <bengunn at compuserve dot com>. ## Committee of the Society [The printed version carries contact details of committee members] Chairman Ernest Morris FBCS Vice-Chairman Tony Sale FBCS Secretary Hamish Carmichael FBCS Treasurer Dan Hayton Science Museum representative Doron Swade CEng MBCS Chairman, Elliott 803 Working Party John Sinclair Chairman, Elliott 401 Working Party Chris Burton CEng FIEE FBCS Chairman, Pegasus Working Party Len Hewitt MBCS Chairman, DEC Working Party Dr Adrian Johnstone CEng MIEE MBCS Chairman, S100 bus Working Party Robin Shirley Chairman, Turing Bombe Working Party John Harper CEng MIEE MBCS Chairman, North West Group Professor Frank Sumner FBCS Meetings Secretary George Davis CEng FBCS Editor, Resurrection Nicholas Enticknap Archivist Harold Gearing FBCS Dr Martin Campbell-Kelly Professor Sandy Douglas CBE FBCS Dr Dave Holdsworth MBCS CEng Dr Roger Johnson FBCS Eric Jukes Graham Morris FBCS Professor Simon Lavington FBCS FIEE CEng Brian Oakley CBE FBCS John Southall FBCS ## Aims and objectives The Computer Conservation Society (CCS) is a co-operative venture between the British Computer Society, the Science Museum of London and the Museum of Science and Industry in Manchester. The CCS was constituted in September 1989 as a Specialist Group of the British Computer Society (BCS). It thus is covered by the Royal Charter and charitable status of the BCS. The aims of the CCS are to • Promote the conservation of historic computers and to identify existing computers which may need to be archived in the future • Develop awareness of the importance of historic computers • Encourage research on historic computers and their impact on society Membership is open to anyone interested in computer conservation and the history of computing. The CCS is funded and supported by a grant from the BCS, fees from corporate membership, donations, and by the free use of Science Museum facilities. Membership is free but some charges may be made for publications and attendance at seminars and conferences. There are a number of active Working Parties on specific computer restorations and early computer technologies and software. Younger people are especially encouraged to take part in order to achieve skills transfer. Resurrection is the bulletin of the Computer Conservation Society and is distributed free to members. Additional copies are £3.00 each, or £10.00 for an annual subscription covering four issues. Editor - Nicholas Enticknap Typesetting - Nicholas Enticknap Typesetting design - Adrian Johnstone Cover design - Tony Sale Printed by the British Computer Society Â© Copyright Computer Conservation Society	2020-03-28 20:02:11	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.1924779862165451, "perplexity": 3435.058734248971}, "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/1585370493120.15/warc/CC-MAIN-20200328194743-20200328224743-00334.warc.gz"}
http://alamos.math.arizona.edu/math577/	Essentially, all models are wrong, but some are useful. — George E. P. Box ## Course Description Information theory is a young branch of mathematics created to study digital data and digital communications. One problem is that of data compression: how to rewrite a digitally encoded message so that it occupies less physical media, such as disk space or memory? Another problem is that of error correction: given a lossy communication channel how to rewrite a digital message so that the message can be accurately transmitted with high probability? The answers to both problems revolve around Shannon entropy'', a single number based on the distribution of symbols in the message, which determines how efficiently both problems can be solved. This course aims at covering topics in probability theory, mathematical modeling and algebra of finite fields. It will be demonstrated how these branches of mathematics work together in solving problems in compression and error correction. Every aspect of the course will be illustrated with short programs written in MATLAB. The topics covered will include: • Fundamentals of Data Compression. • Fundamentals of Error-Correcting Codes. • Relevant Topics of Applied Probability. • Relevant Topics in Finite Fields.	2017-11-18 14:09:23	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.26836276054382324, "perplexity": 709.8932431260739}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1510934804965.9/warc/CC-MAIN-20171118132741-20171118152741-00749.warc.gz"}
https://mothur.org/wiki/Npshannon	# Npshannon The npshannon calculator returns a non-parametric estimate of the classical Shannon diversity index for an OTU definition. This calculator can be used in the summary.single, collect.single, and rarefaction.single commands. The calculations for the non-parametric Shannon index are implemented as described by Chao and Shen. $\hat{H}_{shannon}=\sum_{i=1}^{S_t}\frac{\hat{C}\pi_i \ln\left( \hat{C}\pi_i\right)}{1-\left(1-\hat{C}\pi_i\right)^N}$ $var \left(\hat{H}_{shannon} \right ) {\approx} \sum_{j=1}^{n} \sum_{i=1}^{n} \frac{{\partial}\hat{H}}{{\partial}n_i} \frac{{\partial}\hat{H}}{{\partial}n_j}$ $cov \left( f_i, f_j \right) = f_i \left(1-f_i / S_{ACE} \right ), i = j$ $cov\left ( f_i, f_j \right) = -f_i f_j / {S_{ACE}}, i\ne j$ where, $\hat{C} = 1-\frac{n_1}{N}$ $\pi_i = \frac{n_i}{N}$ n_i = the number of individuals in the ith OTU $N \mbox{ = the number of individuals in the sample}$ $S_t \mbox{ = the total number of OTUs}$ $S_{ACE}$ = the richness estimated using the ace calculator Open the file 98_lt_phylip_amazon.fn.sabund generated using the Amazonian dataset with the following commands: mothur > cluster(phylip=98_lt_phylip_amazon.dist, cutoff=0.10) The 98_lt_phylip_amazon.fn.sabund file is also outputted to the terminal window when the cluster() command is executed: unique 2 94 2 0.00 2 92 3 0.01 2 88 5 0.02 4 84 2 2 1 0.03 4 75 6 1 2 0.04 4 69 9 1 2 0.05 4 55 13 3 2 0.06 4 48 14 2 4 0.07 4 44 16 2 4 0.08 7 35 17 3 2 1 0 1 0.09 7 35 14 3 3 0 0 2 0.10 7 34 13 3 2 0 0 3 The first column is the label for the OTU definition and the second column is an integer indicating the number of sequences in the dominant OTU. The third column indicates the number of OTUs with only one indivdiual, the fourth the number of OTUs with two individuals, etc. The non-parametric Shannon index is then calculated using the values found in the subsequent columns. For demonstration we will calculate the non-parametric Shannon index for an OTU definition of 0.03: $\hat{C} = 1-\frac{75}{98} = 0.2347$ $\hat{H}_{shannon}= 75\frac{\ 0.2347 \left(\frac{1}{98}\right) \ln\left( 0.2347\left(\frac{1}{98}\right)\right)}{1-\left(1-0.2347\left(\frac{1}{98}\right)\right)^{98}} +6\frac{\ 0.2347 \left(\frac{2}{98}\right) \ln\left( 0.2347\left(\frac{2}{98}\right)\right)}{1-\left(1-0.2347\left(\frac{2}{98}\right)\right)^{98}} +1\frac{\ 0.2347 \left(\frac{3}{98}\right) \ln\left( 0.2347\left(\frac{3}{98}\right)\right)}{1-\left(1-0.2347\left(\frac{3}{98}\right)\right)^{98}} +2\frac{\ 0.2347 \left(\frac{4}{98}\right) \ln\left( 0.2347\left(\frac{4}{98}\right)\right)}{1-\left(1-0.2347\left(\frac{4}{98}\right)\right)^{98}}$ $\hat{H}_{shannon}=5.801$ Running... mothur > summary.single(calc=npshannon) ...and opening 98_lt_phylip_amazon.fn.summary gives: label NPShannon unique 7.768419 0.00 7.355786 0.01 6.831284 0.02 6.344819 0.03 5.800593<--- 0.04 5.559488 0.05 5.090494 0.06 4.853388 0.07 4.776910 0.08 4.495760 0.09 4.390298 0.10 4.297894 These are the same values that we found above for a cutoff of 0.03. At this point we have not implemented the 95% confidence interval calculations. As a point of reference it is worth noting that the classical Shannon index gave a value of 4.353.	2019-07-23 09:09:05	{"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": 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": 0.7913858890533447, "perplexity": 1510.6970476940407}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1563195529175.83/warc/CC-MAIN-20190723085031-20190723111031-00208.warc.gz"}
http://openstudy.com/updates/55b6dbd5e4b04559507b8a60	## anonymous one year ago Simplify completely quantity x squared minus 3 x minus 54 over quantity x squared minus 18 x plus 81 times quantity x squared plus 12 x plus 36 over quantity x plus 6. 1. anonymous I am unsure as to whether it is x+6/x-9 or (x+6)^2/x-9 2. AakashSudhakar We start with: $\frac{ x^2 - 3x - 54 }{ x^2 - 18x + 81 }\times \frac{ x^2 + 12x + 36 }{ x + 6 }$ Factoring each of these polynomial expressions gets us the following: $\frac{ (x-9)(x+6)(x+6)(x+6) }{ (x-9)(x-9)(x+6) }$ which quickly condenses to: $\frac{ (x-9)(x+6)^3 }{ (x-9)^2(x+6) }$ Because of all the common factors in the numerator and denominator, we can reduce this further to our final and most reduced form: $\frac{ (x+6)^2 }{ x-9 }$ which is our final answer.	2017-01-18 10:45:08	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6378509402275085, "perplexity": 676.5106234351524}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560280266.9/warc/CC-MAIN-20170116095120-00509-ip-10-171-10-70.ec2.internal.warc.gz"}
https://stats.stackexchange.com/questions/250462/under-the-null-versus-under-the-alternative/250474	# Under the null versus under the alternative I am new to statistics and I am trying to understand conceptually (and hopefully visually!) what is the difference when someone states "under the null" or "under the alternative". I guess this may stem from my lack of understanding of whether or not the null and alternative hypothesis can be based on two different likelihoods. • It would help if you could provide some context. What is it that someone is claiming, eg, "under the null". The meaning could differ depending on the larger claim. More narrowly, by "under the ____", people typically mean under the assumption that the ______ is true. – gung Dec 8 '16 at 21:44 • I just want to add something that hasn't been mentioned yet. Often the null hypothesis is the uninteresting hypothesis to the researcher. Consequently he or she wants to reject it. We deal with an observed statistic. It is common to say that as when discussing a p-value that under the null hypothesis the probability of observing a value as extreme or more extreme than what was observed (the definition of p-value). We have two types of errors, Type I (falsely rejecting the null hypothesis) and type II not rejecting the null hypothesis when the alternative is true. – Michael Chernick Dec 8 '16 at 22:33 • Continuation: In this set up since we control the Type I error (significance level) when we cannot reject the null hypotheses the common statement is "cannot reject" rather that "accept" the null hypothesis. As discussed in other recent posts hypothesis tests can be one-sided or two-sided and so too for the p-value. – Michael Chernick Dec 8 '16 at 22:40 • Another point I want to raise is that in medical research the null and alternative hypotheses are reversed because the uninteresting hypothesis becomes the important one. This occurs when testing that a new treatment is either "non-inferior" or "equivalent " to a standard treatment. In clinical trials this standard in some circumstances is sufficient to register a new drug or a generic drug to be approved as effective by the US Food and Drug Adminstration. See for example the work of William Blackwelder. – Michael Chernick Dec 8 '16 at 22:53 The exact specification of the null and alternative hypotheses depend on the test, but they all share general features. Each hypothesis is associated with a particular probability distribution. For example, suppose you have a sample population with two variables: a type that can take values A or B, and a continuous variable. In this example, you might want to ask whether two groups of people (men and women) have different mean heights. In this example, lets assume that height always follows a normal distribution. Our null hypothesis is that the heights for both men and women follow the same distribution. This is specified as meanwomen = meanmen meanwomen $\ne$ meanmen	2019-06-26 00:16:37	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.6866480112075806, "perplexity": 456.16179486689526}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627999964.8/warc/CC-MAIN-20190625233231-20190626015231-00427.warc.gz"}
https://crypto.stackexchange.com/questions/48118/second-preimages-on-n-bit-hash-functions-by-john-kelsey-and-bruce-schneier	# Second Preimages on n-bit Hash Functions by John Kelsey and Bruce Schneier And found there: «3.2 A Generic Technique: Multicollisions of Different Lengths» «Finding a Collision on Two Messages of Different Lengths.» «ALGORITHM:» «Steps:» THIS: "Build lists A and B as follows:" – for i = 0 to 2n/2 − 1: • A[i] = F(hin,M(i)) • B[i] = F(htmp,M(i)) There «i» takes value from 0 to 2^(n/2), BUT numbers of M’s blocks much less «2^(n/2)» Maybe correct go through the cycle [0, 2^k] ? Otherwise, many of M(i) simply don’t exist. or I'm wrong and something i don't understand? P.S. Is there a chance, that won't find a collision in a range [0, 2^k]? Thanks! • Ok, i figured out. I thought "M" - original message, but it's random data necessary for generation "Expandable Messages". In this case, range [ 0, 2^(n/2) ] is right. – user41204 Jun 9 '17 at 8:54 • If you answered your own question, would you mind taking it out of the comments and submitting (and accepting) it as an answer? – user47922 Jun 9 '17 at 15:14	2019-11-15 20:36:55	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.4176064431667328, "perplexity": 3892.87016170115}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1573496668712.57/warc/CC-MAIN-20191115195132-20191115223132-00347.warc.gz"}
https://proxieslive.com/tag/problem/	## Problem Lotka Volterra Model – Modelling & Plotting in Python I urgently need your help. Currently I’m conducting a research in regards of the revenue calculation and the dynamics within revenue calculation for my masters. I thought of revenue/profit margin as of a dynamical system – Lotka Volterra differential equations. I thought of a contribution margin calculation as within this simple formula: In consequence my idea was the following, but I receive an error: • Can anyone help me? • Do you think it’s a bad idea to use Lotka Volterra equations for this nonlinear purpose of Sales/Cost/Margin Simulation? • How would you model it and why? Am I missing equations or does the simple system already fit the requirements? • How do I generate a valid plot out of these results (Phase Portrait etc.)? ## Gravatar problem in WordPress I do not know what is happening but this is weird. When i log out and when i post comment in one of my posts,then i see gravatar image,near my comment. Why that image appear if i not have setup any gravatar image in my profile? And i am log out too. ## Efficient algorithm for this combinatorial problem [closed] $$\newcommand{\argmin}{\mathop{\mathrm{argmin}}\limits}$$ I am working on a combinatorial optimization problem and I need to figure out a way to solve the following equation. It naturally popped up in a method I chose to use in my assignment I was working on. Given a fixed set $$\Theta$$ with each element $$\in (0,1)$$ and total $$N$$ elements ($$N$$ is about 25), I need to find a permutation of elements in $$\Theta$$ such that $$\vec K = \argmin_{\vec k = Permutation(\Theta)} \sum_{i=1}^N t_i D(\mu_i\|\|k_i)$$ where $$\vec t, \vec \mu$$ are given vectors of length $$N$$ and $$D(p\|\|q)$$ is the KL Divergence of the bernoulli distributions with parameters $$p$$ and $$q$$ respectively. Further, all the $$N$$ elements of $$\vec t$$ sum to 1 and $$\vec \mu$$ has all elements in $$[0,1]$$. It is just impossible to go through all $$N!$$ permutations. A greedy type of algorithm which does not give exact $$\vec K$$ would also be acceptable to me if there is no other apparent method. Please let me know how to proceed! ## Problem with the algorithm I am trying to execute the following algorithm shown in the image. I am trying to get the table shown in the image: 1st Iteration L2: 1:CS=A, SL = A, NSL = A L3: while NSL!=[]: true L4: L6:no children: false L17:NSL =BCDA L18:CS:=B L19:SL=BA L20, L21 2nd Iteration L3:While NSL (true) L4: L6:no children: false L17: NSL=EFBCDA L18: CS:=E L19:SL:= EBA L20, L21 3rd Iteration L3:while NSL (true) L4: L6:no children: false L17:NSL= HIEFBCDA L18: CS:= H L19:SL:=HEBA L20, L21 At this point its fine but when there are no more children of current node, it has to backtrack, so it should execute the while loop, at that point I am losing the track: L3:while NSL(true) L4: L6:no children: true L7:begin L8:while SL is not empty (true) and CS:=H L9: DE=H L10:SL=EBA L11:NSL=IEFBCDA L12:CS=I L14:SL= IEBA Now it should keep traversing the while loop but I am having problem with this. Somebody please correct this algorithm or guide me a better backtracking algorithm which has the contents of table. Zulfi. ## Near identical MySQL deployments behaving very different – High CPU Usage problem So I have five identical websites, running on five machines provisioned in the same way. The only thing that differs between these installations are the language files and the languages of the text stored in MySQL tables. Four of them have no problems what so ever. One is struggling a LOT under the same or somewhat less load than the other four. I cannot understand why this is. Things I’ve done so far: 1. Checked slow queries. All queries uses indexes and are in the realm of 0.0008 Sec execution time i.e. very fast 2. I’ve noticed that the thing that causes most trouble for this MySQL instance is UPDATE and INSERT, so much so, I’ve turned off the UPDATE’s that were there, for this instance. Bear in mind that these UPDATE’s doesn’t cause a blip on the other servers. 3. Tried to eliminate external factors i.e. noisy neighbours (moved host) etc. Worth noticing is that the machines are deployed the same i.e. a vanilla Debian 10 installation with a LEMP stack, nothing out of the ordinary at all. Still, the problem persists. I can see the load of the machine struggling to keep under 1.00. The other machines are in the 0.10 – 0.20 range all the time. Looking at CPU for the MySQL process on this machine (with 2 CPU cores as the other machines have as well) it is quite often above 100%. The other machines are never – EVER – over 60% for the MySQL process. So, any help is much appreciated. Please do let me know if you need me to run a command that you need to see the output from in order to help. Thanks. EDIT Spelling and clarifications ## Interpreting the accuracy of solutions to the correspondence problem I have two pictures of the same object, taken by a car travelling down the road like shown on the right side of the image below. I want to find pixels of the object in each frame that correspond to each other. Now, in the description of the Middlebury Stereo Evaluation v.3 dataset it says Maximum disparities range from 200 to 800 pixels at full resolution. This leads me to my two questions: 1. Do I understand correctly that algorithms working with the Middlebury dataset had to match pixels that were a distance of 200 to 800 pixels apart, like shown on the left side of the image? 2. Consider the leaderboard for the Middlebury Stereo Evaluation. Does an average absolute error metric of 1.4 mean, that above problem could be solved for the images in the dataset with an average accuracy of 1.4 pixels? ## Subset sum problem with a complication I have a sorted list of numbers. I know they can be divided into two parts. I also know the sum of those 2 parts. I want to know what these subsets are. How can i find them? The size of my list can be ~ 10^6. The sum of my subsets can be ~ 10^24. Is there a better way to find my subset with a time complexity of O(nsum) or O(n^2 logn) or is it not possible? ## Problem with “does not evaluate to a numeric scalar at the coordinate {2.75,3/2}, but it is outside domain Problem with "does not evaluate to a numeric scalar at the coordinate {2.75,3/2}, but that point is outside domain. How can I fix it?? Remove["Global"] ; Needs["NDSolveFEM"] HeatTransferModelAxisymmetric[T_, {r_, z_}, k_, ρ_, Cp_,Velocity_, Source_] := Module[{V, Q}, V = If[Velocity === "NoFlow", 0, Velocity.Inactive[Grad][T, {r, z}]]; Q = If[Source === "NoSource", 0, Source];(1 - (r)^2 - ((1 - 0.5^2)/(Log[1/0.5]))Log[1/r])D[T, z] +1/rD[-krD[T, r], r] + D[-kD[T, z], z] + V - Q] op = HeatTransferModelAxisymmetric[T[r, z], {r, z}, k, ρ, Cp,"NoFlow", "NoSource"] parameters = {k -> 10, d -> 10}; Subscript[Γ, flux] = NeumannValue[40(500 - T[r, z]), r == 2.5]; Subscript[Γ, temp] = DirichletCondition[T[r, z] == 1200, r == 1]; Subscript[Γ, enter] = DirichletCondition[T[r, z] == 800, z == 0]; Ω = Rectangle[{1, 0}, {2.5, 3}]; pde = {op == Subscript[Γ, flux],Subscript[Γ, temp],Subscript[Γ, enter]} /. parameters; Tfun =NDSolveValue[pde, T, {r, z} ∈ Ω]; MassTransferModelAxisymmetric[c_, {r_, z_}, d_, Velocity_, Source_] :=Module[{V, Q},V = If[Velocity === "NoFlow", 0, Velocity.Inactive[Grad][c, {r, z}]];Q = If[Source === "NoSource", 0, Source];(1 - (r)^2 - ((1 - 0.5^2)/(Log[1/0.5]))Log[1/r])D[c, z] + 1/rD[-drD[c, r], r] + D[-dD[c, z], z] + 4.67^14Exp[-36635/(Tfun[r, z])]c + V - Q] op2 = MassTransferModelAxisymmetric[c[r, z], {r, z}, d, "NoFlow","NoSource"] Subscript[Γ, enter] = DirichletCondition[c[r, z] == 800, z == 0]; pde2 = {op2 == Subscript[Γ, enter]} /.parameters;cfun=NDSolveValue[pde2, c, {r, z} ∈ Ω]; ## Finding a valid equation for fixed point problem I currently am working on learning more about fixed point method. Finding equations that satisfy the constraints of a g function can sometimes require a bit of engineering. I have come across one that many would consider simple. Yet, I have been stuck on it for some time now. Here it is $$f(x) = x^2 – x – 2 = 0$$ on $$[1.5,3]$$. I have tried many things; however, I have yet to successfully discover one that maps domain to range for both $$g$$ and $$g^\prime$$. Would anyone be able to give me a guiding hand? ## Problem with return 2 libc in 64 bit arch Good day guys I want to perform return to libc in 64 bit architecture using execve. I found a gadget with /bin/sh in it (the /bin/sh offset is 18a143): cbcd8: 00 00 cbcda: 4c 89 ea mov rdx,r13 cbcdd: 4c 89 e6 mov rsi,r12 cbce0: 48 8d 3d 5c e4 0b 00 lea rdi,[rip+0xbe45c] # 18a143 <_libc_intl_domainname@@GLIBC_2.2.5+0x17e> cbce7: e8 94 f9 ff ff call cb680 <execve@@GLIBC_2.2.5> -- cbd92: 48 85 c0 test rax,rax	2020-09-23 15:32:59	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 22, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.47733044624328613, "perplexity": 2289.3665212638166}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400211096.40/warc/CC-MAIN-20200923144247-20200923174247-00633.warc.gz"}
https://zbmath.org/?q=an%3A1093.53044	Einstein metrics on spheres.(English)Zbl 1093.53044 In this interesting paper the authors demonstrate existence theorems for many families of Einstein metrics on spheres and exotic spheres. More precisely, they prove: (i) On $$S^5$$ there exist 68 inequivalent families of Sasakian-Einstein metrics. (ii) All 28 oriented diffeomorphism classes on $$S^7$$ admit inequivalent families of Sasakian-Einstein structures. (iii) For $$n\geq 2$$, the $$(4n+1)$$-dimensional standard and Kervaire spheres both admit many families of inequivalent Sasakian-Einstein metrics. Thus, the authors answer in an affirmative way the long standing open question about the existence of Einstein metrics on exotic spheres. The main steps of the proof are the following. For a sequence $${\mathbf a}= (a_1,\ldots,a_m)\in\mathbb Z^m_+$$ the authors consider the Brieskorn-Pham singularity $Y({\mathbf a}):=\left\{\sum_{i=1}^m z_i^{a_i}=0\right\}\subset\mathbb C^m\quad \text{ and\;its\;link} \;\;L({\mathbf a}):=Y({\mathbf a})\cap S^{2m-1}(1),$ which is a smooth, compact, $$(2m-3)$$-dimensional manifold. Moreover, $$Y({\mathbf a})$$ has a natural $$\mathbb C^\ast$$-action and $$L({\mathbf a})$$ a natural $$S^1$$- action. In the first step, basing on Kobayashi’s circle bundle construction, the authors observe that a positive Kähler-Einstein metric on the base space of a circle bundle gives an Einstein metric on the total space. This result was generalized to orbifolds by C. P. Boyer and K. Galicki [Int. J. Math. 11, No. 7, 873–909 (2000; Zbl 1022.53038)], giving Sasaki-Einstein metrics. Thus, a positive Kähler-Einstein orbifold metric on $$(Y({\mathbf a})\setminus\{0\})/\mathbb C^\ast$$ yields a Sasaki-Einstein metric on $$L({\mathbf a})$$. In the second step, for a sequence $$\mathbf a$$ satisfying certain numerical conditions, the authors use the continuity method developed by Aubin, Siu and Tian to construct Kähler-Einstein metrics on orbifolds. The final step is to get examples, partly through computer searches, partly through writing down well chosen sequences. The authors also formulate the conjecture that all odd-dimensional homotopy spheres which bound parallelizable manifolds admit Sasaki-Einstein metrics. MSC: 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 32Q20 Kähler-Einstein manifolds Zbl 1022.53038 Full Text:	2022-05-28 17:58:46	{"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": 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": 0.8419023752212524, "perplexity": 782.8094170536947}, "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/1652663016949.77/warc/CC-MAIN-20220528154416-20220528184416-00529.warc.gz"}
https://hal.archives-ouvertes.fr/hal-00923208	HAL will be down for maintenance from Friday, June 10 at 4pm through Monday, June 13 at 9am. More information # On the continuity of the eigenvalues of a sublaplacian Abstract : We study the behavior of the eigenvalues of a sublaplacian $\Delta_b$ on a compact strictly pseudoconvex CR manifold $M$, as functions on the set ${\mathcal P}_+$ of positively oriented contact forms on $M$ by endowing ${\mathcal P}_+$ with a natural metric topology. Keywords : Document type : Journal articles Cited literature [11 references] https://hal.archives-ouvertes.fr/hal-00923208 Contributor : Ahmad El Soufi Connect in order to contact the contributor Submitted on : Thursday, January 2, 2014 - 8:58:26 AM Last modification on : Friday, April 22, 2022 - 2:22:03 PM Long-term archiving on: : Saturday, April 8, 2017 - 9:28:18 AM ### Files ContinuitySpectrum.pdf Files produced by the author(s) ### Citation Amine Aribi, Sorin Dragomir, Ahmad El Soufi. On the continuity of the eigenvalues of a sublaplacian. Canadian Mathematical Bulletin, Cambridge University Press, 2014, 57 (1), pp.12--24. ⟨10.4153/CMB-2012-026-9⟩. ⟨hal-00923208⟩ Record views	2022-05-22 17:56:47	{"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": 0.17713914811611176, "perplexity": 3511.0714168688464}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1652662545875.39/warc/CC-MAIN-20220522160113-20220522190113-00389.warc.gz"}
https://en-academic.com/dic.nsf/enwiki/58939	# Small population size Small population size Populations with small population size behave differently to larger populations. Often this has various harmful consequences for the survival of that population. Demographic effects The influence of stochastic (random) variation in demographic (reproductive and mortality) rates is much higher for small populations than large ones. Stochastic variation in demographic rates causes small populations to fluctuate randomly in size. The smaller the population the greater the probability that fluctuations will lead to extinction. They are subject to a higher chance of extinction because they are more vulnerable to genetic drift, resulting in stochastic variation in their gene pool, their demography and their environment. One demographic consequence of a small population size, the probability that all offspring in a generation are of the same sex, and where males and females are equally likely to be produced (see sex ratio), is easy to calculate: it is given by $1/2^\left\{n-1\right\}$ (The chance of all animals being females is $1/2^n$; the same holds for all males, thus this result). This can be a problem in very small populations. In 1977, the last 18 Kakapo on a Fiordland island in New Zealand were all male, though the probability of this was only 0.0000076. With a population of just 3 individuals the probability of them all being the same sex is 0.25. Put another way, for every 4 species reduced to 3 individuals (or more precisely 3 individuals in the effective population), one will go extinct within one generation just because they are all the same sex. If the population remains at this size for several generations, such an event becomes almost inevitable. Environmental effects Stochastic variation in the environment (year to year variation in rainfall, temperature) can produce temporally correlated birth and death rates (i.e. 'good' years when birth rates are high and death rates are low and 'bad' years when birth rates are low and death rates are high) that lead to fluctuations in the population size. Again, smaller populations are more likely to go extinct due to these environmentally generated population fluctuations than are large populations. Genetic consequences Conservationists are often worried about a loss of genetic variation in small populations. There are two types of genetic variation that are important when dealing with small populations. The degree of homozygosity within individuals in a population; i.e. the proportion of an individual's loci that contain homozygous rather than heterozygous alleles. Many deleterious alleles are only harmful in the homozygous form. The degree of monomorphism/polymorphism within a population; i.e. how many different forms of the same allele exist in the gene pool of a population. Polymorphism may be particularly important at loci involved in the immune response. There are two mechanisms operating in small populations that influence these two types of genetic variation. Genetic drift - Genetic variation is determined by the joint action of natural selection and genetic drift (chance). In small populations the relative importance of genetic drift (chance) is higher; deleterious alleles can become more frequent and 'fixed' in a population due to chance. Any allele, deleterious, beneficial or neutral is more likely to be lost from a small population (gene pool) than a large one. This results in a reduction in the number of forms of alleles in a small population and in extreme cases to monomorphism where there is only one form of the allele. Continued fixation of deleterious alleles in small populations is called Muller's ratchet, and can lead to mutational meltdown. Inbreeding - In a small population, related individuals are more likely to breed together. The offspring of related parents have a far higher number of homozygous loci than the offspring of unrelated parents. There are two types of potential consequence of loss of genetic variation: Inbreeding depression - Inbreeding depression is usually taken to mean any immediate harmful effect, on individuals or the population, of a decrease in either type of genetic variation. Inbreeding depression can almost never be found in declining populations that were not very large to begin with; it is somewhat common in large populations "becoming" small though. The reason is purging selection, most efficient in populations that are strongly but not dangerously inbred. The ability of the population to adapt/evolve to changing conditions, “without variability evolution is impossible”who. It is obvious that the absolute size of a population limits the absolute degree of allelic diversity. On the other hand, should an advantageous mutation arise, it is likely to show its effect sooner and more thoroughly. The effective population size is commonly lower than the actual population size. ee also Critical depensation Decline in amphibian populations Founder effect Minimum viable population Muller's ratchet Mutational meltdown Pollinator decline Population genetics Gene pool Genetic pollution Genetic erosion Wikimedia Foundation. 2010. ### Look at other dictionaries: • Population size — In population genetics and population ecology, population size (usually denoted N ) is the number of individual organisms in a population.The effective population size (Ne) is defined as the number of breeding individuals in an idealized… … Wikipedia • Effective population size — In population genetics, the concept of effective population size N e was introduced by the American geneticist Sewall Wright, who wrote two landmark papers on it (Wright 1931, 1938). He defined it as the number of breeding individuals in an… … Wikipedia • Population bottleneck — followed by recovery or extinction A population bottleneck (or genetic bottleneck) is an evolutionary event in which a significant percentage of a population or species is killed or otherwise prevented from reproducing.[1] A slightly different… … Wikipedia • Population density — (people per km2) by country, 2006 … Wikipedia • Population genetics — is the study of the allele frequency distribution and change under the influence of the four evolutionary forces: natural selection, genetic drift, mutation and gene flow. It also takes account of population subdivision and population structure… … Wikipedia • Population dynamics of fisheries — A fishery is an area with an associated fish or aquatic population which is harvested for its commercial or recreational value. Fisheries can be wild or farmed. Population dynamics describes the ways in which a given population grows and shrinks… … Wikipedia • Small-toothed sportive lemur — Conservation status Data Deficient … Wikipedia • Population control — is the practice of limiting population increase, usually by reducing the birth rate. The practice has sometimes been voluntary, as a response to poverty, environmental concerns, or out of religious ideology, but in some times and places it has… … Wikipedia • small-scale — ˈsmall scale adjective small in size or limited in degree: • small scale enterprises • small scale industrial activities * * small scale UK US /ˌsmɔːlˈskeɪl/ adjective [before noun] ► not involving a lot of people, things, or activity, or… … Financial and business terms • population ecology — Introduction study of the processes that affect the distribution and abundance of animal and plant populations. A population is a subset of individuals of one species that occupies a particular geographic area and, in sexually… … Universalium	2023-02-07 01:08:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 2, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.644188404083252, "perplexity": 2503.186411333977}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1674764500368.7/warc/CC-MAIN-20230207004322-20230207034322-00309.warc.gz"}
https://socratic.org/questions/how-do-you-expand-7-4b-2	# How do you expand 7(4b-2)? $28 b - 14$ Distribute $7$ across each term in parenthesis: $\left(7 \cdot 4 b\right) - \left(7 \cdot 2\right) \implies$ $28 b - 14$	2020-02-29 02:08:00	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 4, "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": 0.6363619565963745, "perplexity": 4332.911166017443}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875148163.71/warc/CC-MAIN-20200228231614-20200229021614-00412.warc.gz"}
https://www.samabbott.co.uk/thesis/6-beneficial-bcg-out.html	# Chapter 6 Exploring the effects of BCG vaccination in patients diagnosed with tuberculosis: observational study using the Enhanced Tuberculosis Surveillance system ## 6.1 Introduction Bacillus Calmette–Guérin (BCG) primarily reduces the progression from infection to disease, however there is evidence that BCG may provide additional benefits. In this chapter I aimed to investigate whether there is evidence in routinely-collected surveillance data (see Chapter 4) that BCG vaccination impacts outcomes for tuberculosis (TB) cases in England. Any impact on TB outcomes could add additional weight to vaccination policies with wider population coverage, as these policies would have benefits beyond reducing TB incidence rates. To conduct this study, I first obtained all TB notifications for 2009-2015 in England from the Enhanced Tuberculosis surveillance (ETS) system (see Chapter 4). I then considered five outcomes: All-cause mortality, death due to TB (in those who died), recurrent TB, pulmonary disease, and sputum smear status. I used logistic regression, with complete case analysis, to investigate each outcome with BCG vaccination, years since vaccination and age at vaccination, adjusting for potential confounders. All analyses were repeated using multiply imputed data. This work was adapted from [77]20 (also available as a preprint21) supervised by Hannah Christensen and Ellen Brooks-Pollock. Collaborators at Public Health England including Maeve K Lalor, Dominik Zenner, Colin Campbell, and Mary E Ramsay provided the data and commented on multiple versions of this paper. ## 6.2 Background Bacillus Calmette–Guérin (BCG) is one of the mostly widely-used vaccines and the only vaccine that protects against TB disease. BCG was first used in humans in 1921 and was introduced into the WHO Expanded Program on Immunization in 1974.[38] BCG vaccination has been controversial due to its variable efficacy and possibility of causing a false positive result with the standard skin test for TB.[5] However, the lack of a more effective vaccine and the emergence of drug-resistant TB strains means that BCG vaccination remains an important tool for reducing TB incidence and mortality rates. BCG’s primary mode of action is to directly prevent the development of active, symptomatic disease. Its efficacy in adults is context specific, with estimates ranging between 0% and 78% (see Chapter 2).[25] It has been shown to highly efficacious in England and there is some evidence that efficacy increases with distance from the equator. Efficacy has been shown to be dependent on previous exposure, with unexposed individuals receiving the greatest benefit.[69] Unlike in adults, BCG has consistently been shown to be highly protective against TB and TB meningitis in children.[23,24] For this reason the majority of countries that use BCG, vaccinate at birth.[27] Adult vaccination is no longer common in the UK, where universal BCG vaccination of adolescents was stopped in 2005 in favour of a targeted neonatal programme aimed at high risk children. Vaccination policy has been primarily based on reducing the incidence of TB disease, and mitigating disease severity, with little attention having been given to any additional effects of BCG vaccination on TB outcomes.[30,31] There is some evidence that BCG vaccination induces innate immune responses which may provide non-specific protection,[32] TB patients with BCG scars were found to respond better to treatment with earlier sputum smear conversion,[36] and there is evidence to suggest that BCG vaccination is associated with reduced all-cause neonatal mortality[33,34] and both reduced TB[28] and all-cause[35] mortality in the general population. Given that the immunology behind TB immunity is not fully understood these findings suggest that BCG may play a more important role in improving TB outcomes than previously thought. I aimed to quantify the effects of BCG vaccination on outcomes for individuals with notified TB in England using routinely collected surveillance data (see Chapter 4) to provide evidence for appropriate public health action and provision. Where I found an association, I additionally explored the role of years since vaccination, and age at vaccination. ## 6.3 Method ### 6.3.1 Enhanced TB Surveillance (ETS) system I extracted all notifications from the ETS system from January 1, 2009 to December 31, 2015 (Chapter 4). BCG vaccination status and year of vaccination have been collected since 2008. The outcomes I considered were: all-cause mortality, death due to TB (in those who died), recurrent TB, pulmonary disease, and sputum smear status. These outcomes were selected based on: their availability in the ETS; evidence from the literature of prior associations with BCG vaccination; associations with increased case infectiousness; or severe outcomes for patients. All-cause mortality was defined using the overall outcome recorded in ETS, this is based on up to 36 months of follow up starting from date of starting treatment. Follow up ends when a case is recorded as completing treatment, with treatment status evaluated at 12, 24, and 36 months from starting treatment. Where the treatment start date was not available the notification date was used if appropriate. The date of death was validated against Office for National Statistics (ONS) data. Those that were lost to follow up, or not evaluated were treated as missing. In cases with a known cause of death, death due to TB was defined as those that died from TB, or where TB had contributed to their death. Cause of death was recorded by case managers. TB cases who had recurrent episodes were identified using probabilistic matching. Positive sputum smear status was given to cases that had a sputum sample shown to contain Acid-Fast Bacilli. A positive sputum smear status indicates that cases are more likely to be infectious. Cases were defined as having pulmonary TB if a positive sputum smear sample was recorded, if a positive culture was grown from a pulmonary laboratory specimen, or if they were clinically assessed as having pulmonary TB. ### 6.3.2 Exposure variables relating to BCG I included three exposure variables related to BCG: BCG status (vaccinated, yes/no), years since vaccination and age at vaccination. BCG status was collected and recorded in ETS by case managers. Information on BCG vaccination status may have come from vaccination records, patient recall or the presence of a scar. When cases are uncertain, and there is no evidence of a scar, no BCG status is given. Year of vaccination was collected similarly. Years since BCG vaccination was defined as year of notification minus year of vaccination and categorised into two groups (0 to 10 and 11+ years). This was based on: evidence that the average duration of BCG protection is at least 10-15 years;[28] increasing recall bias with time since vaccination, and any association between years since vaccination and TB outcomes may be non-linear (see Chapter 4). I calculated age at vaccination as year of vaccination minus year of birth. I categorized age at vaccination into $$0$$ to $$< 1$$, $$1$$ to $$< 12$$, $$12$$ to $$< 16$$ and 16+ years because the distribution was bimodel with modes at 0 and 12 years. This categorization captures the current UK policy of vaccination at birth, historic policy of vaccination at 13-15 years and catch up vaccination for high risk children. ### 6.3.3 Statistical Analysis R was used for all statistical analysis.[56] The analysis was conducted in two stages. Firstly, I calculated proportions for all demographic and outcome variables, and compared vaccinated and unvaccinated TB cases using the $$\chi ^2$$ test. Secondly, I used logistic regression, with complete case analysis, to estimate the association between exposures and outcome variables, both with and without adjustment for confounders. In the multivariable models, I adjusted for sex,[7880] age,[81] Index of Multiple Deprivation (2010) categorised into five groups for England (IMD rank),[15] ethnicity,[78,82] UK birth status,[45,83] and year of notification. As the relationship between age and outcomes was non-linear, I modelled age using a natural cubic spline with knots at the 25%, 50% and 75% quantiles. I conducted sensitivity analyses to assess the robustness of the results, by dropping each confounding variable in turn and assessing the effect on the adjusted Odds Ratios (aORs) of the exposure variable. I repeated the analysis excluding duplicate recurrent cases, and restricting the study population to those eligible for the BCG schools scheme (defined as UK born cases that were aged 14 or over in 2004) to assess the comparability of the BCG vaccinated and unvaccinated populations. To mitigate the impact of missing data I used multiple imputation, with the MICE package.[51] I imputed 50 data sets (for 20 iterations) using all outcome and explanatory variables included in the analysis as predictors along with Public Health England centre. The model results were pooled using the small sample method,[84] and effect sizes compared with those from the main analysis. All code for this analysis is available online22. ## 6.4 Results ### 6.4.1 Description of the data There were 51,645 TB notifications between 2009-2015 in England. Reporting of vaccination status and year of vaccination improved over time: 64.9% (20865/32154) of notifications included vaccination status for 2009 to 2012, increasing to 70% (13647/19491) from 2013 to 2015. The majority of cases that had a known vaccination status were vaccinated (70.6%, 24354/34512), and where age and year of vaccination was known, the majority of cases were vaccinated at birth (60%, 5979/10066). Vaccinated cases were younger than unvaccinated cases on average (median age 34 years (IQR 26 to 45) compared to 38 years (IQR 26 to 62)). A higher proportion of non-UK born cases were BCG vaccinated, (72.7%, 18297/25171) compared to UK born cases (65.2%, 5787/8871, P: < 0.001) and, of those vaccinated, a higher proportion of non-UK born cases were vaccinated at birth compared to UK born cases (68%, 4691/6896 vs. 40.5%, 1253/3096 respectively, P: < 0.001). See Table 6.1 for the breakdown of outcome variables and Table 6.2 for the breakdown of confounding variables. See Chapter 4 for an extended discussion of the epidemiology of TB in England. Table 6.1: Outcomes for individuals in England notified with TB between 2009-2015, stratified by BCG vaccination status. BCG status Outcome Total Vaccinated Unvaccinated Unknown vaccine status Total, all cases 51645 24354 {47} 10158 {20} 17133 {33} All-cause mortality 45588 (88) 21685 (89) 9061 (89) 14842 (87) No 43024 [94] 21291 [98] 8495 [94] 13238 [89] Yes 2564 [6] 394 [2] 566 [6] 1604 [11] Death due to TB (in those who died) 1373 (3) 276 (1) 320 (3) 777 (5) No 572 [42] 129 [47] 146 [46] 297 [38] Yes 801 [58] 147 [53] 174 [54] 480 [62] Recurrent TB 48497 (94) 23963 (98) 9991 (98) 14543 (85) No 44869 [93] 22592 [94] 9256 [93] 13021 [90] Yes 3628 [7] 1371 [6] 735 [7] 1522 [10] Pulmonary TB 51432 (100) 24289 (100) 10121 (100) 17022 (99) Extra-pulmonary (EP) only 24280 [47] 12085 [50] 4573 [45] 7622 [45] Pulmonary, with or without EP 27152 [53] 12204 [50] 5548 [55] 9400 [55] Sputum smear status - positive 19551 (38) 9768 (40) 3910 (38) 5873 (34) Negative 11060 [57] 5694 [58] 2231 [57] 3135 [53] Positive 8491 [43] 4074 [42] 1679 [43] 2738 [47] {% all cases}(% complete within vaccine status)[% complete within category] Death due to TB in those who died and where cause of death was known Table 6.2: Potential confounders for individuals in England notified with TB between 2009-2015, stratified by BCG vaccination status. BCG status Confounder Total Vaccinated Unvaccinated Unknown vaccine status Total, all cases 51645 24354 {47} 10158 {20} 17133 {33} Age 51645 (100) 24354 (100) 10158 (100) 17133 (100) Mean [SD] 40 [19] 36 [16] 44 [22] 45 [20] Median [25%, 75%] 36 [27, 52] 34 [26, 45] 38 [26, 62] 41 [29, 59] Sex 51535 (100) 24320 (100) 10136 (100) 17079 (100) Female 22066 [43] 10791 [44] 4312 [43] 6963 [41] Male 29469 [57] 13529 [56] 5824 [57] 10116 [59] IMD rank (with 1 as most deprived and 5 as least deprived) 43525 (84) 21240 (87) 8866 (87) 13419 (78) 1 16800 [39] 7779 [37] 3665 [41] 5356 [40] 2 13057 [30] 6836 [32] 2564 [29] 3657 [27] 3 6838 [16] 3459 [16] 1259 [14] 2120 [16] 4 4045 [9] 1893 [9] 836 [9] 1316 [10] 5 2785 [6] 1273 [6] 542 [6] 970 [7] UK born 49820 (96) 24084 (99) 9958 (98) 15778 (92) Non-UK Born 36988 [74] 18297 [76] 6874 [69] 11817 [75] UK Born 12832 [26] 5787 [24] 3084 [31] 3961 [25] Ethnic group 50416 (98) 24074 (99) 10024 (99) 16318 (95) White 10194 [20] 3560 [15] 2695 [27] 3939 [24] Black-Caribbean 1112 [2] 559 [2] 242 [2] 311 [2] Black-African 8942 [18] 4620 [19] 1602 [16] 2720 [17] Black-Other 462 [1] 261 [1] 80 [1] 121 [1] Indian 12994 [26] 7176 [30] 2061 [21] 3757 [23] Pakistani 8237 [16] 3512 [15] 1720 [17] 3005 [18] Bangladeshi 2025 [4] 918 [4] 480 [5] 627 [4] Chinese 601 [1] 289 [1] 101 [1] 211 [1] Mixed / Other 5849 [12] 3179 [13] 1043 [10] 1627 [10] Calendar year 51645 (100) 24354 (100) 10158 (100) 17133 (100) {% all cases}(% complete within vaccine status)[% complete within category] * Death due to TB in those who died and where cause of death was known ### 6.4.2 All-cause mortality In the univariable analysis the odds of death from any cause were lower for BCG vaccinated TB cases compared to unvaccinated cases, with an OR of 0.28 (95% CI 0.24 to 0.32, P: <0.001) (Table 6.3, Table 6.4); an association remained after adjusting for confounders, but was attenuated with an aOR of 0.76 (95% CI 0.64 to 0.89, P: 0.001). I estimate that if all unvaccinated cases had been vaccinated there would have been on average 19 (95% CI 9 to 29) fewer deaths per year during the study period (out of 81 deaths per year on average in unvaccinated cases). Whilst there was evidence in univariable analyses to suggest all-cause mortality was higher in persons vaccinated more than 10 years prior to notification of TB and that all-cause mortality increased with increasing age group, these disappeared after adjusting for potential confounders (Table 6.5, Table 6.6). Similar results to the multivariable analysis were found using multiply imputed data for the association between vaccination status and all-cause mortality (aOR: 0.76 (95% CI 0.61 to 0.94), P: 0.013), but not for time since vaccination with a greatly increased risk of all-cause mortality estimated for those vaccinated more than 10 years before case notification, compared to those vaccinated more recently (aOR: 12.19 (95% CI 3.48 to 42.64), (see Table 6.5, Table 6.7)). For age at vaccination results for the multivariable analysis using multiply imputed data were comparable to those found using complete case analysis, except that there was some evidence that vaccination in adolescence, compared to under 1, was associated with increased, rather than decreased, all-cause mortality (aOR: 1.57 (95% CI 1.13 to 2.19), Table 6.9). Table 6.3: Summary of logistic regression model output with BCG vaccination as the exposure and all-cause mortality as the outcome. Univariable Multivariable Variable Total All-cause mortality OR (95% CI) P-value aOR (95% CI) P-value Total cases 25993 807 (3) BCG vaccination <0.001 0.001 No 7620 473 (6) 1 1 Yes 18373 334 (2) 0.28 (0.24 to 0.32) 0.76 (0.64 to 0.89) Age <0.001 <0.001 Sex <0.001 <0.001 Female 11502 296 (3) 1 1 Male 14491 511 (4) 1.45 (1.34 to 1.58) 1.48 (1.26 to 1.73) IMD rank (with 1 as most deprived and 5 as least deprived) <0.001 0.001 1 9891 298 (3) 1 1 2 8136 219 (3) 0.85 (0.76 to 0.95) 0.86 (0.70 to 1.04) 3 4100 120 (3) 1.06 (0.93 to 1.20) 0.66 (0.52 to 0.84) 4 2341 98 (4) 1.47 (1.28 to 1.70) 0.72 (0.55 to 0.93) 5 1525 72 (5) 1.70 (1.45 to 1.99) 0.64 (0.47 to 0.85) UK born <0.001 0.136 Non-UK Born 19115 442 (2) 1 1 UK Born 6878 365 (5) 2.62 (2.40 to 2.85) 1.25 (0.93 to 1.67) Ethnic group <0.001 0.171 White 4699 380 (8) 1 1 Black-Caribbean 634 25 (4) 0.45 (0.35 to 0.58) 0.95 (0.59 to 1.53) Black-African 4681 62 (1) 0.14 (0.12 to 0.17) 0.87 (0.59 to 1.29) Black-Other 247 2 (1) 0.13 (0.06 to 0.26) 0.40 (0.10 to 1.69) Indian 7041 168 (2) 0.28 (0.25 to 0.31) 0.80 (0.58 to 1.10) Pakistani 4067 103 (3) 0.30 (0.27 to 0.34) 0.65 (0.46 to 0.92) Bangladeshi 1079 18 (2) 0.21 (0.16 to 0.27) 0.69 (0.40 to 1.22) Chinese 286 7 (2) 0.34 (0.23 to 0.51) 0.69 (0.30 to 1.62) Mixed / Other 3259 42 (1) 0.16 (0.13 to 0.19) 0.59 (0.39 to 0.91) Calendar year 1.06 (1.04 to 1.08) <0.001 1.10 (1.05 to 1.15) <0.001 OR (95% CI): unadjusted odds ratio with 95% confidence intervals, aOR (95% CI): adjusted odds ratios with 95% confidence intervals ### 6.4.3 Deaths due to TB (in those who died) There was little evidence of any association between BCG vaccination and deaths due to TB (in those who died and where cause of death was known) in the univariable analysis (Table 6.4). The adjusted point estimate indicated an association between BCG vaccination and reduced deaths due to TB (in those who died) although the confidence intervals remained wide with a similar result found using multiply imputed data (see Table 6.7). There were insufficient data to robustly estimate an association between deaths due to TB (in those who died) and years since vaccination or age at vaccination (Table 6.5, Table 6.6). ### 6.4.4 Recurrent TB In both the univariable and multivariable analysis there was some evidence that BCG vaccination was associated with reduced recurrent TB, although the strength of the evidence was weakened after adjusting for confounders (Table 6.4). In the adjusted analysis, the odds of recurrent TB were lower for BCG vaccinated cases compared to unvaccinated cases, with an aOR of 0.90 (95% CI 0.81 to 1.00, P: 0.056). The strength of the evidence for this association was comparable in the analysis using multiply imputed data (see Table 6.7). There was little evidence in the adjusted analysis of any association between recurrent TB and years since vaccination (Table 6.5) or age at vaccination (Table 6.6). ### 6.4.5 Other Outcomes After adjusting for confounders there was little evidence for any association between BCG vaccination and pulmonary disease or positive sputum smear status (Table 6.4); similar results were found using multiply imputed data (see Table 6.7). Table 6.4: Summary of associations between BCG vaccination and all outcomes Univariable Multivariable Outcome BCG vaccinated Cases Cases with outcome (%) OR (95% CI) P-value Cases* Cases with outcome (%) aOR (95% CI) P-value All-cause mortality No 9061 566 (6) 1 <0.001 7620 473 (6) 1 0.001 Yes 21685 394 (2) 0.28 (0.24 to 0.32) 18373 334 (2) 0.76 (0.64 to 0.89) Death due to TB (in those who died) No 320 174 (54) 1 0.786 270 143 (53) 1 0.177 Yes 276 147 (53) 0.96 (0.69 to 1.32) 236 126 (53) 0.76 (0.51 to 1.13) Recurrent TB No 9991 735 (7) 1 <0.001 8502 615 (7) 1 0.056 Yes 23963 1371 (6) 0.76 (0.70 to 0.84) 20584 1177 (6) 0.90 (0.81 to 1.00) Pulmonary TB No 10121 5548 (55) 1 <0.001 8595 4685 (55) 1 0.769 Yes 24289 12204 (50) 0.83 (0.79 to 0.87) 20784 10342 (50) 0.99 (0.94 to 1.05) Sputum smear status - positive No 3910 1679 (43) 1 0.187 3367 1435 (43) 1 0.730 Yes 9768 4074 (42) 0.95 (0.88 to 1.02) 8351 3447 (41) 1.02 (0.93 to 1.11) OR (95% CI): unadjusted odds ratio with 95% confidence intervals aOR (95% CI): adjusted odds ratios with 95% confidence intervals Death due to TB in those who died and where cause of death was known Univariable sample size for outcomes ordered as in table (% of all cases) = 30746 (60%), 596 (23%), 33954 (66%), 34410 (67%), 13678 (26%) * Multivariable sample size with outcomes ordered as in table (% of all cases) = 25993 (50%), 506 (20%), 29086 (56%), 29379 (57%), 11718 (23%) Table 6.5: Summary of associations between years since vaccination and all outcomes in individuals who were vaccinated. The baseline exposure is vaccination $$\leq 10$$ years before diagnosis compared to vaccination $$11+$$ years before diagnosis. Deaths due to TB (in those who died) had insufficient data for effect sizes to be estimated in both the univariable and multivariable analysis Univariable Multivariable Outcome Years since BCG Cases Cases with outcome (%) OR (95% CI) P-value Cases* Cases with outcome (%) aOR (95% CI) P-value All-cause mortality $$\leq$$ 10 718 5 (1) 1 0.004 554 4 (1) 1 0.897 11+ 8106 166 (2) 2.98 (1.22 to 7.28) 7171 148 (2) 0.91 (0.24 to 3.54) Death due to TB (in those who died) $$\leq$$ 10 2 2 (100) 1 2 2 (100) 1 11+ 108 59 (55) $$\textit{Insufficient data}$$ 98 53 (54) $$\textit{Insufficient data}$$ Recurrent TB $$\leq$$ 10 780 22 (3) 1 0.005 613 14 (2) 1 0.515 11+ 9172 451 (5) 1.78 (1.15 to 2.75) 8194 406 (5) 1.24 (0.63 to 2.44) Pulmonary TB $$\leq$$ 10 770 480 (62) 1 <0.001 601 382 (64) 1 0.309 11+ 9248 4757 (51) 0.64 (0.55 to 0.74) 8254 4232 (51) 0.87 (0.67 to 1.14) Sputum smear status - positive $$\leq$$ 10 157 81 (52) 1 0.941 122 61 (50) 1 0.920 11+ 3064 1590 (52) 1.01 (0.73 to 1.40) 2734 1405 (51) 1.02 (0.68 to 1.54) OR (95% CI): unadjusted odds ratio with 95% confidence intervals aOR (95% CI): adjusted odds ratios with 95% confidence intervals Death due to TB in those who died and where cause of death was known Univariable sample size for outcomes ordered as in table (% of vaccinated cases) = 8824 (36%), 110 (28%), 9952 (41%), 10018 (41%), 3221 (13%) * Multivariable sample size with outcomes ordered as in table (% of vaccinated cases) = 7725 (32%), 100 (25%), 8807 (36%), 8855 (36%), 2856 (12%) Table 6.6: Summary of associations between age at vaccination and all outcomes in individuals who were vaccinated - the baseline exposure is vaccination at birth compared to vaccination from 1 to < 12, 12 to < 16, and 16+ years of age. Univariable Multivariable Outcome Age at BCG Cases Cases with outcome (%) OR (95% CI) P-value Cases* Cases with outcome (%) aOR (95% CI) P-value All-cause mortality < 1 5234 45 (1) 1 <0.001 4626 43 (1) 1 0.127 1 to < 12 1915 58 (3) 3.60 (2.43 to 5.34) 1678 52 (3) 1.36 (0.85 to 2.16) 12 to < 16 1267 41 (3) 3.86 (2.51 to 5.91) 1094 32 (3) 0.81 (0.45 to 1.46) $$\geq$$ 16 408 27 (7) 8.17 (5.01 to 13.32) 327 25 (8) 1.41 (0.76 to 2.63) Death due to TB (in those who died) < 1 27 20 (74) 1 0.118 27 20 (74) 1 0.543 1 to < 12 43 20 (47) 0.30 (0.11 to 0.87) 39 18 (46) 0.36 (0.08 to 1.51) 12 to < 16 23 13 (57) 0.46 (0.14 to 1.50) 17 9 (53) 0.40 (0.06 to 2.52) $$\geq$$ 16 17 8 (47) 0.31 (0.09 to 1.12) 17 8 (47) 0.35 (0.06 to 2.16) Recurrent TB < 1 5909 284 (5) 1 0.463 5275 258 (5) 1 0.246 1 to < 12 2174 105 (5) 1.01 (0.80 to 1.26) 1928 92 (5) 0.84 (0.65 to 1.09) 12 to < 16 1421 58 (4) 0.84 (0.63 to 1.12) 1242 51 (4) 0.70 (0.48 to 1.02) $$\geq$$ 16 448 26 (6) 1.22 (0.81 to 1.85) 362 19 (5) 0.82 (0.49 to 1.37) Pulmonary TB < 1 5946 2828 (48) 1 <0.001 5305 2510 (47) 1 0.005 1 to < 12 2194 1159 (53) 1.23 (1.12 to 1.36) 1941 1033 (53) 1.15 (1.02 to 1.29) 12 to < 16 1425 971 (68) 2.36 (2.09 to 2.67) 1245 846 (68) 1.09 (0.92 to 1.29) $$\geq$$ 16 453 279 (62) 1.77 (1.45 to 2.15) 364 225 (62) 1.47 (1.15 to 1.88) Sputum smear status - positive < 1 1753 836 (48) 1 <0.001 1557 742 (48) 1 0.862 1 to < 12 755 394 (52) 1.20 (1.01 to 1.42) 682 348 (51) 0.96 (0.79 to 1.17) 12 to < 16 556 357 (64) 1.97 (1.62 to 2.40) 486 308 (63) 1.06 (0.81 to 1.39) $$\geq$$ 16 157 84 (54) 1.26 (0.91 to 1.75) 131 68 (52) 0.93 (0.63 to 1.37) OR (95% CI): unadjusted odds ratio with 95% confidence intervals aOR (95% CI): adjusted odds ratios with 95% confidence intervals Death due to TB in those who died and where cause of death was known Univariable sample size for outcomes ordered as in table (% of vaccinated cases) = 8824 (36%), 110 (28%), 9952 (41%), 10018 (41%), 3221 (13%) * Multivariable sample size with outcomes ordered as in table (% of vaccinated cases) = 7725 (32%), 100 (25%), 8807 (36%), 8855 (36%), 2856 (12%) ### 6.4.6 Sensitivity analysis of the missing data using multiple imputation As discussed in the previous sections, I found that repeating the analysis with an imputed data set had some effect on the results from the complete case analysis. There was a decrease in the accuracy of effect size estimates for BCG vaccination, some increase in p-values (Table 6.7). However, none of the estimated effects changed their direction, and there were no detectable systematic changes in the results. For the secondary exposure variables (years since vaccination and age at vaccination, (Table 6.8 and Table 6.9), I found a change in direction of the point estimate between years since vaccination and all-cause mortality and recurrent TB, but similar results for age at vaccination and outcomes. Table 6.7: Summary of associations between BCG vaccination and all outcomes, using pooled imputed data. Univariable Multivariable Outcome OR (95% CI) P-value fmi aOR (95% CI) P-value fmi All-cause mortality 0.44 (0.35 to 0.56) <0.001 90 0.76 (0.61 to 0.94) 0.013 85 Death due to TB (in those who died) 0.94 (0.57 to 1.56) 0.810 85 0.89 (0.52 to 1.51) 0.651 85 Recurrent TB 0.83 (0.75 to 0.92) <0.001 56 0.90 (0.81 to 1.00) 0.058 54 Pulmonary TB 0.84 (0.79 to 0.90) <0.001 70 0.99 (0.93 to 1.06) 0.814 62 Sputum smear status - positive 0.88 (0.82 to 0.94) <0.001 65 1.01 (0.94 to 1.08) 0.886 60 OR: odds ratio with 95% confidence intervals aOR: adjusted odds ratio with 95% confidence intervals fmi: fraction of missing information Death due to TB in those who died and where cause of death was known Table 6.8: Summary of associations between years since vaccination and all outcomes, using pooled imputed data. There was insufficient data to estimate an effect for deaths due to TB (in those who died) Univariable Multivariable Outcome OR (95% CI) P-value fmi aOR (95% CI) P-value fmi All-cause mortality 3.28 (1.85 to 5.79) <0.001 50 12.19 (3.48 to 42.64) <0.001 70 Death due to TB (in those who died) 0.00 (0.00 to Inf) 0.974 0 0.00 (0.00 to Inf) 0.972 0 Recurrent TB 1.29 (1.00 to 1.66) 0.050 39 0.81 (0.59 to 1.11) 0.187 44 Pulmonary TB 0.58 (0.52 to 0.66) <0.001 33 0.99 (0.84 to 1.17) 0.913 40 Sputum smear status - positive 0.99 (0.82 to 1.19) 0.891 70 0.95 (0.77 to 1.18) 0.648 60 OR: odds ratio with 95% confidence intervals aOR: adjusted odds ratio with 95% confidence intervals fmi: fraction of missing information Death due to TB in those who died and where cause of death was known Table 6.9: Summary of associations between age at vaccination and all outcomes, using pooled imputed data (reference is vaccination at <1 year). Univariable Multivariable Outcome Age group OR (95% CI) P-value fmi aOR (95% CI) P-value fmi All-cause mortality 1 to < 12 6.48 (4.71 to 8.91) <0.001 70 1.69 (1.18 to 2.40) 0.004 68 12 to < 16 3.33 (2.50 to 4.43) <0.001 78 1.57 (1.13 to 2.19) 0.008 79 $$\geq$$ 16 3.36 (2.56 to 4.41) <0.001 69 1.01 (0.70 to 1.46) 0.948 71 Death due to TB (in those who died) 1 to < 12 0.45 (0.22 to 0.92) 0.028 62 0.47 (0.21 to 1.04) 0.063 62 12 to < 16 0.41 (0.22 to 0.75) 0.004 67 0.40 (0.20 to 0.78) 0.008 67 $$\geq$$ 16 0.53 (0.28 to 1.00) 0.051 54 0.47 (0.20 to 1.12) 0.088 62 Recurrent TB 1 to < 12 1.39 (1.11 to 1.73) 0.004 41 1.04 (0.82 to 1.32) 0.736 41 12 to < 16 1.01 (0.88 to 1.16) 0.892 45 0.86 (0.75 to 1.00) 0.052 44 $$\geq$$ 16 0.95 (0.79 to 1.15) 0.598 53 0.77 (0.61 to 0.98) 0.034 55 Pulmonary TB 1 to < 12 1.83 (1.59 to 2.10) <0.001 46 1.36 (1.17 to 1.58) <0.001 44 12 to < 16 1.28 (1.19 to 1.36) <0.001 35 1.12 (1.04 to 1.21) 0.002 36 $$\geq$$ 16 2.28 (2.10 to 2.48) <0.001 34 1.10 (0.98 to 1.23) 0.107 40 Sputum smear status - positive 1 to < 12 1.49 (1.21 to 1.84) <0.001 74 1.08 (0.85 to 1.37) 0.549 76 12 to < 16 1.29 (1.17 to 1.43) <0.001 65 1.09 (0.97 to 1.22) 0.158 67 $$\geq$$ 16 2.40 (2.16 to 2.66) <0.001 58 1.20 (1.04 to 1.37) 0.011 59 OR: odds ratio with 95% confidence intervals aOR: adjusted odds ratio with 95% confidence intervals fmi: fraction of missing information Death due to TB in those who died and where cause of death was known ### 6.4.7 Sensitivity analysis Dropping duplicate recurrent TB notifications increased the magnitude, and precision, of the effect sizes for recurrent TB, all-cause mortality, and deaths due to TB (in those who died) (see Table 6.10). Restricting the analysis to only cases that were eligible for the BCG schools scheme reduced the sample size of the analysis (from an initial study size of 51645, of which 12832 were UK born, to 9943 cases that would have been eligible for the BCG schools scheme). With this reduced sample size, there was strong evidence in adjusted analyses of an association between BCG vaccination and reduced recurrent TB, and evidence of an association with decreased all-cause mortality (see Table 6.10). Table 6.10: Summary of associations between BCG vaccination and all outcomes; cases that have no recurrent flag in the ETS (n=50407), and cases that would have been eligible for the BCG schools scheme (n=9943). Those defined to be eligible for the schools scheme are the UK born, that were aged 14 or over in 2004 Univariable Multivariable Study population Outcome BCG OR (95% CI) P-value aOR (95% CI) P-value Recurrent cases dropped All-cause mortality No 1 <0.001 1 <0.001 Yes 0.27 (0.23 to 0.31) 0.73 (0.61 to 0.86) Death due to TB (in those who died) No 1 0.709 1 0.147 Yes 0.94 (0.68 to 1.31) 0.74 (0.49 to 1.11) Recurrent TB No 1 <0.001 1 <0.001 Yes 0.61 (0.55 to 0.69) 0.76 (0.66 to 0.87) Pulmonary TB No 1 <0.001 1 0.672 Yes 0.83 (0.79 to 0.87) 0.99 (0.93 to 1.04) Sputum smear status - positive No 1 0.141 1 0.871 Yes 0.94 (0.88 to 1.02) 1.01 (0.92 to 1.10) Cases eligible for the schools scheme All-cause mortality No 1 <0.001 1 0.018 Yes 0.24 (0.19 to 0.29) 0.72 (0.55 to 0.95) Death due to TB (in those who died) No 1 0.893 1 0.987 Yes 0.96 (0.57 to 1.63) 0.99 (0.49 to 2.03) Recurrent TB No 1 <0.001 1 <0.001 Yes 0.51 (0.42 to 0.61) 0.66 (0.52 to 0.84) Pulmonary TB No 1 0.017 1 0.417 Yes 0.87 (0.78 to 0.98) 0.94 (0.82 to 1.08) Sputum smear status - positive No 1 0.613 1 0.588 Yes 1.04 (0.89 to 1.22) 1.05 (0.87 to 1.27) OR: odds ratio with 95% confidence intervals aOR: adjusted odds ratio with 95% confidence intervals fmi: fraction of missing information * Death due to TB in those who died and where cause of death was known ## 6.5 Discussion Using TB surveillance data collected in England I found that BCG vaccination, prior to the development of active TB, was associated with reduced all-cause mortality and fewer recurrent TB cases, although the evidence for this association was weaker. There was some suggestion that the association with all-cause mortality was due to reduced deaths due to TB (in those who died), though the study was underpowered to definitively assess this. I did not find evidence of an association between BCG status and positive smear status or pulmonary TB. Analysis with multiply imputed data indicated that notification 10+ years after vaccination was associated with increased all-cause mortality compared to notification wihtin 10 years. In separate analyses, there was some evidence that vaccination at birth, compared to at any other age, was associated with reduced all-cause mortality, and increased deaths due to TB (in those who died). This study used a large detailed dataset, with coverage across demographic groups, and standardized data collection from notifications and laboratories. The use of routine surveillance data means that this study would be readily repeatable with new data. The surveillance data contained multiple known risk factors, this allowed us to adjust for these confounders in the multivariable analysis, which attenuated the evidence for an association with BCG vaccination for all outcomes. However, there are important limitations to consider. The study was conducted within a population of active TB cases, therefore the association with all-cause mortality cannot be extrapolated to the general population. Additionally, vaccinated and unvaccinated populations may not be directly comparable because vaccination has been targeted at high-risk neonates in the UK since 2005. I mitigated this potential source for bias by conducting a sensitivity analysis including only those eligible for the universal school age scheme, and whilst the strength of associations were attenuated there remained some evidence of improved outcomes. Sensitivity analysis excluding recurrent cases indicated their inclusion may have biased our results towards the null. Variable data completeness changed with time, with both BCG vaccination status and year of vaccination having a high percentage of missing data, which may not be missing completely at random. I therefore checked the robustness of our results with multiple imputation including regional variability, however an unknown missing not at random mechanism, or unmeasured confounding may still have introduced bias. I found a greatly increased risk of all-cause mortality for those vaccinated more than 10 years ago in the analysis with multiply imputed data, compared to the complete case analysis. This is likely to be driven by a missing not at random mechanism for years since vaccination, with older cases being both more likely to have been vaccinated more than 10 years previously and to also have an unknown year of vaccination. The high percentage of missing data also means that I was likely to be underpowered to detect an effect of BCG vaccination on sputum smear status and deaths due to TB (in those who died), with years since vaccination, and age at vaccination likely to be underpowered for all outcomes. I was not able to adjust for either tuberculin skin test (TST) stringency, or the latitude effect, although I was able to adjust for UK birth status.[85] However, the bias induced by these confounders is likely to be towards the null, meaning that our effect estimates are likely to be conservative. Finally, BCG vaccination status, and year of vaccination, may be subject to misclassification due to recall bias; validation studies of the recording of BCG status in the ETS would be required to assess this. Little work has been done to assess the overall effect of BCG on outcomes for active TB cases although the possible non-specific effects of BCG are an area of active research.[34,86,87] Whilst multiple studies have investigated BCG’s association with all-cause mortality, it has been difficult to assess whether the association continues beyond the first year of life.[87] The effect size of the association I identified between BCG and all-cause mortality in active TB cases was comparable to that found in a Danish case-cohort study in the general population (adjusted Hazard ratio (aHR): 0.58 (95% CI 0.39 to 0.85).[35] A recent systematic review also found that BCG vaccination was associated with reduced all-cause mortality in neonates, with an average relative risk of 0.70 (95% CI 0.49 to 1.01) from five clinical trials and 0.47 (95% CI 0.32 to 0.69) from nine observational studies at high risk of bias.[34] I found some weak evidence that BCG vaccination was associated with reduced deaths due to TB (in those who died), although our point estimate had large confidence intervals. Several meta-analyses have found evidence supporting this association,[24,28] with one meta-analysis estimating a 71% (Risk ratio (RR): 0.29 95% CI 0.16 to 0.53) reduction in deaths due to TB in individuals vaccinated with BCG.[24] The meta-analysis performed by Abubakar et al. also found consistent evidence for this association, with a Rate ratio of 0.22 (95% CI 0.15 to 0.33).[28] In contrast to our study, both of these meta-analyses estimated the protection from TB mortality in BCG vaccinated individuals rather than in BCG vaccinated cases who had died from any cause. Additionally, neither study explored the association between BCG vaccination and all-cause mortality or recurrent TB. This study could not determine the possible causal pathway for the association between BCG vaccination all-cause mortality, and recurrent TB. These are important to establish in order to understand the effect of BCG vaccination on TB outcomes. I found that BCG vaccination was associated with reduced all-cause mortality, with some weaker evidence of an association with reduced recurrent TB. A plausible mechanism for this association is that BCG vaccination improves treatment outcomes,[36] which then results in decreased mortality, and reduced recurrent TB. However, these effects may also be independent and for all-cause mortality may not be directly related to active TB. In this case, a possible mechanism for the association between BCG vaccination and all-cause mortality is that BCG vaccination modulates the innate immune response, resulting in non-specific protection.[32] For low incidence countries, where the reduction in TB cases has been used as evidence to scale back vaccination programs,[27] these results suggest that BCG vaccination may be more beneficial than previously thought. In countries that target vaccination at those considered to be at high risk of TB the results from this study could be used to help drive uptake by providing additional incentives for vaccination. The evidence I have presented should be considered in future cost-effectiveness studies of BCG vaccination programs. Several Chapters (Chapter 5, Chapter 7, and Chapter 10) in this thesis assess the impact of moving from universal school age vaccination to selective high risk neonatal vaccination. The reduction in BCG coverage that this implies means that on top of any potential increase in TB incidence rates there may also have been a reduction in the benefical effects from the BCG vaccine discussed in this Chapter. However, as outlined in the previous paragraph, the evidence of reductions in both all-cause, and TB specific mortality, is strongest in the early years of life. This means that the move to neonatal vaccination may have led to an increase in the non-specific benefits. Further work is required to determine whether years since vaccination and age at vaccination are associated with TB outcomes as this study was limited by low sample size, missing data for year of vaccination, and the relative rarity of some TB outcomes. However, due to the continuous collection of the surveillance data used in this analysis, this study could be repeated once additional data have been collected. If this study were to be repeated with a larger sample size, particular attention should be given to the functional form of any decay in protection from negative TB outcomes. Additionally, a larger sample size would allow investigation of the associations identified between TB outcomes and BCG vaccination stratified by pulmonary, extrapulmonary, and disseminated TB disease. The results from this study require validation in independent datasets and the analysis should be reproducible in other low incidence countries that have similarly developed surveillance systems. If validated in low incidence countries, similar studies in medium to high incidence countries should be conducted because any effect would have a greater impact in these settings. ## 6.6 Summary • I found evidence of an association between BCG vaccination and reduced all-cause mortality (aOR:0.76 (95%CI 0.64 to 0.89), P:0.001) and weak evidence of an association with reduced recurrent TB (aOR:0.90 (95%CI 0.81 to 1.00), P:0.056). Analyses using multiple imputation suggested that the benefits of vaccination for all-cause mortality were reduced after 10 years. • There was some suggestion that the association with all-cause mortality was due to reduced deaths due to TB (in those who died), though the study was underpowered to definitively assess this. • There was little evidence for other associations. • The code for the analysis contained in this chapter can be found at: doi.org/10.5281/zenodo.121379923 ### References 5 Zwerling A, Behr MA, Verma A et al. The BCG world atlas: A database of global BCG vaccination policies and practices. PLoS medicine 2011;8:e1001012. 15 Bhatti N, Law MR, Morris JK et al. Increasing incidence of tuberculosis in England and Wales: a study of the likely causes. BMJ (Clinical research ed) 1995;310:967–9. 23 Rodrigues LC, Diwan VK, Wheeler JG. Protective effect of BCG against tuberculous meningitis and miliary tuberculosis: a meta-analysis. International journal of epidemiology 1993;22:1154–8. 24 Colditz GA, Brewer TF, Berkey CS et al. Efficacy of BCG Vaccine in the Prevention of Tuberculosis. JAMA 1994;271:698. 25 Mangtani P, Abubakar I, Ariti C et al. Protection by BCG Vaccine Against Tuberculosis: A Systematic Review of Randomized Controlled Trials. Clinical infectious diseases : an official publication of the Infectious Diseases Society of America 2014;58:470–80. 27 Zwerling A, Behr MA, Verma A et al. The BCG World Atlas: a database of global BCG vaccination policies and practices. PLoS medicine 2011;8:e1001012. 28 Abubakar I, Pimpin L, Ariti C et al. Systematic review and meta-analysis of the current evidence on the duration of protection by bacillus Calmette-Guérin vaccination against tuberculosis. Health technology assessment 2013;17:1–372, v–vi. 30 Fine P. Stopping routine vaccination for tuberculosis in schools. BMJ (Clinical research ed) 2005;331:647–8. 31 Teo SSS, Shingadia DV. Does BCG have a role in tuberculosis control and prevention in the United Kingdom? Archives of Disease in Childhood 2006;91:529–31. 32 Kleinnijenhuis J, Quintin J, Preijers F et al. Bacille Calmette-Guerin induces NOD2-dependent nonspecific protection from reinfection via epigenetic reprogramming of monocytes. Proceedings of the National Academy of Sciences of the United States of America 2012;109:17537–42. 33 Garly ML, Martins CL, Balé C et al. BCG scar and positive tuberculin reaction associated with reduced child mortality in West Africa: A non-specific beneficial effect of BCG? Vaccine 2003;21:2782–90. 34 Higgins JPT, Soares-weiser K, López-lópez JA et al. Association of BCG , DTP , and measles containing vaccines with childhood mortality : systematic review. BMJ (Clinical research ed) 2016;i5170. 35 Rieckmann A, Villumsen M, Sørup S et al. Vaccinations against smallpox and tuberculosis are associated with better long-term survival: a Danish case-cohort study 19712010. International journal of epidemiology 2016;0:1–11. 36 Jeremiah K, Praygod G, Faurholt-Jepsen D et al. BCG vaccination status may predict sputum conversion in patients with pulmonary tuberculosis: a new consideration for an old vaccine? Thorax 2010;65:1072–6. 38 The World Health Organization. BCG Vaccine. Weekly epidemiological record 2004;79:27–48. 45 French CE, Antoine D, Gelb D et al. Tuberculosis in non-UK-born persons, England and Wales, 2001-2003. Int J Tuberc Lung Dis 2007;11:577–84. 51 van Buuren S, Groothuis-Oudshoorn K. mice: Multivariate imputation by chained equations in r. Journal of Statistical Software 2011;45:1–67.https://www.jstatsoft.org/v45/i03/ 56 R Core Team. R: A Language and Environment for Statistical Computing. Vienna, Austria: 2016. 69 Barreto ML, Pilger D, Pereira SM et al. Causes of variation in BCG vaccine efficacy: Examining evidence from the BCG REVAC cluster randomized trial to explore the masking and the blocking hypotheses. Vaccine 2014;32:3759–64. 77 Abbott S, Christensen H, Lalor MK et al. Exploring the effects of BCG vaccination in patients diagnosed with tuberculosis: Observational study using the Enhanced Tuberculosis Surveillance system. Vaccine 2019;1–6. 78 Parslow R, El-Shimy NA, Cundall DB et al. Tuberculosis, deprivation, and ethnicity in Leeds, UK, 1982-1997. Archives of disease in childhood 2001;84:109–13. 80 Aaby P, Nielsen J, Benn CS et al. Sex-differential and non-specific effects of routine vaccinations in a rural area with low vaccination coverage: An observational study from Senegal. Transactions of the Royal Society of Tropical Medicine and Hygiene 2014;109:77–84. 81 Teale C, Goldman JM, Pearson SB. The association of age with the presentation and outcome of tuberculosis: a five-year survey. Age and ageing 1993;22:289–93. 82 Abubakar I, Laundy MT, French CE et al. Epidemiology and treatment outcome of childhood tuberculosis in England and Wales: 1999-2006. Archives of Disease in Childhood 2008;93:1017–21. 83 Djuretic T, Herbert J, Drobniewski F et al. Antibiotic resistant tuberculosis in the United Kingdom : 2002;477–82. 84 Barnard J, Rubin DB. Miscellanea. Small-sample degrees of freedom with multiple imputation. Biometrika 1999;86:948–55. 85 Roy a, Eisenhut M, Harris RJ et al. Effect of BCG vaccination against Mycobacterium tuberculosis infection in children: systematic review and meta-analysis. BMJ (Clinical research ed) 2014;349:g4643–3. 86 Kandasamy R, Voysey M, McQuaid F et al. Non-specific immunological effects of selected routine childhood immunisations: systematic review. BMJ (Clinical research ed) 2016;355:i5225. 87 Pollard AJ, Finn A, Curtis N. Non-specific effects of vaccines: plausible and potentially important, but implications uncertain. Archives of Disease in Childhood 2017;102:archdischild–2015–310282.	2021-06-24 03:44:52	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.3858819603919983, "perplexity": 8212.061563641364}, "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-2021-25/segments/1623488550571.96/warc/CC-MAIN-20210624015641-20210624045641-00278.warc.gz"}
https://math.stackexchange.com/questions/1222131/has-a-compact-subset-a-proper-distance-from-a-disjoint-locally-finite-collection	# Has a compact subset a proper distance from a disjoint locally finite collection of compact subsets? Let $n\in\mathbb{R}^n$ and $K\subseteq\mathbb{R}^n$ a compact subset. Let $\{U_i\}_{i\in I}$ be a locally finite collection (every point has a neighborhood, which intersects only finitly many $U_i$) of open subsets $U_i\subseteq\mathbb{R}^n$, which are disjoint from $K$, i.e. $K\cap U_i=\emptyset$. Let furthermore $K_i\subseteq U_i$ be a compact subset for each $i\in I$. Since $K$ is disjoint from $U_i$, it is disjoint from $K_i$ for each $i$. Does their exist a neighborhood of $K$, which is disjoint from all the $K_i$? My thoughts so far: Since $K$ and $K_i$ are compact and disjoint, they have a proper distance $0<\varepsilon_i=dist(K,K_i)$, but as $I$ can be inifite, $\epsilon_i$ might get arbitrary small. (?) Take for instance $K=\{0\}$, and $U_i=\left(\frac{1}{i+1},\frac{1}{i}\right)$. Since $K_i\subset U_i$, we have $x<\frac{1}{i}$ for all $x\in K_i$ and all $i\in\mathbb{N}$. If $U$ is an open neighborhood of $K$, then there certainly exists some integer $i$, such that $\left(0,\frac{1}{i}\right)\subset U$, but then $U$ also contains all elements of $K_i$.	2021-06-18 00:18:14	{"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": 0.9877336025238037, "perplexity": 67.36995534038962}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487634576.73/warc/CC-MAIN-20210617222646-20210618012646-00390.warc.gz"}
http://codeforces.com/problemset/problem/300/E	E. Empire Strikes Back time limit per test 5 seconds memory limit per test 512 megabytes input standard input output standard output In a far away galaxy there is war again. The treacherous Republic made k precision strikes of power ai on the Empire possessions. To cope with the republican threat, the Supreme Council decided to deal a decisive blow to the enemy forces. To successfully complete the conflict, the confrontation balance after the blow should be a positive integer. The balance of confrontation is a number that looks like , where p = n! (n is the power of the Imperial strike), . After many years of war the Empire's resources are low. So to reduce the costs, n should be a minimum positive integer that is approved by the commanders. Help the Empire, find the minimum positive integer n, where the described fraction is a positive integer. Input The first line contains integer k (1 ≤ k ≤ 106). The second line contains k integers a1, a2, ..., ak (1 ≤ ai ≤ 107). Output Print the minimum positive integer n, needed for the Empire to win. Please, do not use the %lld to read or write 64-but integers in С++. It is preferred to use the cin, cout streams or the %I64d specificator. Examples Input 21000 1000 Output 2000 Input 12 Output 2	2018-05-25 10:57:29	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.1917198747396469, "perplexity": 2523.4084891367647}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794867085.95/warc/CC-MAIN-20180525102302-20180525122302-00003.warc.gz"}
https://math.stackexchange.com/questions/3114092/in-triangle-abc-ad-perp-bc-and-ge-is-the-extended-line-of-dg-wher/3114226	In $\triangle ABC$, $AD$ $\perp$ $BC$ and $GE$ is the extended line of $DG$ where $G$ is centroid. Prove that $GD$ = $\frac{EG}{2}$ Let $$ABC$$ be a triangle and in $$\triangle ABC$$, $$AD$$ $$\perp$$ $$BC$$ and three median lines intersect at point $$G$$ where $$G$$ is the centroid of $$\triangle ABC$$. The extension of $$DG$$ intersects the circumcircle of $$\triangle ABC$$ at point $$E$$. Prove that $$GD = \frac{EG}{2}$$ I found this as an isolated problem. My attempt: Nothing speciality I discovered from the diagram. I only connected segment $$AE$$ and drew $$GI$$, where $$GI$$ $$\perp$$ $$AD$$. From the above diagra, $$G$$ is the centroid. So, $$\frac{AG}{GF}$$ = $$\frac{1}{2}$$. And then from right angled triangle $$\triangle AGI$$ and $$\triangle ADF$$, We get $$AI:ID$$ = $$1:2$$ (as $$\triangle AGI$$ $$\sim$$ $$\triangle ADF$$). Right then, if $$\triangle ADE$$ can be showed as a right angled triangle ($$\angle EAD$$ = 90$$^\circ$$) and $$\triangle ADE$$ $$\sim$$ $$\triangle GID$$, we can also likewise show that $$\frac{DG}{GE}$$ = $$\frac{1}{2}$$. But my reverse effort went into vain. I can't anyhow show that $$\angle EAD$$ = 90$$^\circ$$. So, how to solve for that case? Can it be solved by vector? Thanks in advance. • The answer you accepted is deficient. – Aqua Feb 16 '19 at 0:27 What you only have to prove is that $$[CE]\parallel [AB]$$. From there it's trivial (observe the ratio $$CG:GF$$) that $$[GD]=\frac{[EG]}{2}$$ So here goes my proof for $$[CE]\parallel [AB]$$ (I guess there might be a simpler one, however...) Here $$FK$$ is the perpendicular bisector of $$[AB]$$, $$K=DE\cap FK$$ and $$I$$ is the midpoint of $$[CD]$$. Now since $$CD\parallel KF$$ $$\frac{[CD]}{[FK]}=\frac{[CG]}{[GF]}\implies [CD]=2·[FK]$$ Therefore $$[KI]\parallel [AB]$$, which implies that $$[CK]=[DK]$$. Denote by $$O$$ the circumcenter. Simple angle-chasing shows that $$\angle CKO=\angle OKE$$. From the congruence criterion SAS we obtain $$\Delta OKC\cong \Delta OEK\implies \angle KOC=\angle EOK$$ Thus the triangle $$\Delta OEC$$ is isosceles; the angle bisector of $$\angle EOC$$ is $$OK\perp CE$$. We can now conclude that $$CE\parallel AB$$ • @Doctor I didn't understand a fact. How can point $I$ be the midpoint $CD$? If you please explain that, it will be too much better for me. – Anirban Niloy Feb 15 '19 at 17:16 • It's not a conclusion but the designation. It's just like saying 'I will denote with $I$ the midpoint of $[CD]$ – Dr. Mathva Feb 15 '19 at 17:22 • And sorry for changing the letters of the vertices. I noticed that when the answer was finish... – Dr. Mathva Feb 15 '19 at 17:23 • No problem. Main fact is that I understood the total process. I have to say you tnx cordially for a commendable approach. And there is no reason for being sorry. – Anirban Niloy Feb 15 '19 at 17:26 • Why is that: $[KE]$ is the reflection of $[CK]$ over $KF$? This is true if $AB\|\|CE$ but that is the essence of the prove. – Aqua Feb 16 '19 at 0:22 Before solving the problem, let's put ourselves in this situation (see attached image) If $$HD=DJ, HG = 2(GO)$$ and $$JO=OI$$ then you have to necessarily $$J, O$$ and $$I$$ are collinear. A simple test is by contradiction. Suppose they are not collinear, so let's place $$L$$ in $$HO$$ such that $$DL // JO$$ then we would have $$HL = 3a$$, $$LG = a$$ and $$GO = 2a$$ ($$a$$ is a constant), in addition $$DL = 2 (JO) = 2 (OI)$$. With all this, we would have that $$DL // OI$$ (the triangles $$DLG$$ and $$GOI$$ are similar). Let's call $$\theta = \angle JOL$$, then $$\angle DLH = \theta$$ and $$\angle DLO = \angle LOI = 180 - \theta$$, therefore $$J,O,I$$ are collinear. In the problem, let's call $$H$$ the orthocenter of the triangle $$ABC$$ and $$O$$ to its circumcenter. It is known that $$H,G,O$$ are collinear and $$HG = 2GO$$. Now let $$J$$ be the intersection of the extension of $$AH$$ with the circumscribed circumference, so it is easy to see that $$HD = DJ$$. Finally, we would have by the initial observation that necessarily J, O, I are collinear. Then $$G$$ is the centroid of the triangle $$HIJ$$ and $$GI=2DG$$. Sorry for having changed the letter $$E$$ for the $$I$$, I hope it is understood in the same way Let $$E'$$ be such that $$ABCE'$$ is isosceles trapezoid ($$AB = CE'$$). Then $$E'$$ is on circle through $$A,B,C$$. Let $$DE'$$ meet $$AF$$ at $$G'$$. Since triangle $$DFG'\sim E'AG'$$ and $$AE' = 2DF$$ we have $${AG'\over G'F} = {E'G'\over G'D} = {2\over 1}$$ so $$G'=G$$ and $$E'=E$$ and the claim is proved. • You provided me with an efficient solution. But without diagram, my brain doesn't work. Moreover, my desired condition that I have asked for is to show the $CE$ $\parallel$ $AB$. Any idea? Then please add that to your post if Dr. Mathva has done mistake. – Anirban Niloy Feb 16 '19 at 3:25 • I thought you accept my answer? – Aqua Feb 24 '19 at 10:38	2020-01-29 11:32:24	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 102, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8936758637428284, "perplexity": 225.62855788194983}, "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-2020-05/segments/1579251796127.92/warc/CC-MAIN-20200129102701-20200129132701-00375.warc.gz"}
https://datascience.stackexchange.com/tags/regression/hot	# Tag Info 3 Think carefully before you do this. You have no idea what the underlying height distribution is. Here are four possibilities. If you were building a regression model, each of these sets of height data would be interpreted differently. However, if replaced by your ordinal variable, they all look numerically equivalent. If you use this variable as a ... 2 You should not use Label Encoding for Categorical data unless there is a known ranking and that also in the specified ratio between the level values. In this case, the model will assume 10 as 2 times of 5. One-hot will add a lot of dimensions as I can see in your data. You must try other Categorical encoding techniques esp. Sum Coding Or Helmert. You ... 2 Taking the log doesn't result in a normally-distributed target; it would tend to if the target was log-normally distributed, and you have something normalish there, not quite. But, this distribution isn't actually what matters. What taking the log does is change your model of how errors arise when fitting a regressor. You're now saying that the target ... 2 I am working almost on the same problem these days: I have tried two options using XGB Regression with different objective functions including: Using a linear regression objectiive function ("reg:linear" or "reg:squarederror") and transformed the target to the log space Using the gamma objective function ("reg:gamma"), which is useful for a skewed target ... 2 It's impossible in general, simply because a particular value or range for feature A might correspond to class 'good' if feature B has a certain value/range but correspond to class 'bad' otherwise. In other words, the features are inter-dependent so there's no way to be sure that a certain range for a particular feature is always associated with a particular ... 1 Writing a custom loss function could be handy, but it may be simpler to just try to treat this as a class balance problem for your regression model. For starters, just try undersampling all of the higher and medium grades until they're close to balanced with your failing students. Given your number of data points and features you can probably still just ... 1 Try writing a custom loss function for a regression model! Keras' neural networks support this, for example. See https://stackoverflow.com/q/43818584/745868 (But many other libraries give support for this as well) The only special thing about your custom loss function is that it doesn't add up the error of a datapoint if min(pred_y, actual_y) >= THRESHOLD 1 The first question about missing data is always why is it missing? Have you checked or know why the data is missing and whether it is MAR, MCAR or not missing at random? If your data is MCAR imputation is generally fine and your lower test metric might simply indicate a suboptimal imputation strategy. In this case you could try MICE or similar more advanced ... 1 First you should define a metric that suits the problem $R^2$ in your case. Do a correct cross-validation and train test splits. And then choose in the cross validation which option has the best results for your model (imputing missing or xgboost no imputing). This way you are doing an empirical experiment and selecting the best result. Probably you want to ... 1 Flier values/skewed predictors will have a high influence on the regression model. If you want to counteract that, you have a few choices. 1) If your target is always non-zero, and if you expect the regression to be close to linear, you can try to use a log(), sqrt() or even boxcox() conversion transform on the target variable. This will help keep the large ... 1 @Ben Norris found out that the relaimpo packages has a hard minimum number of observations, so if I wanted to pursue this path I have to up my sample size. As I only have the data that I have, I pursued a "hacky" solution which I am going to describe for completionists sake. The steps were as follows: Assign each individual to one of k groups randomly, so ... 1 Of course your error rate is going to decrease. Remember that your changes in MAE values may come from the fact that the scales of your original variable and that variable transformed by a logarithm ARE NOT THE SAME, and mean is scale-dependant. About your second question, is exactly that! If you would like to compare the use (or not) of logarithms. You ... 1 I think it's complete alright. In fact, the second model mathematical expression is given by y=x3f(x1, x2, x3), which is just like the first model but with some specific feature engineering. I don't see any possibility for data leakage. Only top voted, non community-wiki answers of a minimum length are eligible	2020-07-02 10:11:17	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5702550411224365, "perplexity": 762.4741501149985}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1593655878639.9/warc/CC-MAIN-20200702080623-20200702110623-00144.warc.gz"}
https://math.stackexchange.com/questions/817102/proof-that-if-f-is-function-continuous-on-an-interval-i-then-fi-is-also-an-in	# Proof that if f is function, continuous on an interval I then f(I) is also an interval The theorem would be: Let $f:E\to\mathbb{R}$ a continuous function and $I$ and interval, $I \subseteq E$. Then $f(I)$ is also an interval. I'm not sure if I've understood completely what I have to prove. So, I need to prove that f takes all the values in $f(I)$, which is easy, using the intermediate value theorem. But how do I know that $\forall x \in f(I)$, then $f(x) \in I$ ? Isn't it necesary to also prove this? I know for sure that $\forall \lambda \in f(I)$ , there is an $x_{\lambda}$ such that $f(x_{\lambda})=\lambda$, given the fact that $I=[a,b]$ , and $f(I)=[f(a),f(b)]$. That is, forbsure, every value in $f(I)$ is taken by f. So, what I don't know how to prove is how do I know that for any value in $I$ , the function sends me for sure in $f(I)$?? • It is not true that if $I=[a,b]$, then $f(I)=[f(a),f(b)]$. Consider for example $f(x)=x^2$ and $I=[-1,1]$. – Servaes Jun 1 '14 at 15:23 • Noe that in general $f([a,b])\ne [f(a),f(b)]$. – Hagen von Eitzen Jun 1 '14 at 15:23 • Indeed, you guys are right – Bardo Jun 1 '14 at 15:36 • This follows for the extreme value theorem and intermediate value theorem. – Gamma Function Jun 1 '14 at 16:04 • Note that this requires singleton sets such as $\{ a \}$ (or $[a,a]$) to be considered intervals. Some definitions of "interval" exclude them. – Jeppe Stig Nielsen Jun 1 '14 at 22:54 The comments already answer the questions raised in your last paragraph so I'm not sure my answer adds much. But I can always delete it later. As for proving that $f(I)$ is an interval if $I$ is, here is how I would do it: First I'd prove that in $\mathbb R$ a set is connected if and only if it is an interval. But this was probably done in the book you are reading or the lecture you are taking. Next I'd prove that a continuous function maps connected sets to connected sets: By contradiction assume that not. Then there are open disjoint sets $U,V$ such that $f(I) = U \cup V$. Then $I = f^{-1}U \cup f^{-1}V$ is disjoint, a contradiction. Note that this is exactly what you are asked to show when the statement is restricted to $\mathbb R$. So perhaps what I wrote earlier is not true and this exercise is exactly asking you to prove that a set in $\mathbb R$ is connected if and only it is an interval. Since if you assumed both that and the intermediate value theorem there is nothing left to prove. • Well, here's the answer my textbook gives, and I'm quoting word by word: We need to show that if $\alpha,\beta in f(I)$ then it exsits $a,b\in I$ such that $f(a)=\alpha, f(b)=\beta, a \neq b$. But f is continuous and $\lambda$ is an intermediate value, then there is $x_{\lambda} \in (a,b)$ such that $f(x_{\lambda})=\lambda$, that is $\lambda \in f(I)$ . But I am not conviced at all – Bardo Jun 1 '14 at 15:37 • Honestly, in my opinion this doesn't prove the theorem at all, and it's just some words that don't make sense, but probably I'm wrong. – Bardo Jun 1 '14 at 15:44 • @Bardo No, I agree with you, as stated it doesn't make much sense. But it's clear to me what it's trying to do: it wants to show that $f(I)$ is an interval by applying the intermediate value theorem. It completely fails to mention what lambda is and that is must be between $\alpha$ and $\beta$. On top of that, the English is broken. Maybe it would be helpful to you if (in addition) you could get yourself a better book from the library. – Rudy the Reindeer Jun 1 '14 at 16:02 • Wait, I found a proof!! – Bardo Jun 1 '14 at 16:11 • I posted itt. I actually asked uou to do thaf, but in a comment of mine at the answer od @Gamma. Thank you! – Bardo Jun 1 '14 at 16:35 It seems that the biggest problem I had proving this, is that I didn't know how to rigurously define an interval. So, let me first state the rigurous definition of an interval, for those who may end up here, and who don't already know it, but they are also asking themselves how to prove the theorem in my question. We say that the set $I$ is an interval if for any $a,b\in I$, with $a<b$ we have that if $a\leq c\leq b$, for a number $c$ , then also $c\in I$. Now, if $\alpha,\beta \in f(I)$ , there is $a,b\in I$, such that $f(a)=\alpha$ and $f(b) = \beta$. Let $\lambda$ , with $f(\alpha) \leq \lambda \leq f(\beta)$ . We want to show that $\lambda \in f(I)$. If $\lambda=f(\alpha)=f(\beta)$, then obsiously $\lambda \in f(I)$, so we now have to deal only with the case when the inequality is strict: $f(\alpha) < \lambda < f(\beta)$ But we know from the intermediate value theorem that that there is an $x_{\lambda}\in (a,b)$ such that $f(x_{\lambda})=\lambda$, that is $\lambda \in f(I)$, because obsiously $f(x_{\lambda})\in f(I)$. Q.E.D. Is this correct? I haven't been able to find any flaw in my proof. • Just a minor thing: If you have $f(\alpha)<\lambda< f(\beta)$ you can't have $\lambda=f(\alpha)=f(\beta)$. You'd have to slightly rephrase this. And I'd change $x_\lambda \in I$ into $x_\lambda \in (a,b)$ just to be more precise. But it's not necessary. Otherwise this looks good to me! – Rudy the Reindeer Jun 1 '14 at 16:41 • I modified the proof, rephrasing that part. Thank you for your observation. I love the elegance of mathematical analysis. This is the first year I study it. In my classroom, we don't bother with proofs of such theorems, just with how we vcan apply it in problems. The teacher always gives us the proof, but it underlines every time that we don't need to know it. The reason is because I'm in highschool, and we don't actually need to do formal proofs like this in our exams, just to solve limits, and other problems. – Bardo Jun 1 '14 at 17:30 • But I like this branch of math so much, that I decided to retake all the manual by myself and prove everything (or almost everything) in there, to get a better understanding of the subject. – Bardo Jun 1 '14 at 17:31	2019-08-22 01:01:06	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8355994820594788, "perplexity": 176.24504671438962}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027316555.4/warc/CC-MAIN-20190822000659-20190822022659-00143.warc.gz"}