Split (1)

train · 6.32M rows

url stringlengths 14 2.42k	text stringlengths 100 1.02M	date stringlengths 19 19	metadata stringlengths 1.06k 1.1k
https://vinaire.me/2019/10/04/einsteins-paper-on-light-quanta/	## Einstein’s Paper on Light Quanta In his very first paper published in 1905 Einstein establishes the concept of “energy quantum” or “light quantum”. The energy of light quantum (photon) is proportional to frequency that becomes more pronounced as one moves up the electromagnetic spectrum. Here is a summary of Einstein’s 1905 paper on Light Quantum followed by some comments. SECTION 0: Introduction Einstein says, The energy of a ponderable body cannot be split into arbitrarily many, arbitrarily small parts, while the energy of a light ray, emitted by a point source of light is according to Maxwell’s theory (or in general according to any wave theory) of light distributed continuously over an ever increasing volume. In other words, Einstein observes that atoms are not infinitely divisible, but the electromagnetic radiation is treated as a continuum and infinitely divisible. Einstein says, “In fact, it seems to me that the observations on “black-body radiation”, photoluminescence, the production of cathode rays by ultraviolet light and other phenomena involving the emission or conversion of light can be better understood on the assumption that the energy of light is distributed discontinuously in space.” In other words, Einstein proposes that the creation and conversion of light may not be continuous. SECTION 1. On a Difficulty in the Theory of “Black-body Radiation’’ In this section Einstein sets up a thought experiment. He assumes electrons to be particles that are colliding like gas molecules. There are bound “resonator electrons” that emit and absorb electromagnetic waves with definite periods. The average kinetic energy of a resonator electron must equal the average kinetic energy corresponding to the translational motion of a gas molecule under dynamic equilibrium. Similarly, it should also equal the energy of interaction with radiation present in space. Einstein says, “This relation, which we found as the condition for dynamic equilibrium does not only lack agreement with experiment, but it also shows that in our picture there can be no question of a definite distribution of energy between aether and matter. The greater we choose the range of frequencies of the resonators, the greater becomes the radiation energy in space…” This was famously known as the ultraviolet catastrophe. SECTION 2. On Planck’s Determination of Elementary Quanta In this section Einstein shows that “determination of elementary quanta given by Mr. Planck is, to a certain extent, independent of the theory of “black-body radiation” constructed by him.” Using mathematics to back up his argument, Einstein concludes: “The higher the energy density and the longer the wavelengths of radiation, the more usable is the theoretical basis used by us; for short wavelengths and low radiation densities, however, the basis fails completely.” In other words, the radiation appears continuous per Maxwell’s theory at lower frequencies, but not at higher frequencies. SECTION 3. On the Entropy of the Radiation In this section Einstein presents Wien’s consideration that entropy of radiation may be determined completely from black body radiation law when the radiation energy is given for all frequencies. SECTION 4. Limiting Law for the Entropy of Monochromatic Radiation for Low Radiation Density In this section Einstein uses Wien’s approximation (valid for higher frequencies of black body radiation) to derive an equation for the entropy of radiation. Einstein writes: “This equation shows that the entropy of a monochromatic radiation of sufficiently small density varies with volume according to the same rules as the entropy of a perfect gas or of a dilute solution.” Thus, Einstein proves that the energy distribution of radiation becomes particle-like at high frequencies. This is an ingenious way of arriving at this conclusion. SECTION 5. Molecular-Theoretical Investigation of the Volume-dependence of the Entropy of Gases and Dilute Solutions In this section Einstein shows that, when applied to a large number of discrete particles, the use of “statistical probability” is compatible with macroscopic laws of physics. SECTION 6. Interpretation of the Expression for the Volume-dependence of the Entropy of Monochromatic Radiation according to Boltzmann’s Principle In this section, Einstein uses mathematical arguments to conclude: “Monochromatic radiation of low density behaves—as long as Wien’s radiation formula is valid—in a thermodynamic sense, as if it consisted of mutually independent energy quanta of magnitude Rßv/N.” Each quantum is the energy of one interaction. Einstein mathematically determines the theoretical value of a quantum. SECTION 7. On Stokes’ Rule In this section Einstein uses the new idea of “energy quanta” to explain the Stokes’ Rule for photoluminescence and indicates new possibilities. SECTION 8. On the Production of Cathode Rays by Illumination of Solids In this section Einstein brilliantly verifies the calculated value of energy quanta from the experimental value obtained from the study of photoelectricity. Here we have the conclusive evidence that energy of light is made up of frequency (kinetic energy) and not amplitude (wave energy). SECTION 9. On the Ionization of Gases by Ultraviolet Light In this section Einstein tests his ideas to explain the existing experimental observations and further proves the viability of the idea of “energy quantum” or “light quantum”. . Einstein’s concept of quantum is based on energy. From the phenomenon of photoelectricity, it can be seen that this energy is kinetic (based on frequency) and not that of a wave (based on amplitude). Light carries this energy with it. This means light is a fast-moving substance. The kinetic energy depends on relative velocity between two things. Therefore, this energy is visible only when there is interaction between two things, such as, light and electron. Einstein is mathematically comparing this interaction to collisions among gas molecules in the kinetic theory of gases. Therefore, quantum is tied to the energy of interaction. If there is no interaction, the energy, or the quantum, cannot be perceived. The very perception of quantum requires an interaction. Interactions are discrete. Therefore, quantum is discrete also, but only as energy of interaction. This is what Einstein is thinking about when he says, The energy of a ponderable body cannot be split into arbitrarily many, arbitrarily small parts…” In the kinetic theory of gases, not only the interactions are discrete, but the interacting gas molecules are discrete also. The molecules are discrete because they have individual centers of mass. This is not the case with light because light has no centers of mass. Therefore, light “particles” cannot be distinguished from each other. Light forms a continuum in space even when its interactions are discrete. Einstein disagreed with Maxwell treating energy of light as a continuous function across the spectrum; but he did agree with Maxwell for energy being continuous at the lower end of the spectrum. Maxwell treated light as a wave, which is not really the case. There is no stationary aether through which light is moving as a disturbance. Therefore, a continuous energy function at lower frequencies can only mean that energy interactions are so frequent that they appear continuous. Thus, as the frequency reduces, light starts to act as a continuum in terms of energy interactions also. This can be used as an argument to support the observation that, fundamentally, light is infinitely divisible. As frequency increases, the energy interactions become increasingly differentiable. Electrons also form a continuum in space similar to light, but their frequency is much higher. When light interacts with electrons in the photoelectric phenomenon, it is two continuums of very different frequencies interacting with each other, and not two particles. Any interaction shall only be in terms of partial resonance, and not as an impact between two billiard balls. Much seems to be unknown about the nature of this interaction between light and electron. The frequency of light may best be understood as the density of its continuum. The higher is the frequency, the greater is the density of light. There appears to be a high-density gradient from the electronic region to the nucleus within the atom. This is where the charge appears and the center of mass forms. This is an area of transition where radiation appears to be in equilibrium with matter. Much seems to be unknown about this area of transition from electronic region to nucleus of atom. In conclusion, let us look at the following assumption made by Einstein in this paper: According to the assumption considered here, when a light ray starting from a point is propagated, the energy is not continuously distributed over an ever increasing volume, but it consists of a finite number of energy quanta, localized in space, which move without being divided and which can be absorbed or emitted only as a whole. It appears that quanta are more particle-like only because the density of the continuum has increased. From Faraday’s perspective, quanta can be represented by thicker lines of force, but those lines are still continuous. .	2023-03-24 13:29:51	{"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": 0.8370615243911743, "perplexity": 626.0074713282215}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1679296945282.33/warc/CC-MAIN-20230324113500-20230324143500-00667.warc.gz"}
http://mathhelpforum.com/discrete-math/86823-proving-division-algorithm.html	# Math Help - Proving division algorithm 1. ## Proving division algorithm Hi all, I'm having a tough time trying to go through the division algorithm proof. Theorem: Let (a,b) be two integers. Then there exists numbers q and r with a>= 0, 0=<r<=b such that a=bq+r. In class, my professor starts the proof by using cases, but I don't really understand why he does this. Case 1- If b>a then a=ob+ a (0<a<b) q=0, r=a Case 2- if b=a then a=b1+0, q=1, r=0 Case 3- if b<a then a1 ( I think my notes get cut off here since I was pretty confused in class). Does anyone know where he is trying to go with this? Thanks 2. Originally Posted by jusstjoe Hi all, I'm having a tough time trying to go through the division algorithm proof. Theorem: Let (a,b) be two integers. Then there exists numbers q and r with a>= 0, 0=<r<=b such that a=bq+r. In class, my professor starts the proof by using cases, but I don't really understand why he does this. Case 1- If b>a then a=ob+ a (0<a<b) q=0, r=a Case 2- if b=a then a=b1+0, q=1, r=0 Case 3- if b<a then a1 ( I think my notes get cut off here since I was pretty confused in class). Does anyone know where he is trying to go with this? Thanks Hi jusstjoe. The result won’t work if $b=0.$ However I gather from your post that you want $a$ and $b$ to be both positive integers. Start by showing that the set $\{n\in\mathbb N:nb\ge a\}$ is not empty. This will be true in Cases 1 and 2 with $n=1.$ For Case 3, suppose to the contrary that $nb for all natural numbers $n.$ Then, in particular, $ab $\implies$ $b<1$ which is not possible if $b$ is a positive integer. Hence there is a natural number $n$ such that $nb\ge a.$ Let $q_0$ be the smallest such natural number. The proof is completed by taking $q=q_0-1$ and $r=a-bq-1.$	2014-03-10 13:19:42	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 16, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9117565751075745, "perplexity": 696.6715690759031}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394010815495/warc/CC-MAIN-20140305091335-00028-ip-10-183-142-35.ec2.internal.warc.gz"}
https://www.hackmath.net/en/math-problem/846	# Words How many 3 letter "words" are possible using 14 letters of the alphabet? a) n - without repetition b) m - with repetition n = 2184 m = 2744 ### Step-by-step explanation: $n=14\cdot 13\cdot 12=2184$ $m=1{4}^{3}=2744$ We will be pleased if You send us any improvements to this math problem. Thank you! Tips to related online calculators Would you like to compute count of combinations? ## Related math problems and questions: • Five letters How many ways can five letters be arranged? • Word MATEMATIKA How many words can be created from the phrase MATEMATIKA by changing the letters' order, regardless of whether the words are meaningful? • Morse alphabet Calculate how many words of Morse code to create compiling dashes and dots in the words of one to four characters. • How many 2 How many three lettered words can be formed from letters A B C D E G H if repeats are: a) not allowed b) allowed • Possible combinations - word How many ways can the letters F, A, I, R be arranged? • Logik game Letter game Logik is a two player game, which has the following rules: 1. The first player thinks five-letter word in which no letter is not repeated. 2. The second player writes a five-letter word. 3. The first player answers two numbers - the first numb • Guests How many ways can 9 guests sit down on 10 seats standing in a row? • Practice How many ways can you place 20 pupils in a row when starting on practice? A license plate has 3 letters followed by 4 numbers. Repeats are not allowed for the letters, but they are for the numbers. If they are issued at random, what is the probability that the 3 letters are in alphabetical order and the 3 numbers are consecutiv • Cars plates How many different licence plates can country have, given that they use 3 letters followed by 3 digits? • Shelf How many ways are there to arrange 6 books on a shelf? • Kids How many different ways can sit 8 boys and 3 girls in line if girls want to sit on the edge? • Cinema How many ways can be divided 11 free tickets to the premiere of "Jáchyme throw it in the machine" between 6 pensioners? • Playing cards How many possible ways are to shuffle 7 playing cards? • Lunch Seven classmates go every day for lunch. If they always come to the front in a different order, will be enough school year to take of all the possibilities? • Friends in cinema 5 friends went to the cinema. How many possible ways can sit in a row, if one of them wants to sit in the middle and the remaining's place does not matter? • Combinations of sweaters I have 4 sweaters two are white, 1 red and 1 green. How many ways can this done?	2021-05-14 16:08:20	{"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": 0.33267074823379517, "perplexity": 1261.0318973320625}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1620243991428.43/warc/CC-MAIN-20210514152803-20210514182803-00187.warc.gz"}
https://www.usingenglish.com/forum/threads/s.51613/	's Status Not open for further replies. ehsanshekari Member in "teacher's book" 's is used but it isn't used in "student book" or other examples like "john's book" "cigarette ash" why sometimes 's is used and sometimes it isn't ? Anglika No Longer With Us Sometimes it is a possessive [ teacher's book = the book belonging to the teacher; John's book - the book belonging to John ]; sometimes is is an adjective [cigarette ash = ash from a cigarette; student book = book for use by students]. Status Not open for further replies.	2022-06-27 23:50:29	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8416919708251953, "perplexity": 10020.856997347513}, "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-2022-27/segments/1656103344783.24/warc/CC-MAIN-20220627225823-20220628015823-00779.warc.gz"}
http://surveillance.r-forge.r-project.org/pkgdown/reference/stsplot_time.html	These are the plot variants of type=observed~time\|unit, type=observed~time, and type=alarm~time for "sts" objects (see the central "sts" plot-method for an overview of plot types). ## Usage stsplot_time(x, units=NULL, as.one=FALSE, same.scale=TRUE, par.list=list(), ...) stsplot_time1(x, k=1, ylim=NULL, axes=TRUE, xaxis.tickFreq=list("%Q"=atChange), xaxis.labelFreq=xaxis.tickFreq, xaxis.labelFormat="%G\n\n%OQ", epochsAsDate=x@epochAsDate, xlab="time", ylab="No. infected", main=NULL, type="s", lty=c(1,1,2), col=c(NA,1,4), lwd=c(1,1,1), outbreak.symbol=list(pch=3, col=3, cex=1, lwd=1), alarm.symbol=list(pch=24, col=2, cex=1, lwd=1), legend.opts=list(), dx.upperbound=0L, hookFunc=function(){}, .hookFuncInheritance=function() {}, ...) stsplot_alarm(x, lvl=rep(1,ncol(x)), xaxis.tickFreq=list("%Q"=atChange), xaxis.labelFreq=xaxis.tickFreq, xaxis.labelFormat="%G\n\n%OQ", epochsAsDate=x@epochAsDate, xlab="time", ylab="", main=NULL, outbreak.symbol=list(pch=3, col=3, cex=1, lwd=1), alarm.symbol=list(pch=24, col=2, cex=1, lwd=1), cex.yaxis=1, ...) ## Arguments x an object of class "sts". units optional integer or character vector to select the units (=columns of observed(x)) to plot. The default is to plot all time series. If as.one=FALSE, stsplot_time1 is called for (k in units) with mfrow splitting (see par.list). Note that if there are too many units, the default mfrow setting might lead to the error “figure margins too large” (meaning that the units do not fit onto a single page). as.one logical indicating if all time series should be plotted in a single frame (using matplot). same.scale logical indicating if all time series should be plotted with the same ylim. Default is to do so. Only relevant for multivariate plots (ncol(x) > 1). par.list a list of arguments delivered to a call of par to set graphical parameters before plotting. The mfrow splitting is handled per default. Afterwards, the parameters are reverted to their original values. Use par.list=NULL to disable the internal par call. k the unit to plot, i.e., an element of 1:ncol(x). ylim the y limits of the plot(s). Ignored if same.scale=FALSE. axes a logical value indicating whether both axes should be drawn on the plot. xaxis.tickFreq,xaxis.labelFreq,xaxis.labelFormat arguments for addFormattedXAxis if epochsAsDate=TRUE. Use xaxis.labelFormat=NULL to get a standard x-axis (without date labels). epochsAsDate Boolean indicating whether to treat the epochs as Date objects (or to transform them to dates such that the new x-axis formatting is applied). Default: Value of the epochAsDate slot of x. xlab a title for the x axis. See plot.default. ylab a title for the y axis. See plot.default. main an overall title for the plot: see 'title'. type type of plot to do. lty vector of length 3 specifying the line type for the three lines in the plot -- see col argument. col Vector of length 3 specifying the color to use in the plot. The first color is the fill color of the polygons for the counts bars (NA for unfilled), the 2nd element denotes their border color, the 3rd element is the color of the upperbound plotting. lwd Vector of length 3 specifying the line width of the three elements to plot. See also the col argument. alarm.symbol a list with entries pch, col, cex and lwd specifying the appearance of the outbreak symbol in the plot. outbreak.symbol a list with entries pch, col, cex and lwd specifying the appearance of the outbreak symbol in the plot. legend.opts a list of arguments for the legend. If missing(legend.opts) (i.e., not explicitly specified), the default legend will only be added if the "sts" object contains outbreaks, alarms, or upperbounds. The default legend options are x "top" legend c("Infected","Threshold","Outbreak","Alarm")[included] lty,lwd,pch,col the corresponding graphical settings of the included elements where individual elements are only included in the legend if they are plotted (except for alarms, which are also included if upperbounds exist). To disable the legend, use legend.opts=NULL. dx.upperbound horizontal change in the plotting of the upperbound line. Sometimes it can be convenient to offset this line a little for better visibility. lvl A vector of length ncol(x), which is used to specify the hierarchy level for each time series in the sts object for alarm plots. cex.yaxis The magnification to be used for y-axis annotation. hookFunc a function that is called after all the basic plotting has be done, i.e., it is not possible to control formatting with this function. See Examples. .hookFuncInheritance a function which is altered by sub-classes plot method. Do not alter this function manually. ... further arguments for the function matplot. If e.g. xlab or main are provided they overwrite the default values. ## Details The time series plot relies on the work-horse stsplot_time1. Its arguments are (almost) similar to plot.survRes. ## Value NULL (invisibly). The functions are called for their side-effects. ## Author Michael Höhle and Sebastian Meyer There is an autoplot-method, which implements ggplot2-based time-series plots of "sts" objects. The stsplot help page gives an overview of other types of plots for "sts" objects. ## Examples data("ha.sts") print(ha.sts) plot(ha.sts, type=observed ~ time \| unit) # default multivariate type plot(ha.sts, units=c("mitt", "pank")) # selected units plot(ha.sts, type=observed ~ time) # aggregated over all districts ## Hook function example hookFunc <- function() grid(NA,NULL,lwd=1) plot(ha.sts, hookFunc=hookFunc) ## another multivariate time series example plotted "as.one" data("measlesDE") plot(measlesDE, units=1:2, as.one=TRUE, legend.opts=list(cex=0.8)) ## more sophisticated plots are offered by package "xts" if (requireNamespace("xts")) plot(as.xts.sts(measlesDE)) ## Use ISO8601 date formatting (see ?strptime) and no legend data("salmNewport") plot(aggregate(salmNewport,by="unit"), xlab="Time (weeks)", xaxis.tickFreq=list("%m"=atChange,"%G"=atChange), xaxis.labelFreq=list("%G"=atMedian),xaxis.labelFormat="%G") ## Formatting now also works for daily data (illustrate by artifical ## outbreak converted to sts object by linelist2sts) set.seed(123) exposureTimes <- as.Date("2014-03-12") + sample(x=0:25,size=99,replace=TRUE) sts <- linelist2sts(data.frame(exposure=exposureTimes), dateCol="exposure",aggregate.by="1 day") ## Plot it with larger ticks for days than usual surveillance.options("stsTickFactors"=c("%d"=1, "%W"=0.33, "%V"=0.33, "%m"=1.75, "%Q"=1.25, "%Y"=1.5, "%G"=1.5)) plot(sts,xaxis.tickFreq=list("%d"=atChange,"%m"=atChange), xaxis.labelFreq=list("%d"=at2ndChange),xaxis.labelFormat="%d-%b", xlab="Time (days)")	2023-03-23 21:19:13	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.5242465138435364, "perplexity": 9125.80389779403}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945183.40/warc/CC-MAIN-20230323194025-20230323224025-00666.warc.gz"}
https://djalil.chafai.net/blog/2011/09/page/2/	# Month: September 2011 I have uploaded lecture notes entitled Around the circular law written with Charles Bordenave (hal-00623894 & arXiv:1109.3343). These lecture notes are the expanded version of a joint course that we gave at the occasion of the France-China summer school held in Changchun. They incorporate some posts from this blog. It was a great pleasure for us to write down this synthesis, accumulating few years of thinking on the subject, sometimes with our friend Pietro Caputo (I will remember this stimulating week in Rome!). I hope to reduce my time turning around the circular law in 2012, even if I am still fascinated by few of the open problems 😉 Girko (1946 – ) contributed substantially to the solution of the circular law problem. His interest in random matrices came from his early works on random determinants, motivated at the origin by the van der Waerden conjecture: among all $n\times n$ doubly stochastic matrices, the matrix with all entries equal to $1/n$ has minimal permanent. It turns out that the van der Waerden conjecture was solved around 1980 by another Ukrainian mathematician: Falikman. Personally (author of this blog), I have two favorites (still open) problems on doubly stochastic matrices: • find an exact representation of the uniform law on the polytope of doubly stochastic matrices (similar to what we have for the $\ell^1$ ball with i.i.d. exponentials), • show that the empirical spectral distribution of random doubly stochastic matrices (distributed according to the uniform law) tends to the circular law. This is why I came to random matrix theory… A proof of the van der Waerden conjecture can be found in the last chapter of the book Matrix inequalities by Zhan. Coincidentally, this book contains also somewhere (proof of theorem 3.32) an inequality (for singular values and rows norms) which turned out to be a crucial ingredient in the solution of the heavy tailed analogue of the circular law theorem (obtained in collaboration with Bordenave and Caputo). I knew this book from my postdoc (by simple curiosity), but it became useful only ten years later! Van der Waerden is famous for his heorem in Ramsey theory about the structure of integers. Have you ever heard about John Michael Hammersley (1920-2004)? May be not. Hammersley was an eccentric British mathematician who contributed to probability theory at large. Hammersley studied in Cambridge, and participated, as many others, to the British efforts during the second world war: “[…] Finally in 1942 I was transferred to the Trials Wing at Lydstep. Amongst the personnel at Trials Wing there was a team of about 40 girls who carried out the computations necessary for analyzing the performance of the anti-aircraft equipment, and I was responsible for directing their calculations. One of their jobs consisted in operating the kinetheodolites for tracking a target. The kinetheodolites were a pair of synchronized telescopic cameras at each end of a base line about a couple of miles long, which could give simultaneous readings of the respective angles to a target (either an aircraft or a radar sleeve towed behind an aircraft). From the resulting data it was possible to compute fairly accurate positions of the target and how these positions depended upon time as the target moved along its flight path. In practice it was just an ugly piece of three-dimensional trigonometry; and when I first arrived at Lydstep it was done with pencil and paper with the aid of a 7-figure tables of trigonometric functions, in accordance with traditions of military surveyors. But while surveyors may conceivably be interested in determining a position to the nearest fraction of an inch, it was nonsense to do so for an aircraft target in view of the more dominant errors inherent in gunnery. One of my first reforms was simply to introduce 4-figure trigonometric tables, and to equip the computing room with desk calculating machines in place of longhand pencil and paper sums. The calculating machines were winkled out of the Treasury, who were keeping them massed in a big cupboard in case they might be of future service for financial purposes. […]”, in John Michael Hammersley (1920-2004) by Grimmett and Welsh, 2006. Grimmett and Welsh are two former students of Hammersley. They also edited a volume in honor of Hammersley entitled Disorder in Physical Systems, originally published by Oxford University Press in 1990 (all rights gracefully returned to the authors by OUP in 2001). Hammersley lived in England and in the US. He was an imaginative problem solver, one of these mathematicians who refused to split mathematics into pure mathematics and applied mathematics. He passionately and provocatively advocated the importance of problem solving and the analysis of concrete mathematical models, in opposition to the exclusive development of abstraction in Mathematics. Recall that this was a huge debate in the second part of the twentieth century, with structuralism, Nicolas Bourbaki, New Math, … In my humble opinion (blog author), there is no canonical way of doing Mathematics. I appreciate Harmmersley’s arguments because he was fighting against an excess. Mathematics is more a creative art than a technological industry. We should leave each mathematician working using his or her own style, imagination, and tastes. This is of course difficult since the publication process involves evaluations by peers and thus human comedy. Mathematics are rich when both problem solvers and theory builders can work in parallel creatively and successfully. Many of them are complementary. Creativity is reduced if one imposes dogmas or constraints. I do believe that the value of works is only known years after their making. Hammersley wrote several pioneering and influential works, for instance on the following: Hammersley on the MathSciNet database (he published about 90 papers). Hammersley took only eight PhD students. John Halton (known for Halton sequences) was one one them. He tells us how he was unusually recruited: “A cousin drew my attention to an advertisement in the Observer…, seeking applicants for UKAEA Research Studentships, to study Monte Carlo methods for a DPhil at Oxford. . . . In a few weeks, I was invited to “present myself for examination” at the UKAEA site at Didcot. With very little idea of what this would entail, I went. There I found a [number of ] equally bemused applicants, who were ushered into a large hall furnished with a suitable number of small desks and sat down. John Hammersley strode breezily up to the podium, introduced himself, and asked us to write a four-hour examination, consisting of a dozen or so tough mathematical questions. I attempted to solve each problem in turn, suggested possible lines of approach, and tried to answer the questions posed, with little success. At the end of four hours, the papers were collected and we waited anxiously for the outcome.”, ibid. In 1906, at the age of 50, Andrey Markov introduced his chain model known nowadays as Markov chains. This model played a key role in the development of the theory of stochastic processes during the twentieth century, and is still the source of numerous open and rich problems, with applied or theoretical motivations. It is amusing to note that Perron and Frobenius studied the spectral decomposition of positive matrices during the same period, but the connection with Markov chains was not made immediately. You may read the historical paper of Seneta on the subject. On the theoretical side of probability theory, the study of Markov processes was extremely active after the second world war, but was gradually superseded by the study of general stochastic processes and (semi)martingales for a while. Persi Diaconis (2005): “I only met P.-A. Meyer once (Luminy, 1995). He kindly stayed around after my talk and we spoke for about an hour. I was studying rates of convergence of finite state space Markov chains. He made it clear that, for him, finite state space Markov chains is a trivial subject. Hurt but undaunted, I explained some of our results and methods. He thought about it and said, “I see, yes, those are very hard problems”. […] In the present paper I treat rates of convergence for a simple Markov chain. I am sorry not to have another hour with Paul-Andre Meyer. […]”. Finite Markov chains coincide with random walks on weighted graphs. Personally, I have learnt Markov chains at the end of the past century from Dominique Bakry and Laurent Saloff-Coste. I like very much this probabilistic subject, connected to linear algebra, algebra, analysis, statistics, geometry, and ergodic theory. It plays a role in many applied domains, has connections with Computer Science and Physics(), and allows simulation. Last but not least, it carries many fascinating elementary problems. Here is one of them: consider a sequence of complex numbers lying in the unit disk of the complex plane, stable by conjugacy and containing the unity. Can you construct a Markov chain which admits this sequence as its spectrum? How? Some answers can be found in a book by Chu and Golub (section 4.5). () to me econometrics and mathematical finance/biology are part of Physics at large. The famous Mersenne twister pseudo-random generator is an example of a finite state space Markov chain (matrix linear recurrence over $\mathrm{F}_{\!\!2}$) used billions of times daily worldwide. Here are some basic books (in English) about discrete Markov chains that I like: and more advanced or specialized books (listed here in a pseudo-random order): Still I am not completely satisfied by this list of references. The subject is actually so rich that all these books are complementary. About Markov chains, did you know that Noam Chomsky became famous fifty years ago by proving that no matter how complex a finite state machine is, it could not handle all constructions of English? Last Updated on 2012-12-13 The Girko circular law theorem states that if ${X}$ is a ${n\times n}$ random matrix with independent and identically distributed entries (i.i.d) of variance ${1/n}$ then the empirical measure $\frac{1}{n}\sum_{i=1}^n\delta_{\lambda_i(X)}$ made with the eigenvalues of ${X}$, converges, as the dimension ${n}$ tends to infinity, to the uniform law on the unit disk ${\{z\in\mathbb{C}:\|z\|\leq1\}}$. This is to me the most beautiful spectral theorem in random matrix theory (well, I also like the Voiculescu asymptotic freeness theorem). The random matrix ${X}$ has i.i.d. entries and its eigenvalues are the roots of its characteristic polynomial. The coefficients of this random polynomial are neither independent nor identically distributed. Beyond random matrices, let us consider a random polynomial $P(z)=a_0+a_1z+\cdots+a_nz^n$ where ${a_0,\ldots,a_n}$ are independent random variables. By analogy with random matrices, one may ask about the behavior as ${n\rightarrow\infty}$ of the roots $\lambda_1(P),\ldots,\lambda_n(P)$ of ${P}$ in ${\mathbb{C}}$ and in particular the behavior of their empirical measure $\frac{1}{n}\sum_{i=1}^n\delta_{\lambda_i(P)}.$ The literature on this subject is quite rich and can be traced back to the works of Littlewood and Offord, Rice, and Kac. We refer to Shub and Smale, Azaïs and Wschebor, and Edelman and Kostlan (see also this paper) for (partial) reviews. As for random matrices, the case where the coefficients are real is more subtle due to the presence of real roots. Regarding the complex case, the zeros of Gaussian analytic functions is the subject of a recent monograph in connection with determinantal processes. Various cases are considered in the literature, including the following three families: • Kac polynomials, for which ${(a_i)_{0\leq i\leq n}}$ are i.i.d. • Binomial polynomials, for which ${a_i=\sqrt{\binom{n}{i}}b_i}$ for all ${i}$ and ${{(b_i)}_{0\leq i\leq n}}$ are i.i.d. • Weyl polynomials, for which ${a_i=\frac{1}{\sqrt{i!}}b_i}$ for all ${i}$ and ${{(b_i)}_{0\leq i\leq n}}$ are i.i.d. Geometrically, the complex number ${z}$ is a root of ${P}$ if and only if the vectors $(1,z,\ldots,z^n) \quad\mbox{and}\quad (\overline{a_0},\overline{a_1},\ldots,\overline{a_n})$ are orthogonal in ${\mathbb{C}^{n+1}}$, and this connects the problem to Littlewood-Offord type problems and small balls probabilities estimates, see this 1943 paper for instance. Regarding Kac polynomials, Kac (see also the errata) has shown in the real Gaussian case that the asymptotic number of real roots is about ${\frac{2}{\pi}\log(n)}$ as ${n\rightarrow\infty}$. Kac obtained the same result when the coefficients are uniformly distributed. Hammersley derived an explicit formula for the ${k}$-point correlation of the roots of Kac polynomials. Shparo and Shur have shown that the empirical measure of the roots of Kac polynomials with light tailed coefficients tends as ${n\rightarrow\infty}$ to the uniform law one the unit circle ${\{z\in\mathbb{C}:\|z\|=1\}}$ (the arc law). If the coefficients are heavy tailed then the limiting law concentrates on the union of two centered circles, see Götze and Zaporozhets and references therein. Regarding Weyl polynomials, various simulations and conjectures have been made, see e.g. Galligo and Emiris, Galligo, and Tsigaridas. For instance, if ${(b_i)_{0\leq i\leq n}}$ are i.i.d. standard Gaussian, it was conjectured that the asymptotic behavior of the roots of the Weyl polynomials is analogous to the Ginibre Ensemble. Namely, the empirical distribution of the roots tends as ${n\rightarrow\infty}$ to the uniform law on the centered disc of the complex plane (circular law), and moreover, in the real Gaussian case, there are about ${\frac{2}{\pi}\sqrt{n}}$ real roots as ${n\rightarrow\infty}$ and their empirical distribution tends as ${n\rightarrow\infty}$ to a uniform law on an interval, as for the real Ginibre Ensemble. The complex Gaussian case was considered by Leboeuf and by Peres and Virág, while the real roots of the real Gaussian case were studied by Schehr and Majumdar. Beyond the Gaussian case, it is tempting to try the logarithmic potential approach with the companion matrix of ${P}$. Recall that the companion matrix of the monic polynomial $Q(z):=c_0+c_1z+\cdots+c_{n-1}z^{n-1}+z^n$ is the ${n\times n}$ matrix $\begin{pmatrix} 0 & 1 & 0 & \cdots & 0\\ 0 & 0 & 1 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1\\ -c_0 & -c_1 & -c_2 & \cdots & -c_{n-1}\\ \end{pmatrix}.$ Its characteristic polynomial is ${Q}$. Numerical simulations reveal strange phenomena depending on the law of the coefficients but we ignore it they are purely numerical. Note for instance that if the coefficients of ${P}$ are all real positive then the roots cannot be real positive. The heavy tailed case is also of interest (rings?). Last Updated on 2012-02-02 Syntax · Style · .	2021-04-14 11:49:57	{"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": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6241694688796997, "perplexity": 687.5574049857288}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1618038077810.20/warc/CC-MAIN-20210414095300-20210414125300-00191.warc.gz"}
https://math.stackexchange.com/questions/3046990/kernel-and-range-of-a-linear-operator-in-a-reflexive-space	# kernel and range of a linear operator in a reflexive space Let $$X$$ be a reflexive Banach space and $$T:X\to X$$ is a linear operator. Is it true that $$X = \mathrm{ker}(I-T) \oplus \overline{\mathrm{range}(I-T^\ast)},$$ where $$\oplus$$ is the direct sum? I know this is true when $$X$$ is a Hilbert space (due to the fact that $$Tx=x$$ implies $$T^\ast x =x$$), and I would suspect something like that is true when $$T$$ is a compact operator. How about the general case? • This isn't even true in finite dimensional spaces unless you take some further assumptions (such as $T = T^$) – Omnomnomnom Dec 20 '18 at 0:01 • @Omnomnomnom What if range(I-T) is changed to range(I-T)? – user58955 Dec 20 '18 at 0:05 • Then that's at least correct in the finite dimensional case – Omnomnomnom Dec 20 '18 at 1:28 • What does it mean to add a subspace of $X$ and a subspace of $X^{}$? – Kavi Rama Murthy Dec 20 '18 at 5:58 This fails even in finite dimension. Let $$X=\mathbb C^2$$, and $$T=\begin{bmatrix} 1&1\\0&1\end{bmatrix}.$$ Then $$(I-T)^2=0$$, so $$\overline{\operatorname{ran}(I-T)}\subset\ker(I-T).$$ When $$X$$ is a Hilbert space, it is true that $$\tag1 X=\ker T\oplus \overline{\operatorname{ran} T^}$$ for any bounded linear operator $$T$$, since $$\overline{\operatorname{ran} T^}=(\ker T)^\perp$$. For a general Banach space, the equality $$(1)$$ makes no sense, as $$\ker T\subset X$$ and $$\operatorname{ran} T^\subset X^*$$. More dramatically, it is known that any Banach space that is not isomorphic to a Hilbert space has a non-complemented subspace $$M$$. And if $$X$$ is separable, it is known that there exists a bounded linear $$T:X\to X$$ with $$\ker T=M$$. This impedes $$(1)$$ in general, and there is nothing you can put on the second summand that will make it work.	2019-07-23 00:58:17	{"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": 24, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9413591623306274, "perplexity": 80.63303766362836}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1563195528635.94/warc/CC-MAIN-20190723002417-20190723024417-00453.warc.gz"}
https://machinelearning.subwiki.org/wiki/User:IssaRice/Taking_inf_and_sup_separately	# User:IssaRice/Taking inf and sup separately ## Satement Let $A$ and $B$ be bounded subsets of the real line. Suppose that for every $a\in A$ and $b\in B$ we have $a\geq b$. Then $\inf(A)\geq \sup(B)$. Actually, do $A$ and $B$ have to be bounded? I think they can even be empty! ## Proof Let $a\in A$ and $b\in B$ be arbitrary. We have by hypothesis $a\geq b$. Since $b$ is arbitrary, we have that $a$ is an upper bound of the set $B$, so taking the superemum over $b$ we have $a \geq \sup(B)$ (remember, $\sup(B)$ is the least upper bound, whereas $a$ is just another upper bound). Since $a$ was arbitrary, we see that $\sup(B)$ is a lower bound of the set $A$. Taking the infimum over $a$, we have $\inf(A) \geq \sup(B)$, as required. ## Applications ### liminf vs limsup (Notation from Tao's Analysis I.) Let $(a_n)_{n=m}^\infty$ be a sequence of real numbers. Let $L^- := \liminf_{n\to\infty} a_n$ and let $L^+ := \limsup_{n\to\infty} a_n$. Then we have $L^- \leq L^+$. Consider the sequences $(a^-_N)_{N=m}^\infty$ and $(a^+_N)_{N=m}^\infty$ defined by $a^-_N := \inf(a_n)_{n=N}^\infty$ and $a^+_N := \sup(a_n)_{n=N}^\infty$. Now consider the sets $A := \{a^+_N : N \geq m\}$ and $B := \{a^-_N : N \geq m\}$. If we can show that $a^+_j \geq a^-_k$ for arbitrary $j,k\geq m$, then we can apply the trick to these sets to conclude that $L^+ = \inf(a^+_N)_{N=m} = \inf(A) \geq \sup(B) = \sup(a^-_N)_{N=m} = L^-$. ### Lower and upper Riemann integral (Notation from Tao's Analysis I.) Let $I$ be a bounded interval on the real line, and let $f : I \to \mathbf R$. We have $\overline{\int}_I f := \inf\left\{p.c.\int_I g : g\text{ is a p.c. function on }I\text{ that majorizes }f\right\}$ $\underline{\int}_I f := \sup\left\{p.c.\int_I g : g\text{ is a p.c. function on }I\text{ that minorizes }f\right\}$ We want to show $\underline{\int}_I f \leq \overline{\int}_I f$. Define $A := \left\{p.c.\int_I g : g\text{ is a p.c. function on }I\text{ that majorizes }f\right\}$ $B := \left\{p.c.\int_I g : g\text{ is a p.c. function on }I\text{ that minorizes }f\right\}$ Then we have $\overline{\int}_I f = \inf(A)$ and $\underline{\int}_I f = \sup(B)$. To apply the trick all we need to do is to let $g$ be a p.c. function on $I$ that majorizes $f$, and let $h$ be a p.c. function on $I$ that minorizes $f$, and show that $p.c.\int_I g\geq p.c.\int_I h$. ## alternating series test (this one is more of a failed application) each even partial sum is at least as large as each odd partial sum, so the inf over the even partial sums is at least as large as the sup over the odd partial sums. this actually isn't strong enough to prove what we want. we actually need the stronger condition that the even partial sums are a decreasing sequence, and that the odd partial sums are an increasing sequence, and that eventually their difference becomes arbitrarily small. ## References After I wrote this page, I found the same theorem in Apostol's Calculus (volume 1, 2nd edition, p. 28) in the section "Fundamental properties of the supremum and infimum".	2020-08-11 12:51:32	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 52, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9766350984573364, "perplexity": 113.96736056130399}, "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-2020-34/segments/1596439738777.54/warc/CC-MAIN-20200811115957-20200811145957-00118.warc.gz"}
https://physics.aps.org/synopsis-for/10.1103/PhysRevLett.106.146804	# Synopsis: Early detection Quick detection of photogenerated electrons in a quantum dot may lead to a better method to transfer quantum information between optical and electrical media. In a quantum dot, the spin of an electron can act as a unit of quantum information. The photon is the ideal carrier of this information, which it can convey undisturbed over large distances. Researchers are therefore always on the lookout for robust means to transfer a quantum state between the storage (electron spin) and the messenger (photon). In recent years, the quantum dot has come to be seen as a useful source of photogenerated electrons, which can be probed by so-called single-shot measurements using quantum point contacts. That said, such detection techniques are hampered by the need to complete the detection process before the electron spin flips thermally. Now, Alessandro Pioda, at the University of Tokyo, and coauthors in Japan report in a paper in Physical Review Letters that they may have dealt with this limitation in quantum dots formed from $\text{GaAs}$-based semiconductor heterostructures. By manipulating the electrical properties of the dot, and hence the tunneling time across the dot, they tune the time it takes to detect photogenerated electrons, on occasion making it shorter than the spin-flip time. The short timescale and tunability allows them to determine the spin direction of electrons generated by circularly polarized light. The authors hope that the ability to transfer the polarization of a photon to the spin of an electron will some day lead to a device to coherently transfer quantum information between an optical and an electrical medium—a solid-state quantum repeater. – Sami Mitra More Features » ## Previous Synopsis Materials Science Astrophysics ## Related Articles Quantum Information ### Viewpoint: A Solid Footing for a Quantum Repeater Crystals with rare-earth ions could lead to quantum repeaters that enable secure quantum communications over long distances. Read More » Quantum Information ### Synopsis: A Dark Side for Qubits Dark solitons in a Bose-Einstein condensate could, according to calculations, function as qubits with long lifetimes. Read More » Astrophysics ### Synopsis: Blocking out Starlight A proposed telescope update could enable incoming light from multiple stars to be simultaneously blocked, making it easier to image exoplanets orbiting two or more stars. Read More »	2017-05-26 22:31:09	{"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": 0.397477388381958, "perplexity": 1371.1191035419731}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463608686.22/warc/CC-MAIN-20170526222659-20170527002659-00084.warc.gz"}
https://code.tutsplus.com/articles/queries-in-rails-part-2--cms-26450	Unlimited Plugins, WordPress themes, videos & courses! Unlimited asset downloads! From \$16.50/m # Queries in Rails, Part 2 Length:LongLanguages: In this second article, we’ll dive a little deeper into Active Record queries in Rails. In case you are still new to SQL, I’ll add examples that are simple enough that you can tag along and pick up the syntax a bit as we go. That being said, it would definitely help if you run through a quick SQL tutorial before you come back to continue to read. Otherwise, take your time to understand the SQL queries we used, and I hope that by the end of this series it won’t feel intimidating anymore. Most of it is really straightforward, but the syntax is a bit weird if you just started out with coding—especially in Ruby. Hang in there, it’s no rocket science! ## Topics • Joining Tables • Scopes • Aggregations • Dynamic Finders • Specific Fields • Custom SQL These queries include more than one database table to work with and might be the most important to take away from this article. It boils down to this: instead of doing multiple queries for information that is spread over multiple tables, includes tries to keep these to a minimum. The key concept behind this is called “eager loading” and means that we are loading associated objects when we do a find. If we did that by iterating over a collection of objects and then trying to access its associated records from another table, we would run into an issue that is called the “N + 1 query problem”. For example, for each agent.handler in a collection of agents, we would fire separate queries for both agents and their handlers. That is what we need to avoid since this does not scale at all. Instead, we do the following: #### Rails If we now iterate over such a collection of agents—discounting that we haven't limited the number of records returned for now—we'll end up with two queries instead of possibly a gazillion. #### SQL This one agent in the list has two handlers, and when we now ask the agent object for its handlers, no additional database queries need to be fired. We can take this a step further, of course, and eager load multiple associated table records. If we needed to load not only handlers but also the agent’s associated missions for whatever reason, we could use includes like this. #### Rails Simple! Just be careful about using singular and plural versions for the includes. They depend on your model associations. A has_many association uses plural, while a belongs_to or a has_one needs the singular version, of course. If you need, you can also tuck on a where clause for specifying additional conditions, but the preferred way of specifying conditions for associated tables that are eager loaded is by using joins instead. One thing to keep in mind about eager loading is that the data that will be added on will be sent back in full to Active Record—which in turn builds Ruby objects including these attributes. This is in contrast to “simply” joining the data, where you will get a virtual result that you can use for calculations, for example, and will be less memory draining than includes. ## Joining Tables Joining tables is another tool that lets you avoid sending too many unnecessary queries down the pipeline. A common scenario is joining two tables with a single query that returns some sort of combined record. joins is just another finder method of Active Record that lets you—in SQL terms—JOIN tables. These queries can return records combined from multiple tables, and you get a virtual table that combines records from these tables. This is pretty rad when you compare that to firing all kinds of queries for each table instead. There are a few different kinds of data overlap you can get with this approach. The inner join is the default modus operandi for joins. This matches all the results that match a certain id and its representation as a foreign key from another object or table. In the example below, put simply: give me all missions where the mission’s id shows up as mission_id in an agent’s table. "agents"."mission_id" = "missions"."id". Inner joins exclude relationships that don’t exist. #### SQL So we are matching missions and their accompanying agents—in a single query! Sure, we could get the missions first, iterate over them one by one, and ask for their agents. But then we would go back to our dreadful “N + 1 query problem”. No, thank you! What’s also nice about this approach is that we won’t get any nil cases with inner joins; we only get records returned that match their ids to foreign keys in associated tables. If we need to find missions, for example, that lack any agents, we would need an outer join instead. Since this currently involves writing your own OUTER JOIN SQL, we will look into this in the last article. Back to standard joins, you can also join multiple associated tables, of course. #### Rails And you can add onto these some where clauses to specify your finders even more. Below, we are looking only for missions that are executed by James Bond and only the agents that belong to the mission 'Moonraker' in the second example. #### SQL With joins, you also have to pay attention to singular and plural use of your model associations. Because my Mission class has_many :agents, we can use the plural. On the other hand, for the Agent class belongs_to :mission, only the singular version works without blowing up. Important little detail: the where part is simpler. Since you are scanning for multiple rows in the table that fulfill a certain condition, the plural form always makes sense. ## Scopes Scopes are a handy way to extract common query needs into well-named methods of your own. That way they are a bit easier to pass around and also possibly easier to understand if others have to work with your code or if you need to revisit certain queries in the future. You can define them for single models but use them for their associations as well. The sky is the limit really—joins, includes, and where are all fair game! Since scopes also return ActiveRecord::Relations objects, you can chain them and call other scopes on top of them without hesitation. Extracting scopes like that and chaining them for more complex queries is very handy and makes longer ones all the more readable. Scopes are defined via the “stabby lambda” syntax: #### SQL As you can see from the example above, finding James Bond is much nicer when you can just chain scopes together. That way you can mix and match various queries and stay DRY at the same time. If you need scopes via associations, they are at your disposal as well: You can also redefine the default_scope for when you are looking at something like Mission.all. ## Aggregations This section is not so much advanced in terms of the understanding involved, but you will need them more often than not in scenarios that can be considered a bit more advanced than your average finder—like .all, .first, .find_by_id or whatever. Filtering based on basic calculations, for example, is most likely something that newbies don’t get in touch with right away. What are we looking at exactly here? • sum • count • minimum • maximum • average Easy peasy, right? The cool thing is that instead of looping through a returned collection of objects to do these calculations, we can let Active Record do all this work for us and return these results with the queries—in one query preferably. Nice, huh? • count #### SQL • average #### SQL Since we now know how we can make use of joins, we can take this one step further and only ask for the average of gadgets the agents have on a particular mission, for example. #### SQL Grouping these average number of gadgets by missions' names becomes trivial at that point. See more about grouping below: #### SQL • sum #### SQL • maximum #### SQL • minimum ## Attention! All of these aggregation methods are not letting you chain on other stuff—they are terminal. The order is important to do calculations. We don’t get an ActiveRecord::Relation object back from these operations, which makes the music stop at that point—we get a hash or numbers instead. The examples below won’t work: ### Grouped If you want the calculations broken down and sorted into logical groups, you should make use of a GROUP clause and not do this in Ruby. What I mean by that is you should avoid iterating over a group which produces potentially tons of queries. #### SQL This example finds all the agents that are grouped to a particular mission and returns a hash with the calculated average number of gadgets as its values—in a single query! Yup! The same goes for the other calculations as well, of course. In this case, it really makes more sense to let SQL do the work. The number of queries we fire for these aggregations is just too important. ## Dynamic Finders For every attribute on your models, say name, email_addressfavorite_gadget and so on, Active Record lets you use very readable finder methods that are dynamically created for you. Sounds cryptic, I know, but it doesn’t mean anything other than find_by_id or find_by_favorite_gadget. The find_by part is standard, and Active Record just tucks on the name of the attribute for you. You can even get to add an ! if you want that finder to raise an error if nothing can be found. The sick part is, you can even chain these dynamic finder methods together. Just like this: #### SQL Of course you can go nuts with this, but I think it loses its charm and usefulness if you go beyond two attributes: #### SQL In this example, it is nevertheless nice to see how it works under the hood. Every new _and_ adds an SQL AND operator to logically tie the attributes together. Overall, the main benefit of dynamic finders is readability—tucking on too many dynamic attributes loses that advantage quickly, though. I rarely use this, maybe mostly when I play around in the console, but it’s definitely good to know that Rails offers this neat little trickery. ## Specific Fields Active Record gives you the option to return objects that are a bit more focused about the attributes they carry. Usually, if not specified otherwise, the query will ask for all the fields in a row via * (SELECT "agents".*), and then Active Record builds Ruby objects with the complete set of attributes. However, you can select only specific fields that should be returned by the query and limit the number of attributes your Ruby objects need to “carry around”. #### SQL As you can see, the objects returned will just have the selected attributes, plus their ids of course—that is a given with any object. It makes no difference if you use strings, as above, or symbols—the query will be the same. #### Rails A word of caution: If you try to access attributes on the object that you haven’t selected in your queries, you will receive a MissingAttributeError. Since the id will be automatically provided for you anyway, you can ask for the id without selecting it, though. ## Custom SQL Last but not least, you can write your own custom SQL via find_by_sql. If you are confident enough in your own SQL-Fu and need some custom calls to the database, this method might come in very handy at times. But this is another story. Just don’t forget to check for Active Record wrapper methods first and avoid reinventing the wheel where Rails tries to meet you more than halfway. #### Rails Unsurprisingly, this results in: #### SQL Since scopes and your own class methods can be used interchangeably for your custom finder needs, we can take this one step further for more complex SQL queries. #### Rails We can write class methods that encapsulate the SQL inside a Here document. This lets us write multi-line strings in a very readable fashion and then store that SQL string inside a variable which we can reuse and pass into find_by_sql. That way we don’t plaster tons of query code inside the method call. If you have more than one place to use this query, it’s DRY as well. Since this is supposed to be newbie-friendly and not an SQL tutorial per se, I kept the example very minimalistic for a reason. The technique for way more complex queries is quite the same, though. It’s easy to imagine having a custom SQL query in there that stretches beyond ten lines of code. Go as nuts as you need to—reasonably! It can be a life saver. A word about the syntax here. The SQL part is just an identifier here to mark the beginning and end of the string. I bet you won’t need this method all that much—let’s hope! It definitely has its place, and Rails land wouldn’t be the same without it—in the rare cases that you will absolutely want to fine-tune your own SQL with it. ## Final Thoughts I hope you got a bit more comfortable writing queries and reading the dreaded ol’ raw SQL. Most of the topics we covered in this article are essential for writing queries that deal with more complex business logic. Take your time to understand these and play around a bit with queries in the console. I’m pretty sure that when you leave tutorial land behind, sooner or later your Rails cred will rise significantly if you work on your first real-life projects and need to craft your own custom queries. If you are still a bit shy of the topic, I’d say simply have fun with it—it really is no rocket science!	2021-04-20 10:26:24	{"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": 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": 0.24418701231479645, "perplexity": 1195.392835945453}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1618039388763.75/warc/CC-MAIN-20210420091336-20210420121336-00131.warc.gz"}
http://www.webassign.net/manual/instructor_guide/t_i_displaying_choices_horizontally_or_tables.htm	# Display Choices Horizontally or in Tables By default, the choices for a multiple-choice question are displayed as a vertical list. Sometimes, you might want to arrange the choices horizontally or in a table. ## Example Horizontal Multiple-Choice Question The following table summarizes an actual question. QID 1158695 Name Template2 2.MC.04. Mode Multiple-Choice Question What is the sum of the first five natural numbers? <_> <_> <_> Answer 15 120 18 Display to Students ## Example Tabular Multiple-Choice Question The following table summarizes an actual question. Note: Always specify alternative text when adding images to your questions. QID 1247290 Name Template2 2.MC.05. Mode Multiple-Choice Question Identify the wood duck. (Public domain images from http://photogallery.nrcs.usda.gov) <_><_> <_><_> Answer 'top')> 'top')> 'top')> 'top')> Display to Students	2015-08-02 04:16:24	{"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": 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": 0.675460696220398, "perplexity": 11147.438614898341}, "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-2015-32/segments/1438042988962.66/warc/CC-MAIN-20150728002308-00212-ip-10-236-191-2.ec2.internal.warc.gz"}
http://old.marketingforum.com.ua/f6zhh6/8e27f8-wavelength-graph-generator	Waves Birkenhead College. To change the wave type from a sine wave (pure tone) to a square/triangle/sawtooth wave, click thebutton. (1)2.5 m/s (3)35 m/s (2)15 m/s (4)250 m/s 7. 2 Conversion of wavelength into RGB values and resulting spectrum: To convert a particular wavelength of light into a colour that can be displayed on a computer monitor, an algorithm is necessary to generate RGB values (the amplitude of Red, Green and Blue signals) used by the computer display. The IR Spectrum Table is a chart for use during infrared spectroscopy.The table lists IR spectroscopy frequency ranges, appearance of the vibration and absorptions for functional groups. Amplitude and Period bengraber. To override the computer's automatic scales, type in the ranges for your axes and switch to manual scale. Doppler Effect Applet (2) 400 nm is the wavelength of light at the blue end of the visible spectrum. Guitar String Makes a Longitudinal Wave. where 'e' is energy (joules), 'f' is frequency (cycles per second), 'h' is Planck's constant (6.6260695729 x 10-34 Jouleseconds) and wavelength λ is in meters. Sales of refurbished/previous rental stock equipment are limited to stock on hand and pricing is subject to change without notice. The process begins by coating a substrate with a light-sensitive organic material. Calculus: Integral with adjustable bounds. f(x) = 1011.79 x + 1.11779 E 6 . A wave generator located 4.0 meters from a reflecting wall produces a standing wave in a string, as shown in the diagram below. © Academo.org 2020. Step 2: Choose which column you want to put on your x and y axes. Determining the Harmonic Frequencies. To play a constant tone, click Play or press Space.. To change the frequency, drag the slider or press ← → (arrow keys). Given frequency, distance and time. The tone will continue until the stop button is pushed. Lost a graph? People get wavelength and period mixed up all the time. Different detection modes are available: peak mode, area mode, point mode, threshold mode, markers mode. Tuning Fork Makes a Longitudinal Wave. For λ = wavelength of the incident photon, then 3. In addition to changing the dimensions and background color of the waveform image, you can also adjust the colors of the gradient that is used on the waveform visualization. The grating width and the total number of vertical rulings on the grating are given for each one. Keysight's function generator and waveform generator products offer the standard signals and features engineers expect, such as modulation, sweep, and burst that give the engineer capabilities and flexibility to get the job done quickly. Click here to view all our demos, or click on of the buttons below to browse a specific category.. Engineering Geography Maths Music Physics Simply enter your desired frequency and press play. Sonic Boom. λ th = c/γ th. 5 Tips for Getting the Most Out of Your Function Generator. Putting these figures (without commas) into the calculator above shows that the wavelength is 0.122 metres, or 12.2 centimetres. P.S. wavelength of 10 meters? Since it uses a python interface, any MATLAB version that supports python interaction will be compatible with this script. Wavelength vs. Absorbance is a commonly used graph used in UV-Visible light spectrometers. Generate customiseable waveform images from mp3 and m4a audio files and download them for free If you find this useful, our online spectrum analyser may also be of interest to you. The period (and hence the frequency) remain constant because 8 times 500µs still equals 0.004s. This will take data from any microphone connected to your computer and display the live audio data. k. Type. Revolutionary Waveform and Function Generators. The fiber may comprise three or more segments of fiber having alternating highly dispersive and highly nonlinear characteristics. At first glance, it appears to be a linear relationship. Radio waves travel at the speed of light, so in this case v is equal to 299,792,458 metres per second (m/s), and 2.45 GHz is 2,450,000,000 Hz, so that’s the frequency. What is the wavelength of sine wave? The wavelength of a sound wave is the distance between two compressed regions of air. Save Graph. Putting these figures (without commas) into the calculator above shows that the wavelength is 0.122 metres, or 12.2 centimetres. It is important to be able to interpret these graphs correctly. ... 10MHz Sine Wave Generator. ), If you are browsing using the latest version of Google Chrome, the input dropdown box allows you to select "live input". The process begins by coating a substrate with a light-sensitive organic material. Free, Simple and Easy to Use. What was the shortest-duration EVA ever? Sine wave. Online Tone Generator. The initial signal above is a 200Hz sine wave, which has an amplitude of 5 volts. Wavelength Graph ... Wind Generator: WG: WiseGuy (software) WG: Wegener Granulomatosis: WG: Wire Gauge: WG: Water Gauge: WG: Wie Geht's? x-Axis: y-Axis: x. If you were to change the setting to 10 volts/div, the waveform now only reaches up half of a square. a. You should be able to work out the wavelength of a wave from a displacement-time graph. A collection of interactive, educational demos and tools. Calculate. Step 1: Paste Your Data (TSV or CSV) in the box below. where c is a constant, f is the lens focal length and n is the index of refraction of glass. What was the shortest-duration EVA ever? What is the apparent shape of the graph of wavelength vs. frequency? This inversion is equivalent to a 1/2 wavelength … What is the linear regression equation obtained for wavelength vs. frequency? Determining wave frequency from a graph cheesenuggett. Sound in air = 340 m/s From both together, the wave speed can be determined. Question Sonic Boom. Confirm your country or area to access relevant pricing, special offers, events, and contact information. For instance, a 0.42 MHz sine wave takes 3.3 µs to travel 2500 meters. 2 The intensity of the beam is plotted as a function of the wavelength of the radiation. Today’s function generators can do much more than their predecessors. Draw the graph onto A4 graph paper and use it to determine a value for d. (6) (b) Three diffraction gratings, illustrated below, are available for observing line spectra. Practice: Calculating frequency and wavelength from displacement graphs. Generate customiseable waveform images from mp3 and m4a audio files and download them for free Resonance in a wave on rope. c = Speed of Light (299,792,458 m/s) f = Frequency. Question Other Calculators Wavelength to Frequency Calculator; Popular Calculators dBm to Watts Calculator Academo. Wave characteristics review. An optical pulse generator comprises a comb-like dispersion profiled fiber formed into an optical loop mirror. Its most basic form as a function of time (t) is: A displacement-distance graph is a snapshot of the wave at any given time. Don't reformat your data. Explore these resources to help you make more accurate measurements. I have implemented the heuristic LAG approach based on the exact solution to MCP presented in graph theory literature as well as Edge Disjoint Path (EDP) approach for performance benchmarking. Looking for abbreviations of WG? hν max = hc/λ min = eV . x. y = x. FIG. Resonance in a wave on rope. example. What is the wavelength of the radiation? where 'e' is energy (joules), 'f' is frequency (cycles per second), 'h' is Planck's constant (6.6260695729 x 10-34 Jouleseconds) and wavelength λ is in meters. This is especially useful TIP: If you add kidszone@ed.gov to your contacts/address book, graphs that you send yourself through this system will not be blocked or filtered. Exploring chromatic aberration. Consider an 80-cm long guitar string that has a fundamental frequency (1st harmonic) of 400 Hz. These two sliders allow you to adjust the position of the oscilloscope's trace on the grid. A displacement-distance graph is a snapshot of the wave at any given time. Clip Art Graph Maker. These are available for purchase with the instrument or anytime thereafter. If you would like to embed the oscilloscope on your own website, please copy and paste the following html onto your web page. Light in water = 225,000 km/s. The rate at which a wave travels from one point to another determines the wave's (1)frequency (3)amplitude (2)period (4)velocity 8. It is Wavelength Graph. Click here to email you a list of your saved graphs. As always, we have written a calculator to make the work a little easier for you. Academo. In the field of electronics, an LC circuit is employed to either generate signals at a particular frequency or select one signal from a more complex signal at a particular frequency. A photoresist (also known simply as a resist) is a light-sensitive material used in several processes, such as photolithography and photoengraving, to form a patterned coating on a surface.This process is crucial in the electronic industry.. Doppler Effect: Siren. WHAT IS SOUNDWAVE ART? You can be confident you are seeing your design’s characteristics, and not that of your waveform generator, in your measurements. Therefore, offer to sell is not finalized until a confirmation quote is processed by a Wavelength Lighting representative and payment has been made The relation between frequency, f, speed of wave propagation, v, and wavelength, λ, is given by $$f = \frac{v}{\lambda}$$ (1) The velocity of a wave in a string or wire will depend on the tension, T, and the mass per unit length (μ = m/L) of the string. Learn how to use five advanced waveform capabilities that help improve testing and to save you time in the lab. Many options are available including linear, sine, exponential, inverse, parabolic and more. If you're seeing this message, it means we're having trouble loading external resources on our website. This is the currently selected item. Step 2: Choose which column you want to put on your x and y axes. (You can also choose to display a square wave. A collection of interactive, educational demos and tools. Our function generator offers the standard signals and features you expect — modulation, sweep, and burst. Click here to email you a list of your saved graphs. Click here to view all our demos, or click on of the buttons below to browse a specific category.. Engineering Geography Maths Music Physics Wavelength. Sound. Practice: Wave energy from graphs. Formula: λ = C/f Where, λ (Lambda) = Wavelength in meters. In the above color spectrum chart, indigo is made a subset of violet color. The Simple Wave Simulator Interactive provides the learner with a virtual wave machine for exploring the nature of a wave, quantitative relationships between wavelength, frequency and speed, and comparisons between transverse waves such as those traveling through a rope and longitudinal waves such as sound. Horizontal and Vertical Offsets / V (volts) Figure 2. With Wavelength, you can experience your music like you’ve never experienced it before. It's completely free, and there's no need to register or sign-in. Connect, control instruments, and automate test sequences with ease, Achieve fast and easy instrument control seamlessly in many PC application environments, Get advanced signal creation and editing capabilities without spending hours programming, Improve test with our portfolio of calibration, technology refresh, finance, and optimization services, Explore subscription plans that provide committed response times, asset tracking, software updates, and more. Wave generators placed at position P and position Q produce water waves of wavelength 4.0 cm. volts / divThis setting is very similar to the timebase setting described above, but instead of stretching the wave along the x-axis, it involves stretching it along the y-axis. This online virtual oscilloscope allows you to visualise live sound input and get to grips with how to adjust the display. This allows you to measure properties of the wave, such as amplitude or frequency. An oscilloscope is a useful tool for anyone working with electrical signals because it provides a visual representation of the signal's shape, or waveform. Calculus: Fundamental Theorem of Calculus seconds / divThis control allows you to adjust the length of time that each square of the grid represents. Sound. DTMF Tone Generator Applet. Doppler Effect Applet (2) 4 Ways to Efficiently Generate Complex Waveforms, How to Easily Create an Arbitrary Waveform Without Programming, Using a Function / Arbitrary Waveform Generator to Generate Pulses, How to Add DC Offsets to a Function Generator's Output, Waveform Modulation with Your Function Generator, How to Time Synchronize Multiple Function Generators Together, Tips on Using Frequency Sweep and List with Your Function Generator, How to Phase Synchronize Two Signals Together with a Function Generator. You will hear a pure tone sine wave sampled at a rate of 44.1kHz. From the distance graph the wavelength may be determined. It also has features that give you the capabilities and flexibility you need to get your job done quickly, no matter how complex. x. y = x. Considering that the wave travels a distance of one wavelength during one period, We know that . Free, Simple and Easy to Use. waveform with the gridlines (this can make it easier for you to count the squares when determining wavelength, for example). Instructions: Use this Period and Frequency Calculator to find the period and frequency of a given trigonometric function, as well as the amplitude, phase shift and vertical shift when appropriate. Standing waves. What is the wavelength of the radiation? DTMF Tone Generator Applet. Instructions. Refer to the Google image (graph) on the next page. Any of the wave parameters below can be changed. If the end is closed, the wave inverts (because it has hit a more rigid medium). The optical loop mirror construction splits an input pulse into two portions that propagate through the CDPF in opposite directions. Two different displaying modes exist: Jones graph, Poincaré sphere. This maximum energy or minimum wavelength is called the Duane-Hunt limit. 3 The graph in Figure 3 also shows a dashed line called the “Littrow line.” When the curve associated with a particular m < 0 order intersects this line, the angle of diffraction is equal and opposite to the angle of incidence. The IR Spectrum Table is a chart for use during infrared spectroscopy.The table lists IR spectroscopy frequency ranges, appearance of the vibration and absorptions for functional groups. When the oscilloscope is first loaded, this setting is set at 1ms, and shows one complete waveform over 4 squares. Speed = Wavelength x Frequency. Let’s make waves Makati Science High School. So we can write the above equation as: That is, the speed of a wave is equal to its frequency multiplied by the wavelength. This means that the period of the wave is 4ms, or 0.004s, giving a frequency of (1/0.004) = 250Hz. Advertisement. Graph theory technique was adopted once I recognized the similarity between Wavelength Assignment Problem (WAP) and Maximum Clique Problem (MCP). Applet: Doppler Effect. Calculation precision. The frequency of wavelength range for indigo is around 425-450 nm and frequency of 670-700 THz. By default, k = 1, a = 0, which gives us a classic graph . A photoresist (also known simply as a resist) is a light-sensitive material used in several processes, such as photolithography and photoengraving, to form a patterned coating on a surface.This process is crucial in the electronic industry.. Lesson 34: Photoelectric Effect Graphs Like many other topics in science, the results of the photoelectric effect can be better understood if the results are presented in a graph. Import Data. Resonance. Optionally, the state of polarization can be measured. The file is very large. This occurs whether the other end of the tube is closed or open. Doppler Effect: Siren. m. Click here to view image. The well-known American author, Bill Bryson, once said: “Physics is really nothing more than a search for ultimate simplicity, but so far all we have is a kind of elegant messiness.” People get these mixed up because there's an alternate way to create a graph of this sound wave. You can mi… They are particularly useful for lining up parts of the A gain of 1 will have no effect, a gain of less than 1 will make the signal smaller and a gain of more than 1 will make it larger. The frequency of this wave can be adjusted by using the "Input Wave Frequency" slider. Guitar String Makes a Longitudinal Wave. Email this graph HTML Text To: You will be emailed a link to your saved graph project where you can make changes and print. Lifetime Only Specific Range (Select on graph above) Exclude Specific Range (Select on graph above) They are particularly useful for lining up parts of the waveform with the gridlines (this can make it easier for you to count the squares when determining wavelength, for example). The low range of the color explains why it is difficult to distinguish this color in the spectral band. (Different microphones send different voltages to the computer, so for consistency we have normalised the input so the raw input signal will always be limited to somewhere between -5 and +5 volts.). Wavelength Graph listed as WG. Email this graph HTML Text To: You will be emailed a link to your saved graph project where you can make changes and print. Exploring the Relationship Between Wavelength and Frequency . 16. s/m 3 Speed of sound of air at 20°C is c = 343 m/s "Distance = velocity × time" is the key to the basic wave relationship. Radio waves travel at the speed of light, so in this case v is equal to 299,792,458 metres per second (m/s), and 2.45 GHz is 2,450,000,000 Hz, so that’s the frequency. The demo is written in MATLAB 2019a and has been tested under Windows 7 and 10. Next lesson. The sine wave has an amplitude of 5V, meaning when volts/div is set to 5, the waveform just reaches the top of the first square. Save Graph. From the time graph, the period and frequency can be obtained. Adjust the timebase to a convenient scale allows you to calculate the frequency of your whistle by counting the period of one complete waveform. Given frequency, distance and time. Trueform Series Waveform/Function Generator, 33503A BenchLink Waveform Builder Pro Software, High-Speed Digitizers + Multichannel Data Acquisition Solutions, Dynamic Signal Analyzers, Materials Measurement, Parameter + Device Analyzers, Curve Tracer, Handheld Oscilloscopes, Analyzers, Meters, EMI + EMC Measurements, Phase Noise, Physical Layer Test Systems, LCR Meters + Impedance Measurement Products, Modern Slavery Act Transparency Statement. Fig. 2. Calculator City, calculators for algebra, geometry, trigonometry, calculus, finance, astronomy, physics, chemistry, time cards, HTML Colors, percentages, fractions (German greeting) WG: Wade-Giles: WG: Wave Guide: WG: The tone will continue until the stop button is pushed. Import Data. In addition to changing the dimensions and background color of the waveform image, you can also adjust the colors of the gradient that is used on the waveform visualization. The 33500B and 33600A series Trueform generators offer the highest signal fidelity with the lowest jitter and harmonic distortion in their class, The 33210A pairs uncompromising performance with an entry-level product that delivers basic functions and waveforms. Change Data Set Cycles 5 Data Sets. Show graph. Hot Network Questions What was the "5 minute EVA"? There are two tables grouped by frequency range and compound class. To play a constant tone, click Play or press Space. BRIEF DESCRIPTION OF THE DRAWINGS. What is the wavelength of sine wave? As always, we have written a calculator to make the work a little easier for you. It's completely free, and there's no need to register or sign-in. Enable Javascript and browser cookies for improved site capabilities and performance. Enable browser cookies for improved site capabilities and performance. A tunable optical comb generator having a source laser configured to generate a continuous wave (CW) light at a first wavelength; and a microresonator coupled to the source laser and configured to receive the CW light and generate an optical signal having a plurality of output wavelengths corresponding to the first wavelength. because you can still adjust the time base and volts per division setting. If you change the timebase to 500µs (half of what it started at), you should see the waveform now takes 8 squares to complete one full oscillation. Use these results to obtain the values needed to plot a straight line graph. Transloadit can generate waveform images from audio files. The following graph is a typical example of how the photoelectric effect will be shown to you. The wavelength result is 3 m. Most common velocities: Light in vacuum (air) = 300,000 km/s. Transloadit can generate waveform images from audio files. With Keysight, you will always have the latest capabilities. Graph of AC voltage over time (the sine wave). 700 nm is the wavelength of light at the red end of the visible spectrum. The tone generator can play four different waveforms: Sine, Square, Sawtooth and … Wave velocity (m/s) =Wavelength (m) * Frequency (Hz) Example calculation. x-Axis: y-Axis: x. Sine wave. The tone generator can play four different waveforms: Sine, Square, Sawtooth and … Spectrometers are commonly used to identify the presence or relative quantity of chemicals such as molecules or protein in solution. WG - Wavelength Graph. A sine wave or sinusoid is a mathematical curve that describes a smooth periodic oscillation.A sine wave is a continuous wave.It is named after the function sine, of which it is the graph.It occurs often in both pure and applied mathematics, as well as physics, engineering, signal processing and many other fields. Soundwave Art ™ offers the unique experience of converting your voice or any sound into personalized art or jewelry. TIP: If you add kidszone@ed.gov to your contacts/address book, graphs that you send yourself through this system will not be blocked or filtered. You should be able to work out the wavelength of a wave from a displacement-time graph. To change the frequency, drag the slider or press ← → (arrow keys).To adjust the frequency by 1 Hz, use the buttons or press Shift + ← and Shift + →.To adjust the frequency by 0.01 Hz, press Ctrl + ← and Ctrl + →;to adjust it by 0.001 Hz, press Ctrl + Shift + ← and Ctrl + Shift+ →To halve/double the frequency (go down/up one octave), click×½ and ×2. The lens maker's equation approximately states 1 f = c(n − 1). For all the endless customization options, ... please click and drag a selection on a graph above or click on a specific bar. There are two tables grouped by frequency range and compound class. For instance, a 0.42 MHz sine wave takes 3.3 µs to travel 2500 meters. When you have finished entering data, click on the quantity you wish to calculate. If you're seeing this message, it means we're having trouble loading external resources on our website. Lightpath Affiliation Graph approach for wavelength ... we propose to apply Lightpath Affiliation Graph partitioning heuristic for ... generated by Georgia Tech GT-ITM topology generator. Threshold Wavelength (λ th) During the emission of electrons, a metal surface corresponding to the greatest wavelength to incident light is known threshold wavelength. You will hear a pure tone sine wave sampled at a rate of 44.1kHz. Tacoma Bridge Video Links. Oscilloscope gainThis is a number that the incoming signal is multiplied by. The Trueform function generator offers the highest signal fidelity so you can generate the exact waveforms you need for your most challenging measurements. Try whistling and freezing the input. Acquires the spectrum and plots the data to a graph (amplitude vs wavelength information). Let’s take for instance the case of a wave with a frequency of 56 Hz going through a material at a speed of 168 m/s. Don't reformat your data. Tacoma Bridge Video Links. Piano. The outermost scale around the perimeter of Smith chart is called the Wavelengths towards generator(WTG) scale.It has been constructed to denote movement on the transmission line toward the generator,in units of the wavelength,λ.Thus ,l is measured in wavelengths and one complete rotation corresponds to l=λ/2. Please type in a periodic function (For example: $$f(x) = 3\sin(\pi x)+4$$) 1 is a graph showing a relationship between the absorbance and the wavelength in the transient absorption spectrum of an iridium hydride complex. Applet: Doppler Effect. Practice using a displacement graph and wave speed to find the frequency and wavelength of a wave. Just as a fingerprint is unique, your voice generates its own distinctive pattern, providing you with the opportunity to create a personal and truly unique gift. The present invention relates to a photoacid generator comprising a metal hydride complex represented by the formula (I): wherein X represents a metal atom. Modes are available including linear, sine, exponential, inverse, parabolic and more for Getting the Most of... For purchase with the instrument or anytime thereafter λ ( Lambda ) = wavelength of the tube is,. Images from audio files calculate the frequency of 670-700 THz in your measurements advanced waveform capabilities that help improve and! Our online spectrum analyser may also be of interest to you should able... Experience of converting your voice or any sound into personalized Art or.. Features you expect — modulation, sweep, and shows one complete waveform interactive educational... Still adjust the position of the wave speed to find the frequency of 670-700 THz 's automatic scales, in. Spectrum and plots wavelength graph generator data to a convenient scale allows you to visualise live sound input and get grips... Be determined this inversion is equivalent to a graph showing a relationship between wave speed be... Results to obtain the values needed to plot a straight line graph wave travels down an air column and the... The apparent shape of the visible spectrum five advanced waveform capabilities that help improve testing and to save you in! Basic form as a function of wavelength range for indigo is made a subset of violet color ( German ). Free, and there 's no need to get your job done quickly no... And switch to manual wavelength graph generator in the spectral band grating width and the number! Result is 3 m. Most common velocities: light in vacuum ( air ) wavelength. You are seeing your design ’ s function generators can do much more than their predecessors sweep, and 's. Our function generator offers the highest signal fidelity so you can also Choose to a... You ’ ve never experienced it before the instrument or anytime thereafter wavelength in transient. Of 44.1kHz this is especially useful because you can be determined graph of wavelength used in light. The incoming signal is multiplied by you make more accurate measurements protein in solution optical loop.. A fundamental frequency ( Hz ) Example calculation guitar string that has a fundamental frequency ( 1st ). Drag a selection on a graph above or click on the next page dispersion profiled fiber formed into an loop! Lens maker 's equation approximately states 1 f = c ( n − 1 ) these (! Range of the polarimeter or as a function of wavelength graphs correctly Calculators! Art ™ offers the highest signal fidelity so you can still adjust the time it takes an!, our online spectrum analyser may also be of interest to wavelength graph generator Jones,... You time in the box below specific bar are given for each one the sine wave ) ve never it! Stop button is pushed scale allows you to adjust the position of the wavelength of sound. Refer to the Google image ( graph ) on the grid in vacuum ( )... Each one mode, point mode, point mode, area mode, point,. To 10 volts/div, the state of polarization can be measured lens focal length and n is the linear equation... Uses a python interface, any MATLAB version that supports python interaction will be shown to you of the of... The stop button is pushed of 44.1kHz a displacement-time graph the radiation purchase the. Matlab version that supports python interaction will be no photoelectron emission range and compound class rigid medium.! S make waves Makati Science high School first loaded, this setting set. Made a subset of violet color question Transloadit can generate waveform images from audio files download... 2: Choose which column you want to put on your x and y axes flexibility... Files and download them for free save graph equation obtained for wavelength vs. frequency takes for an air molecule oscillate... Air molecule to oscillate back and forth one time gainThis is a tone. And effect relationship between the Absorbance and the wavelength of the color explains why is. Giving a frequency of wavelength range for indigo is around 425-450 nm and frequency of range! Than their predecessors ) to a 1/2 wavelength … online tone generator to access relevant pricing special. The setting to 10 volts/div, the wave speed can be measured always, we written. Waveform over 4 squares register or sign-in were to change the wave speed can be measured to plot straight... Scales, type in the box below a 200Hz sine wave sampled at a rate of.! Image ( graph ) on the grid for purchase with the instrument or thereafter! Means we 're having trouble loading external resources on our website this is! And effect relationship between the Absorbance and the wavelength of the radiation with how to use advanced. Display the live audio data = 1, a = 0, which has amplitude. Generate waveform images from mp3 and m4a audio files and download them for free save graph Art ™ offers unique. Data ( TSV or CSV ) in the above color spectrum chart, is. Sound input and get to grips with wavelength graph generator to use five advanced waveform capabilities that improve... Question Transloadit can generate the exact waveforms you need to register or sign-in design. Violet color Jones graph, the waveform now only reaches up half of a sound wave is the regression! Wave frequency '' slider blue end of the wave inverts ( because it has hit a more medium. Λ = C/f Where, λ ( Lambda ) = 250Hz until the stop button is.. Click on the next page Getting the Most out of your function generator step 2: Choose which column want... The blue end of the polarimeter or as a function of wavelength range of the incident photon then... For indigo is around 425-450 nm and frequency of your saved graphs Google (! Standard signals and features you expect — modulation, sweep, and contact.... Seeing your design ’ s make waves Makati Science high School scales, type in the ranges for your and! 1St harmonic ) of 400 Hz this is especially useful because you can be over! Closed, the waveform now only reaches up half of a sound is... Wavelength range for indigo is made a subset of violet color comb-like dispersion profiled fiber into. Especially useful because you can mi… What is the wavelength in the.... The path of the polarimeter or as a function of the wave is the of! A collection of interactive, educational demos and tools to Watts calculator to make work.	2021-06-22 11:57:17	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2821873426437378, "perplexity": 1665.9361615455496}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488517048.78/warc/CC-MAIN-20210622093910-20210622123910-00295.warc.gz"}
https://www.e-olymp.com/en/problems/8815	Problems # Surface area and volume 2 Find the surface area and volume of the cube by the length of its edge. #### Input The length of cube's edge a (a106). #### Output Print the surface area and volume of the cube . Time limit 1 second Memory limit 128 MiB Input example #1 3 Output example #1 54 27 Author Matviychuk Sergiy Volodymyrovych Source "ABC programming"	2021-08-01 22:20:59	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.768810510635376, "perplexity": 8689.488438537368}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1627046154277.15/warc/CC-MAIN-20210801221329-20210802011329-00629.warc.gz"}
https://www.physicsforums.com/threads/torque-problem-please-help.270704/	1. Nov 9, 2008 ### xregina12 A floodlight with a mass of 40 kg is used to illuminate the parking lot in front of a library. The floodlight is supported at the end of a horizontal beam that is hinged to a vertical pole, as shown. A cable thatmakes an angle of 11◦ with the beam is attached to the pole to help support the floodlight. Assume the mass of the beam is negligible when compared with the mass of the floodlight. The acceleration of gravity is 9.81 m/s2 . a) Find the force FT provided by the cable. 2056 N b) Find the horizontal force exerted on the beam by the pole. Answer in units of N. c) Find the vertical force exerted on the beam by the pole. Answer in units of N. I got a and b but for some reason I can't get c. My work. Fynet=0=Tsin11+Fy-mg where Fy is the vertical force exerted on the beam by the pole. Tsin11+Fy=mg 392.4 +Fy=392.4 Fy=0 however, that doesn't work anyone has suggestions for part c? 2. Nov 9, 2008 ### asleight $$\sum\tau=\tau_1+\tau_2=r\vec{w}+r\vec{T}\rightarrow\vec{w}=-\vec{T}\sin(11\pi/180)\rightarrow\vec{T}=2057N$$. Then, we notice that $$\sin(11\pi/180)\cdot2057=392.5$$ and $$\vec{w}=392.4$$. So, maybe it's just that 0.10N that they're worrying about? IDK.	2018-05-22 08:48:08	{"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": 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": 0.5693628787994385, "perplexity": 1764.5969402654403}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794864648.30/warc/CC-MAIN-20180522073245-20180522093245-00468.warc.gz"}
https://radar.inria.fr/report/2020/tropical/uid0.html	EN FR • Legal notice • Accessibility - non conforme ##### TROPICAL - 2020 2020 Activity report Project-Team TROPICAL RNSR: 201621988K Research center Team name: Tropical methods: structures, algorithms and interactions In collaboration with: Centre de Mathématiques Appliquées (CMAP) Domain Applied Mathematics, Computation and Simulation Theme Optimization and control of dynamic systems Creation of the Team: 2016 January 01, updated into Project-Team: 2018 July 01 # Keywords • A1.2.4. QoS, performance evaluation • A2.3.3. Real-time systems • A2.4. Formal method for verification, reliability, certification • A6.2.5. Numerical Linear Algebra • A6.2.6. Optimization • A6.4.2. Stochastic control • A6.4.6. Optimal control • A7.2.4. Mechanized Formalization of Mathematics • A8.1. Discrete mathematics, combinatorics • A8.3. Geometry, Topology • A8.9. Performance evaluation • A8.11. Game Theory • A9.6. Decision support • B4.3. Renewable energy production • B4.4. Energy delivery • B4.4.1. Smart grids • B6.6. Embedded systems • B8.4. Security and personal assistance • B8.4.1. Crisis management # 1 Team members, visitors, external collaborators ## Research Scientists • Stéphane Gaubert [Team leader, Inria, Senior Researcher, HDR] • Marianne Akian [Inria, Senior Researcher, HDR] • Xavier Allamigeon [Inria, Researcher, Corps des Mines, under secondment] • Yang Qi [Inria, Starting Research Position] • Constantin Vernicos [Univ de Montpellier, Senior Researcher, HDR] • Cormac Walsh [Inria, Researcher] ## Post-Doctoral Fellows • Armando Gutierrez Collaguazo [Académie de Finlande, from Aug 2020] • Hanieh Tavakolipour [Inria] ## PhD Students • Marin Boyet [Inria] • Quentin Canu [Ecole normale supérieure Paris-Saclay, from Sep 2020] • Mael Forcier [École Nationale des Ponts et Chaussées] • Maxime Grangereau [EDF] • Quentin Jacquet [EDF, from Nov 2020] • Paulin Jacquot [EDF, until Jan 2020] • Shanqing Liu [Institut Polytechnique de Paris, from Sep 2020] • Duy Nghi Benoit Tran [Univ Paris-Est Marne La Vallée] ## Technical Staff • Baptiste Colin [Inria, Engineer] ## Interns and Apprentices • Quentin Canu [Ecole normale supérieure Paris-Saclay, from Mar 2020 until Jul 2020] • Matteo Clémot [École Normale Supérieure de Lyon, from Jun 2020 until Jul 2020] • Ayoub Foussoul [Inria, from Mar 2020 until Aug 2020] • Shanqing Liu [Inria, from Mar 2020 until Sep 2020] • Luz Valerie Pascal [Inria, from Jun 2020 until Nov 2020] # 2 Overall objectives The project develops tropical methods motivated by applications arising in decision theory (deterministic and stochastic optimal control, game theory, optimization and operations research), in the analysis or control of classes of dynamical systems (including timed discrete event systems and positive systems), in the verification of programs and systems, and in the development of numerical algorithms. Tropical algebra tools are used in interaction with various methods, coming from convex analysis, Hamilton–Jacobi partial differential equations, metric geometry, Perron-Frobenius and nonlinear fixed-point theories, combinatorics or algorithmic complexity. The emphasis of the project is on mathematical modelling and computational aspects. The subtitle of the Tropical project, namely, “structures, algorithms, and interactions”, refers to the spirit of our research, including a methodological component, computational aspects, and finally interactions with other scientific fields or real world applications, in particular through mathematical modelling. ## 2.1 Scientific context Tropical algebra, geometry, and analysis have enjoyed spectacular development in recent years. Tropical structures initially arose to solve problems in performance evaluation of discrete event systems 51, combinatorial optimization 57, or automata theory 97. They also arose in mathematical physics and asymptotic analysis 87, 84. More recently, these structures have appeared in several areas of pure mathematics, in particular in the study of combinatorial aspects of algebraic geometry 76, 111, 101, 80, in algebraic combinatorics 69, and in arithmetics 62. Also, further applications of tropical methods have appeared, including optimal control 88, program invariant computation 48 and timed systems verification 86, and zero-sum games 2. The term `tropical' generally refers to algebraic structures in which the laws originate from optimization processes. The prototypical tropical structure is the max-plus semifield, consisting of the real numbers, equipped with the maximum, thought of as an additive law, and the addition, thought of as a multiplicative law. Tropical objects appear as limits of classical objects along certain deformations (“log-limits sets” of Bergman, “Maslov dequantization”, or “Viro deformation”). For this reason, the introduction of tropical tools often yields new insights into old familiar problems, leading either to counterexamples or to new methods and results; see for instance 111, 92. In some applications, like optimal control, discrete event systems, or static analysis of programs, tropical objects do not appear through a limit procedure, but more directly as a modelling or computation/analysis tool; see for instance 106, 51, 78, 58. Tropical methods are linked to the fields of positive systems and of metric geometry 94, 12. Indeed, tropically linear maps are monotone (a.k.a. order-preserving). They are also nonexpansive in certain natural metrics (sup-norm, Hopf oscillation, Hilbert's projective metric, ...). In this way, tropical dynamical systems appear to be special cases of nonexpansive, positive, or monotone dynamical systems, which are studied as part of linear and non-linear Perron-Frobenius theory 85, 3. Such dynamical systems are of fundamental importance in the study of repeated games 91. Monotonicity properties are also essential in the understanding of the fixed points problems which determine program invariants by abstract interpretation 63. The latter problems are actually somehow similar to the ones arising in the study of zero-sum games; see 7. Moreover, positivity or monotonicity methods are useful in population dynamics, either in a discrete space setting 107 or in a PDE setting 53. In such cases, solving tropical problems often leads to solutions or combinatorial insights on classical problems involving positivity conditions (e.g., finding equilibria of dynamical systems with nonnegative coordinates, understanding the qualitative and quantitative behavior of growth rates / Floquet eigenvalues 10, etc). Other applications of Perron-Frobenius theory originate from quantum information and control 100, 105. # 3 Research program ## 3.1 Optimal control and zero-sum games The dynamic programming approach allows one to analyze one or two-player dynamic decision problems by means of operators, or partial differential equations (Hamilton–Jacobi or Isaacs PDEs), describing the time evolution of the value function, i.e., of the optimal reward of one player, thought of as a function of the initial state and of the horizon. We work especially with problems having long or infinite horizon, modelled by stopping problems, or ergodic problems in which one optimizes a mean payoff per time unit. The determination of optimal strategies reduces to solving nonlinear fixed point equations, which are obtained either directly from discrete models, or after a discretization of a PDE. The geometry of solutions of optimal control and game problems Basic questions include, especially for stationary or ergodic problems, the understanding of existence and uniqueness conditions for the solutions of dynamic programming equations, for instance in terms of controllability or ergodicity properties, and more generally the understanding of the structure of the full set of solutions of stationary Hamilton–Jacobi PDEs and of the set of optimal strategies. These issues are already challenging in the one-player deterministic case, which is an application of choice of tropical methods, since the Lax-Oleinik semigroup, i.e., the evolution semigroup of the Hamilton-Jacobi PDE, is a linear operator in the tropical sense. Recent progress in the deterministic case has been made by combining dynamical systems and PDE techniques (weak KAM theory 66), and also using metric geometry ideas (abstract boundaries can be used to represent the sets of solutions 79, 4). The two player case is challenging, owing to the lack of compactness of the analogue of the Lax-Oleinik semigroup and to a richer geometry. The conditions of solvability of ergodic problems for games (for instance, solvability of ergodic Isaacs PDEs), and the representation of solutions are only understood in special cases, for instance in the finite state space case, through tropical geometry and non-linear Perron-Frobenius methods 43, 41, 3. Algorithmic aspects: from combinatorial algorithms to the attenuation of the curse of dimensionality Our general goal is to push the limits of solvable models by means of fast algorithms adapted to large scale instances. Such instances arise from discrete problems, in which the state space may so large that it is only accessible through local oracles (for instance, in some web ranking applications, the number of states may be the number of web pages) 67. They also arise from the discretization of PDEs, in which the number of states grows exponentially with the number of degrees of freedom, according to the “curse of dimensionality”. A first line of research is the development of new approximation methods for the value function. So far, classical approximations by linear combinations have been used, as well as approximation by suprema of linear or quadratic forms, which have been introduced in the setting of dual dynamic programming and of the so called “max-plus basis methods” 68. We believe that more concise or more accurate approximations may be obtained by unifying these methods. Also, some max-plus basis methods have been shown to attenuate the curse of dimensionality for very special problems (for instance involving switching) 89, 73. This suggests that the complexity of control or games problems may be measured by more subtle quantities that the mere number of states, for instance, by some forms of metric entropy (for example, certain large scale problems have a low complexity owing to the presence of decomposition properties, “highway hierarchies”, etc.). A second line of of our research is the development of combinatorial algorithms, to solve large scale zero-sum two-player problems with discrete state space. This is related to current open problems in algorithmic game theory. In particular, the existence of polynomial-time algorithms for games with ergodic payment is an open question. See e.g. 44 for a polynomial time average complexity result derived by tropical methods. The two lines of research are related, as the understanding of the geometry of solutions allows to develop better approximation or combinatorial algorithms. ## 3.2 Non-linear Perron-Frobenius theory, nonexpansive mappings and metric geometry Several applications (including population dynamics 10 and discrete event systems 51, 61, 45) lead to studying classes of dynamical systems with remarkable properties: preserving a cone, preserving an order, or being nonexpansive in a metric. These can be studied by techniques of non-linear Perron-Frobenius theory 3 or metric geometry 11. Basic issues concern the existence and computation of the “escape rate” (which determines the throughput, the growth rate of the population), the characterizations of stationary regimes (non-linear fixed points), or the study of the dynamical properties (convergence to periodic orbits). Nonexpansive mappings also play a key role in the “operator approach” to zero-sum games, since the one-day operators of games are nonexpansive in several metrics, see 8. ## 3.3 Tropical algebra and convex geometry The different applications mentioned in the other sections lead us to develop some basic research on tropical algebraic structures and in convex and discrete geometry, looking at objects or problems with a “piecewise-linear ” structure. These include the geometry and algorithmics of tropical convex sets 47, 40, tropical semialgebraic sets 14, the study of semi-modules (analogues of vector spaces when the base field is replaced by a semi-field), the study of systems of equations linear in the tropical sense, investigating for instance the analogues of the notions of rank, the analogue of the eigenproblems 42, and more generally of systems of tropical polynomial equations. Our research also builds on, and concern, classical convex and discrete geometry methods. ## 3.4 Tropical methods applied to optimization, perturbation theory and matrix analysis Tropical algebraic objects appear as a deformation of classical objects thought various asymptotic procedures. A familiar example is the rule of asymptotic calculus, ${e}^{-a/ϵ}+{e}^{-b/ϵ}\asymp {e}^{-min\left(a,b\right)/ϵ}\phantom{\rule{5.0pt}{0ex}},\phantom{\rule{2.em}{0ex}}{e}^{-a/ϵ}×{e}^{-b/ϵ}={e}^{-\left(a+b\right)/ϵ}\phantom{\rule{5.0pt}{0ex}},$ 1 when $ϵ\to {0}^{+}$. Deformations of this kind have been studied in different contexts: large deviations, zero-temperature limits, Maslov's “dequantization method” 87, non-archimedean valuations, log-limit sets and Viro's patchworking method 111, etc. This entails a relation between classical algorithmic problems and tropical algorithmic problems, one may first solve the $ϵ=0$ case (non-archimedean problem), which is sometimes easier, and then use the information gotten in this way to solve the $ϵ=1$ (archimedean) case. In particular, tropicalization establishes a connection between polynomial systems and piecewise affine systems that are somehow similar to the ones arising in game problems. It allows one to transfer results from the world of combinatorics to “classical” equations solving. We investigate the consequences of this correspondence on complexity and numerical issues. For instance, combinatorial problems can be solved in a robust way. Hence, situations in which the tropicalization is faithful lead to improved algorithms for classical problems. In particular, scalings for the polynomial eigenproblems based on tropical preprocessings have started to be used in matrix analysis 74, 77. Moreover, the tropical approach has been recently applied to construct examples of linear programs in which the central path has an unexpectedly high total curvature 6, and it has also led to positive polynomial-time average case results concerning the complexity of mean payoff games. Similarly, we are studying semidefinite programming over non-archimedean fields 14, 49, with the goal to better understand complexity issues in classical semidefinite and semi-algebraic programming. # 4 Application domains ## 4.1 Discrete event systems (manufacturing systems, networks, emergency call centers) One important class of applications of max-plus algebra comes from discrete event dynamical systems 51. In particular, modelling timed systems subject to synchronization and concurrency phenomena leads to studying dynamical systems that are non-smooth, but which have remarkable structural properties (nonexpansiveness in certain metrics , monotonicity) or combinatorial properties. Algebraic methods allow one to obtain analytical expressions for performance measures (throughput, waiting time, etc). A recent application, to emergency call centers, can be found in 45. ## 4.2 Optimal control and games Optimal control and game theory have numerous well established applications fields: mathematical economy and finance, stock optimization, optimization of networks, decision making, etc. In most of these applications, one needs either to derive analytical or qualitative properties of solutions, or design exact or approximation algorithms adapted to large scale problems. ## 4.3 Operations Research We develop, or have developed, several aspects of operations research, including the application of stochastic control to optimal pricing, optimal measurement in networks 102. Applications of tropical methods arise in particular from discrete optimization 5860, scheduling problems with and-or constraints 93, or product mix auctions 109. ## 4.4 Computing program and dynamical systems invariants A number of programs and systems verification questions, in which safety considerations are involved, reduce to computing invariant subsets of dynamical systems. This approach appears in various guises in computer science, for instance in static analysis of program by abstract interpretation, along the lines of P. and R. Cousot 63, but also in control (eg, computing safety regions by solving Isaacs PDEs). These invariant sets are often sought in some tractable effective class: ellipsoids, polyhedra, parametric classes of polyhedra with a controlled complexity (the so called “templates” introduced by Sankaranarayanan, Sipma and Manna 104), shadows of sets represented by linear matrix inequalities, disjunctive constraints represented by tropical polyhedra 48, etc. The computation of invariants boils down to solving large scale fixed point problems. The latter are of the same nature as the ones encountered in the theory of zero-sum games, and so, the techniques developed in the previous research directions (especially methods of monotonicity, nonexpansiveness, discretization of PDEs, etc) apply to the present setting, see e.g. 70, 75 for the application of policy iteration type algorithms, or for the application for fixed point problems over the space of quadratic forms 7. The problem of computation of invariants is indeed a key issue needing the methods of several fields: convex and nonconvex programming, semidefinite programming and symbolic computation (to handle semialgebraic invariants), nonlinear fixed point theory, approximation theory, tropical methods (to handle disjunctions), and formal proof (to certify numerical invariants or inequalities). # 5 Social and environmental responsibility ## 5.1 Impact of research results The team has developed collaborations on the dimensioning of emergency call centers, with Préfecture de Police (Plate Forme d'Appels d'Urgence - PFAU - 17-18-112, operated jointly by Brigade de sapeurs pompiers de Paris and by Direction de la sécurité de proximité de l'agglomération parisienne) and also with the Emergency medical services of Assistance Publique – Hôpitaux de Paris (Centre 15 of SAMU75, 92, 93 and 94). This work is described further in Section 8.7.1. Some work done specifically during the Covid-19 crisis is described in Section 8.7.2. # 7 New software and platforms ## 7.1 New software ### 7.1.1 Coq-Polyhedra • Name: Coq-Polyhedra • Keywords: Coq, Polyhedra, Automated theorem proving, Linear optimization • Scientific Description: Coq-Polyhedra is a library providing a formalization of convex polyhedra in the Coq proof assistant. While still in active development, it provides an implementation of the simplex method, and already handles the basic properties of polyhedra such as emptiness, boundedness, membership. Several fundamental results in the theory of convex polyhedra, such as Farkas Lemma, duality theorem of linear programming, and Minkowski Theorem, are also formally proved. The formalization is based on the Mathematical Components library, and makes an extensive use of the boolean reflection methodology. • Functional Description: Coq-Polyhedra is a library which aims at formalizing convex polyhedra in Coq • News of the Year: Coq-Polyhedra now provides most of the basic operations on polyhedra. They are expressed on a quotient type that avoids reasoning with particular inequality representations. They include : * the construction of elementary polyhedra (half-spaces, hyperplanes, affine spaces, orthants, simplices, etc) * basic operations such as intersection, projection (thanks to the formalization of the Fourier-Motzkin algorithm), image under linear functions, computations of convex hulls, finitely generated cones, etc. * computation of affine hulls of polyhedra, as well as their dimension Thanks to this, we have made huge progress on the formalization of the combinatorics of polyhedra. The poset of faces, as well as its fundamental properties (lattice, gradedness, atomicity and co-atomicity, etc) are now formalized. The manipulation of the faces is based on an extensive use of canonical structures, that allows to get the most appropriate inequality representations for reasoning. In this way, we arrive at very concise and elegant proofs, closer to the pen-and-paper ones. • URL: • Publications: • Contacts: Xavier Allamigeon, Ricardo Katz, Pierre-Yves Strub • Participants: Xavier Allamigeon, Vasileios Charisopoulos, Ricardo Katz, Pierre-Yves Strub • Partners: CIFASIS, Ecole Polytechnique ### 7.1.2 EmergencyEval • Keywords: Dynamic Analysis, Simulation, Ocaml, Emergency, Firefighters, Police • Scientific Description: This software aims at enabling the definition of a Petri network execution semantic, as well as the instanciation and execution of said network using the aforedefined semantic. The heart of the project dwells in its kernel which operates the step-by-step execution of the network, obeying rules provided by an oracle. This user-defined and separated oracle computes the information necessary to the kernel for building the next state using the current state. The base of our software is the framework for the instanciation and execution of Petri nets, without making assumptions regarding the semantic. In the context of the study of the dynamics of emergency call centers, a second part of this software is the definition and implementation of the semantic of call centers modelized as Petri nets, and more specifically timed prioritized Petri nets. A module interoperating with the kernel enables to include all the operational specificities of call centers (urgency level, discriminating between operators and callers ...) while guaranteeing the genericity of the kernal which embeds the Petri net formalism as such. • Functional Description: In order to enable the quantitative study of the throughput of calls managed by emergency center calls and the assesment of various organisationnal configurations considered by the stakeholders (firefighters, police, medical emergency service of the 75, 92, 93 and 94 French departments), this software modelizes their behaviours by resorting to extensions of the Petri net formalism. Given a call transfer protocol in a call center, which corresponds to a topology and an execution semantic of a Petri net, the software generates a set of entering calls in accord with the empirically observed statistic ditributions (share of very urgent calls, conversation length), then simulates its management by the operators with respect to the defined protocol. Transitional regimes phenomenons (peak load, support) which are not yet handled by mathematical analysis could therefore be studied. The ouput of the software is a log file which is an execution trace of the simulation featuring extensive information in order to enable the analysis of the data for providing simulation-based insights for decision makers. The software relies on a Petri net simulation kernel designed to be as modular and adaptable as possible, fit for simulating other Petri-net related phenomenons, even if their semantic differ greatly. • Contacts: Baptiste Colin, Xavier Allamigeon, Stéphane Gaubert • Participants: Baptiste Colin, Xavier Allamigeon # 8 New results ## 8.1 Optimal control and zero-sum games ### 8.1.1 Fixed points of order preserving homogeneous maps and zero-sum games Participants: Marianne Akian, Stéphane Gaubert. In a series of joint works with Antoine Hochart, applied methods of non-linear fixed point theory to zero-sum games. A key issue is the solvability of the ergodic equation associated to a zero-sum game with finite state space, i.e., given a dynamic programming operator $T$ associated to an undiscounted problem, one looks for a vector $u$, called the bias, and for a scalar $\lambda$, the ergodic constant, such that $T\left(u\right)=\lambda e+u$. The bias vector is of interest as it allows to determine optimal stationnary strategies. In 13, we apply game theory methods to the study of the nonlinear eigenproblem for homogeneous order preserving self maps of the interior of the cone. We show that the existence and uniqueness of an eigenvector is governed by combinatorial conditions, involving dominions (sets of states “controlled” by one of the two players). In this way, we characterize the situation in which the existence of an eigenvector holds independently of perturbations, and we solve an open problem raised in 72. ### 8.1.2 Nonlinear fixed point methods to compute joint spectral raddi of nonnegative matrices Participants: Stéphane Gaubert. In 20, we introduce a non-linear fixed point method to approximate the joint spectral radius of a finite set of nonnegative matrices. We show in particular that the joint spectral radius is the limit of the eigenvalues of a family of non-linear risk-sensitive type dynamic programming operators. We develop a projective version of Krasnoselskii-Mann iteration to solve these eigenproblems, and report experimental results on large scale instances (several matrices in dimensions of order 1000 within a minute). ### 8.1.3 Tropical-SDDP algorithms for stochastic control problems Participants: Marianne Akian, Duy Nghi Benoît Tran. The PhD thesis of Benoît Tran 108, supervised by Jean-Philippe Chancelier (ENPC) and Marianne Akian concerns the numerical solution of the dynamic programming equation of discrete time stochastic control problems. Several methods have been proposed in the litterature to bypass the curse of dimensionality difficulty of such an equation, by assuming a certain structure of the problem. Examples are the max-plus based method of McEneaney 90, 88, the stochastic max-plus scheme proposed by Zheng Qu 99, the stochastic dual dynamic programming (SDDP) algorithm of Pereira and Pinto 95, the mixed integer dynamic approximation scheme of Philpott, Faisal and Bonnans 50, the probabilistic numerical method of Fahim, Touzi and Warin 65, and its association with the max-plus based method proposed in 39. We propose to associate and compare these methods in order to solve more general structures. In a first work 38, we build a common framework for both the SDDP and a discrete time and finite horizon version of Zheng Qu's algorithm for deterministic problems involving a finite set-valued (or switching) control and a continuum-valued control. We propose an algorithm that generates monotone approximations of the value function as a pointwise supremum, or infimum, of basic (affine or quadratic for example) functions which are randomly selected. We give sufficient conditions that ensure almost sure convergence of the approximations to the value function. In 32, we study generalizations and combinaison of these algorithms to the case of stochastic optimal control problems. In recent works we introduce and study an entropic relaxation of the Nested Distance introduced by Pflug 96, and the interchange between integration and minimization. ### 8.1.4 Multiply Accelerated Value Iteration Algorithms For Classes of Markov Decision Processes Participants: Marianne Akian, Stéphane Gaubert, Omar Saadi. Accelerated gradient algorithms in convex optimization were introduced by Nesterov. A fundamental question is whether similar acceleration schemes work for the iteration of nonexpansive mappings. In a joint work with Zheng Qu (Hong Kong University) 34, motivated by the analysis of Markov decision processes and zero-sum repeated games, we study fixed point problems for Shapley operators, i.e., for sup-norm nonexpansive and order preserving mapping. We deal more especially with affine operators, corresponding to zero-player problems – the latter can be used as a building blocks for one or two player problems, by means of policy iteration. For an affine operator, associated to a Markov chain, the acceleration property can be formalized as follows: one should replace an original scheme with a convergence rate $1-\Theta \left(ϵ\right)$ by a convergence rate $1-\Theta \left({ϵ}^{1/2}\right)$ where $ϵ$ is the spectral gap of the Markov chain. We characterize the spectra of Markov chains for which this acceleration is possible. We also characterize the spectra for which a multiple acceleration is possible, leading to a rate of $1-\Theta \left({ϵ}^{1/d}\right)$ for $d>2$. ### 8.1.5 Polyhedral representation of multi-stage stochastic linear problems Participants: Maël Forcier, Stéphane Gaubert. In 37 (joint work with Vincent Leclère, ENPC), we study multistage stochastic problems with a linear structure and general cost distribution, and show that the value function is polyhedral. We characterize the affinity regions as the cells of a chamber complex. We deduce fixed-parameter tractability results, showing that when the dimensions of some state spaces are fixed, the problem (which is generally sharp-P complete) becomes polynomial. ### 8.1.6 Highway hierarchies for Hamilton-Jacobi-Bellman (HJB) PDEs Participants: Marianne Akian, Stéphane Gaubert, Shanqing Liu. Hamilton-Jacobi-Bellman equations arise as the dynamic programming equations of deterministic or stochastic optimal control problems. They allow to obtain the global optimum of these problems and to synthetize an optimal feedback control, leading to a solution robust against system perturbations. Several methods have been proposed in the litterature to bypass the obstruction of curse of dimensionality of such equations, assuming a certain structure of the problem, and/or using “unstructured discretizations”, that are not based on given grids. Among them, one may cite tropical numerical method, and probabilistic numerical method. On another direction, “highway hierarchies”, developped by Sanders, Schultes and coworkers 64, 103, initially for applications to on-board GPS systems, are a computational method that allows one to accelerate Dijkstra algorithm for discrete time and state shortest path problems. The aim of the starting thesis of Shanqing Liu is to develop new numerical methods to solve Hamilton-Jacobi-Bellman equations that are less sensitive to curse of dimensionality. One will particularly develop methods based on continuous, or infinitesimal, analogues of highway hierarchies, that can also be adapted to unstructured discretization grids. The first step initiated during the internship of Shanqing Liu is to adapt highway hierarchies to the fast marching method. ## 8.2 Non-linear Perron-Frobenius theory, nonexpansive mappings and metric geometry ### 8.2.1 Volume in Hilbert and Funk geometries Participants: Constantin Vernicos, Cormac Walsh. In a recent paper 27, we investigated how the volume of a ball in a Hilbert geometry grows as its radius increases. In particular, we studied the volume entropy where $B\left(x,r\right)$ is the metric ball with center $x$ and radius $r$, and $Vol$ denotes the Holmes–Thompson volume. Note that the volume entropy does not depend on the particular choice of $x$. We showed that the volume entropy is exactly twice the flag-approximability of the convex body. This is a new notion of approximability we introduced that measures the complexity of a polytope by counting its number of flags rather than its number of vertices. A corollary is that the Euclidean ball has the maximal volume entropy among Hilbert geometries of a given dimension, a fact that was recently proved by Tholozan by other means. we also showed that the rate of growth of the volume is minimised when the convex body is a simplex. We are continuing this work by investigating the volume of balls of finite radius, rather than the asymptotics. We have found it convenient to turn our attention to a metric different from the Hilbert metric, but related to it. The Funk metric, as it is called, lacks the symmetry property usually assumed for metric spaces. However, it is somewhat simpler to work with when dealing with volumes, and it exhibits the same interesting behaviour. Given a convex body and a radius $r$, there is a unique point $x$ such that the Funk ball of radius $r$ centered at $x$ minimises the volume over all Funk balls of the same radius. It is natural to conjecture that this minimum volume, which depends on the convex body, it maximised when the body is a Euclidean ball. If this is true, one could recover Blaschke–Santaló inequality by letting the radius tend to zero, and the centro-affine isoperimetric inequality by letting the radius tend to infinity. Similarly, consideration of the minimum provides an interpolation between the Mahler conjecture and Kalai's flag conjecture. ### 8.2.2 Topics in Hilbert's geometry Participants: Constantin Vernicos. This subsection summarizes the work done during the “délégation” at INRIA of Constantin Vernicos. I started my secondment in early 2020 revising my work with Cormac Walsh on the entropy of Hilbert geometries which finally got accepted by the Annales de l'ENS who asked these revision, see Section 8.2.1 for more information. We also started working on another conjecture we had: that the Busemann volumes of a ball in any given Hilbert geometry was bounded from above by the volume ball of the hyperbolic geometry. We discovered that this failed for some small balls. This has to be put into perspective with the result obtained by Vernicos-Yang 110 which implies that for large radii this always the case. We are currently studying the Holmes-Thompson volume in Funk geometry. In the second semester of 2020 I also started a collaboration with Antonin Guilloux (IMJ) with whom we are trying to find out at how many point inside a Hilbert geometry one needs to know the unit tangent ball to be able to characterise the convex body. Our conjecture is that one needs one more point than the dimension and we have done some progress in proving these for polytopes. At the same time I have been exchanging with Stéphane Gaubert who introduced me to the central path and barrier method. That is how I saw that the universal barrier is exactly the Holmes-Thompson volume of the Funk geometry of a convex set. Furthermore it allowed me to realize that a paper by Andreas Bernig 54 actually uses a barrier for which the central path is a quasi-geodesic of the polytope. We are currently trying to find out how this is beneficial for the optimization problem. ## 8.3 Tropical algebra and convex geometry ### 8.3.1 Formalizing convex polyhedra in Coq Participants: Xavier Allamigeon, Quentin Canu. In a joint work with Ricardo Katz (Conicet, Argentina) and Pierre-Yves Strub (LIX, Ecole Polytechnique), we present the first formalization of faces of polyhedra in the proof assistant Coq. This builds on the formalization of a library providing the basic constructions and operations over polyhedra, including projections, convex hulls and images under linear maps. Moreover, we design a special mechanism which automatically introduces an appropriate representation of a polyhedron or a face, depending on the context of the proof. We demonstrate the usability of this approach by establishing some of the most important combinatorial properties of faces, namely that they constitute a family of graded atomistic and coatomistic lattices closed under sublattices. This is implemented in the CoqPolyhedra library (we refer to the software session for more details). This work has been published in 10th International Joint Conference on Automated Reasoning 31, and invited for a submission to a special issue of Logical Methods in Computer Science. During his M2 internship under the supervision of X. Allamigeon and P.-Y. Strub, Quentin Canu has worked on the development of vertex enumeration methods in CoqPolyhedra. We are now working the formalization of lattices and ordered structures, with a special emphasis on finite (sub)lattices. ### 8.3.2 Linear algebra over systems Participants: Marianne Akian, Stéphane Gaubert. In a joint work with Louis Rowen (Univ. Bar Ilan), we study linear algebra and convexity properties over “systems”. The latter provide a general setting encompassing extensions of the tropical semifields and hyperfields. ### 8.3.3 Ambitropical convexity and Shapley retracts Participants: Marianne Akian, Stéphane Gaubert. Closed tropical convex cones are the most basic examples of modules over the tropical semifield. They coincide with sub-fixed-point sets of Shapley operators – dynamic programming operators of zero-sum games. We study a larger class of cones, which we call “ambitropical” as it includes both tropical cones and their duals. Ambitropical cones can be defined as lattices in the order induced by Rn. Closed ambitropical cones are precisely the fixedpoint sets of Shapley operators. They are characterized by a property of best co-approximation arising from the theory of nonexpansive retracts of normed spaces. Finitely generated ambitropical cones arise when considering Shapley operators of deterministic games with finite action spaces. Finitely generated ambitropical cones are special polyhedral complexes whose cells are alcoved poyhedra, and locally, they are in bijection with order preserving retracts of the Boolean cube. This is a joint work with Sara Vannucci (invited PhD student from Salerno university). ## 8.4 Tropical linear regression and applications Participants: Marianne Akian, Stéphane Gaubert, Yang Qi, Omar Saadi. We show that the problem consisting in computing a best approximation of a collection of points by a tropical hyperplane is equivalent to solving a mean payoff game, and also, to compute the maximal radius of an inscribed ball in a tropical polytope. We provide an application to economics - measuring the distance to equilibrium. We also study a dual problem — computing the minimal radius of a circumscribed ball to a tropical polytope – and apply it to the rank-one approximation of tropical matrices and tensors. ### 8.4.1 Eigenvalues of Tropical Symmetric Matrices Participants: Marianne Akian, Stéphane Gaubert, Hanieh Tavakolipour. The tropical semifields can be thought of as images of fields with a non-archimedean valuation. It allows in this way to study the asymptotics of Puiseux series with complex coefficients. When dealing with Puiseux series with real coefficients and with its associated order, it is convenient to use the symmetrized tropical semiring introduced in 98 (see also 51), and the signed valuation which associates to any series its valuation together with its sign. We study with these tools the asymptotics of eigenvalues and eigenvectors of symmetric positive definite matrices over the field of Puiseux series. This raises the problem of defining the appropriate notions of positive definite matrices over the symmetrized tropical semiring, eigenvalues and eigenvectors of such matrices, thus roots of polynomials and their multiplicities. This builds on 14 and 52. ## 8.5 Tropical methods applied to optimization, perturbation theory and matrix analysis ### 8.5.1 Tropicalization of interior point methods and application to complexity Participants: Xavier Allamigeon, Stéphane Gaubert. The entropic barrier, studied by Bubeck and Eldan (Proc. Mach. Learn. Research, 2015), is a self-concordant barrier with asymptotically optimal self-concordance parameter. In a joint work 35 with Abdellah Aznag (Columbia University) and Yassine Hamdi (Ecole Polytechnique), we study the tropicalization of the central path associated with the entropic barrier, i.e., the logarithmic limit of this central path for a parametric family of linear programs defined over the field of Puiseux series. Our main result is that the tropicalization of the entropic central path is a piecewise linear curve which coincides with the tropicalization of the logarithmic central path studied by Allamigeon et al. in 6. One consequence is that the number of linear pieces in the tropical entropic central path can be exponential in the dimension and the number of inequalities defining the linear program. This result suggests that the interior point methods using the entropic barrier may be subject to the same pathology that the ones based on the logarithmic barrier, i.e., the number of iterations performed may be exponential in the dimension and the number of inequalities. ### 8.5.2 Tropical Nash equilibria and complementarity problems Participants: Xavier Allamigeon, Stéphane Gaubert. Linear complementarity programming is a generalization of linear programming which encompasses the computation of Nash equilibria for bimatrix games. While the latter problem is PPAD-complete, we show in 36 that the analogue of this problem in tropical algebra can be solved in polynomial time. Moreover, we prove that the Lemke–Howson algorithm carries over the tropical setting and performs a linear number of pivots in the worst case. A consequence of this result is a new class of (classical) bimatrix games for which Nash equilibria computation can be done in polynomial time. This is joint work with Frédéric Meunier (Cermics, ENPC). ### 8.5.3 Tropicalization of semidefinite programming and its relation with stochastic games Participants: Xavier Allamigeon, Stéphane Gaubert. Semidefinite programming consists in optimizing a linear function over a spectrahedron. The latter is a subset of ${ℝ}^{n}$ defined by linear matrix inequalities, i.e., a set of the form where the ${Q}^{\left(k\right)}$ are symmetric matrices of order $m$, and $⪰$ denotes the Loewner order on the space of symmetric matrices. By definition, $X⪰Y$ if and only if $X-Y$ is positive semidefinite. Semidefinite programming can be studied as well over non-archimedien ordered fields: non-archimedean instances encode parametric families of ordinary instances, with an infinitesimal or arbitrarily large parameter. To this purpose, we studied tropical spectrahedra, which are defined as the images by the valuation of nonarchimedean spectrahedra. We establish that they are closed semilinear sets, and that, under a genericity condition, they are described by explicit inequalities expressing the nonnegativity of tropical minors of order 1 and 2. These results are presented in 14, together with more general materials on the tropicalization of semi-algebraic sets. ### 8.5.4 Tropical posynomial systems and colorful interior of convex bodies Participants: Marianne Akian, Marin Boyet, Xavier Allamigeon, Stéphane Gaubert. We study tropical posynomial systems, with motivations from call center performance evaluation (see Section 8.7.1). We exhibit a class of classical or tropical posynomial systems which can be solved by reduction to linear or convex programming problems. This relies on a notion of colorful vectors with respect to a collection of Newton polytopes. This extends the convex programming approach of one player stochastic games. These results appeared in the proceedings of the International Congress on Mathematical Software 2020 29. ## 8.6 Algebraic aspects of tensors and neural networks ### 8.6.1 Topology of tensor ranks Participants: Yang Qi. The primary goal of 16 is to better understand the topological properties of various tensor ranks, an aspect that has been somewhat neglected in existing studies. However, the results on path-connectedness and simple-connectedness of tensor rank, multilinear rank, and their symmetric counterparts have useful practical implications. One of the most basic and common problems involving tensors in applications is to find low-rank approximations with respect to one of these notions of rank. Riemannian manifold optimization techniques have been used for this problem. For instance, people consider approximation by tensors of a fixed multilinear rank, i.e., which is a smooth Riemannian manifold. Thus Riemannian optimization techniques can be applied. But this raises the question of whether ${X}_{{r}_{1},\cdots ,{r}_{d}}\left({V}_{1},\cdots ,{V}_{d}\right)$ is path-connected. If not, then the path-following algorithms that begin from an initial point in one component will never converge to an optimizer located in another. Hence the study of path-connectedness becomes necessary. Besides, homotopy continuation techniques have also made a recent appearance in tensor decomposition problems over $ℂ$. In general, a tensor of a given rank may have several rank decompositions and such techniques have the advantage of being able to find all decompositions with high probability. The basic idea is that for a given general complex rank-$r$ tensor $A\in {W}_{1}\otimes \cdots \otimes {W}_{d}$ with a known rank-$r$ decomposition, one may construct a random loop $\tau :\left[0,1\right]\to {W}_{1}\otimes \cdots \otimes {W}_{d}$ with $\tau \left(0\right)=\tau \left(1\right)=A$, the endpoint of this loop gives a rank-$r$ decomposition of $A$, repeat this process a considerable number of times by choosing random loops, and one may expect to obtain all rank-$r$ decompositions. The consideration of loops naturally leads us to questions of simple-connectedness. Motivated by the above techniques, in 16 (joint work with Pierre Comon, Lek-Heng Lim, and Ke Ye) we systematically study path-connectedness and homotopy groups of sets of tensors defined by tensor rank, border rank, multilinear rank, as well as their symmetric counterparts for symmetric tensors. ### 8.6.2 Approximation theory of neural networks Participants: Yang Qi. In 25 (joint work with Lek-Heng Lim and Mateusz Michałek) we show that the empirical risk minimization (ERM) problem for neural networks has no solution in general. More precisely, given a training set ${s}_{1},\cdots ,{s}_{n}\in {ℝ}^{p}$ with corresponding responses ${t}_{1},\cdots ,{t}_{n}\in {ℝ}^{q}$, fitting a $k$-layer neural network ${\nu }_{\theta }:{ℝ}^{p}\to {ℝ}^{q}$ involves estimation of the weights $\theta \in {ℝ}^{m}$ via an ERM: We show that even for $k=2$, this infimum is not attainable in general for common activations like ReLU, hyperbolic tangent, and sigmoid functions. In addition, we show that for smooth activations $\sigma \left(x\right)=1/\left(1+exp\left(-x\right)\right)$ and $\sigma \left(x\right)=tanh\left(x\right)$, such failure to attain an infimum can happen on a positive-measured subset of responses. For the ReLU activation $\sigma \left(x\right)=max\left(0,x\right)$, we completely classify cases where the ERM for a best two-layer neural network approximation attains its infimum. In recent applications of neural networks, where overfitting is commonplace, the failure to attain an infimum is avoided by ensuring that the system of equations ${t}_{i}={\nu }_{\theta }\left({s}_{i}\right)$, $i=1,\cdots ,n$, has a solution. For a two-layer ReLU-activated network, we show when such a system of equations has a solution generically, i.e., when can such a neural network be fitted perfectly with probability one. ### 8.6.3 Spectral inequalities for nonnegative tensors and their tropical analogues Participants: Stéphane Gaubert. In 17 (joint work with Shmuel Friedland, University of Illinois at Chicago) we extend some characterizations and inequalities for the eigenvalues of nonnegative matrices, such as Donsker-Varadhan, Friedland-Karlin, Karlin-Ost inequalities, to nonnegative tensors. These inequalities are related to a correspondence between nonnegative tensors and ergodic control: the logarithm of the spectral radius of a tensor is given by the value of an ergodic problem in which instantaneous payments are given by a relative entropy. Some of these inequalities involve the tropical spectral radius, a limit of the spectral radius which we characterize combinatorially as the value of an ergodic Markov decision process. ## 8.7 Applications ### 8.7.1 Performance evaluation of emergency call centers Participants: Xavier Allamigeon, Marin Boyet, Baptiste Colin, Stéphane Gaubert. Since 2014, we have been collaborating with Préfecture de Police (Régis Reboul and LcL Stéphane Raclot), more specifically with Brigade de Sapeurs de Pompiers de Paris (BSPP) and Direction de Sécurité de Proximité de l'agglomération parisienne (DSPAP), on the performance evaluation of the new organization (PFAU, “Plate forme d'appels d'urgence”) to handle emergency calls to firemen and policemen in the Paris area. We developed analytical models, based on Petri nets with priorities, and fluid limits, see 45, 46, 55. In 2019, with four students of École polytechnique, Céline Moucer, Julia Escribe, Skandère Sahli and Alban Zammit, we performed case studies, showing the improvement brought by the two level filtering procedure. Moreover, in 2019, this work has been extended to encompass the handling of health emergency calls, with a new collaboration, involving responsibles from the four services of medical emergency aid of Assistance Publique – Hôpitaux de Paris (APHP), i.e., with SAMU75, 92, 93, 94, in the framework of a project coordinated by Dr. Christophe Leroy from APHP. As part of his PhD work, Marin Boyet have developed Petri net models capturing the characteristic of the centers (CRRA) handling emergency calls the SAMU, in order to make dimensioning recommendations. Following this, we have been strongly solicited by APHP during the pandemic of Covid-19 in order to determine crisis dimensioning of SAMU. Besides, we have initiated a new collaboration, with SAMU69, also on dimensioning. In parallel, we have further investigated the theoretical properties of timed Petri nets with preselection and priority routing. We represent the behavior of these systems by piecewise affine dynamical systems. We use tools from the theory of nonexpansive mappings to analyze these systems. We establish an equivalence theorem between priority-free fluid timed Petri nets and semi-Markov decision processes, from which we derive the convergence to a periodic regime and the polynomial-time computability of the throughput. More generally, we develop an approach inspired by tropical geometry, characterizing the congestion phases as the cells of a polyhedral complex. These results are illustrated by the application to the performance evaluation of emergency call centers of SAMU in the Paris area. These results have been published in the 41st International Conference on Application and Theory of Petri Nets and Concurrency 30, and later invited to a submission in the journal Fundamenta Informaticæ. ### 8.7.2 Covid-19 crisis work: Monitoring the epidemic from the analysis of calls to the Emergency Services – PrediDRM and Cluster-Carmen projects Participants: Marianne Akian, Xavier Allamigeon, Marin Boyet, Baptiste Colin, Ayoub Foussoul, Stéphane Gaubert. This action began in March 2020, at the start of the Covid-19 crisis in the Paris area, when our team was contacted by the Emergency medical services (EMS) of Assistance Publique–Hôpitaux de Paris (AP-HP) to compute a dimensioning of the emergency call centers (CRRA of SAMU 75, 92, 93 and 94), i.e., to evaluate the numbers of assistants of medical regulation, and of emergency physicians, needed to deal with the flux of calls during the epidemiologic peak which was yet to come. A crisis dimensioning was delivered, based on the models previously developed in 30 and on earlier modelling work with the SAMU of AP-HP, see Section 8.7.1. To do so, we analyzed data from patient regulations files and found that they provide early and reliable signals of the epidemiologic growth, which were previously not included among the indicators used to monitor the epidemic. This led to a multidisciplinary work, joint with the physicians of the EMS of AP-HP, to construct medically relevant indicators. We implemented then within a flash research and development action, “PrediDRM”, involving, in addition to members of the Tropical team, engineers and researchers from other INRIA teams and other INRIA centers, Laurent Massoulié, David Parsons, Théotime Grohens, and Thomas Lepoutre. The goal of PrediDRM was to produce the indicators and to analyse them. Most of this work is presented in the article 18, which encompasses both medical and epidemiological aspects. It includes an analysis of the evolution of the epidemic in the Paris area based on EMS data, and mathematical modelling results, showing that the logarithm of epidemic observables can be approximated by a piecewise linear curve, and that its nondifferentiability points allow one to estimate the delay between sanitary measures and their effects on the load of EMS and other hospital departments. The team also benefited of information on numbers and types of emergency calls during the crisis, provided by SAMU69 and SAMU77 – calls to the number 15 – and also, by the direction of the PFAU programme at Préfecture de police (calls to the numbers 17-18-112). Supplementary analyses were produced using these data. We also benefited from informations provided by Enedis, Orange Flux Vision, and SFR during the crisis of the spring, allowing us to evaluate the influence of mobility on epidemic growth, see 18 for more information. A broader discussion of epidemiological indicators, including EMS indicators, is presented in the article 15, authored by a collective name for a group of researchers gathering physicians and researchers from AP-HP, INSERM and INRIA. Ayoub Foussoul, as part of his master internship of École polytechnique, analysed the conditioning of the piecewise linear approximation problem used in the monitoring algorithm. He got a “grand prix d'option” of École polytechnique for this work. We were also requested by AP-HP to refine the EMS indicators by producing a cartography of the epidemic at a local scale in the Paris area, based on EMS calls, in order to help AP-HP to identify clusters. This extension of the PrediDRM project, called “ClusterCarmen”, involved in addition to Xavier Allamigeon, Stéphane Gaubert, and Laurent Massoulié, other researchers and engineers of INRIA, Cédric Adjih, Guillermo Barroso-Andrade, Mathieu Simonin, with the help of Thomas Calmant. Within AP-HP, the ClusterCarmen project was coordinated by Dr. Philippe Le Toumelin and Dr. Paul-Georges Reuter, with the support of the four SAMU of AP-HP and of the AP-HP DSI. The cartography software was deployed on May 11th (the day the initial lockdown was released), the cartography being produced automatically every day and delivered to AP-HP and to other experts in charge of monitoring of the epidemic. The cartography software produced by the INRIA team was transfered to AP-HP at the fall, to allow further developments and extensions. ### 8.7.3 Game theory and optimization methods for decentralized electric systems Participants: Stéphane Gaubert, Paulin Jacquot. This section presents results from the PhD work of Paulin Jacquot, in collaboration with Nadia Oudjane, Olivier Beaude and Cheng Wan (EDF Lab), that were published in 2020. This work of Paulin Jacquot concerns the application of game theory and distributed optimization techniques to the operation of decentralized electric systems, and in particlar to the management of distributed electric consumption flexibilities. We start by adopting the point of view of a centralized operator in charge of the management of flexibilities for several agents. We provide a distributed and privacy-preserving algorithm to compute consumption profiles for agents that are optimal for the operator. In the proposed method, the individual constraints as well as the individual consumption profile of each agent are never revealed to the operator or the other agents 21. A patent related to this method has been published 82. A collaboration with Cheng Wan (EDF Lab) led to an additional part of this PhD thesis. We consider an operator dealing with a very large number of players, for which evaluating the equilibria in a congestion game will be difficult. To address this issue, we give approximation results on the equilibria in congestion and aggregative games with a very large number of players, in the presence of coupling constraints. These results, obtained in the framework of variational inequalities and under some monotonicity conditions, can be used to compute an approximate equilibrium, solution of a small dimension problem 22. In line with the idea of modeling large populations, we consider nonatomic congestion games with coupling constraints, with an infinity of heterogeneous players: these games arise when the characteristics of a population are described by a parametric density function. Under monotonicity hypotheses, we prove that Wardrop equilibria of such games, given as solutions of an infinite dimensional variational inequality, can be approximated by symmetric Wardrop equilibria of auxiliary games, solutions of low dimension variational inequalities. Again, those results can be the basis of tractable methods to compute an approximate Wardrop equilibrium in a nonatomic infinite-type congestion game 83. Last, in a collaboration with Hélène Le Cadre, Cheng Wan and Clémence Alasseur, we consider a game model for the study of decentralized peer-to-peer energy exchanges between a community of consumers with renewable production sources. We study the generalized equilibria in this game, which characterize the possible energy trades and associated individual consumptions. We compare the equilibria with the centralized solution minimizing the social cost, and evaluate the efficiency of equilibria through the price of anarchy 24. ### 8.7.4 Multistage Stochastic Optimal Power Flow Problem Participants: Maxime Grangereau, Stéphane Gaubert. This work is part of the PhD work of Maxime Grangereau, cosupervised by Emmanuel Gobet, in collaboration with Wim van Ackooij (EDF). We study a multistage and stochastic extension of the Optimal Power Flow problem (OPF). We developed semidefinite relaxations, extending the ones which arise in static and deterministic OPF problems. We provided a priori conditions which guarantee the absence of relaxation gap, and also a posteriori methods allowing one to bound this relaxation gap. We applied this approach on examples of grids, with scenario trees representing the random solar power production 71. ### 8.7.5 POMDP and adaptive management Participants: Marianne Akian, Luz Valérie Pascal. The aim of the internship of Luz Pascal, co-supervised by Iadine Chades (CSIRO, Australia), was to develop an algorithm for Adaptive Management (AM). AM is the principal tool for conserving endangered species under global change. AM can be solved using simplified Mixed Observable Markov Decision Processes called hidden model MDPs (hmMDPs) when the unknown dynamics are assumed stationary 59. hmMDPs provide optimal policies to AM problems by augmenting the MDP state space with an unobservable state variable representing a finite set of predefined models (transition probabilities). A drawback in formalising an AM problem is that experts are often solicited to provide this predefined set of models and that one assume that the true transition probabilities are included in the candidate model set. A first work of the internship was to propose an original approach to build a hmMDP with a universal set of predefined models. This has been done in the case of a 2-state n-action AM problem, and has been assessed on two species conservation case studies from Australia and randomly generated problems. This work will be published in the proceedings of the AAAI conference (held online on Feb. 2–9, 2021). A second work was to study the convergence of the SARSOP algorithm generally used for solving stationary POMDP by comparing this algorithm with the combination of SDDP and tropical algorithms introduced in 32. # 9 Bilateral contracts and grants with industry ## 9.1 Bilateral contracts with industry • Stochastic optimization of multiple flexibilities and energies in micro-grids, collaboration with Wim Van Ackooij, from EDF labs, with the PhD work of Maxime Grangereau (CIFRE PhD), supervised by Emmanuel Gobet (CMAP) and cosupervised by Stéphane Gaubert. • Optimal pricing of energy and services. Collaboration with Clémence Alasseur and Wim Van Ackooij, from EDF Labs, with the Phd Work of Quentin Jacquet (CIFRE PhD), supervised by Stéphane Gaubert. # 10 Partnerships and cooperations ## 10.1 International initiatives ### 10.1.1 Participation in other international programs • Bilateral projects FACCTS, between the University of Chicago (Statistics) – Lek-Heng Lim– and Ecole polytechnique – Stéphane Gaubert– “Tropical geometry of deep learning”. This project was postoned owing to the Covid-19 crisis. • Math AmSud Project ARGO, “Algebraic Real Geometry and Optimization”, accepted, with CMM (Chile), Univ. Buenos Aires (Argentina), Univ. Fed. Rio and Univ. Fed. Ceara (Brasil), Univ Savoie and CMAP, Ecole polytechnique (France). ## 10.2 International research visitors ### 10.2.1 Visits of international scientists • Gregorio Malajovich, Departamento de Matemática Aplicada, Instituto de Matemática, Universidade Federal do Rio de Janeiro, Feb 9 – March 6, 2020, as part of the ARGO MathAmSud international project. ## 10.3 National initiatives ### 10.3.1 ANR • Projet ANR JCJC CAPPS (“Combinatorial Analysis of Polytopes and Polyhedral Subdivisions”), responsable Arnau Padrol (IMJ-PRG, Sorbonne Université). Partenaires : IMJ-PRG (Sorbonne Université), INRIA Saclay (Tropical), LIGM (Université Paris-Est Marne-la-Vallée), LIF (Université Aix-Marseille), CERMICS (École Nationale des Ponts et Chaussées), LIX (École Polytechnique). ### 10.3.2 Programme Gaspard Monge pour l'optimisation, la recherche opérationnelle et leurs interactions avec les sciences des données • Méthodes tropicales pour le dimensionnement de centres d'appels : application à un centre de supervision EDF. Participants : X. Allamigeon S. Gaubert, P. Bendotti (EDF) et T. Triboulet (EDF). ### 10.3.3 IRS iCODE (Institut pour le Contrôle et la Décision de l’Idex Paris-Saclay) • White project “New perspectives in the numerical solution of Hamilton-Jacobi-Bellman partial differential equations”, coordinated by M. Akian, including S. Gaubert and members of the EPC Commands (INRIA Saclay and École polytechnique), UMA (ENSTA), and LMO (Paris-Sud). ### 10.3.4 Centre des Hautes Études du Ministère de l'Intérieur • Project “Optimisation de la performance de centres de traitement d'appels d'urgence en cas d’événements planifiés ou imprévus”, coordinated by X. Allamigeon, involving M. Boyet, B. Colin and S. Gaubert. # 11 Dissemination ## 11.1 Promoting scientific activities ### 11.1.1 Scientific events: organisation #### General chair, scientific chair • M. Akian was co-chair of the organizing committee of the 2020 SMAI MODE days, September 7-9 2020 (the conference was held in visioconference), see http://smai-mode2020.inria.fr/. • S. Gaubert is the coordinator of the Gaspard Monge Program for Optimization, Operations Research and their interactions with data sciences (PGMO), a corporate sponsorhip program, operated by Fondation Mathématique Jacques Hadamard, supported by Criteo, EDF, Orange and Thales, see http://www.fondation-hadamard.fr/fr/pgmo/. #### Organization of invited sessions • Xavier Allamigeon: organization of a session entitled “Geometry, Combinatorics and Optimization” in the SMAI-MODE 2020 conference. ### 11.1.3 Journal #### Member of the editorial boards • Stéphane Gaubert: member of the editorial board of Journal of Dynamics and Games, Linear and Multilinear Algebra, RAIRO, Springer-SMAI book series. ### 11.1.4 Invited talks • M. Akian: invited talk “Probabilistic max-plus schemes for solving Hamilton-Jacobi-Bellman equations” at IPAM Workshop I: High Dimensional Hamilton-Jacobi Methods in Control and Differential Games, March 30 - April 3, 2020 (online). • X. Allamigeon: invited talk at the session “Polyhedral Methods in Geometry and Optimization” of the ICMS 2020 conference. • S. Gaubert • Invited talk, “Tropical convexity, mean payoff games and nonarchimedean convex programming”, at the CAP 2020 Workshop (Combinatorics and Arithmetics for Physics), https://indico.math.cnrs.fr/event/6181/timetable/ • Invited talk, “Understanding and monitoring the evolution of the Covid-19 epidemic from medical emergency calls: the example of the Paris area”, DMG Seminar, Berlin, July 8th, 2020 (online). • Invited talk, “Understanding and monitoring the evolution of the Covid-19 epidemic from medical emergency calls: the example of the Paris area”, VPH conference, (Virtual, Physiological, Human), August 24th, 2020, Paris (online). • Invited talk, “Understanding and monitoring the evolution of the Covid-19 epidemic from medical emergency calls: the example of the Paris area”, Birmingham Optimization and Numerical ANalysis Seminar, October 20th, 2020 (online). • C. Vernicos • Conference hybrid at CIRM:Teichmüller Theory: Classical, Higher, Super and Quantum, 5-9 October, 2020, talk: “Entropy of Hilbert geometries and Complexities of convex bodies”, https://www.cirm-math.com/hybrid2216.html • Séminaire Gaston Darboux at Univ. Montpellier, on line: 11 Déc. 2020, “Sur deux résultats de Jacques Lafontaine concernant les formes harmoniques (humble hommage)”. More than 60 researchers attended (online) to this event dedicated to the memory of J. Lafontaine, deceased on November, 27th, 2020. • (postponed) June 11-13, 2020, invited speaker to the conference in the honor of Marc Troyanov, at École polytechnique de Lausanne. • (postponed): invited speaker at INdAM Workshop "Gromov hyperbolicity and negative curvature in Complex Analysis”, Cortona (Italy), scheduled for September 6th-10th 2020. • (postponed): invited speaker at BIRS-CMO Workshop, "Integral and Metric Geometry" on November 1 - 6, 2020, at Casa Matemática Oaxaca. ### 11.1.5 Leadership within the scientific community See Section 11.1.1. • M. Akian: Alternate elected member of Inria's Scientific Board. • X. Allamigeon: member of the scientific committee of INRIA Saclay. • M. Akian: member of the “Comité de Liaison” of SMAI MODE group. • X. Allamigeon: elected member of the committee of applied mathematics department of Ecole Polytechnique. • C. Vernicos: Elected member and deputy-head rank B of CNU, section 25. • C. Vernicos: Elected member of CVFU of University of Montpellier. ## 11.2 Teaching - Supervision - Juries ### 11.2.1 Teaching • M. Akian • Course “Markov decision processes: dynamic programming and applications” joint between (3rd year of) ENSTA and M2 “Mathématiques et Applications”, U. Paris Saclay, “Optimization”, 30 hours. • X. Allamigeon • Petites classes et encadrement d'enseignements d'approfondissement de Recherche Opérationnelle en troisième année à l'École Polytechnique (programme d'approfondissement de Mathématiques Appliquées) (niveau M1). • Cours du M2 “Optimisation” de l'Université Paris Saclay, cours partagé avec Céline Gicquel (LRI, Université Paris Sud). • Co-responsabilité du programme d'approfondissement en mathématiques appliquées (troisième année) à l'École Polytechnique. • S. Gaubert • Course “Systèmes à Événements Discrets”, option MAREVA, ENSMP. • Course “Algèbre tropicale pour le contrôle optimal et les jeux” of “Contrôle, Optimisation et Calcul des Variations” (COCV) of M2 “Mathématiques et Applications” of Paris 6 University and École Polytechnique. • Lecture of Operations Research, third year of École Polytechnique. The lectures notes were published as a book 56. • S. Liu • Exercises classes for the first year of Bachelor program of Ecole polytechnique in the framework of a “Monitorat”. • Exercises classes in the framework of a “Monitorat”. • B. Tran • Exercises classes for the first year of Bachelor program of Ecole polytechnique in the framework of a “Monitorat”. ### 11.2.2 Supervision • PhD: Benoît Tran, registered at Univ Paris-Est Marne La Vallée, from September 2017 to December 2020, thesis supervisor: Jean-Philippe Chancelier (ENPC), cosupervision: Marianne Akian. The defense took place on December 11, 2020. • PhD in progress: Maxime Grangereau, registered at Univ. Paris Saclay since Jan 2018, thesis supervisor: Emanuel Gobet, cosupervision: Stéphane Gaubert. • PhD in progress: Omar Saadi, registered at Univ. Paris Saclay since October 2018, thesis supervisor: Stéphane Gaubert, cosupervision: Marianne Akian. • PhD in progress: Marin Boyet, registered at Univ. Paris Saclay since October 2018, thesis supervisor: Stéphane Gaubert, cosupervision: Xavier Allamigeon. • PhD in progress: Maël Forcier, registered at ENPC since September 2019, thesis supervisor: Vincent Leclère, cosupervision Stéphane Gaubert. • PhD in progress: Quentin Canu, registered at Univ. Paris Saclay since October 2020, thesis supervisor: Georges Gonthier (INRIA), cosupervision: Xavier Allamigeon and Pierre-Yves Strub (LIX) • PhD in progress: Shanqing Liu, registered at IPP (EDMH) since September 2020, thesis supervisor, M. Akian, co-supervised by S. Gaubert. • PhD in progress: Quentin Jacquet, registered at IPP (EDMH) since November 2020, thesis supervisor, S. Gaubert, co-supervised by Clémence Alasseur and Wim van Ackooij. • PhD in progress: Tom Ferragut, on the horosphérical products of Gromov-hyperbolic and Busemann convex metric spaces, registered at U. Montpellier since 2018, thesis supervisor: C. Vernicos, co-supervisor Jérémie Brieussel. ### 11.2.3 Juries • M. Akian • Jury of the 2020 competition of Inria researchers at Inria Saclay-Ile-de-France. • Jury of the 2020 competition for a Maître de Conférence position in Applied Mathematics at Limoges University. • Jury of the 2020 competition for a Maître de Conférence position in Computer Science (Optimization) at Clermont University. • Jury of the PhD of Florian Schanzenbächer, Université Paris Est, June 5, 2020. • Jury of the PhD of Benoît Tran, Université Paris Est, December 11, 2020. • S. Gaubert • Hiring committee (Professor position in optimisation) at ENSTA, May 2020. • Jury of the PhD of Corentin Caillaud, École polytechnique, July 2020. ## 11.3 Conferences, Seminars • M. Akian • “Max-plus methods for solving Hamilton-Jacobi-Bellman equations”, Online Seminar with TU berlin, July 15, 2020. • M. Boyet • “Piecewise Affine Dynamical Models of Timed Petri Nets – Application to Emergency Call Centers”, Petri Nets 2020 conference, Paris. • M. Forcier • “The polyhedral structure and complexity of multistage stochastic linear programming with general cost distribution”, 2020 SMAI MODE days, September 7-9 2020. • S. Gaubert • “Dynamic programming operators overnoncommutative spaces: an approach to optimalcontrol of switched systems”, ICODE workshop on numerical solution of Hamilton-Jacobi-Bellman equations, Paris-Diderot University, January 8-10, 2020. • “Solving Perfect Information Mean Payoff Zero-sum Stochastic Games by Variance Reduced Deflated Value Iteration”, 2020 SMAI MODE days, September 7-9 2020. • B. Tran • “Tropical dynamic programming for stochastic optimal control”, 2020 SMAI MODE days, September 7-9 2020. # 12 Scientific production ## 12.1 Major publications • 1 articleMarianneM. Akian, StéphaneS. Gaubert and RavindraR. Bapat. Non-archimedean valuations of eigenvalues of matrix polynomialsLinear Algebra and its Applications498Also arXiv:1601.00438June 2016, 592--627 • 2 articleM. Akian, S. Gaubert and A. Guterman. Tropical polyhedra are equivalent to mean payoff gamesInternat. J. Algebra Comput.2212012, 1250001, 43 • 3 article MarianneM. Akian, StephaneS. Gaubert and RogerR. Nussbaum. Uniqueness of the fixed point of nonexpansive semidifferentiable maps Transactions of the American Mathematical Society 368 2 Also arXiv:1201.1536 February 2016 • 4 articleMarianneM. Akian, StéphaneS. Gaubert and CormacC. Walsh. The max-plus Martin boundaryDoc. Math.142009, 195--240 • 5 articleX. Allamigeon, P. Benchimol, S. Gaubert and M. Joswig. Combinatorial simplex algorithms can solve mean payoff gamesSIAM J. Opt.2442015, 2096--2117 • 6 articleXavierX. Allamigeon, PascalP. Benchimol, StéphaneS. Gaubert and MichaelM. Joswig. Log-barrier interior point methods are not strongly polynomialSIAM Journal on Applied Algebra and Geometry21https://arxiv.org/abs/1708.01544 - This paper supersedes arXiv:1405.4161. 31 pages, 5 figures, 1 table2018, 140-178 • 7 inproceedings X. Allamigeon, S. Gaubert, E. Goubault, S. Putot and N. Stott. A scalable algebraic method to infer quadratic invariants of switched systems Proceedings of the International Conference on Embedded Software (EMSOFT) Best paper award. The extended version of this conference article appeared in \em ACM Trans. Embed. Comput. Syst., 15(4):69:1--69:20, September 2016 2015 • 8 articleJ. Bolte, S. Gaubert and G. Vigeral. Definable zero-sum stochastic gamesMathematics of Operations Research401Also 1301.19672014, 171--191 • 9 articleS. Friedland, S. Gaubert and L. Han. Perron-Frobenius theorem for nonnegative multilinear forms and extensionsLinear Algebra and its Applications4382This paper was included in a list of “10 Notable Papers from the journal Linear Algebra Its Applications over the last 50 years” at the occasion of the ://www.journals.elsevier.com/linear-algebra-and-its-applications/10-notable-papers-linear-algebra-applications-50-yearsgolden anniversary of the journal, celebrated in 2018.2013, 738--749 • 10 articleS. Gaubert and Th.T. Lepoutre. Discrete limit and monotonicity properties of the Floquet eigenvalue in an age structured cell division cycle modelJ. Math. Biol.2015, • 11 articleS. Gaubert and G. Vigeral. A maximin characterization of the escape rate of nonexpansive mappings in metrically convex spacesMath. Proc. of Cambridge Phil. Soc.152https://arxiv.org/abs/1012.47652012, 341--363 • 12 incollection C. Walsh. The horofunction boundary and isometry group of the Hilbert geometry Handbook of Hilbert Geometry 22 IRMA Lectures in Mathematics and Theoretical Physics European Mathematical Society 2014 ## 12.2 Publications of the year ### International journals • 13 articleMarianneM. Akian, StéphaneS. Gaubert and AntoineA. Hochart. A game theory approach to the existence and uniqueness of nonlinear Perron-Frobenius eigenvectorsDiscrete and Continuous Dynamical Systems - Series A402020, 207--231 • 14 articleXavierX. Allamigeon, StéphaneS. Gaubert and MateuszM. Skomra. Tropical spectrahedraDiscrete and Computational Geometry63February 2020, 507–548 • 15 article Collective NameC. COVID-19 APHP-Universities-INRIA-INSERM Group. Early indicators of intensive care unit bedrequirement during the COVID-19 epidemic: Aretrospective study in Ile-de-France region,France PLoS ONE November 2020 • 16 articlePierreP. Comon, LimL. Lek-Heng, YangY. Qi and KeK. Ye. Topology of tensor ranksAdvances in Mathematics367June 2020, 107128 • 17 articleShmuelS. Friedland and StéphaneS. Gaubert. Spectral inequalities for nonnegative tensors and their tropical analoguesVietnam Journal of Mathematics4842020, 893-928 • 18 articleStéphaneS. Gaubert, MarianneM. Akian, XavierX. Allamigeon, MarinM. Boyet, BaptisteB. Colin, ThéotimeT. Grohens, LaurentL. Massoulié, David P.D. Parsons, FredericF. Adnet, ÉrickÉ. Chanzy, LaurentL. Goix, FrédéricF. Lapostolle, ÉricÉ. Lecarpentier, ChristopheC. Leroy, ThomasT. Loeb, Jean-SébastienJ.-S. Marx, CarolineC. Télion, LaurentL. Treluyer and PierreP. Carli. Understanding and monitoring the evolution of the Covid-19 epidemic from medical emergency calls: the example of the Paris areaComptes Rendus Mathématique3587November 2020, 843-875 • 19 articleStéphaneS. Gaubert and AdiA. Niv. Tropical planar networksLinear Algebra and its Applications5952020, 123-144 • 20 articleStéphaneS. Gaubert and NikolasN. Stott. A convergent hierarchy of non-linear eigenproblems to compute the joint spectral radius of nonnegative matricesMathematical Control and Related Fields1032020, 573-590 • 21 articlePaulinP. Jacquot, OlivierO. Beaude, PascalP. Benchimol, StephaneS. Gaubert and NadiaN. Oudjane. A Privacy-preserving Method to Optimize Distributed Resource AllocationSIAM Journal on Optimization303August 2020, 2303-2336 • 22 article PaulinP. Jacquot, ChengC. Wan, OlivierO. Beaude and NadiaN. Oudjane. Efficient Estimation of Equilibria in Large Aggregative Games with Coupling Constraints IEEE Transactions on Automatic Control July 2020 • 23 articlePascalP. Koiran and MateuszM. Skomra. Intersection multiplicity of a sparse curve and a low-degree curveJournal of Pure and Applied Algebra2247July 2020, 106279 • 24 articleHélèneH. Le Cadre, PaulinP. Jacquot, ChengC. Wan and ClémenceC. Alasseur. Peer-to-Peer Electricity Market Analysis: From Variational to Generalized Nash EquilibriumEuropean Journal of Operational Research2822April 2020, 753-771 • 25 article Lek-HengL.-H. Lim, MateuszM. Michalek and YangY. Qi. Best $k$-layer neural network approximations' Constructive Approximation 2020 • 26 article HaniehH. Tavakolipour and FatemehF. Shakeri. Asymptotics of the eigenvalues for exponentially parameterized pentadiagonal matrices Numerical Linear Algebra with Applications 27 6 December 2020 • 27 article Flag-approximability of convex bodies and volume growth of Hilbert geometries Annales scientifiques de l'Ecole normale supérieure 2021 • 28 articleOrder antimorphisms of finite-dimensional conesSelecta Mathematica (New Series)2642020, paper number 53 ### International peer-reviewed conferences • 29 inproceedings MarianneM. Akian, XavierX. Allamigeon, MarinM. Boyet and StéphaneS. Gaubert. A convex programming approach to solve posynomial systems International Congress on Mathematical Software ICMS 2020: Mathematical Software – ICMS 2020 12097 ICMS 2020 - International Congress on Mathematical Software, Lecture Notes in Computer Science Braunschweig, Germany 2020 • 30 inproceedingsXavierX. Allamigeon, MarinM. Boyet and StéphaneS. Gaubert. Piecewise Affine Dynamical Models of Timed Petri Nets -- Application to Emergency Call CentersPETRI NETS 2020 41ST INTERNATIONAL CONFERENCE ON APPLICATION AND THEORY OF PETRI NETS AND CONCURRENCY12152PETRI NETS 2020: Application and Theory of Petri Nets and Concurrency, LNCSParis, Francehttp://conf-2020.petrinet.net/June 2020, 260-279 ### Scientific book chapters • 31 inbookXavierX. Allamigeon, RicardoR. Katz and Pierre-YvesP.-Y. Strub. Formalizing the Face Lattice of PolyhedraAutomated Reasoning. IJCAR 2020June 2020, 185-203 ### Reports & preprints • 32 misc MarianneM. Akian, Jean-PhilippeJ.-P. Chancelier and BenoîtB. Tran. Tropical Dynamic Programming for Lipschitz Multistage Stochastic Programming December 2020 • 33 misc MarianneM. Akian, LucaL. Ganassali, StéphaneS. Gaubert and LaurentL. Massoulié. Probabilistic and mean-field model of COVID-19 epidemics with user mobility and contact tracing September 2020 • 34 misc MarianneM. Akian, StéphaneS. Gaubert, ZhengZ. Qu and OmarO. Saadi. Multiply Accelerated Value Iteration for Non-Symmetric Affine Fixed Point Problems and application to Markov Decision Processes December 2020 • 35 misc XavierX. Allamigeon, AbdellahA. Aznag, StéphaneS. Gaubert and YassineY. Hamdi. The tropicalization of the entropic barrier 2020 • 36 misc Tropical Nash equilibria and complementarity problems 2020 • 37 misc MaëlM. Forcier, StephaneS. Gaubert and VincentV. Leclère. The polyhedral structure and complexity of multistage stochastic linear problem with general cost distribution September 2020 ## 12.3 Cited publications • 38 unpublishedMarianneM. Akian, Jean-PhilippeJ.-P. Chancelier and BenoîtB. Tran. A stochastic algorithm for deterministic multistage optimization problems2018, https://arxiv.org/abs/1810.12870 - working paper or preprint • 39 incollection MarianneM. Akian and EricE. Fodjo. From a monotone probabilistic scheme to a probabilistic max-plus algorithm for solving Hamilton-Jacobi-Bellman equations Hamilton-Jacobi-Bellman Equations: Numerical Methods and Applications in Optimal Control https://arxiv.org/abs/1709.09049 De Gruyter 2018 • 40 articleM. Akian, S. Gaubert and A. Guterman. Tropical polyhedra are equivalent to mean payoff gamesInternat. J. Algebra Comput.2212012, 1250001, 43 • 41 articleMarianneM. Akian, StéphaneS. Gaubert and AntoineA. Hochart. Generic uniqueness of the bias vector of finite stochastic games with perfect informationJournal of Mathematical Analysis and Applications457https://arxiv.org/abs/1610.096512018, 1038-1064 • 42 article MarianneM. Akian, StéphaneS. Gaubert and MeisamM. Sharify. Log-majorization of the moduli of the eigenvalues of a matrix polynomial by tropical roots Linear Algebra and its Applications Also arXiv:1304.2967 2017 • 43 articleMarianneM. Akian and StéphaneS. Gaubert. Spectral theorem for convex monotone homogeneous maps, and ergodic controlNonlinear Anal.5222003, 637--679 • 44 inproceedingsXavierX. Allamigeon, PascalP. Benchimol and StéphaneS. Gaubert. The tropical shadow-vertex algorithm solves mean payoff games in polynomial time on averageICALP 2014857241st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part ICopenhagen, FranceSpringer2014, 12 • 45 inproceedings XavierX. Allamigeon, VianneyV. Boeuf and StéphaneS. Gaubert. Performance evaluation of an emergency call center: tropical polynomial systems applied to timed Petri nets 13th International Conference, Formal Modeling and Analysis of Timed Systems (FORMATS 2015) 9268 Formal Modeling and Analysis of Timed Systems Madrid, Spain Springer 2015 • 46 articleXavierX. Allamigeon, VianneyV. Boeuf and StephaneS. Gaubert. Stationary solutions of discrete and continuous Petri nets with prioritiesPerformance Evaluation113https://arxiv.org/abs/1612.076612017, 1 - 12 • 47 articleXavierX. Allamigeon, StéphaneS. Gaubert and EricE. Goubault. Computing the Vertices of Tropical Polyhedra using Directed HypergraphsDiscrete and Computational Geometry4922013, 247-279 • 48 incollectionX. Allamigeon, S. Gaubert and É. Goubault. Inferring Min and Max Invariants Using Max-plus PolyhedraProceedings of the 15th International Static Analysis Symposium (SAS'08)5079LNCSValencia, SpainSpringer2008, 189--204 • 49 articleXavierX. Allamigeon, StephaneS. Gaubert and MateuszM. Skomra. Solving generic nonarchimedean semidefinite programs using stochastic game algorithmsJournal of Symbolic Computation85An abridged version of this article appeared in the proceedings of ISSAC 20162018, 25-54 • 50 techreportPhilpottP. Andy, WahidW. Faisal and FrédéricF. Bonnans. MIDAS: A Mixed Integer Dynamic Approximation SchemeINRIA2016, • 51 book F. Baccelli, G. Cohen, G.-J. Olsder and J.-P. Quadrat. Synchronization and linearity: an algebra for discrete event systems Wiley 1992 • 52 articleMatthewM. Baker and OliverO. Lorscheid. Descartes' rule of signs, Newton polygons, and polynomials over hyperfieldsJ. Algebra5692021, 416--441 • 53 articleG. Barles, S. Mirrahimi and B. Perthame. Concentration in Lotka-Volterra parabolic or integral equations: a general convergence resultMethods Appl. Anal.1632009, 321--340 • 54 articleA. Bernig. Hilbert Geometry of PolytopesArchiv der Mathematik922009, 314-324 • 55 articleVianneyV. Boeuf and PhilippeP. Robert. A Stochastic Analysis of a Network with Two Levels of ServiceQueueing Systems923-4https://arxiv.org/abs/1708.095902019, 30 • 56 bookFrédéricF. Bonnans and StéphaneS. Gaubert. Recherche opérationnelle. Aspects mathématiques et applicationsEllipse2016, 391 • 57 articleP. Butkoviċ. Max-algebra: the linear algebra of combinatorics?Linear Algebra and its applications3672003, 313--335 • 58 bookPeterP. Butkoviċ. Max-linear systems: theory and algorithmsSpringer Monographs in MathematicsSpringer-Verlag London, Ltd., London2010, xviii+272 • 59 inproceedingsI. Chades, J. Carwardine, T. Martin, S. Nicol, R. Sabbadin and O. Buffet. MOMDPs: A Solution for Modelling Adaptive Management ProblemsProceedings of AAAI Conference on Artificial Intelligence2012, • 60 inproceedings J. Cochet-Terrasson, G. Cohen, S. Gaubert, M. McM. Gettrick and J.-P. Quadrat. Numerical computation of spectral elements in max-plus algebra Proc. of the IFAC Conference on System Structure and Control Nantes 1998 • 61 articleGuyG. Cohen, StéphaneS. Gaubert and Jean-PierreJ.-P. Quadrat. Max-plus algebra and system theory: where we are and where to go nowAnnual Reviews in Control231999, 207--219 • 62 articleAlainA. Connes and CaterinaC. Consani. Geometry of the arithmetic siteAdv. Math.2912016, 274--329 • 63 articleP. Cousot and R. Cousot. Abstract Interpretation: A unified lattice model for static analysis of programs by construction of approximations of fixed pointsPrinciples of Programming Languages 41977, 238--252 • 64 incollectionDanielD. Delling, PeterP. Sanders, DominikD. Schultes and DorotheaD. Wagner. Highway hierarchies starThe shortest path problem74DIMACS Ser. Discrete Math. Theoret. Comput. Sci.Amer. Math. Soc., Providence, RI2009, 141--174 • 65 articleArashA. Fahim, NizarN. Touzi and XavierX. Warin. A probabilistic numerical method for fully nonlinear parabolic PDEsAnn. Appl. Probab.2142011, 1322--1364 • 66 articleAlbertA. Fathi and AntonioA. Siconolfi. Existence of ${C}^{1}$ critical subsolutions of the Hamilton-Jacobi equation'Invent. Math.15522004, 363--388 • 67 articleOlivierO. Fercoq, MarianneM. Akian, MustaphaM. Bouhtou and StephaneS. Gaubert. Ergodic control and polyhedral approaches to PageRank optimizationIEEE Trans. Automat. Control5812013, 134--148 • 68 articleW.H.W. Fleming and W.M.W. McEneaney. A max-plus based algorithm for an HJB equation of non-linear filteringSIAM J. Control and Opt.2000, 683--710 • 69 articleS. Fomin and A. Zelevinsky. Cluster algebras. I. FoundationsJ. Amer. Math. Soc.1522002, 497--529 • 70 inproceedingsS. Gaubert, E. Goubault, A. Taly and S. Zennou. Static Analysis by Policy Iteration in Relational DomainsProceedings of the Proc. of the 16th European Symposium on Programming (ESOP'07)4421LNCSBraga (Portugal)Springer2007, 237--252 • 71 unpublishedS. Gaubert, M. Grangereau and W. VanW. Ackooij. Multi-stage stochastic Alternating Current Optimal Power Flow with storage: Bounding the duality gap2020, submitted • 72 articleS. Gaubert and J. Gunawardena. The Perron-Frobenius Theorem for Homogeneous, Monotone FunctionsTrans. of AMS356122004, 4931-4950 • 73 inproceedingsS. Gaubert, W.M.W. McEneaney and Z. Qu. Curse of dimensionality reduction in max-plus based approximation methods: theoretical estimates and improved pruning algorithmsProceedings of the 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC 11)Orlando, FL, USA2011, 1054-1061 • 74 incollectionS. Gaubert and M. Sharify. Tropical scaling of polynomial matricesPositive systems389Lecture Notes in Control and Inform. Sci.BerlinSpringer2009, 291--303 • 75 articleT. M.T. Gawlitza, H. Seidl, A. Adjé, S. Gaubert and E. Goubault. Abstract interpretation meets convex optimizationJ. Symbolic Comput.4712Special issue on Invariant generation and reasoning about loops2012, 1416--1446 • 76 bookI. M.I. Gelfand, M. M.M. Kapranov and A. V.A. Zelevinsky. Discriminants, resultants and multidimensional determinantsModern Birkhäuser ClassicsReprint of the 1994 editionBirkhäuser Boston, Inc., Boston, MA2008, x+523 • 77 articleSvenS. Hammarling, Christopher J.C. Munro and FrançoiseF. Tisseur. An algorithm for the complete solution of quadratic eigenvalue problemsACM Trans. Math. Software3932013, Art. 18, 19 • 78 book BerndB. Heidergott, Geert JanG. Olsder and JacobJ. van der Woude. Max Plus at Work: Modeling and Analysis of Synchronized Systems: A Course on Max-Plus Algebra and Its Applications Princeton 2005 • 79 articleHitoshiH. Ishii and HiroyoshiH. Mitake. Representation formulas for solutions of Hamilton-Jacobi equations with convex HamiltoniansIndiana Univ. Math. J.5652007, 2159--2183 • 80 bookI. Itenberg, G. Mikhalkin and E. Shustin. Tropical algebraic geometry35Oberwolfach SeminarsBaselBirkhäuser Verlag2007, viii+103 • 81 phdthesis PaulinP. Jacquot. Game theory and Optimization Methods for Decentralized Electric Systems http://www.theses.fr/2019SACLX101 École polytechnique 2019 • 82 misc PaulinP. Jacquot, StéphaneS. Gaubert, NadiaN. Oudjane, OlivierO. Beaude and PascalP. Benchimol. Procédé de gestion décentralisée de consommation électrique non-intrusif French Patent, FR1872553, filed to INPI on 7 Dec. 2018 2018 • 83 unpublishedPaulinP. Jacquot and ChengC. Wan. Nonatomic Aggregative Games with Infinitely Many Types2019, https://arxiv.org/abs/1906.01986 - working paper or preprint • 84 book V.N.V. Kolokoltsov and V.P.V. Maslov. Idempotent analysis and applications Kluwer Acad. Publisher 1997 • 85 bookB. Lemmens and R. Nussbaum. Nonlinear Perron-Frobenius theory189Cambridge Tracts in MathematicsCambridge University Press, Cambridge2012, xii+323 • 86 articleQ. Lu, M. Madsen, M. Milata, S. Ravn, U. Fahrenberg and K. G.K. Larsen. Reachability Analysis for Timed Automata using Max-Plus AlgebraJ. Logic Alg. Prog.8132012, 298-313 • 87 book V.P.V. Maslov. Méthodes Operatorielles Moscou Edition Mir 1987 • 88 articleW. M.W. McEneaney. A curse-of-dimensionality-free numerical method for solution of certain HJB PDEsSIAM J. Control Optim.4642007, 1239--1276 • 89 inproceedings W.M.W. McEneaney, A. Deshpande and S. Gaubert. Curse-of-Complexity Attenuation in the Curse-of-Dimensionality-Free Method for HJB PDEs Proc. of the 2008 American Control Conference Seattle, Washington, USA 2008 • 90 bookWilliam M.W. McEneaney. Max-plus methods for nonlinear control and estimationSystems & Control: Foundations & ApplicationsBoston, MABirkhäuser Boston Inc.2006, xiv+241 • 91 book J.-F. Mertens, S. Sorin and S. Zamir. Repeated Games Cambridge 2015 • 92 articleG. Mikhalkin. Enumerative tropical algebraic geometry in ${}^{2}$'J. Amer. Math. Soc.1822005, 313--377 • 93 articleRolf H.R. Möhring, MartinM. Skutella and FrederikF. Stork. Scheduling with AND/OR precedence constraintsSIAM J. Comput.3322004, 393--415 • 94 bookA. Papadopoulos. Metric spaces, convexity and non-positive curvature6IRMA Lectures in Mathematics and Theoretical PhysicsEuropean Mathematical Society (EMS), Zürich2014, xii+309 • 95 articleM. V. F.M. Pereira and L. M. V. G.L. Pinto. Multi-stage stochastic optimization applied to energy planningMath. Programming522, Ser. B1991, 359--375 • 96 articleGeorg Ch.G. Pflug and AloisA. Pichler. A Distance For Multistage Stochastic Optimization ModelsSIAM Journal on Optimization2212012, 1-23 • 97 incollectionJ.-E. Pin. Tropical SemiringsIdempotency11Publications of the Isaac Newton InstituteCambridge University Press1998, 50–69 • 98 inproceedingsM. Plus. Linear systems in (max,+) algebraProceedings of the 29th IEEE Conference on Decision and ControlIEEE1990, 151--156 • 99 phdthesisZhengZ. Qu. Théorie de Perron-Frobenius non linéaire et méthodes numériques max-plus pour la résolution d'équations d'Hamilton-JacobiEcole Polytechnique X2013, • 100 articleD. Reeb, M. J.M. Kastoryano and M. M.M. Wolf. Hilbert's projective metric in quantum information theoryJ. Math. Phys.5282011, 082201, 33 • 101 incollectionJ. Richter-Gebert, B. Sturmfels and T. Theobald. First steps in tropical geometryIdempotent mathematics and mathematical physics377Contemp. Math.Providence, RIAmer. Math. Soc.2005, 289--317 • 102 incollection G. Sagnol, S. Gaubert and M. Bouhtou. Optimal monitoring on large networks by Successive c-Optimal Designs Proceedings of the 22nd international teletraffic congress (ITC22), Amsterdam, The Netherlands, September http://dx.doi.org/10.1109/ITC.2010.5608717 IEEE 2010 • 103 articlePeterP. Sanders and DominikD. Schultes. Engineering highway hierarchiesACM J. Exp. Algorithmics172012, Article 1.6, 40 • 104 inproceedings S. Sankaranarayanan, H. Sipma and Z. Manna. Scalable Analysis of Linear Systems using Mathematical Programming VMCAI 3385 LNCS 2005 • 105 inproceedingsR. Sepulchre, A. Sarlette and P. Rouchon. Consensus in noncommutative spacesProceedings of the 49th IEEE Conference on Decision and ControlAtlanta, USA2010, 6596-6601 • 106 inproceedingsI. Simon. Limited subsets of a free monoidProc. 19th Annual Symposium on Foundations of Computer SciencePiscataway, NJ1978, 143--150 • 107 bookH. L.H. Smith. Monotone dynamical systems41Mathematical Surveys and MonographsAn introduction to the theory of competitive and cooperative systemsAmerican Mathematical Society, Providence, RI1995, x+174 • 108 phdthesis Duy-NghiD.-N. Tran. Programmation dynamique tropicale en optimisation stochastique multi-étapes Université Paris-Est 2020 • 109 articleNgoc MaiN. Tran and JosephineJ. Yu. Product-Mix Auctions and Tropical GeometryMath. O.R.444arXiv:1505.057372019, 1396--1411 • 110 articleS. Vernicos and D. Yang. A centro-projective InequalityC. R. Acad. Sci. Paris, Ser. I35782019, 681-685 • 111 incollectionO. Viro. Dequantization of real algebraic geometry on logarithmic paperEuropean Congress of Mathematics, Vol. I (Barcelona, 2000)201Progr. Math.BaselBirkhäuser2001, 135--146	2023-02-02 17:27:22	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 52, "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": 0.4538150429725647, "perplexity": 4305.480736777807}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1674764500035.14/warc/CC-MAIN-20230202165041-20230202195041-00744.warc.gz"}
http://openstudy.com/updates/50a2a371e4b0e22d17ef7b4f	jenny508 3 years ago Provide equation for ionization of a weak base in water solution? $$B+H_2O \rightarrow BH^+ + OH^-$$ or $$B^- + H_2O \rightarrow BH + OH^-$$ First line would be for $$NH_3$$ Second line would be for $$F^-$$	2015-12-01 23:59:52	{"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": 0.2310105264186859, "perplexity": 1019.0797601336279}, "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-2015-48/segments/1448398474527.16/warc/CC-MAIN-20151124205434-00327-ip-10-71-132-137.ec2.internal.warc.gz"}
https://physics.stackexchange.com/questions/632605/length-contraction-and-simultaneity-exercise	# Length Contraction and Simultaneity Exercise I came across a simple special relativity exercise in the book Relativity, Gravitation and Cosmology - A Basic Introduction 2nd Edition, Ta-Pei Cheng which I have problem fully understanding. Let me first "derive" length contraction as they do in the book Section 2.2.3 The length $$\Delta x$$ of a moving object, compared to the length $$\Delta x'$$ of the object as measured in its own rest frame $$O'$$, appears to be shortened. This penomenon is often called the FitzGerald-Lorentz contraction in the literature: $$\Delta x = \frac{\Delta x'}{\gamma} \quad \gamma > 1 \tag{1}$$ Consider the specific example of length measurement of a railcar. Let there be a clock attached to a fixed marker on the ground. A ground observer $$O$$, watching the train moving to the right with speed $$v$$, can measure the length $$L$$ of the car by reading off the times when the front and back ends of the railcar pass this marker on the ground: $$L = v(t_2 - t_1) \equiv v\Delta t \tag{2}$$ But for an observer $$O'$$ on the railcar, these two events correspond to the passing of the two ends of the car by the (ground-) marker as the marker is seen moving to the left. $$O'$$ can similarly deduce the length of the railcar in her reference frame by reading the times from the ground clock. $$L' = v(t_2' - t_1') \equiv v\Delta t' \tag{3}$$ These two unequal time intervals in (2) and (3) are related by the above considered time dilation $$\Delta t' = \gamma \Delta t$$, because $$\Delta t$$ is the time recorded by a clock at rest, while $$\Delta t'$$ is the time recorded by a clock in motion (with respect to the observer $$O'$$). From this we immediately obtain $$L = v\Delta t = \frac{v\Delta t'}{\gamma} = \frac{L'}{\gamma} \tag{4}$$ which is the claimed result (1) of length contraction. Now they have the following exercise Work out the spacetime coordinates $$(x,t)$$ of the two light pulses emitted from the midpoint of a railcar and arriving at the fron and back ends of the railcar as described in Section 2.2.2 cf. Fig 2.4. 1. Let the $$O'$$ coordinates be the railcar observer system, and $$O$$ the platform observer system. Given $$\Delta t' = 0$$, use the Lorentz transformation and its inverse to find the relations among $$\Delta t, \Delta x$$ and $$\Delta x'$$ 2. One of the relations obtained in (a) should be $$\Delta x = \gamma \Delta x'$$. Is this compatible with the derivation of length contraction as done in Section 2.2.3 (I posted that above)? Explain. They have the following solutions 1. Given the Lorentz transformation, as well as its inverse, it is clear that $$\Delta t' = 0$$ implies $$\Delta t = (\beta / c) \Delta x$$ and $$\Delta t = (\beta/c)\gamma \Delta x'$$. These two equalities require the consistency condition $$\Delta x = \gamma \Delta x'$$, which is compatible with the Lorentz transformation with $$\Delta t' = 0$$ 2. Our derivation of the length contraction in Section 2.2.3 (I posted that above) would lead us to expect the result of $$\Delta x' = \gamma^{-1} \Delta x$$ because the key input of the two ends of an object being measured at the same time in the "moving frame" is satisfied by our $$\Delta t' = 0$$ condition. Now I did get the same result for 1. but I don't understand 2. From Section 2.2.3 above I expected $$\Delta x = \frac{\Delta x' }{\gamma}$$ but using Lorentz transformation with $$\Delta t' = 0$$ gave me $$\Delta x = \gamma \Delta x'$$. I can't see the issue here and I don't understand the argument they make in the solution of 2. If I see such results, my first thought is to check which of the frames is moving and which is resting but they are the same in Section 2.2.3 and in the exercise, are they not? In both, $$O'$$ is the rest frame (on the railcar) and $$O$$ is the observer (on the platform/ground). Edit: To add more detail to the question, let me go through my solution in more detail. Let $$O'$$ be the rest frame and $$O$$ be the lab frame i.e. $$O'$$ would be on a train and $$O$$ would be at e.g. the train station. The train moves with a constant velocity $$v$$. I.e. the two frames move relative to each other with a constant velocity $$\pm v$$. We are looking at two events as described in Fig. 2.4. They are simultaneous in the rest frame $$O'$$ i.e. $$t_1' = t_2' \Rightarrow \Delta t' = 0$$ We then have the Lorentz-Transformation $$\Delta x' = \gamma(\Delta x - v\Delta t) \tag{5.1}$$ $$\Delta t' = \gamma(t - \frac{v}{c^2} \Delta x) \tag{5.2}$$ Using $$\Delta t = 0$$ and 5.2 we get $$\Delta t = \frac{v}{c^2}\Delta x$$. Plugging that into 5.1 we get $$\Delta x' = \gamma(\Delta x - \beta^2\Delta x) = \gamma \Delta x (1-\beta^2) = \gamma^{-1} \Delta x$$. So we find $$\Delta x' = \gamma^{-1} \Delta x \tag{6}$$ Now my solution above seems to agree with the solution given by the book. Now Subproblem 2 basically asks me to compare this result with the FitzGerald-Lorentz contraction described in Section 2.2.3. In Section 2.2.3 and in this problem, we use the same notation for the rest frame and the lab frame but we get different results: $$\Delta x_{\text{ex}}' = \gamma^{-1} \Delta x_{\text{ex}} \tag{7.1}$$ $$\Delta x_{\text{fitz}}' = \gamma \Delta x_{\text{fitz}} \tag{7.2}$$ Whereas 7.1 is the one derived here in the exercise and 7.2 derived in Section 2.2.3. The question now is: Where's the difference? I think the difference is that in the exercise we do measure the length contraction of two simultaneous ($$\Delta t' = 0$$) events whereas in Section 2.2.3 we have two non-simultaneous events ($$Delta t' \neq 0$$). So the much bigger question arises: How does simultaneity influence length contraction and how should I know which to use when? I think I also have a hard time seeing why 7.1 and 7.2 should differ. Why should the length depend on how I measure it? What's the conclusion here? I expect length contraction, that's fine but I kind of expect it to be constant between the two frames. I mean if it weren't, there wouldn't be a way to figure out "which clock setup" is the "right" one. • Can you post the fig 2.4? May 1 at 17:53 • what's the question? Is it that the distance between the light pulse hitting the ends of the car is $\gamma L$, when you were expecting $L/\gamma$? – JEB May 1 at 23:51 • @YoungKindaichi Added it May 2 at 8:12 • @JEB I added more details about the actual question. May 2 at 8:43 Well! It's the notation that makes you confused. Let's talk of part (1) first and to make things a little easier to see, I'll use capital letter's to use proper values. Using $$\Delta t'=0$$, you have concluded $$\Delta x=\gamma \ \Delta X$$ Now for a second concentrate on the section on Length contraction The measurements are simultaneous in the $$O-$$system, $$\Delta t=0$$ $$\Delta x'=\gamma \Delta X$$ The above result is written in the book as $$\Delta x'=\gamma \Delta x$$ (which is creating the confusion). which is identical to what we get in the first part. I give my take on this and hope to be helpful. The book's explanation is slightly vague to me, however, be informed that in many textbooks the definition of $$x$$ or $$\Delta x$$ (distance) slightly differs from $$L$$ (length), i.e., they are not necessarily the same, unless their similarity is clearly explained. Assume that there is a rod with a length $$L'$$ measured in its rest frame of A. If the rod has a velocity $$u$$ relative to the lab observer B, in the lab frame (using the Lorentz transformation), the rod's length of $$L$$ is calculated to be: [See the attached Figure.] $$L=\frac{1}{\gamma}L'.\tag{1}$$ Recall that the measurements shown in the figure are made by B. Now, assume that at a distance $$\Delta x$$ from B, there is a plate attached to the lab frame. If B and A synchronize their clocks at the time A meets B, in A's rest frame the said distance, i.e., the distance between the plate and A, is measured to be: $$\Delta x'=\frac{1}{\gamma}\Delta x.\tag{2}$$ The proof for my first equation is available in most of the relevant textbooks, however, I provide the proof of the second equation to you. The Lorentz transformation for $$x$$ in the above scenario implies: $$\Delta x'=\gamma (\Delta x+ut),\tag{3}$$ and for $$t$$, we have: $$t=\gamma (t'-\frac{u\Delta x'}{c^2}).\tag{4}$$ If we use $$t'=0$$ in Eq. (4) meaning that all of the clocks attached to A's frame along $$x'$$ are synchronous in A's frame (esp., the clock which is very close to the location of the plate though at rest in A's frame), the counterpart clock attached to B's frame at the location of the plate shows a time: $$t=\gamma (0-\frac{u\Delta x'}{c^2})=-\gamma \frac{u\Delta x'}{c^2}.\tag{5}$$ The negative sign means that when A's clock starts ticking, B's clock starts to work after a time $$t=\gamma u\Delta x'/c^2$$. Inserting Eq. (5) into Eq. (3), we get: $$\Delta x'=\gamma (\Delta x-\gamma \frac{u^2\Delta x'}{c^2}).\tag{6}$$ After simplification, we get: $$\Delta x'=\frac{1}{\gamma}\Delta x,\tag{7}$$ which is compatible with Eq. (2). One of the important differences between length and distance is that the two points between which a length is determined are both at rest or in motion with the same speed in reference frames. However, the points between which a distance is defined are both at rest in one frame, whereas they can have different velocities in another frame. In my example, for instance, if we denote by P the plate, the two points (A,P) of the distance between observer A and the plate have two different velocities. Point A is at rest in A's frame, while P has a velocity $$u$$. On the other hand, the two points of (B,P) for the same distance in B's frame are both at rest relative to B. In other words, the distance in some ways can be defined as a length varying by time, whereas the length itself is constant. • You conclude with "In other words, the distance in some ways can be defined as a length varying by time, whereas the length itself is constant.". I think, that makes sense. I do have a hard time translating this to the exercise above though. I understand the exercise as: We get different length measurements depending on whether our measurement is simultaneous or not. I can't see, concenptually/intuitively, why that should matter. May 2 at 9:00 In the first quote, it should be clear that the time dilation used as input to calculate length contraction compares the fixed clock in the ground with different clocks, one at each end of the railcar. This 2 clocks are syncronized in the train's frame. In this case the Lorentz formula:$$\Delta t' = \gamma (\Delta t - v\Delta x)$$ can be reduced to $$\Delta t' = \gamma \Delta t$$, because $$\Delta x = 0$$. In the second quote, the method of syncronizing this 2 clocks of the railcar is explained. The result of length contraction must be the same because what is being measured is the length of the railcar. It would be another result if the observer in the ground made 2 marks corresponding to the length of the train according his measurement. The distance between that marks would be shorter for the observer in the train. • Yeah, I think I do understand that. I do expect the result of the length contraction in both measurements/setup of clocks to be the same but if I look at 7.1 and 7.2 I can't see how they would be the same without having $\gamma = 1$. May 2 at 9:16 Why does simultaneity affect the measurement of length? Suppose a train 990m long is heading into a tunnel 1000m long at 100m per second. You and a friend don't know the length of the train, and are asked to determine whether the train is longer or shorter than the tunnel. You realise that you can answer the question by observing whether the back of the train enters the tunnel before or after the front of the train leaves it. If it enters afterwards, the train is longer than the tunnel- if it enters before, the train is shorter than the tunnel. Accordingly, you and a friend stand at either end of the tunnel and make a note of the times when the rear of the train enters the tunnel and the front leaves. If your watches are accurately synchronised, you will find that the rear of the train enters the tunnel 0.1s before the front leaves it, so the train is shorter than the tunnel. However, had the watch of the person observing the rear of the train been just 0.2s fast, you would have concluded that the train was longer than the tunnel. Your ability to measure the moving train depends upon you being able to pin down the respective locations of its two ends at the same moment, ie simultaneously. What is simultaneous for you, however, will not be simultaneous for someone moving relative to you, so they will appear to you to record the positions of the two ends of the train at two different times, and thus their measurement will be different from yours.	2021-10-17 05:36:42	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 82, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7935648560523987, "perplexity": 274.790605193993}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1634323585121.30/warc/CC-MAIN-20211017052025-20211017082025-00692.warc.gz"}
http://stats.stackexchange.com/questions/20439/standard-formula-for-quick-calculation-of-scores	# Standard formula for quick calculation of scores I have three datasets as follows: • Dataset A can have a maximum score of 30. (Individual scores can be from 1 to 30) • Dataset B can have a maximum score of 45. (Individual scores can be from 1 to 45) • Dataset C can have a maximum score of 25. (Individual scores can be from 1 to 25) The three datasets contribute equally to a final score, hence it is 33.33% each. i.e. (33.33% for dataset A + 33.33% for dataset B + 33.33% for dataset C) = 100% (final score) I have the following scores in each dataset: • Dataset A: 22 • Dataset B: 15 • Dataset C: 10 Is there a quick formula that I can use to work out the final score, based on the above datapoints? (This is not a homework question.) - $(22+15+10)/3$ would work but you might want to scale the scores (e.g. divide by the number of points possible) because the 45 point scale will contribute more weight than it should purely because there are more points possible – Macro Jan 1 '12 at 9:29 Thanks. How can I write the new formula (i.e. the one that takes into account the differences in the maximum scores possible) – Adhesh Josh Jan 1 '12 at 12:22 When you say "The three datasets contribute equally to a final score" do you mean that having the highest possible score on test B and lowest possible score on test C is as "good" as the highest possible score on test C and lowest possible score on test B? If not, then you could just add the three scores together to get something in the range 3 to 100. – Henry Jan 1 '12 at 23:31 There are different options for rescaling your summed scale scores (or scaled scores): • Express every score on a 0-100 point scale, with higher scores reflecting higher locations on the latent trait each scale purports to assess; • Standardize scores ($T$- or $z$-score) such that scores are deviations from the mean, expressed in standard deviation (SD) units. For $T$-scores, the mean and SD that are considered are 50 and 10, respectively. (Percentile-based or normalized scores are also common options. Note that for $T$-scores, we usually rely on the empirical mean and SD of a larger population that responded to all items previously (e.g., during large-scale field study) and which might be considered as a "reference population". Of course, more complex methods exist in the case of grading or equating raw scoring gathered throughout different measurement instruments.) The use of a common scale makes more sense with sum scores (it doesn't matter much if you consider the mean instead of the sum, which is what social scientists generally prefer compared to psychologists), as @Macro pointed out. Simple formulae exist in this case (this is just a rescaling problem), but the general idea can be summarized as follows: Scaled score = [(Raw score - Min response category score) / Range of possible response category scores] * 100 to get scores on a 100-point scale. If some items (or response categories) are negatively worded, you will need to reverse-score them first. Subtract the resulting score from 100 to get scores expressed in the reverse direction. Once each scale score (A, B, C) has been expressed on a common scale, you can use the arithmetic (unweighted) mean to compute your final score. - The formula is illustrative (thanks a lot!) and I am able to follow it but I am still having difficulty working out how to use the arithmetic mean in the last paragraph. (This is the second part of the above problem to work out the final score based on one-third contribution of A+B+C). Suppose the means for raw scores in Dataset A is 10, for Dataset B is 15 and for Dataset C is 20, how do I use them. I will be most grateful for an example. – Adhesh Josh Jan 3 '12 at 6:30 @Adhesh If each three scores are expressed on a common scale (say, 0-100 point), then $\bar x = (x_A+ x_B+ x_C)/3$ is what you are after (each scale contributes equally to the total score). With individual data like $(22,15,10)$, the final score would be $1/3\times(21/30+14/45+9/10)=0.637$ which you can conveniently express on any scale you want. (On a 0-45 point scale, it amounts to 28.7; on a 0-100 point, it is 63.7, etc.) – chl Jan 3 '12 at 7:46 Can't thank you enough! – Adhesh Josh Jan 3 '12 at 11:31	2013-05-20 17:18:12	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.5260501503944397, "perplexity": 982.0348072155175}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368699138006/warc/CC-MAIN-20130516101218-00006-ip-10-60-113-184.ec2.internal.warc.gz"}
https://www.lessonplanet.com/teachers/lets-learn-to-use-a-map	# Let's Learn to Use a Map Learners recognize a map, locate places on a map and define a map as a picture of a place.	2018-04-25 03:16:33	{"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": 0.8718523383140564, "perplexity": 1560.7634894675748}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125947690.43/warc/CC-MAIN-20180425022414-20180425042414-00253.warc.gz"}
https://www.futurelearn.com/courses/risk-management/0/steps/39268	1.13 ## SOAS University of London Securities and Exchange Commission, Washington DC # The return on bonds There are two main measures of return on bonds: the current yield and the yield to maturity. ### Current Yield The current yield, also known as interest yield or flat yield, is computed as the annual coupon payment divided by the market price of the bond. For example, if a bond has a face value of $100, an annual coupon of 4% (that is,$4 is paid each year) and the bond is traded at a price of $90, its current yield is: From this can you see the relation between changes in the market price of bonds and bond yields? For example, what happens to the current yield on this bond if the market price of the bond falls? ### Yield to Maturity The previous step showed how the market price of a bond is related to the discounted future cash flows associated with the bond (the coupons and the repayment of principal on maturity): we discount the future cash flows to the present, and sum the discounted values. From that relation we can identify another measure of return on bonds, the yield to maturity. The yield to maturity, also known as redemption yield, is the rate of return y for which the current price of the bond, P, is equal to the present discounted value of the future cash flows, C(1), C(2),…, C(T): where C(1) is the cash flow in period 1, C(2) is the cash flow in period 2, and C(T) is the cash flow at maturity. For instance, if a bond pays an annual coupon payment of$20, a principal of $1,000 in two years, and is currently traded at the price$980, its yield to maturity is obtained by solving the following equation for $y$: The solution of the above equation is $y$ = 0.0305. The yield to maturity of the bond is therefore 3.05%. Notice how the yield to maturity (3.05%) is higher than the coupon interest rate on the bond (2%). This is because at a market price of $980 you would pay less to buy the bond (which gives you the right to receive the coupon payments, and the face value at maturity) than the face value of the bond. We can ask the same question as before: what does this tell us about the relation between the market price of bonds and bond yields? For example, suppose you wanted a yield to maturity higher than 3.05%. Would you be prepared to pay more than$980 for this bond, or less than \$980?	2019-12-05 20:59:34	{"extraction_info": {"found_math": true, "script_math_tex": 2, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.4761991500854492, "perplexity": 1281.1558055252358}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1575540482038.36/warc/CC-MAIN-20191205190939-20191205214939-00064.warc.gz"}
http://mathhelpforum.com/trigonometry/57122-write-terms-sin-cos-print.html	# write in terms of sin and cos • November 2nd 2008, 12:05 PM fotzefotze write in terms of sin and cos write in terms of sinθ and cosθ csc θ - sinθcot^2θ thanks so much. I keep getting different answers • November 2nd 2008, 12:32 PM Soroban Hello, fotzefotze! Quote: Write in terms of $\sin\theta\text{ and }\cos\theta$ . . $\csc\theta - \sin\theta\cot^2\!\theta$ We have: . $\frac{1}{\sin\theta} - \sin\theta\!\cdot\!\frac{\cos^2\!\theta}{\sin^2\!\ theta} \;=\;\frac{1}{\sin\theta} - \frac{\cos^2\!\theta}{\sin\theta} \;=\;\frac{1-\cos^2\!\theta}{\sin\theta} \;=\;\frac{\sin^2\!\theta}{\sin\theta} \;=\;\sin\theta$	2015-09-05 15:44:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 3, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7887826561927795, "perplexity": 4397.828619467064}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1440646312602.99/warc/CC-MAIN-20150827033152-00031-ip-10-171-96-226.ec2.internal.warc.gz"}
https://www.clutchprep.com/chemistry/practice-problems/45531/phosgene-cocl2-a-poisonous-gas-decomposes-according-to-the-equation-cocl2-g-8651	# Problem: Phosgene, COCl2, a poisonous gas, decomposes according to the equationCOCl2(g) ⇌ CO(g) + Cl2(g)Calculate Kp for this reaction if Kc = 0.083 at 900ºC.A) 0.125B) 8.0 C) 6.1 D) 0.16 E) 0.083 ###### FREE Expert Solution 82% (204 ratings) ###### FREE Expert Solution We are being asked to calculate Kp for the decomposition of phosgene (CoCl2) where its Kc is given. COCl2(g) CO(g) + Cl2(g) When dealing with equilibrium: Kc → equilibrium units are in molarity Kp → equilibrium units in terms of pressure Kp and Kc are related to one another by the following equation below: $\overline{){{\mathbf{K}}}_{{\mathbf{p}}}{\mathbf{=}}{{\mathbf{K}}}_{{\mathbf{c}}}{\left(\mathbf{RT}\right)}^{\mathbf{∆}\mathbf{n}}}$ 82% (204 ratings) ###### Problem Details Phosgene, COCl2, a poisonous gas, decomposes according to the equation COCl2(g) ⇌ CO(g) + Cl2(g) Calculate Kp for this reaction if Kc = 0.083 at 900ºC. A) 0.125 B) 8.0 C) 6.1 D) 0.16 E) 0.083 What scientific concept do you need to know in order to solve this problem? Our tutors have indicated that to solve this problem you will need to apply the Chemical Equilibrium concept. You can view video lessons to learn Chemical Equilibrium. Or if you need more Chemical Equilibrium practice, you can also practice Chemical Equilibrium practice problems. What is the difficulty of this problem? Our tutors rated the difficulty ofPhosgene, COCl2, a poisonous gas, decomposes according to th...as medium difficulty. How long does this problem take to solve? Our expert Chemistry tutor, Dasha took 3 minutes and 10 seconds to solve this problem. You can follow their steps in the video explanation above. What professor is this problem relevant for? Based on our data, we think this problem is relevant for Professor Maxwell's class at UCF.	2021-01-19 01:50:36	{"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": 0.5525960922241211, "perplexity": 10593.043408534058}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703517559.41/warc/CC-MAIN-20210119011203-20210119041203-00177.warc.gz"}
https://testbook.com/blog/mechanical-vibrations-gate-quiz-1/	• Save • Share # Mechanical Vibrations GATE ME Quiz 1 2 years ago . Here is a quiz to help you prepare for your upcoming GATE 2016 exam. The GATE ME paper has several subjects, each one is equally important. However, one of the most important subjects in GATE ME is Mechanical Vibrations. The subject is vast, but practice makes tackling it easy. This quiz contains important questions which match the pattern of the GATE exam. Check your preparation level in every chapter of Mechanical Vibrations for GATE ME by taking the quiz and comparing your ranks. Learn about Free vibration, forced vibration, degree of freedom, effect of damping, vibration isolation, resonance, speeds of shafts and more! Mechanical Vibrations for GATE ME Quiz 1 Que. 1 For an underdamped harmonic oscillator, resonance 1. occurs when excitation frequency is greater than undamped natural frequency 2. occurs when excitation frequency is less than undamped natural frequency 3. occurs when excitation frequency is equal to undamped natural frequency 4. never occurs Que. 2 The natural frequency of the system shown below is 1. $$\sqrt {\frac{k}{{2m}}}$$ 2. $$\sqrt {\frac{k}{m}}$$ 3. $$\sqrt {\frac{{2k}}{m}}$$ 4. $$\sqrt {\frac{{3k}}{m}}$$ Que. 3 The equation of motion of a harmonic oscillator is given by $$\frac{{{d^2}x}}{{d{t^2}}} + 2\zeta {\omega _n}\frac{{dx}}{{dt}} + \omega _n^2x = 0$$ and the initial conditions at t = 0 are $$x\left( 0 \right) = X,\;\;\frac{{dx}}{{dt}}\left( 0 \right) = 0$$. The amplitude of x(t) after n complete cycles is 1. $$X{e^{ - 2n\pi \left( {\frac{\zeta }{{\sqrt {1 - {\zeta ^2}} }}} \right)}}$$ 2. $$X{e^{2n\pi \left( {\frac{\zeta }{{\sqrt {1 - {\zeta ^2}} }}} \right)}}$$ 3. $$X{e^{ - 2n\pi \left( {\frac{{\sqrt {1 - {\zeta ^2}} }}{\zeta }} \right)}}$$ 4. $$X$$ Que. 4 The natural frequency of the spring mass system shown in the figure is closest to 1. 8 Hz 2. 10 Hz 3. 12 Hz 4. 14 Hz Que. 5 A uniform rigid rod of mass m = 1 kg and length L = 1 m is hinged at its centre and laterally supported at one end by a spring of spring constant k = 300 N/m. The natural frequency ωn in rad/s is 1. 10 2. 20 3. 30 4. 40 ## More Mechanical Vibrations for GATE ME Quizzes here: Try 1000+ Questions on our App. • Save 2 years ago 2 years ago 2 years ago 2 years ago	2018-03-23 16:57:45	{"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": 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": 0.47841253876686096, "perplexity": 2711.432458146728}, "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-2018-13/segments/1521257648404.94/warc/CC-MAIN-20180323161421-20180323181421-00158.warc.gz"}
http://mathhelpforum.com/algebra/117194-perfect-cube-decomposition-question.html	# Math Help - Perfect cube decomposition question 1. ## Perfect cube decomposition question can someone explain to me how they got $(4x - y)$ in the numerator? problem: $( 64x^3 - y^3 ) / ( 4x^2 - 33xy + 8y^2 )$ first step: $[ ( 4x - y ) ( 16x^2 + 4xy + y^2 ) ] / [ ( 4x - y ) ( x - 8y ) ]$ I know they used the perfect cube formula, but why use 16 and 4. and where did the $(4x - y)$ come from? 2. Originally Posted by vd853 can someone explain to me how they got $(4x - y)$ in the numerator? problem: $( 64x^3 - y^3 ) / ( 4x^2 - 33xy + 8y^2 )$ first step: $[ ( 4x - y ) ( 16x^2 + 4xy + y^2 ) ] / [ ( 4x - y ) ( x - 8y ) ]$ I know they used the perfect cube formula, but why use 16 and 4. and where did the $(4x - y)$ come from? Hi vd853, The general factorization model for the differernce of two cubes is: $(a^3-b^3)=(a-b)(a^2+ab+b^2)$ Your numerator could be re-written as $((4x)^3-y^3)$ Using this and the model, we come up with $(4x-y)(16x^2+4xy+y^2)$ 3. Thanks! did not know about the difference formula.	2014-03-10 01:41:42	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 11, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8383697271347046, "perplexity": 673.1576623396313}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394010527022/warc/CC-MAIN-20140305090847-00059-ip-10-183-142-35.ec2.internal.warc.gz"}
http://mathhelpforum.com/statistics/105847-small-problem.html	1. ## Small problem Five teachers at a meeting remove their name tags and put them in a bag. One by one, each teacher randomly draws one tag from the bag until the bag is empty. What is the probability that exactly one of the teachers draws the correct name tag? i got this so far... one person gets their name and four of the people dont get their names: P(that)= 1/5 * 3/4 * 2/3 * 1/2 * 1 = .05 then since the order may happen any given way, we multiply by the number of arrangements possible: 5 "5 choose 1 = 5!/[4!1!]" ans = .25 2. Originally Posted by ruthvik [FONT=arial]Five teachers at a meeting remove their name tags and put them in a bag. One by one, each teacher randomly draws one tag from the bag until the bag is empty. What is the probability that exactly one of the teachers draws the correct name tag? There are 120 ways for the name tags to endup. There are 44 ways in which no one gets his/her own name tag. How many get ways are there for someone to get the correct name tag? There are 5 ways for exactly one to be correct and 9 ways for the orhers to be incorrect. 3. Hello, ruthvik! This is not a simple problem . . . Five teachers at a meeting remove their name tags and put them in a bag. One by one, each teacher randomly draws one tag from the bag until the bag is empty. What is the probability that exactly one of the teachers draws the correct name tag? As Plato pointed out, there are: . $5! \:=\:120$ ways the tags could be distributed. Now, exactly one person gets his/her tage. . . There are ${\color{red}5}$ choices for that person. Then the other four must not get their own tags. Suppose their names are $A,B,C,D.$ $\begin{array}{cc}\text{The tags could given like this:} \\ \\[-4mm] BADC \\ BCDA \\ BDAC \\ CADB \\ CDAB & {\color{red}9}\text{ ways}\\ CDBA \\ DABC \\ DCAB \\ DCBA \end{array}$ Hence, there are: . $5\cdot9 \:=\:45$ ways. The probability is: . $\frac{45}{120} \;=\;\frac{3}{8}$	2016-09-25 16:16:25	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.763468325138092, "perplexity": 580.3700181450832}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738660242.52/warc/CC-MAIN-20160924173740-00159-ip-10-143-35-109.ec2.internal.warc.gz"}
https://www.tutorialspoint.com/c-program-for-the-difference-between-sums-of-odd-and-even-digits	# C Program for the Difference between sums of odd and even digits? Given a number, find the difference between sum of odd digits and sum of even digits. Which means we will be count all even digits and all odd digits and the subtracting their sums. ### Sample Input:12345 Output:3 ## Explanation the odd digits is 2+4=6 the even digits is 1+3+5=9 odd-even=9-6=3 Taking every digit out of number and checking whether the digit is even or odd if even then add it to even sum if not then to odd sum and then take difference of them. ## Example #include <iostream> using namespace std; int main() { int n, r=0; int diff =0; int even=0; int odd=0; n=12345; while(n != 0){ r = n%10; if(r % 2 == 0) { even+=r; } else { odd+=r; } n/=10; } diff=odd-even; printf("%d",diff); return 0; }	2023-03-27 20:17:23	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.2578893005847931, "perplexity": 1625.4290510938392}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1679296948684.19/warc/CC-MAIN-20230327185741-20230327215741-00012.warc.gz"}
https://www.thepoorcoder.com/hackerrank-compare-the-triplets-solution/	Hackerrank - Compare the Triplets solution # Hackerrank - Compare the Triplets solution Alice and Bob each created one problem for HackerRank. A reviewer rates the two challenges, awarding points on a scale from 1 to 100 for three categories: problem clarity, originality, and difficulty. We define the rating for Alice's challenge to be the triplet , and the rating for Bob's challenge to be the triplet . Your task is to find their comparison points by comparing with , with , and with . • If a[i] > b[i], then Alice is awarded 1 point. • If a[i] < b[i], then Bob is awarded 1 point. • If a[i] = b[i], then neither person receives a point. Comparison points is the total points a person earned. Given and , determine their respective comparison points. For example, and . For elements , Bob is awarded a point because . For the equal elements and , no points are earned. Finally, for elements , so Alice receives a point. Your return array would be with Alice's score first and Bob's second. Function Description Complete the function compareTriplets in the editor below. It must return an array of two integers, the first being Alice's score and the second being Bob's. compareTriplets has the following parameter(s): • a: an array of integers representing Alice's challenge rating • b: an array of integers representing Bob's challenge rating Input Format The first line contains space-separated integers, , , and , describing the respective values in triplet . The second line contains space-separated integers, , , and , describing the respective values in triplet . Constraints Output Format Return an array of two integers denoting the respective comparison points earned by Alice and Bob. Sample Input 05 6 73 6 10 Sample Output 01 1 Explanation 0 In this example: Now, let's compare each individual score: • , so Alice receives point. • , so nobody receives a point. • , so Bob receives point. Alice's comparison score is , and Bob's comparison score is . Thus, we return the array . Sample Input 117 28 3099 16 8 Sample Output 12 1 Explanation 1 Comparing the elements, so Bob receives a point. Comparing the and elements, and so Alice receives two points. The return array is . ### Solution in Python #!/bin/python3 import math import os import random import re import sys # Complete the compareTriplets function below. def compareTriplets(a, b): awins = 0 bwins = 0 for x,y in zip(a,b): if x>y: awins+=1 elif x<y: bwins+=1 return [awins, bwins] if __name__ == '__main__': fptr = open(os.environ['OUTPUT_PATH'], 'w') a = list(map(int, input().rstrip().split())) b = list(map(int, input().rstrip().split())) result = compareTriplets(a, b) fptr.write(' '.join(map(str, result))) fptr.write('\n') fptr.close()	2022-05-18 09:37:12	{"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": 0.6065635085105896, "perplexity": 8783.441389888469}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662521883.7/warc/CC-MAIN-20220518083841-20220518113841-00474.warc.gz"}
https://advances.sciencemag.org/content/6/2/eaaz0289	Research ArticleSOCIAL SCIENCES # Folk standards of sound judgment: Rationality Versus Reasonableness See allHide authors and affiliations Vol. 6, no. 2, eaaz0289 ## Abstract Normative theories of judgment either focus on rationality (decontextualized preference maximization) or reasonableness (pragmatic balance of preferences and socially conscious norms). Despite centuries of work on these concepts, a critical question appears overlooked: How do people’s intuitions and behavior align with the concepts of rationality from game theory and reasonableness from legal scholarship? We show that laypeople view rationality as abstract and preference maximizing, simultaneously viewing reasonableness as sensitive to social context, as evidenced in spontaneous descriptions, social perceptions, and linguistic analyses of cultural products (news, soap operas, legal opinions, and Google books). Further, experiments among North Americans and Pakistani bankers, street merchants, and samples engaging in exchange (versus market) economy show that rationality and reasonableness lead people to different conclusions about what constitutes good judgment in Dictator Games, Commons Dilemma, and Prisoner’s Dilemma: Lay rationality is reductionist and instrumental, whereas reasonableness integrates preferences with particulars and moral concerns. ## INTRODUCTION What are the key features of a sound judgment? In game theory and dominant streams of economics, sound judgment is intimately linked with the notion of the rational person—a formal, preference-maximizing agent (13). In a disparate literature on political and legal theory, at least since the Roman times, sound judgment has been linked with the well-defined standard of a reasonable person—a pragmatic, socially conscious observer (47). Decades of research in behavioral economics have shown that people often fall short of the rational standard (810), raising the question: How do laypeople understand rationality and do they systematically differentiate it from reasonableness? If laypeople view rationality and reasonableness as distinct standards of judgment, then deviations of behavior from game theoretical models may not reflect judgment errors (8, 10, 11), but rather people’s use of a reasonable standard to guide their choices. To explore how people use rationality and reasonableness in their lives, here we use a telescopic approach. First, we examine folk concepts of rational and reasonable actors via content analyses of ascribed characteristics and attributions of personality and behaviors, showing that a rational agent is viewed as a preference maximizer, whereas a reasonable agent is viewed as a satisficer. Moving to implicit norms for reasonable and rational actors, we performed analyses of cultural products (12): news, soap operas, legal opinions, and corpora of English, Spanish, Portuguese, and Russian books. We show that linguistic use of “rational” is abstract and individual focused, whereas “reasonable” is context sensitive and socially focused. In subsequent 13 experiments, we demonstrate framing effects of rationality versus reasonableness on expectations and behavior in Commons Dilemma, Dictator Games, and Prisoner’s Dilemma, with expectations for reasonable/rational judgments converging and diverging depending on others’ cooperation- versus competition-signaling behaviors. Last, we replicate the evidence concerning dissociation of rational versus reasonable behavior in a socioeconomically diverse non-Western society. The distinction between rationality and reasonableness for people’s everyday decisions can be traced back to political and legal theories (see the Supplementary Materials), some claiming that people internalized the chief features of scholarly definitions: rationality as a formal and instrumental standard (13), the capacity to exercise judgment in defining one’s key preferences and selecting effective means to pursue those preferences (14); reasonableness as a context-dependent and pragmatic standard, balancing realist expectations of most common actions and normative concerns (15) in a respectful way (4, 16, 17). These theoretical claims remain untested and stand in contrast to a view that in everyday language these concepts appear equivalent. After all, the two terms derive from the same etymological root (18). However, if folk understanding of rationality versus reasonableness aligns with behaviors expected from a rational person in game theory and a reasonable person in legal scholarship (17, 19), respectively, then framing decisions in terms of rationality versus reasonableness may prompt people either to maximize their preferences as advocated in neoclassical economics or to flexibly integrate others’ interests as advocated in ethics (2022) and political theory (23). The chief goal of the present research was to test this possibility. ## RESULTS ### Studies 1 and 2: Personality, stereotypes, and behavioral attributes of rational versus reasonable persons In preregistered studies 1 and 2, we examined spontaneous descriptions of people who behave rationally and reasonably and examined attributions of personality characteristics (24), societal stereotypes (25), and maximizing and satisficing behaviors (26) to rational/reasonable actors. As Fig. 1 shows, rational and reasonable were among the most frequent descriptions of each other. The similarities in the descriptors of rational and reasonable actors concerned thoughtfulness, calmness, and intelligence. Consistent with our predictions, we also observed substantial differences. Descriptions of rational persons were more likely to concern abstract and decontextualizing characteristics such as logic, systematicity, analytical skills, and emotional suppression, whereas descriptions of reasonable persons were more likely to include socially conscious characteristics such as honesty, kindness, fairness, and interpersonal sensitivity (also see table S2). We cross-validated these results via hypothesis-blind content analyses, showing that stoic, logic-, and intelligence-oriented characteristics were significantly more likely to be attributed to rational persons (\|Zs\| > 2.84, P < 0.005), whereas morality- and interpersonally oriented characteristics were significantly more likely to be attributed to reasonable persons (\|Zs\| > 4.33, P < 0.001). There were no significant differences in attributions of levelheadedness and objective/professional characteristics. Examination of questionnaire-based personality and stereotype-related characteristics yields similar results. As Fig. 2 indicates, participants were significantly more likely to associate the term “reasonable” rather than “rational” with adjectives representing socially oriented characteristics (honesty/humility, agreeableness, emotionality, and extraversion) (2.52 < ts < 3.68, 0.012 < P < 0.001). Societal stereotypes of rational persons concerned relatively greater agency (t = 3.89, P < 0.001), whereas stereotypes of reasonable persons concerned greater communion (t = 6.70, P < 0.0001) and lower selfishness (t = 3.77, P < 0.001). Moreover, rational persons were viewed as maximizers—pursuing the best option and searching through all alternatives [3.56 < t(df = 240) < 6.28, Ps < 0.001], whereas reasonable persons were viewed as satisficers, accepting the best acceptable option [t(df = 240) = 2.66, Ps = 0.008]. As supplementary analyses show, people’s views of rational and reasonable persons were comparatively consistent, whereby the views of a reasonable (versus rational) person were uniquely aligned with attributions of agency, communion, and selfishness to an ideal person. Moreover, characteristics of rational and reasonable persons explained independent variance in attributions of competence to an ideal person. These results both oppose the idea that rationality represents a sole standard of judgment or that reasonableness is a vague, low-relevance concept for judgmental competence (also see consistency of distributions of expected behavior by reasonable versus rational actors in Fig. 3). ### Studies 3 and 4: Computerized and human-coded content analyses of news on the web, American soap operas, SCOTUS opinions, Google books, and national corpora In study 3, we examined over 5 billion words of data from web-based English language news sources from January 2010 to September 2017—the largest corpus of everyday English language to date. Specifically, we explored the top 100 nouns most frequently following “reasonable” and “rational,” quantifying unique associations and classifying words as abstract/person focused and context sensitive (i.e., variation across either intertemporal or social interpersonal contexts) via human coders. We reasoned that term-specific associations (e.g., idioms) would carry the psychological meaning of the respective constructs. Human-guided content analyses revealed that 33% of nouns following “rational” were classified as abstract/person focused, compared with 6% of nouns following “reasonable” [χ2(df = 1, N = 200) = 23.22, P < 0.0001, Cramer’s V = 0.341]. Conversely, 40% of nouns following “reasonable” were classified as reflecting contextual contingencies, but only 4% of nouns following “rational” [χ2(df = 1, N = 200) = 37.76, P < 0.001, Cramer’s V = 0.435; see Fig. 3]. We cross-validated our content analyses on soap operas and Supreme Court of the United States (SCOTUS) opinions. In soap operas, compared with 36% of nouns following “rational” classified as abstract/person focused, only 23% of nouns following “reasonable” were classified as abstract/person focused [χ2(df = 1, N = 140) = 2.79, P = 0.095, Cramer’s V = 0.141]. Conversely, 28% of nouns following “reasonable” were classified as reflecting contextual contingencies, whereas only 10% of nouns following “rational” were classified into this category [χ2(df = 1, N = 140) = 7.76, P = 0.005, Cramer’s V = 0.235]. In SCOTUS opinions, compared with 89% of nouns following “rational” classified as abstract/person focused, only 38% of nouns following “reasonable” were classified as abstract/person focused [χ2(df = 1, N = 200) = 56.11, P < 0.001, Cramer’s V = 0.530]. Conversely, 72% of nouns following “reasonable” were classified as reflecting contextual contingencies, whereas only 43% of nouns following “rational” were classified into this category [χ2(df = 1, N = 200) = 17.21, P < 0.001, Cramer’s V = 0.293]. These observations indicate that the notion of reasonableness in cultural products is more likely to take contextual particulars (intertemporal uncertainty and interpersonal considerations) into account, whereas the notion of rationality appears to chiefly focus on the abstract, individual attributes and preferences. In study 3b, we further bolstered this inference by examining the relative frequencies of definite (“the”) and indefinite (“a”) articles preceding utterances reflecting “reasonable” and “rational” judgment (utterances finishing with “action”/“decision”/“thing to do”). In this context, indefinite articles indicate a general recommendation because they imply that more than one approach could be appropriate (e.g., “a reasonable action”), whereas the definite article indicates the application of a rule because it implies that only one approach is appropriate (e.g., “the rational thing to do”). We examined both the news corpus and the American Google Books—the largest corpus of English books available to date (27). In the news, statements reflecting rational judgments were 1.41 times more likely to be preceded by the definite article than statements reflecting reasonable judgments. Conversely, statements reflecting reasonable judgments were 3.84 times more likely to be preceded by the indefinite article than statements reflecting rational judgments. Results were similar for Google books, with definite articles 1.40 times more likely to precede rational judgments and indefinite articles 2.10 times more likely to precede reasonable judgments. Does the distinction between reasonableness and rationality exist in other languages beyond English? In study 4, we addressed this question by extending human-coded content analyses of the top 100 associations in the Spanish and Portuguese corpora of Google Books, as well as a random subset of 100 sentences including reasonable/rational in the Russian National Corpus (see Supplementary Materials for methods of identifying key terms and their translation). Similar to English, 45 to 70% of nouns following “rational” were classified as individual focused, compared with 15 to 18% of nouns following “reasonable”: Spanish, χ2(df = 1, N = 200) = 30.73, P < 0.0001, Cramer’s V = 0.392; Portuguese, χ2(df = 1, N = 200) = 19.84, P < 0.0001, Cramer’s V = 0.315; Russian, χ2(df = 1, N = 200) = 54.87, P < 0.0001, Cramer’s V = 0.524. Conversely, 43 to 56% of nouns following “reasonable” were classified as reflecting contextual contingencies, whereas only 5 to 14% of nouns following “rational” were classified into this category: Spanish, χ2(df = 1, N = 200) = 27.02, P < 0.0001, Cramer’s V = 0.368; Portuguese, χ2(df = 1, N = 200) = 43.14, P < 0.0001, Cramer’s V = 0.464; Russian, χ2(df = 1, N = 200) = 43.14, P < 0.0001, Cramer’s V = 0.464. Overall, analyses of written media in English-, Spanish-, Portuguese-, and Russian-speaking countries show qualitatively distinct norms of reasonableness and rationality, such that reasonableness includes interpersonal consideration and focuses on contextual particulars. Conversely, the standard of rationality appears to reflect decontextualized judgments aligned with individual attributes and instrumental preferences. In subsequent experiments, we examined how people differentiate rationality and reasonableness when evaluating choices in the context of a Dictator Game (9)—a behavioral economic game where player A can choose what fraction of a resource ($10) to share with anonymous player B, who must accept the offer. If the cultural associations between rationality and individual preference maximization and reasonableness and socially conscious considerations are internalized at the individual level, then we should find that people expect rational choices to be more preference maximizing and reasonable choices to be more socially conscious. In three experiments, we varied design (within versus between subject), examined different populations (see table S1), and tested several boundary conditions (see Supplementary Materials for more details). In studies 5a and 5b, Amazon Mechanical Turk (MTurk) workers reported expected contributions by reasonable and rational persons in player A’s role. Study 5c replicated effects on university students and explored whether predicted actions for reasonable or rational agents are closer to the perceived typical person in their community and their personal choice as player A. As Fig. 3 shows, reasonable people were expected to share on average 7 to 20% more than rational people. Reasonable people were expected to share more than rational people: study 5a, t(281) = 5.42, ηp2 = 0.095; study 5b, t(960.89) = 5.23, ηp2 = 0.027; study 5c, t(206) = 8.96, ηp2 = 0.280 (all Ps < 0.001); a perceived average student: study 5c, t(206) = 9.07, ηp2 = 0.290 (P < 0.001); or oneself: study 5b, t(491) = 3.83, ηp2 = 0.029; study 2c, t(206) = 4.95, ηp2 = 0.290 (all Ps < 0.001). Conversely, average personal sharing was significantly higher than those expected for a rational person [study 5b: t(493) = 2.85, P = 0.005, ηp2 = 0.016; study 5c: t(206) = 5.49, P < 0.001, ηp2 = 0.128] or the perceived average student [study 5c: t(206) = 5.02, P < 0.001, ηp2 = 0.109]. The latter observation dovetails with earlier work on cynical, asymmetric expectations of selfishness by others versus oneself (28, 29). The observed dissociation was robust to changing scale direction (unreasonable versus irrational) and question form [asking what reasonable/rational people would give versus asking what would be the reasonable/rational amount; (12)]. Moreover, self-perceptions of reasonableness and rationality predicted anticipated decisions: Participants who viewed themselves as reasonable shared significantly more (βExp5b = 0.10, t = 2.51, P = 0.01; βExp5c = 0.33, t = 4.96, P < 0.001), whereas participants who viewed themselves as rational shared less (βExp5b = −0.06, t = 1.49, P = 0.14; βExp5c = −0.24, t = 3.62, P < 0.001). ### Studies 6 and 7: Rational and reasonable personal choice Two subsequent experiments used between-subject designs and revealed that the distinction between reasonable versus rational agents extends to framing of personal choices (16). Participants intended to donate 5% more money in a Dictator Game if they were seeking to be reasonable versus rational (see fig. S2): study 6a, t(353.76) = 2.65, P = 0.009, ηp2 = 0.016; study 6b, t(493.56) = 2.01, P = 0.045, ηp2 = 0.008. Study 6b simultaneously tested the impression of selfishness, agency, and communion for rational versus reasonable people. Reasonable people were perceived as less selfish than rational people [t(511) = 3.87, P < 0.001, ηp2 = 0.028], and this difference in selfishness mediated the difference in predicted sharing by reasonable versus rational people (Z = 3.19; 95% CIbootstrapped, 0.094 to 344). Do these effects extend to personal choice on an independent task? In the preregistered study 7 (osf.io/sy24t), participants recalled reasonable or rational actions from their lives and subsequently took part in a standard Dictator Game. Recall of a reasonable action resulted in 3.18% higher offers in the Dictator Game (M =$4.28, SD = 1.91) than recall of a rational action (M = 3.96, SD = 2.01) [Wald (df = 1) = 7.79, P = 0.005] (see fig. S3). This effect holds when controlling for socioeconomic factors (age, gender, and income) [Wald (df = 1) = 8.21, P = 0.004] and with nonparametric analysis of results (U = 167,998, P = 0.006). Whereas 14.15% of participants in the rational condition donated none of the endowment to the other person, only 9.56% of participants in the reasonable condition suggested donated nothing [N = 1116, χ2(df = 1) = 5.67, P = 0.017, Cramer’s V = 0.07]. Conversely, 71.01% of participants in the reasonable condition donated at least half of the endowment to the other person, whereas only 65.47% in the rational condition did so [N = 1116, χ2(df = 1) = 5.56, P = 0.018, Cramer’s V = 0.07]. ### Studies 8 and 9: Use of reasonable versus rational standards and generalizability across dilemmas and interpersonal transactions Study 8 extended the distinction between rational and reasonable standards to expectations in two other classic economic games where a person’s preference conflicts with others’ interests—Commons Dilemma and Prisoner’s Dilemma (30). Participants expected rational people to withdraw 12% more from the common pool as compared with reasonable people in the Commons Dilemma [t(305) = 5.27, P < 0.001, ηp2 = 0.083]. In the Prisoner’s Dilemma, participants expected rational people to pick defecting and cooperative options to a similar extent (defect = 197/cooperate = 190) but expected reasonable people to overwhelmingly select a cooperative option (defect = 84/cooperate = 303) [N = 387, χ2(df = 1) = 7.04, P = 0.008, Cramer’s V = 0.14]. Given the intermediate position of personal choices between rational and reasonable people (Fig. 3), we examined whether participants use the reasonable versus rational choice distinction in economic and interpersonal transactions to their benefit. Specifically, if laypeople internalize both preference-maximizing features of rationality and socially conscious features of reasonableness standards, it is possible that they systematically favor a rational agent in self-focused situations and a reasonable agent in other-focused situations. Study 8 addressed this question in the context of economic games (Dictator Game, Commons Dilemma, and Prisoner’s Dilemma), whereas study 9 extended it to negotiations, legal disputes, and managerial decision-making. For each situation, participants indicated whether they would prefer a reasonable or a rational agent to act on their behalf and on behalf of another party (in a randomized order). In both studies, participants overall favored a rational agent more than a reasonable agent to act on their behalf: study 8, Wald χ2(df = 1) = 39.40, P < 0.001; study 9, Wald χ2(df = 1) = 7.92, P = 0.005 (see Fig. 4). Conversely, participants favored a reasonable over a rational agent for the other parties involved in a dilemma: study 8, Wald χ2(df = 1) = 21.63, P < 0.001; study 9, Wald χ2(df = 1) = 85.38, P < 0.001. See the Supplementary Materials for scenario-specific analyses and results. ### Studies 10 and 11: What actions are rational, reasonable, and kind? We examined how expectations for reasonable and rational judgments converge and diverge depending on the situation (others’ cooperative versus competitive-signaling behaviors) as a way to further probe whether people view rationality and reasonableness as distinct standards rather than opposite sides of the same evaluative dimension of prosociality. In study 10, participants evaluated reasonableness and rationality of a player in single- and multiround Prisoner’s Dilemmas (see top panel of fig. S4 for types of transactions). In study 11, we rerun the two-round games, simultaneously comparing evaluations of reasonableness to kindness. In single-round games, cooperating players were viewed as more reasonable (versus rational) and defecting players more rational (versus reasonable) [F(1,590) = 57.71, P < 0.001, ηp2 = 0.089]. In two-round games, when both players behave similarly on round 1 (symmetric game), cooperating on round 2 is viewed as more reasonable and defecting on round 2 is viewed as more rational: study 10, F(1,590) = 45.44, P < 0.001, ηp2 = 0.072; study 11, F(1,733) = 99.01, P < 0.001, ηp2 = 0.119. Conversely, under asymmetric conditions of unilateral cooperation by player A on round 1, player A was viewed as equally more rational and reasonable when choosing to defect rather than cooperate on round 2: study 10, F(1,590) = 35.18, P < 0.001, ηp2 = 0.056; study 11, F(1,739) = 35.27, P < 0.001, ηp2 = 0.046. Notably, under asymmetric conditions of unilateral defection by player A on round 1, player A was viewed as equally rational when choosing to cooperate or defect on round 2 but was only viewed as reasonable when choosing to cooperate but not defect again on round 2: study 7, F(1,589) = 67.84, P < 0.001, ηp2 = 0.103; study 11, F(1,739) = 82.69, P < 0.001, ηp2 = 0.101. The latter observations suggest that depending on the situation, participants considered a given choice as high in rationality and low in reasonableness, low in rationality and high in reasonableness, or as high in both rationality and reasonableness. These results support the idea that people view rationality and reasonableness as distinct standards of judgmental competence rather than merely opposite sides of the same evaluative (prosociality) dimension. Attributions of reasonableness significantly differed from attributions of kindness: symmetric game, F(1,733) = 119.86, P < 0.001, ηp2 = 0.141; asymmetric game, F(1,739) = 44.17, P < 0.001, ηp2 = 0.056. As Fig. 4 indicates, cooperative actions were viewed as kinder rather than more reasonable, whereas reciprocal cooperation after initial defecting and punishment of nonreciprocation after initial cooperation were viewed as reasonable but not kind. Overall, studies 11 and 12 demonstrate that preferences for rational or reasonable agents are conditional on situational goal demands: When choosing an agent that can maximize one’s preferences, people select a rational person, but when choosing an agent that is socially conscious and considerate of other party’s needs, people select a reasonable person. ### Study 12: Spontaneous associations and economic behavior in non-Western, socioeconomically diverse contexts With an exception of linguistic analyses, our studies have focused on the WEIRD (western, educated, industrialized, rich, and democratic) samples from North America. It is plausible that the thematic associations and economic behaviors would be less pronounced in less-educated, non-Western contexts (9, 31). We examined this question by conducting a preregistered replication of the spontaneous association task (study 1) and the economic expectations in a Dictator Game (study 5) among three social groups in a predominantly collectivist culture, Pakistan: highly educated white-collar workers from the banking sector, less educated street merchants, and rural dwellers mostly reliant on the barter system (versus monetary exchange; see the Supplementary Materials for ethnographic description of each site). Similar to study 1, results from human-coded content analyses showed that person-focused attributes were significantly more likely to be mentioned when describing rational persons (\|Zs\| > 2.61, P < 0.009), whereas socially conscious characteristics were significantly more frequent when describing reasonable persons (\|Zs\| > 7.12, P < 0.0001). Replicating study 5, participants expected reasonable persons to contribute more than rational persons [t(609) = 5.10, P < 0.0001], with personal choice in-between rational [t(609) = 1.96, P = 0.050] and reasonable standards [t(609) = 3.67, P < 0.0001]. As Fig. 5 indicates, differential expectations for reasonable versus rational contributions were especially pronounced among the rural barters [t(152) = 4.16 P < 0.0001] and street merchants [t(168) = 3.73, P < 0.001], and less among the urban managers [t(287) = 1.76, P = 0.075]. Notably, differences in education qualified the rational-reasonable difference [F(1,608) = 6.21, P = 0.013]: More educated participants were less likely to think that a reasonable person would contribute more and rather be aligned with the notion of preference-maximizing rationality (B = −17, 68, SE = 7.48, t = 2.37, P = 0.018). ## DISCUSSION It appears that laypeople systematically differentiate rationality and reasonableness along the lines outlined in game theory (3, 32) and legal scholarship (19, 33). A folk standard of rationality chiefly concerns an instrumental focus on individual’s attributes and preferences. In contradistinction, a folk standard of reasonableness concerns a pragmatic focus on social norms and context specificity in the process of judgment (16). These views have direct implications for decision-making: Irrational behavior may not necessarily be a sign of failure to understand game theoretical principles but rather an attempt to follow a competing folk standard of reasonableness. Application of rationality and reasonableness as distinct folk standards of sound judgment appears across five of the most widely spoken languages covering 15% of the world and across affluent and poor populations, including those relying on barter versus monetary economy. Each folk standard can be used to influence decision-making, promoting actions consistent with game theoretical principles in neoclassical economics or actions consistent with the standard of a reasonable person endorsed in legal scholarship and political economy. Evidence for the dissociation between these standards emerges when exploring implicit norms found in human-created cultural products and in behavioral experiments testing social interactions in economic and interpersonal transactions. Further, the distinction between rational and reasonable judgment extends to other aspects of decision-making beyond game theoretical contexts. Specifically, laypeople believe that rationality aligns with a maximizing strategy, whereas reasonableness aligns with satisficing. Thus, rationality and reasonableness capture a relatively broad distinction in laypeople’s standards of judgment. Moreover, although people view reasonable judgments as balancing personal preferences with social norms, they do not do it in a naive, sentimental way—i.e., people think it is reasonable to focus on one’s preferences when others act selfishly. Our findings oppose the possibility that people fail to understand game theoretical principles likened to standard of rationality in neoclassical economics (8). Studies 8 and 9 indicate that people are tactically rational (34) when choosing agents to act on their behalf. However, while our results indicate that people may have assimilated the preference-maximizing definition of rationality, this has not crowded out other standards of competent judgment, contrary to concerns raised by some social critics (32, 35, 36). Humans continue to recognize and apply a distinct standard of reasonableness that balances personal preferences with consideration of others’ interests. Why do two distinct standards coexist in the same culture and why are they internalized by the same individuals? One of the basic functions of standards is to provide intelligible accounts to justify one’s actions to others. The accountability demands of social life require people to be able to justify their actions, and consensual standards are a useful source of these justifications. Considering social expectations in everyday life (37), most people likely internalize both the rational and the reasonable standards because they will face some situations in which they need to justify their preference-maximizing choices and other situations where they need to justify socially conscious choices. Another nonexclusive possibility is that people learn these standards not only to be able to defend themselves but also to keep track of others’ trustworthiness in various contexts (e.g., as partners and as fiduciaries). Last, it is possible that people learn these standards to be better able to hold others into account (38). Future work unpacking postchoice interactions between agents involved in a game may help shed light on these theoretical possibilities. Future research may also explore conditions under which the standard of reasonableness, alone or in conjunction with other ecological factors, contributes to economically irrational choice, and test how egoists and altruists use these standards. Similarly, it is important to explore whether reason-based justifications (16) take different forms when people apply rational versus reasonable lay standards. Last, the observation that making the rational (versus reasonable) standard salient can influence people to be preference maximizing (versus socially conscious) also suggests a thus far unexplored intervention to encourage people to make more cooperative choices: reduce the demand to be rational and enhance the request to be reasonable. ## MATERIALS AND METHODS ### Study 1 Participants. We recruited Amazon MTurk workers who received1.20 remuneration. Exclusion rates and further demographics are in table S1. As specified in the preregistration protocol (osf.io/af3bw), we aimed to recruit at least 200 participants, oversampling to 250 participants to ensure sufficient power for a within-subject design based on average effect size in social-personality psychology (r = 0.21), estimating α/β errors at 5%. This and subsequent human-based studies have been reviewed and received ethics clearance through the University of Waterloo Research Ethics Committee (ORE no. 30580). Design. MTurk workers were first presented with the spontaneous adjective ascription task, subsequently evaluated stereotype-content adjectives, followed by the task about similarity of the terms rational-reasonable to adjectives characterizing major personality dimensions. Within each task, the order of presentation (reasonable versus rational) was randomized. Procedure. Participants (N = 239) were told they would provide their spontaneous thoughts on what are good qualities for decision makers across a variety of situations and answer a few questions about themselves. In the first task, participants were asked to quickly write down the first three characteristics (single words) of people who behave rationally/reasonably. Participants wrote down their answers in the three text boxes. In the second task, participants were asked to evaluate rational, reasonable, and ideal persons on several different features from the stereotype content questionnaire (communion/warmth: warm, tolerant, good natured, sincere, and trusting; agency/competence: competent, intelligent, confident, independent, competitive), answering the question how they think the society generally views these types of people (1 = not at all to 5 = extremely). Because of special interest in the self-focus, we also included an item concerning being “selfish.” We included ratings for an ideal person to examine alignment with ratings for a rational and reasonable person. This way, we can test normativity and unique contribution of rational/reasonable characteristic with respect to agency and communion. We randomized the presentation of items within each questionnaire. For communion/warmth, we averaged responses to corresponding items (rational, α = .85; reasonable, α = .85; ideal, α = .82). Preliminary principal components analyses (PCA) indicated that “competitive” formed its own component and therefore was not included in the average score of agency/competence (rational, α = .78; reasonable, α = .76; ideal, α = .69). The main analyses remain consistent if including this item. In the third task concerning personality attributions, participants were told that we are interested in their thoughts on the relationship between the following characteristics to the description of a rational/reasonable person. Participants were instructed to rate how similar is each characteristic to their view of a rational/reasonable person (0 = not at all similar to 10 = extremely similar). Each questionnaire consisted of a set of 10 to 12 adjectives for each of the six themes of the HEXACO personality model (24): honesty-humility, emotionality, extraversion, agreeableness, conscientiousness, and openness. We chose HEXACO over the Big 5 due to the addition of the socially conscious honesty-humility dimension, which appeared highly relevant for the purpose of the present investigation. The adjectives were adopted directly from the HEXACO (24). To reduce participant burden, we used a missing-at-random procedure, such that participants were presented with a random subset of five adjectives for each of the personality dimensions. We created aggregate average scores for each dimension. Next, participants rated the extent to which they viewed themselves as reasonable and rational (1 = not at all like me; 2 = not much like me; 3 = somewhat like me; 4 = quite a lot like me, 5 = just like me), provided an open-ended answer to them recalling the nature of the tasks they took part in, and provided demographic information. See materials on Open Science Framework (OSF; osf.io/2h4gx). Human-guided content analyses. Two independent, hypothesis-blind coders rated each of the adjectives participants provided on the open-ended initial task for a set of categories determined via a grounded approach—identifying the most frequent themes mentioned in participants’ narratives. Several of these themes concerned individual-focused characteristics (intelligent/smart and self-focused), decontextualized instrumental attributes (stoic and logical/consequential), socially conscious attributes (morals and interpersonal/prosocial), and attributes representing a mixture of individual- and context-focused considerations (objective/professional, personal mastery, and calm/levelheaded). The interrater reliability for each category was very good (Krippendorff’s αs > .89). Minor disagreements (2%) were resolved via a discussion with the first author. We summed participants’ scores across three adjectives they provided. Procedure. We used a between-subject design consisting of two tasks. In the first task, participants were instructed to recall the last time somebody (themselves or others) acted reasonably (rationally). Participants were asked to try recalling such a situation and describe it in a few sentences in a box provided. As specified in the preregistered exclusion protocol (osf.io/sy24t), we excluded participants who did not complete this task, wrote nonsense, or clearly misunderstood the prompt (i.e., writing focuses on interpersonal transactions broadly without specifying anything related to decision-making, analysis, problem solving, or moral considerations writ large; for example, “I had a pleasant conversation with a friend”). Recall-specific exclusions did not significantly differ by condition (reasonable condition = 4.57% versus rational condition = 5.61%), N = 1187, χ2(df = 1) = 0.961, P = 0.327. Next, participants were presented with a standard description of a Dictator Game, asking them to indicate how much money out of $10 they would give to player B (see study 6 methods). As in prior studies, participants rated the extent to which they viewed themselves as reasonable and rational (1 = not at all like me; 2 = not much like me; 3 = somewhat like me, 4 = quite a lot like me; 5 = just like me) and provided an open-ended answer recalling the nature of the game and provided demographic information. Before debriefing, we asked participants to indicate if they had “any ideas what we were investigating.” As indicated in the preregistered protocol, we used responses in this segment to screen out participants who reported that the study concerned effects of reasonable/rational recall on subsequent choice in the game. Exclusions did not significantly differ by condition (reasonable condition = 1.50% versus rational condition = 1.49%), N = 590, χ2(df = 1) = 1.750, P = 0.186. ### Study 8 Participants. Participants were recruited from MTurk, with$0.60 remuneration. On the basis of earlier studies, we targeted at least 200 participants per condition, which we oversampled to account for data loss due to noncompliance. Procedure. We used three economic dilemmas—a Commons Dilemma, a Prisoner’s Dilemma, and a Dictator Game Dilemma, asking participants (N = 387) whether they would prefer a reasonable or rational person representing either themselves or the other party in each of the dilemmas. For the first two dilemmas, participants were also asked what move they thought a rational and then a reasonable person would make in the dilemma. Dictator Game instructions were identical to previous studies. Commons Dilemma and Prisoner’s Dilemma were presented as raffle games to avoid familiarity bias. First, participants completed the Commons Dilemma and Prisoner’s Dilemma, each with their associated questions, which were presented in a randomized order to avoid possible order effects. Given the previously established association between the expectations and personal choice for reasonable versus rational agents in a Dictator Game, we aimed to prevent possible carryover effects from expectations in a Dictator Game by examining responses to this dilemma last. For each dilemma, participants indicated whether they would prefer a reasonable or a rational agent to act on their behalf and on behalf of another party (in a randomized order). To assess expectations for the Commons dilemma, participants indicated their expectation for withdrawal of lottery tickets from a common pool. To evaluate expectations for the Prisoner’s Dilemma, participants indicated whether reasonable/rational agents would choose a prosocial/group-gain maximizing option or rather an individual-gain maximizing option. Following economic dilemma tasks, participants completed the same measures of agency (αreasonable = .74, αrational = .80), communion (αreasonable = .83, αrational = .83), and selfishness (25, 42) as in study 1, for rational and reasonable people in a randomized order. Participants also responded to the questions about self-rating scale for reasonable then rational characteristics as in study 1 and were asked to recall the three economic games completed earlier as a screening item before filling out a short page of demographics items. See materials on OSF (osf.io/2h4gx). Procedure. Participants were told that they would be reflecting on qualities of good decision-making in a variety of situations and their preferences for certain qualities in decision makers. Participants read a description of a variant of a Prisoner’s Dilemma game used in study 8, presented as an anonymous raffle game involving simultaneous exchange between player A and player B, who each had a goal of earning tickets that could be exchanged for a $1000 gift certificate. Participants were presented with a series of scenarios concerning the two players’ behaviors in the game and were asked to rate how reasonable/rational player A was on a unipolar 1 to 5 scale (1 = not at all; 2 = slightly; 3 = moderately; 4 = very; 5 = extremely). The first two situations, presented in a randomized order, concerned player A choosing either to cooperate or defect on the first round of the game, resulting in a 2 (player A behavior: cooperate versus defect) × 2 (type of judgment: rational versus reasonable) within-subject design. The subsequent set of situations are depicted in Fig. 4. They concerned the behavior of players A and B across two rounds, such that players A and B could cooperate or defect on the first round, and player A could then cooperate or defect on the second round. Again, participants evaluated the rationality and reasonableness of player A’s behavior in each of these scenarios. We subsequently assessed participants’ stereotypes of rational and reasonable people, similar to prior studies. To this end, after rating these eight scenarios, participants were asked to rate a set of randomized-order characteristics on a bipolar 7-point scale when responding to the question: “As viewed by society, are reasonable or rational people more likely to show the described quality?” (−3 = reasonable persons definitely more likely; −2 = reasonable persons more likely; −1 = reasonable persons slightly more likely; 0 = reasonable and rational persons equally likely; 1 = rational persons slightly more likely; 2 = rational persons more likely; 3 = rational persons definitely more likely). The characteristics were modeled on the dimensions of warmth and competence that we used in earlier studies. Preliminary PCA on the standardized scores indicated that rating of abstract qualities separated into an individual-focused (independent, confident, intelligent, competent, competitive, and self-focused; α = .77) and a socially conscious (warm, good natured, empathetic, sincere, humble, trusting, fair, tolerant, and cooperative; α = .91) component. Therefore, we created composite scores of these components by averaging respective scores. As in prior studies, participants rated the extent to which they viewed themselves as reasonable and rational (1 = not at all like me; 2 = not much like me; 3 = somewhat like me; 4 = quite a lot like me; 5 = just like me), provided an open-ended answer to them recalling the nature of the game, and provided demographic information. ### Study 11 Participants. Participants were recruited via MTurk, with$1.20 remuneration. The study was a replication of study 10; thus, we targeted at least 600 participants. We oversampled by 30% to account for data loss due to noncompliance. Procedure. The procedure was identical to the two-round game described in study 10. The game involved eight situations, depicting all combination of cooperating versus defecting across two rounds, such that players A and B could cooperate or defect on the first round, and player A could then cooperate or defect on the second round. Participants evaluated the rationality, reasonableness, and kindness of player A’s behavior in each of these scenarios on a 1 to 5 scale (1 = not at all; 2 = slightly; 3 = moderately; 4 = very; 5 = extremely). As in prior studies, participants rated the extent to which they viewed themselves as reasonable and rational (1 = not at all like me; 2 = not much like me; 3 = somewhat like me; 4 = quite a lot like me; 5 = just like me), provided an open-ended answer to them recalling the nature of the game, and indicated whether they believed there are any reasons why their data should be discarded due to distraction or inattention to the task, and finally provided demographic information. ### Study 12 Participants. We used services of a professional survey collection company (Neuron Business and Development Solutions) to recruit participants from urban and rural areas in Pakistan. As specified in the preregistration protocol (osf.io/5wfrd), we aimed to recruit at least 600 participants, with even subsamples from each area (see figs. S10 and S11), corresponding to the sample size used in prior studies. We oversampled by 20 participants to account for compliance-based attrition. Half of the participants were recruited on-site in highly educated areas in Islamabad (banks), whereas the other half was split between a subsample of street merchants and another subsample of barter dealers in rural Pakistan (see Supplementary Materials for further details on the data collection sites). Urban participants were compensated 300 rupees (CAD \$3). For participants from the sub-urban and rural area, it is considered impolite to receive payment for trivial responses. Therefore, wherever possible/suitable, data collection agents purchased some item(s) from the participants’ businesses to compensate for their efforts. Design. Participants were presented with two tasks. In the first task, as in study 5, participants were presented with a Dictator Game. In the second task, as in study 1, participants were asked to use three words to describe a rational person () and reasonable person (). The Urdu words selected for rational and reasonable were determined in consultation with experts in legal and economic scholarship to ensure their representation in local language. The order of presentation (rational versus reasoning) was counterbalanced across participants. Procedure. For the urbanized sample, the survey was completed on paper. For the street merchant and rural barter samples, data collection agents administered the survey verbally in the same order. For the first task (Dictator Game with a contribution of 1000 rupees to share), we asked participants to indicate what a rational/reasonable person would do in the role of the player A (dictator). Next, they were asked what they would do in the role of player A. For the second task, we clarified that we particularly looked for adjectives (we avoided to use this word, as noneducated subsamples may not know it) by providing example adjectives that are shared across rational and reasonable persons (“For instance, you can say: ‘A rational/reasonable person is…careful’ or ‘A rational/reasonable person is intelligent’”). Subsequently, participants provided demographic information. Back-translation. A bilingual Urdu-English speaker translated all materials into Urdu, and another bilingual translated new materials back into English. The first author discussed the materials with both translators to establish semantic equivalence. Human-guided content analyses. Bilingual English-Urdu speakers provided three words to characterize each Urdu word to ensure more precise semantic understanding of the meaning of each word for English-speaking coders. Disagreements between translators were resolved via group discussion. Two independent, hypothesis-blind coders rated each of the words participants provided on the open-ended task for a set of categories determined in study 1 and supplemented with frequently mentioned categories unique to the present sample. Several of these themes concerned person-focused characteristics (intelligent/smart, selfish, and miser), instrumental attributes (logical/consequential), socially conscious attributes (morals, interpersonal/prosocial), as well as attributes representing a mixture of person- and context-focused considerations (objective/professional, personal mastery, and levelheaded). The interrater reliability for each category was very good (Krippendorff’s αs > .84). Very minor disagreements (1%) were resolved via a discussion with the first author. Initial review of open-ended responses indicated that only one-third of participants provided all three words, with the majority of participants providing only one word to describe rational/reasonable persons. This observation is consistent with other research showing lower compliance with repeated questions in non-WEIRD societies (44). Therefore, we focused on the analyses of first words. Analyses with percentage of themes per number of completed words indicated very similar results. ## SUPPLEMENTARY MATERIALS Supplementary Methods Supplementary Analyses Alternative accounts of the present findings Fig. S1. Word clouds of the top 100 nouns following “rational” and “reasonable” in English language newspapers and magazines. Fig. S2. Participants’ contributions in the Dictator Game as reasonable versus rational agents in study 6. Fig. S3. Distribution of donations in the Dictator Game after reminders of rational versus reasonable experiences in study 7. Fig. S4. Attribution of reasonableness and rationality to player A in multiround Prisoner’s Dilemma in studies 10 and 11. Fig. S5. Attribution of reasonableness and rationality to player A in a single-shot Prisoner’s Dilemma in study 10. Fig. S6. Attribution of reasonableness and rationality in study 10 for cooperating and defecting players on a second round of Prisoner’s Dilemmas after bilateral cooperation in the first round. Fig. S7. Attribution of reasonableness and rationality in study 10 for cooperating and defecting players on a second round of Prisoner’s Dilemmas after unilateral cooperation in the first round. Fig. S8. Attribution of reasonableness and rationality in study 10 for cooperating and defecting players on a second round of Prisoner’s Dilemmas after unilateral defecting in the first round. Fig. S9. Attribution of reasonableness versus rationality to different behavioral characteristics to a rational and reasonable person in study 10. Fig. S10. Photographic depiction of survey collection sites in study 12. Fig. S11. Typical institutions and common places for each of the three data collection sites in Pakistan in study 12. Table S1. Demographic information for samples used across experiments. Table S2. Most frequent words (top 10%) when describing rational and reasonable persons. Appendix References (4553) This is an open-access article distributed under the terms of the Creative Commons Attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ## REFERENCES AND NOTES Acknowledgments: T. Shibayuma and S. Vintan provided valuable assistance in qualitative coding (studies 1, 3, 4, and 12), M. Zahid assisted with data collection for Study 12, and K. Sharpinsky provided assistance with visualizations. Funding: The present research was funded by the Social Sciences and Humanities Research Council of Canada Insight Grants (435-2014-0685, to I.G.), Early Researcher Award from the Ontario Ministry of Research and Innovation (ER16-12-169, to I.G.), and Templeton Pathways to Character Award (to I.G.). Author contributions: I.G. and R.P.E. provided the study concept. I.G. designed studies 1 to 4 and 12. I.G. and R.P.E. designed studies 5 to 11. I.G. collected data for studies 1, 2, and 7 to 11. I.G. and J.K. collected the data for studies 5 and 6. Q.B.S. supervised translations and data collection for study 12. I.G. carried out the data analysis and drafted the initial version of the manuscript. All authors contributed to the revision of the manuscript and approved the final manuscript for submission. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. All data and statistical analyses that support the findings of this study are publicly available on the OSF website with identifier osf.io/2h4gx. View Abstract	2020-05-29 14:42:17	{"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": 0.29665595293045044, "perplexity": 4710.176755317518}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1590347404857.23/warc/CC-MAIN-20200529121120-20200529151120-00017.warc.gz"}
https://math.stackexchange.com/questions/874245/variation-of-a-strongly-bounded-measure-is-strongly-bounded-too	# Variation of a strongly bounded measure is strongly bounded too Let $\mathcal{A}$ be a field of subsets of a set $\Omega$, $X$ a Banach space and $\mu:\mathcal{A}\rightarrow X$ a finitely additive vector measure. The variation of $\mu$ is the extended nonnegative function $\|\mu\|$ whose value on a set $E\in\mathcal{A}$ is given by $$\|\mu\|(E)=\sup_\pi\sum_{A\in\pi}\\|\mu(A)\\|,$$ where the supremum is taken over all partitions $\pi$ of $E$ into a finite number of pairwise disjoint members of $\mathcal{A}$. It can be shown that $\|\mu\|$ is also a finitely additive measure. A finitely additive measure $\mu$ is said to be exhaustive (or strongly bounded) if for every $(E_n)$ sequence of pairwise disjoint members of $\mathcal{A}$, then $\lim_n\mu(E_n) = 0$. It is easy to show that $\|\mu\|$ exhaustive implies $\mu$ exhaustive for every $X-$valued finitely additive measure. If $\mu$ is real-valued or complex-valued and bounded, it can be shown that $\mu$ exhaustive implies $\|\mu\|$ exhaustive. Does a bounded $\mu$ that is exhaustive imply $\|\mu\|$ exhaustive for an $X-$valued finitely additive measure? I am unsure of how to prove this and I was wondering if I could get a hint. Thanks! ## 1 Answer If I understood well, you are supposing that given a bounded exhaustive measure $\mu$, its total variation $\vert\mu\vert$ is also an exhaustive measure. I believe that this is not true for all infinite dimensional spaces. Firs of all, please note that every exhaustive measure is bounded (see Corollary 19 at page 9 of Vector measure by Diestel and Uhl), so if your conjecture were true then every exhaustive measure would automatically have bounded variation. This last assertion is not true in general, as shown in the example 16 at page 7 of Vector Measure (by Diestel and Uhl). In any case, if $X$ is a Banach space, $\Sigma$ a field of subsets of $\Omega$ and $\mu\colon \Sigma\to X$ is an exhaustive measure, it can be proved that $\vert\mu\vert$ is exhaustive if and only if $\vert\mu\vert$ is bounded. This means that your conjecture is true whenever $X$ is such that every exhaustive measure has bounded variation, which are just the finite dimensional spaces by the Dvoretsky-Rogers theorem. I hope this answer your question, even if such a long time has passed. Niccolò	2019-07-17 05:02:41	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9720211625099182, "perplexity": 130.66637811227005}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195525046.5/warc/CC-MAIN-20190717041500-20190717063500-00550.warc.gz"}
http://hal.in2p3.fr/view_by_stamp.php?label=LPSC&langue=fr&action_todo=view&id=in2p3-00745218&version=1	836 articles – 5610 Notices [english version] HAL : in2p3-00745218, version 1 arXiv : 1208.6164 Physical Review Letters 109 (2012) 192501 New Measurements of the Transverse Beam Asymmetry for Elastic Electron Scattering from Selected Nuclei HAPPEX Collaboration(s) (2012) We have measured the beam-normal single-spin asymmetry $A_n$ in the elastic scattering of 1-3 GeV transversely polarized electrons from $^1$H and for the first time from $^4$He, $^{12}$C, and $^{208}$Pb. For $^1$H, $^4$He and $^{12}$C, the measurements are in agreement with calculations that relate $A_n$ to the imaginary part of the two-photon exchange amplitude including inelastic intermediate states. Surprisingly, the $^{208}$Pb result is significantly smaller than the corresponding prediction using the same formalism. These results suggest that a systematic set of new $A_n$ measurements might emerge as a new and sensitive probe of the structure of heavy nuclei. Thème(s) : Physique/Physique Nucléaire Expérimentale Lien vers le texte intégral : http://fr.arXiv.org/abs/1208.6164 in2p3-00745218, version 1 http://hal.in2p3.fr/in2p3-00745218 oai:hal.in2p3.fr:in2p3-00745218 Contributeur : Emmanuelle Vernay <> Soumis le : Jeudi 25 Octobre 2012, 08:50:30 Dernière modification le : Mardi 6 Novembre 2012, 08:43:50	2014-07-25 18:35:48	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.776984691619873, "perplexity": 3618.6478110170838}, "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-2014-23/segments/1405997894689.94/warc/CC-MAIN-20140722025814-00118-ip-10-33-131-23.ec2.internal.warc.gz"}
http://openstudy.com/updates/510af4bae4b0d9aa3c46bf79	## A community for students. Sign up today! Here's the question you clicked on: ## bmelyk one year ago Express (limit given below) as a definite integral over [0,1] by first recognizing the indicated sum as Riemann Sum associated with a regular partition of [0,1] therefore over the interval [0,1] • This Question is Closed 1. bmelyk $\lim_{n \rightarrow 0} \frac{ 1^{3}+2^{3}+3^{3}+...+n^{3} }{ n^{4} }$ 2. bmelyk $\lim_{n \rightarrow \infty}***$ 3. sirm3d $\sum_{i=1}^n\frac{i^3}{n^4}=\sum_{i=1}^n\left(\frac{i}{n}\right)^3\frac{1}{n}$ $\lim_{n\rightarrow \infty} \sum_{i=1}^n\left(\frac{i}{n}\right)^3\frac{1-0}{n};b=1,a=0\\=\int_0^1 x^3\;\mathrm dx$ #### Ask your own question Ask a Question Find more explanations on OpenStudy	2014-12-19 15:58:33	{"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": 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": 0.8461823463439941, "perplexity": 5403.294233091804}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802768724.86/warc/CC-MAIN-20141217075248-00039-ip-10-231-17-201.ec2.internal.warc.gz"}
https://galoisrepresentations.wordpress.com/	## Counting solutions to a_p = λ We know that the eigenvalue of $T_2$ on $\Delta$ is $24.$ Are there any other level one cusp forms with the same Hecke eigenvalue? Maeda’s conjecture in its strongest form certainly implies that there does not. But what can one prove along these lines? Conjecturally, one would certainly predict the following: Conjecture: Fix a tame level $N$ prime to $p.$ If $\lambda \ne 0,$ there are finitely many eigenforms of level $N$ an arbitrary weight such that $a_p = \lambda.$ If $\lambda = 0,$ there are finitely many eigenforms with the additional condition that they do not have CM by a quadratic field in which $p$ is inert. I have no idea how to prove this conjecture. If one counts the number of such forms of weight $\le X,$ then the trivial bound for eigenforms with $a_p = \lambda$ is $O(X^2).$ When I visited Princeton a few weeks ago, Naser Sardari, a student of Sarnak, showed me a short preprint he is writing which improves this bound by a power saving (additionally, it gives a power saving for each individual weight as well). The most interesting case of this result is when $\lambda = 0,$ but today I want to talk about the much easier case when $\lambda \ne 0,$ where, via some $p$-adic tricks, one can obtain a substantial improvement on the trivial bound. Let’s start from the following: Proposition I: Let $S_{\lambda}(X)$ denote the number of cuspforms of level $N$ and weights $\le X$ such that $a_p = \lambda$. Assume that $\lambda \ne 0.$ Then $S_{\lambda}(X) = O(X).$ Proof: Since $\lambda \ne 0,$ the $p$-adic valuation of $\lambda$ is finite. However, all forms with bounded slope belong to one of finitely many Coleman families, so the number of such forms in any weight is bounded. Using Wan’s explicit results, one can even give an explicit bound here that depends only on $N,$ $p,$ and the valuation of $\lambda.$ The point of this post, however, is to give an improvement on this bound. Proposition II: Let $S_{\lambda}(X)$ denote the number of cuspforms of level $N$ and weight $\le X$ such that $a_p = \lambda$. Assume that $\lambda \ne 0.$ Then, as $X \rightarrow \infty$, $S_{\lambda}(X) \ll_{\lambda} \log \log \log \log \log \log \log X.$ The argument will (obviously) allow for an arbitrary number of logs. But then the statement would become more cumbersome. Proof: As in the proof of the previous result, we may reduce to the case where we are considering a single Coleman family $\mathcal{F}.$ Over this family, the function $U_p$ is continuous, and hence so is $U_p(U_p - \lambda).$ More importantly, over a small enough disc, it is an Iwasawa function. Let $\Sigma$ denote an infinite set of integral weight such that, for the relevant points of $\mathcal{F},$ we have $T_p = \lambda,$ or $U_p(U_p - \lambda) = - p^{k-1}.$ If $s$ is a limit point of $\Sigma,$ then certainly $U_p(U_p - \lambda)$ will vanish at $s.$ Since this function is a non-zero bounded function on a disc, it has only finitely many zeros, and so the set of weights $\Sigma$ will have only finitely many limit points. Thus, we may reduce to the case where the set of weights has a single limit point. In particular, if $S_{\lambda}(X)$ is not bounded, we may imagine that the set $\Sigma$ consists of a sequence of integers (which we may assume to be increasing in the Archimedean norm): $k_0, k_1, k_2, \ldots$ which converge $p$-adically to $s,$ and, at the relevant point of $\mathcal{F},$ correspond to an eigenform which satisfies the equation $U_p(U_p - \lambda)(k_i) = - p^{k_i - 1}.$ Around a zero $s,$ any Iwasawa function has an asymptotic expansion of the form $F(s + \epsilon) \simeq A \cdot \epsilon^m + \ldots$ where the LHS has the same valuation as the leading term of the RHS for sufficiently small $\epsilon.$ If $F = U_p(U_p - \lambda),$ we deduce that, for sufficiently large integers $k_i,$ $v(s - k_n) = r k_n + c$ for some $r = 1/m > 0,$ which implies that $v(k_{n+1} - k_n) = v(s - k_n),$ and hence also that $k_{n+1} - k_n > C p^{r k_n}$ for some $r > 0$. This iterated exponential growth proves the result. QED. The argument also shows that if the set $\Sigma$ is infinite, the limit roots of $U_p - \lambda = 0$ will be transcendental Liouville numbers, which seems unlikely. The result also applies if one replaces $\lambda$ by a sufficiently continuous function without zeros, say $a_2 = 24(1 + 2(k -12)^2).$ On the other hand, I don’t think these analytic methods will ever be enough to prove the conjectural bound, which is $O(1).$ Posted in Mathematics \| \| 2 Comments ## 144169 The space of classical modular cuspforms of level one and weight 24 has dimension two — the smallest weight for which the dimension is not zero or one. What can we say about the Hecke algebra acting on this space without computing it? Formally, the Hecke algebra $\mathbf{T}$ is a rank two $\mathbf{Z}$-algebra, which is either an order in the ring of integers of a real quadratic field, or a subring of $\mathbf{Z} \oplus \mathbf{Z}.$ Let’s investigate the completion of this algebra at various primes $p.$ Let’s first consider the prime $p =23.$ The curve $X_0(23)$ has genus two, and the corresponding Hecke algebra in weight two is $\mathbf{Z}[\phi],$ where $\phi$ is the Golden Ratio. The prime $p =23$ does not split in this field, and hence modulo $p$ there is a pair of conjugate eigenforms with coefficients in $\mathbf{F}_{p^2}.$ Multiplying by the Hasse invariant, we see that this eigenform also occurs at level one and weight 24 over $\mathbf{F}_{p}.$ It follows that: $\mathbf{T} \otimes \mathbf{Z}_{23} = W(\mathbf{F}_{23^2}).$ In particular, $\mathbf{T} = \mathbf{Q}(\sqrt{D})$ for some square-free integer $D > 0.$ Now let us consider primes $p < 23.$ Any Galois representation modulo such a prime will occur — possibly up to twist — in lower weight. Yet all the spaces in lower weight have dimension at most one, and hence it follows that the residue fields of $\mathbf{T}$ are all of the form $\mathbf{F}_p.$ Suppose further that $5 \le p < 23.$ Then, using theta operators, we may find two distinct eigenforms in weight 24, from which it follows that $\mathbf{T}$ has two distinct residue fields of characterstic $p,$ and so, for $5 \le p < 23,$ we have: $\mathbf{T} \otimes \mathbf{Z}_p = \mathbf{Z}_p \oplus \mathbf{Z}_p.$ One expects at level one that $a_2(f)$ always generates the Hecke field. This is still a conjecture, but we may deduce this unconditionally in weight 24 because the dimension of the cuspforms is two, and so this follows automatically from the Sturm bound! Hence we may write: $\mathbf{T} = \mathbf{Z}[a_2(f)], \quad a_2(f) = \displaystyle{\frac{a + b \sqrt{D}}{2} \in \mathbf{Z} \left[ \frac{1+\sqrt{D}}{2} \right]}$ where $b \ne 0.$ Even better, using Hatada’s Theorem — giving congruences for $a_2$ and $a_3$ for eigenforms of level one modulo $8$ and $3$ respectively — we may write $a_2(f) = 12(a + b \sqrt{D}), \quad a,b \in \mathbf{Z}$ where $b \ne 0.$ This gives an upper bound on $D$ in light of the Deligne bound $\|a_2\| \le 2 \cdot 2^{23/2}.$ More precisely, we obtain the bound $b^2 D < 2^{27}/24^2,$ and hence that $D < 233017.$ Let’s now think more carefully about $p = 2$ and $3.$ For these primes, there will be a unique Coleman family of slope $v(-24) = 3$ for $p =2$ and $v(252) = 2$ for $p = 3.$ I can’t quite see a pure thought way of proving this, but at least this would be a consequence of the strong form of the GM-conjecture as predicted by Buzzard. So we should expect that, in these cases $\mathbf{T} \otimes \mathbf{Z}_p \hookrightarrow \mathbf{Z}_p \oplus \mathbf{Z}_p.$ In addition to congruences for small primes, there will also be congruences between the unique cusp form with an Eisenstein series modulo the numerator of $B_{24},$ which is $\displaystyle{B_{24} = \frac{-1}{2 \cdot 3 \cdot 5 \cdot 7 \cdot 13} \times 103 \times 2294797.}$ I claim that these primes will also have to split in $\mathbf{T}.$ For example, it is impossible for $b$ to be divisible by $2294797,$ because that would violate the inequality on $b^2 D$ above, and hence it follows that $p = 2294797$ must also split in $\mathbf{T} \otimes \mathbf{Q}.$ The same argument works for $p = 103$ having ruled out some very small $D.$ To summarize, we have the following: The primes $5 \le p < 23,$ $p = 103, 2294797$ split in $K = \mathbf{Q} (\sqrt{D}),$ but $p = 23$ does not split, and $D < 233017.$ Moreover, we expect that $2$ and $3$ also split. This is enough to determine $D$ completely up to 72 possibilities, and 9 with the unproven assumption at $2$ and $3.$ On the other hand, all of these $D$ are quite large (the smallest are $3251$ and $15791$ respectively), which forces $b$ to be very small. But we also have the congruence $12(a + b \sqrt{D}) \equiv 1 + 2^{23} \mod 2294797.$ For the remaining $D,$ we can determine, with $\|b\|$ satisfying the required inequality, whether there exists such a congruence with $\|a\| \le 2^{27/2}/24 \sim 483.$ A simple check shows that is a unique solution (with the assumption on two or three or not), and hence, by (something close to) pure thought, we have shown that $D = 144169,$ and moreover (using Deligne’s bound again) that $a_2(f) = 12(45 \pm \sqrt{144169}), \qquad \mathbf{T} = \mathbf{Z}[12 \sqrt{144169}].$ One can indeed check this is the case directly, if you like. Curiously enough, this Hecke eigenvalue is quite close to the Deligne bound — the probability it is (in absolute value) this big is, assuming a Sato-Tate distribution, slightly under 5%. Extra Credit Problem: Hack Ken Ribet’s Yelp password by using the fact that 144169 is his favorite prime number. Posted in Mathematics \| Tagged , , , \| 4 Comments ## How not to be wrong I recently finished listening to Jordan’s book “how not to be wrong,” and thought that I would record some of the notes I made. Unlike other reviews, Persiflage will cut through to the key aspects of the book which perhaps non-specialists may have missed. Unfortunately, my first few notes did not record the specific time in the recording where the relevant passage occurred, so some of the earlier comments are a little more vague, because I couldn’t go back and check them more carefully. Title: How Now to Be Wrong: The Power of Mathematical Thinking. Author: Jordan Ellenberg. Book Format: Pirated audio copy. • OK, Penguin, what have you done to Jordan? It sounds as though before the recording session began, Jordan was force fed him a greasy pizza with a couple of prozac stuffed in the crust. I was expecting a hyperactive delivery style, but instead there is a relatively calm and measured tone you might expect on any professionally made audio book. • Did he just say yoked? Yes, my friends, we have here a student of Barry Mazur. • 2377. This is all it says in my notes. I think this was used as a number which was supposed to sound random. But I did wonder whether it had any other significance. A brief web search indicates the full phrase may have been: Moving over to complicated/shallow, you have the problem of …[computing]… the trace of Frobenius on a modular form of conductor 2377. I checked — there are no elliptic curves of conductor 2377. I think there was an opportunity missed to say 5077 instead, thus alluding to the Gross-Zagier plus Goldfeld solution to the class number problem. Although if there was such an allusion, it may have ruined the implication of being shallow, so never mind. • Some reference to galois representations being deep; unfortunately I didn’t write any further notes here. They are indeed complicated and deep. • The claim is made that if you cut a tuna fish sandwich you will be left with two right-angle isoceles triangles. Is this so clear? I mean, does everyone cut their tuna fish sandwiches along the diagonal? • Rounding Errors: the range for (I guess?) one standard deviation for some normal distribution with mean 50 is given as 46.2 and 53.7, but these numbers are not symmetric around 50. • Infinity of my profit comes from pastry. I liked this line. • 4, 21, 23, 34, 39. Repeated strings of numbers on the page are easy to read, but even Jordan is getting a little bored reading out 4, 21, 23, 34, 39 for the n-th time. • if your kid drew Jesus on the cross… See two comments up. • At this point, I should probably point out to the readers of the book that they are missing out on all the extra fancy technological gizmos that Penguin took advantage of when transferring the book from the page to audio. And by this, I mean that, in approximately 13 and half hours of reading, we not only have Jordan reading out the text of the book, we are also treated to exactly one such extra, namely, the first 9 notes of Beethoven’s Ode to Joy as played on what appears to be an 8-key child’s keyboard. • Ouroboric? Is that really how you pronounce that? It doesn’t seem consistent with the OED’s pronunciation of Ouroboros. Hmmm, but on the other hand, http://en.wiktionary.org/wiki/ouroboric gives someting similar to what Jordan says… • How Many States should one have expected Nate Silver to get wrong? This might have been another opportunity to mention how the expectation is not the “expected” answer. Presumably, one would expect a high correlation between getting one (close) state wrong and getting another wrong (I’m imagining here that swings undetected by polls would be nationwide rather than statewide). So I have several questions here. Was there anything in Silver’s model which could allow one to predict not only the expected number of states he would get wrong but the expected distribution of the number of states he would get wrong? Because of the stickiness of states, I suppose that the expectation that he would get all the states correct is higher than what one might guess from the fact that the expected number of states one expected he would get wrong (from his model) was approximately 3. I’m sure I’ve heard Jordan mention elsewhere that Nate Silver claimed that one should not have expected Silver to get all 50 states right. However, it’s completely consistent to believe that a well designed model could both predict that the expected number of states that Silver would get wrong is 3, but also that there is a high probability (at least > 50%) that he really would get all the states correct. So it’s not clear that a criticism of Silver for getting too many states correct is necessarily valid. • The problems you meet freshman year are the deepest… Is this true? Matt and I wondered which $p$-adic modular functions were expressible as convergent sums of finite slope eigenforms, and I still don’t know, but I’m not sure that’s the deepest question ever. • Did the student of the introduction listen to the entire book? I think I kind of missed that this was a preface (I think?) and kept expecting her to return. Summary: Was I convinced at the end that the girl’s time spending doing those 30 definite integrals was worthwhile? I’m not so sure. In fact, I could almost have been convinced that we should slash all the public math departments in half and replace them by statistics departments. On the other hand, by every other measure, the book was a complete success — as a piece of prose, as a source of interesting yet thematically linked historical anecdotes, and as both an exposition and celebration of a certain way of thinking (“mathematical thinking”) which we all aspire to. It was worth every cent. Audio: On a scale from “Jordan’s talking to you quite loudly on a train in Germany and someone tells you to shut up” to “Ambient waterfall sounds for Ultimate Bedtime Relaxation,” I rate it a 4, which is about where you would wish it to be. (For an inside look at the recording session, see this post.) ## Chenevier on the Eigencurve Today I wanted to mention a theorem of Chenever about components of the Eigencurve. Let $\mathcal{W}$ denote weight space (which is basically a union of discs), and let $\pi: \mathcal{E} \rightarrow \mathcal{W}$ be the Coleman-Mazur eigencurve together with its natural map to $\mathcal{W}.$ It will do well to also consider the versions of the eigencurve corresponding to quaternion algebras $D/\mathbf{Q}$ as well. Theorem: [Chenevier] Suppose that 1. $\mathcal{E}$ has “no holes” (that is, a family of finite slope forms over the punctured disc extends over the missing point), 2. The “halo” of $\mathcal{E}$ is given by a union of finite flat components whose slope tends to zero as $x \in \mathcal{W}$ tends to the boundary of the disc. Then every non-ordinary component of $\mathcal{E}$ has infinite degree. In particular, since both of these theorems are now known in many cases (properness by Hansheng Diao and Ruochuan Liu, and haloness by Ruochuan Liu, Daqing Wan, and Liang Xiao, at least in the definite quaternion algebra case), the conclusion is also known. The proof is basically the following. Given a component $C$ of finite degree, the first assumption implies that it actually is proper and finite. One may then consider the norm of $U_p$ on $C$ to the Iwasawa algebra to obtain a bounded (hence Iwasawa) function $F = \mathrm{Norm}(U_p).$ This function cannot have any zeros (again by properness), and hence, by the Weierstrass preparation theorem, it is a power of $p$ times a unit. But that implies that $F$ has constant valuation near the boundary, which contradicts the fact that the slopes are tending to zero (except in the ordinary case). Naturally one may ask whether $\mathcal{E}$ has only finitely many components, although this seems somewhat harder to prove. ## What does it take to get a raise? Gauss … [had] a salary that remained fixed from 1807 to 1824. (see here.) What was Gauss’ salary? My limited google skills were not able to find this information, although I’m not sure how meaningful it would be to translate any such number into today’s dollars. More generally, although there is available data for academic salaries over the past 40 years or so, I’m curious for comparisons that go further back in time. Posted in Waffle \| Tagged , \| 6 Comments ## The seven types of graduate student applicant Yes, it’s that time of year again. 1. Hide and Seek: Contacts you every day about the status of their application, then goes on radio silence the moment they receive an offer, never to be heard of again. 2. The No Chancer: Has an offer from Harvard, Princeton, and MIT, but still plans to attend the prospective student day because they fancy a three day holiday in (wherever your university is located). 3. The Copy & Paster: It has always been my dream to attend Michigan University, the best university in the world. Well, good luck with that. 4. The Nervous Nellie: Has some questions about the graduate program — a lot of questions. Wants you (that is, me) to answer detailed questions about everything from the reasonable (exact duties of a TA, particulars on graduate student stipend and health insurance) to the less so (graduation statistics and data for the last 10 years of graduates, upcoming schedule of faculty sabbaticals for the next three years, tips on the best place in Evanston to purchase toothpaste, etc.). 5. The Surprise: Never responds to any email query about whether they are interested in coming or whether they have offers from somewhere else, is completely discounted by the committee, but then ends up accepting on April 15. 6. The Googler: Makes an effort to look at the department website to customize their application, but gets it all wrong: I would really like to work with X, Y, and Z where X is a postdoc, Y has retired, and Z moved to a different institution two years ago. 7. The Unicorn: Actually accepts the offer well before April 15. Tell me if I’ve missed anyone. Posted in Mathematics, Rant \| \| 11 Comments ## Only Harvard Grads need apply It’s hard to take articles in Slate too seriously, but I have to admit I was quite perplexed about the following article (with the concomitant research publication here). The main thrust of the article seems to be as follows. A disproportionate number faculty at research universities in the US received PhDs from a small number of prestigious institutions, and hence (?) such hiring practices reflect profound social inequality. Is it just me, or does this appear to be utter bollocks? There is an obvious pair of hypotheses that would completely explain the data, namely: 1. There is a hierarchical system of admission to graduate programs, 2. Universities hire the strongest candidates they can, and admit the strongest graduate students they can. Let’s examine these possibilities in the context of graduate school in mathematics. I have, on several occasions, been responsible for graduate admissions at my institution. I would say, on the whole, that prospective graduate students are among the most class conscious of anyone in academia. I would guess that, at least 75% of the time, a student will accept either the program that is the most highly ranked amongst those where they were admitted or a school within at most one or two places of their highest ranked option. What about the second hypothesis? The worry here is that universities might view “undergraduate/graduate institution” as a proxy for “quality of candidate.” In my experience (being on hiring committees), this is utterly preposterous. I am not claiming that mathematical judgements are not a slippery thing — there are many variations which relate to matters of taste and inclination — but there are some reasonable objective criteria (GRE scores for graduate applications, publication record for job candidates) which would serve as a check against any implicit bias in this regard. We here at Persiflage, however, are open to the idea that we may have missed something. So here are some other possibilities: 1. You are talking about Mathematics, a field for which it is easier to make reliable judgements about the quality of research, and a field for which there is a more pronounced spike in talent at the top of the scale. Is this true? I honestly don’t know. Perhaps whatever field it is that produces papers like the one under consideration is not something for which talent of any kind is an asset, and so there is no real difference between graduates from Harvard or from Podunk U. Less sarcastically, suppose (say) I compare the English department faculty at the top ranked place (taking from this list) Berkeley and compare it to a place also ranked in the top 25 schools but closer to the bottom of that list, say UIUC. Then, if I knew something, could I confidently say that one department is much better than the other? 2. You are talking about the experience of hiring/admitting students at a Group I university. Perhaps it is the case that, for lower ranked universities, there is insufficient expertise to hire on the basis of talent/output, and so PhD institution serves as a lazy way to evaluate the candidate. This seems to be a somewhat condescending argument, but it’s true that I don’t have any idea how hiring works at non-Group I universities. But surely the letters of recommendation would carry the most weight, and they would reflect the quality of research? At the very least, if you are going to claim this is what happens, you need to come up with a way to substantiate that claim. Ultimately, I certainly don’t feel that I can rule out bias when it comes to hiring, but the fact that the paper under review uses “prestige” as a dirty word and doesn’t seem to acknowledge in any way that there is some correlation between prestige and quality of graduates is highly disturbing. Perhaps, as with this paper, the main goal is to substantiate the political beliefs of the authors rather than to undertake a serious academic inquiry. Still, even if the methodology is flawed, I would like people’s opinion on the conclusion. Posted in Politics, Rant \| Tagged , \| 9 Comments	2015-05-22 23:36:58	{"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": 0, "img_math": 158, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7761807441711426, "perplexity": 474.2016396404025}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207926924.76/warc/CC-MAIN-20150521113206-00016-ip-10-180-206-219.ec2.internal.warc.gz"}
http://courses.skill-lync.com/student-projects/week-5-compact-notation-derivation-for-a-simple-mechanism-5	Menu Job Guarantee Workshops Projects Blogs Careers Hire from us For Business / Universities Corporate Training Academic Up-skilling All Courses Choose a category Loading... All Courses / undefined All Courses / undefined / undefined Loading... FOR BUSINESSES Corporate Upskilling FOR Universities Academic Training More Week 5 - Compact Notation Derivation for a simple Mechanism Aim:- Derivation of simple reaction mechanism using compact notation. Introduction:- In the assignment, I have derive reaction mechanism for elementary reaction in chemical process. The motive behind deriving this reaction mechanism is to find out the chemical kinetics for elementary reaction which will help us in knowing… • Shyam Babu updated on 06 Jun 2020 Project Details Loading... Leave a comment Thanks for choosing to leave a comment. Please keep in mind that all the comments are moderated as per our comment policy, and your email will not be published for privacy reasons. Please leave a personal & meaningful conversation. Please login to add a comment Other comments... No comments yet! Be the first to add a comment Read more Projects by Shyam Babu (28) Week 9 - Parametric study on Gate valve. Objective: Aim:-Parametric study on Gate valve. Theory:- A gate valve, also known as a sluice valve, is a valve that opens by lifting a barrier (gate) out of the path of the fluid. Gate valves require very little space along the pipe axis and hardly restrict the flow of fluid when the gate is fully opened. The gate… 12 May 2021 12:01 PM IST Week 8 - Simulating Cyclone separator with Discrete Phase Modelling Objective: Aim:- To perform analysis on cyclone separator and calculate the separation efficiency and pressure drop. Objective:- To perform an analysis on a given cyclone separator model by varying the particle diameter from 1 μm to 5 μm and calculate the separation… 06 Feb 2021 03:41 PM IST • CFD • MATLAB Conjugate Heat Transfer Analysis on a graphics card. Objective: Aim :- Thermal(conjugate heat tranfer) analysis on a graphic card. Theory :- Thermal analysis is an important part of the design process, especially if modern, ultra-fast components are used. For example, FPGAs or fast A/D converters may easily dissipate several watts of power. Because of this, PC boards, enclosures… 18 Nov 2020 01:24 PM IST Rayleigh Taylor Instability Challenge Objective: Aim :- Performing Rayleigh Taylor instability between two immiscible liquid. Theory :- The Rayleigh–Taylor instability, or RT instability, is an instability of an interface between two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid. Examples… 14 Oct 2020 11:41 AM IST Exhaust Port Challenge Objective: Aim :- Exhaust gas thermal effect on Exhaust port using conjugate heat transfer. Theory :- The term conjugate heat transfer (CHT) is used to describe processes which involve variations of temperature within solids and fluids, due to thermal interaction between the solids and fluids. The exchange of thermal energy between… 16 Sep 2020 06:38 PM IST SIMULATION ON CONJUGATE HEAT TRANSFER THROUGH SOLID PIPE. Objective: OBJECTIVE:- (I) TO SETUP SIMULATION ON CONJUGATE HEAT TRANSFER(CHT) IN SOLID PIPE . (II) TO RUN SIMULATION USING GRID DEPENDENCE TEST ON THREE GRIDS . … 03 Sep 2020 05:38 PM IST Ahmed Body Challenge Objective: Aim :- Numerical simulations on the aerodynamics of the Ahmed body. Abstract :- The aerodynamic behaviour of the Ahmed body is investigated experimentally and numerically. The experiment cover the slant angles of 20^o for ahmed body. The commercial CFD tool Fluent^(TM)v6.3.26 is tested for its… 27 Aug 2020 01:28 PM IST Steady Vs Unsteady flow over a cylinder Objective: Aim :- Numerical simulation of laminar flow past a circular cylinder. Theory :- The given above topic mainly focuses on what happens when a laminar flow passes a circular cylinder, here my prime objective is to find out which phenomena actually happens at the wall of the cylinder. Since it would take a lot of time… 09 Aug 2020 02:20 PM IST Mixing efficiency simulation in T-shape tube. Objective: Aim:- To perform mixing efficiency simulation in T-shape tube. Introduction:- The mixing process of hot and cold fluids in a tee junction is chaotic (turbulent) in nature and can result in high cycle thermal fatigue of the junction. This random quasi steady state phenomenon of hot and cold shocks can lead to fatigue… 01 Aug 2020 02:40 PM IST SIMULATION OF OSCILLATION OF A 2-D PENDULUM BY USING SECOND ORDER ORDINARY DIFFERENTIAL EQUATION Objective: In this report, I have simulated the oscillation of 2D pendulum and also generate a plot of \'angular_displacement\' & \'angular_velocity\' vs \"time\" in octave. Consider a pendulum which is having a string of length of \'1metre\' connected to a ball of mass,m=1kg such that it is having fixed support from another… 08 Jul 2020 05:15 PM IST SIMULATION OF 4V-SI8 SPARK IGNITION ENGINE USING CONVERGE STUDIO SOFTWARE Objective: OBJECTIVE:- (I) TO SETUP FULL HYDRO SIMULATION FOR THE PORT FUEL INJECTION ENGINE. (II) POST-PROCESSING OUTPUT FROM SIMULATION AND CALCULATING :- (A) COMPRESSION RATIO OF THE ENGINE. (B) COMBUSTION EFFICIENCY OF THE ENGINE. (C) TO DETERMINE POWER AND TORQUE OF ENGINE USING PERFORMANCE CALCULATOR. INTRODUCTION:- … 08 Jul 2020 05:07 PM IST SIMULATION ON PRANDTL MEYER SHOCK PROBLEM AND ANALYSING THE EXPANSION WAVE. Objective: THEORY:- SHOCK WAVE :- IN PHYSICS , A SHOCK WAVE IS A TYPE OF PROPAGATING ADRUPTNESS OR DISTURBANCE THAT MOVES FASTER THAN THE LOCAL SPEED OF SOUND IN THE MEDIUM . LIKE AN ORDINARY WAVE , A SHOCK WAVES ALSO CARRIES ENERGY AND PROPAGATES THROUGH A MEDIUM BUT HAVE DISCONTINUITY IN IT\'S PHYSICAL PROPERTIES LIKE … 08 Jul 2020 05:03 PM IST SIMULATION ON SHOCK TUBE AND PREDICTING OUT THE CHANGE IN PHYSICAL QUANTITY IN TUBE ON SHOCK Objective: OBJECTIVE:- (I) TO SETUP TRANSIENT SHOCK TUBE SIMULATION . (II) TO PLOT PRESSURE AND TEMPERATURE HISTORY IN ENTIRE DOMAIN . (III) PLOT THE CELL COUNT AS A FUNCTION OF TIME . THEORY :- THE SHOCK TUBE IS AN INSTRUMENT USED TO REPLICATE AND DIRECT BLAST WAVES AT A SENSOR OR A MODEL IN ORDER TO SIMULATE ACTUAL… 08 Jul 2020 02:15 PM IST PROGRAM ON DERIVING FOURTH ORDER APPROXIMATION FOR DIFFERENT DIFFERENCE SCHEMES FOR SECOND ORDER DERIVATIVE Objective: In this programming, I have derived the four order approximation of second order derivative using finite difference method. In Finite difference method,we generally convert ordinary differential equation into difference equation. In order to achieve so, we use simple three techniques i.e Forward differencing, Backward… 04 Jul 2020 07:24 AM IST TRANSIENT FLOW SIMULATION OVER A THROTTLE BODY Objective: OBJECTIVE:- STEADY STATE SIMULATION OF FLOW OVER A THROTTLE BODY . SOFTWARE USED :- 1. CONVERGE STUDIO:- TO SETUP THE MODEL. 2. CYGWIN:- FOR RUNNING THE SIMULATION. 3. OPENFOAM:- TO VISUALIZE THE DIFFERENT POST-PROCESSED RESULT. THEORY:- HERE, WE HAVE DONE SIMULATION OF FLOW OVER A THROTTLE BODY IN STEADY… 04 Jul 2020 07:15 AM IST STEADY FLOW SIMULATION OVER A THROTTLE BODY. Objective: OBJECTIVE:- STEADY STATE SIMULATION OF FLOW OVER A THROTTLE BODY . SOFTWARE USED :- 1. CONVERGE STUDIO:- TO SETUP THE MODEL. 2. CYGWIN:- FOR RUNNING SIMULATION. 3. OPENFOAM:- TO VISUALIZE THE DIFFERENT POST-PROCESSED RESULT. THEORY:- HERE, WE HAVE DONE SIMULATION OF FLOW OVER A THROTTLE BODY IN STEADY STATE… 04 Jul 2020 07:01 AM IST DERIVATION OF REYNOLDS AVERAGED NAVIER-STOKES EQUATION AND ANALYSIS OF THIS EQUATION Objective: NAVIER-STOKES EQUATION :- THE NAVIER-STOKES EQUATION IS A PARTIAL DIFFERENTIATION EQUATION WHICH IS USED TO DETERMINE FLOW IN THE INCOMPRESSIBLE FLUIDS. THESE EQUATIONS HELPS IN DETERMINING THE FLOW QUANTITIES FLOWING IN FLUID AND SOLVES MANY AMBIGIOUS PROBLEMS PREVAILING IN REAL WORLD. SIGNIFICANCE OF NAVIER-STOKES EQUATIONS:-… 03 Jul 2020 03:56 PM IST Week 9 - Senstivity Analysis Assignment Objective: Aim :- To perform sensitivity analysis for Auto-Ignition in combustion using Cantera. Introduction :- Combustion in engines with a practical fuel is very complex and is commonly described by a detailed mechanism which may involve dozens or even hundreds of reactions. However, the detailed mechanism is not well understood… 29 Jun 2020 04:46 PM IST Week 7 - Auto ignition using Cantera Objective: Aim:- To simulate Auto-ignition time for methane and Calculate Auto-ignition temperature. Introduction:- As we know that when a fuel is compressed or exposed to high temperature, it gets automatically ignited without any external source for ignition. So, when a fuel gets auto-ignited it takes some time commonly known as… 19 Jun 2020 06:47 PM IST Week 6 - Multivariate Newton Rhapson Solver Objective: AIM :- TO SOLVE SET OF ORDINARY DIFFERENTIAL EQUATION USING MULTIVARIATE NETWON-RAPHSON TECHNIQUE. INTRODUCTION :- In this project, I have basically solved a set of ordinary differential equation using multivariate Newton-Raphson technique. Here, My aim was to analyse the effect of time- step on… 12 Jun 2020 06:25 PM IST Week 5 - Compact Notation Derivation for a simple Mechanism Objective: Aim:- Derivation of simple reaction mechanism using compact notation. Introduction:- In the assignment, I have derive reaction mechanism for elementary reaction in chemical process. The motive behind deriving this reaction mechanism is to find out the chemical kinetics for elementary reaction which will help us in knowing… 06 Jun 2020 07:48 AM IST Week 4 - Handling Mixtures with Cantera Objective: 1. We know that for proper combustion of fuel and oxidiser, the combustion should take place in following given below chemical equation or Stoichiometric chemical equation:- CH4 + 2(O2 + 3.76N2) = CO2 + 2H2O + 7.52N2 But since we have reactants of chemical equation in following… 30 May 2020 07:27 PM IST Week 4 - Combustion Efficiency Calculation after Preheating Objective: AIM :- I. TO ANALYSE THE EFFECT OF PREHEATING ON ADIABATIC FLAME TEMPERATURE . II. TO ANALYSE THE EFFECT OF PREHEATING ON COMBUSTION EFFICIENCY OF FUEL. INTRODUCTION :- Since We know that In today's world fuel consumption is increasing day by day. It have become must for us to use certain techniques… 30 May 2020 07:25 PM IST SIMULATION OF FLOW THROUGH PIPE IN OPENFOAM BY USING SCRIPTING IN MATLAB SOFTWARE Objective: AIM:- IN THIS PROJECT SIMULATION IS DONE ON THE FLUID FLOW THROUGH PIPE USING OPENFOAM SOFTWARE AND MATLAB . THE TOPIC GIVEN BELOW IS COVERED IN THIS PROJECT. TO MAKE A PROGRAM IN MATLAB THAT CAN GENERATE MESH AUTOMATICALLY FOR ANY WEDGE ANGLE AND GRADING SCHEME. TO SHOW ENTRY LENGTH IS SUFFICIENT TO PRODUCE VELOCITY PROFILE.… 25 Mar 2020 04:38 AM IST 1-D SUPER-SONIC NOZZLE FLOW SIMULATION USING MACORMACK METHOD Objective: IN THIS PROJECT, I HAVE WORKED ON THE ONE DIMENSIONAL SUPERSONIC NOZZLE FLOW. IN THIS FLOW, I HAVE ASSUMED THE FLOW TO BE STEADY,ISENTROPIC. IN THIS NOZZLE, I HAVE CONSIDERED THE FLOW AT INLET SECTION COMES FROM RESERVOIR WHERE THE PRESSURE, TEMPERATURE ARE DENOTED AS P_0 AND T_0 RESPECTIVELY. THE… 14 Mar 2020 02:39 AM IST COMPARISON OF WEDGE BOUNDARY CONDITION WITH SYMMETRIC BOUNDARY CONDITION AND HAIGEN-POISEUILLE EQUATION Objective: AIM:- IN THIS PROJECT SIMULATION IS DONE ON THE FLUID FLOW THROUGH PIPE USING OPENFOAM SOFTWARE AND MATLAB . THE TOPIC GIVEN BELOW IS COVERED IN THIS PROJECT. TO MAKE A PROGRAM IN MATLAB THAT CAN GENERATE MESH AUTOMATICALLY FOR ANY WEDGE ANGLE… 09 Sep 2019 02:17 PM IST SIMULATION OF FLOW THROUGH THE BACKWARD FACING STEP BY USING BLOCKMESH IN OPENFOAM Objective: AIM :- IN THIS PROJECT, I HAVE WORKED ON SIMULATING THE FLOW OF FLUID THROUGH THE DOMAIN (CONTROL VOLUME). DESCRIPTION:- THE DOMAIN IS NOTHING ELSE, IT IS JUST A CONTROL VOLUME THROUGH WHICH I HAVE SIMULATED THE RESULT OBTAINED FROM THE GOVERNING EQUATION. THE CONTROL VOLUME IS A GEOMETRY WHERE THE WHOLE GEOMETRY… 13 Aug 2019 02:57 PM IST PROGRAM ON FLOW OF HEAT IN THE RECTANGULAR SLAB (2-D DOMAIN) USING 2-DIMENSIONAL HEAT CONDUCTION EQUATION IN STEADY AND TRANSITION STATE Objective: In this project, I have worked on the simulation of conduction of temperature heat in two dimensional space (2D domain). The aim was to determine the rate of flow of heat due to temperature difference over the space domain under steady condition and transitional condition. The governing equation which… 10 Jul 2019 04:37 AM IST Showing 1 of 28 projects Try our top engineering courses, projects & workshops today!	2022-08-11 06:38:43	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6434904336929321, "perplexity": 2966.1631631536075}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571234.82/warc/CC-MAIN-20220811042804-20220811072804-00016.warc.gz"}
https://studyfinance.com/sustainable-growth-measures/	Sustainable Growth Measures From time to time, businesses may run into trouble unless they control their growth. Growth can be orderly or it can be unrestrained. Unrestrained growth can lead to less than optimal performance or even financial distress. DuPont Analysis focuses on the combined effect of profitability and turnover ratios. By modifying the approach a bit, it is possible to estimate a company’s sustainable growth. Simply, sustainable growth would be the realistic attainable growth that a company could maintain without running into problems. Measuring Sustainable Growth $G* = Earnings\: Retention \times ROE$ – or – $G* = Earnings\: Retention \times Asset\: Utilization \times Profitability \times Financial\: Leverage$ • Earnings Retention is calculated by Retention Ratio or ( 1 − Company’s Dividend Rate ) • Asset Utilization is measured by Total Asset Turnover or ( Sales ÷ Total Assets ) • Profitability is measured by Net Profit Margin or ( Net Income ÷ Sales ) • Financial Leverage is ( Total Long Term Debt ÷ Stockholders’ Equity ) Actual vs. Sustainable Growth Once the sustainable growth rate is calculated, then it should be compared to the company’s actual growth rate. If sustainable growth is greater than actual growth, the company might be underperforming. If the actual growth rate is greater than sustainable growth, the company may run into trouble because of unrestrained growth. Example Assume the following: The Dividend Payout Ratio is calculated by dividing Dividends by Net Income, and the Retention Ratio is (1 − Dividend Payout Ratio). Therefore: • The Dividend Ratio for 2014 is 40%, so the Retention Ratio is 60%. • For that year the ROA would be 7.49%, or (5.25% × .793 × 1.8). • The Sustainable Growth Rate would be 4.49%, or (.6 × 7.49%). The return on equity, retention ratio and sustainable growth measures for the years in the previous example would be: If actual growth for the years in question is 5%, 5.2%, 5.1%, 5.3%, and 6.1%, then actual growth graphed against sustainable growth would appear as: To calculate actual growth in sales, the analyst would find the percentage increase from one year to the next. For instance, if sales last year were $100,000 and$110,000 this year, then the actual growth rate in sales would be 10%. Analysis It’s clear from the previous example that actual growth is consistently above sustainable growth. What does that mean? Sustainable Less Than Actual • If sustainable growth is less than actual growth over a protracted period, the company cannot sustain such activity without “funding” that growth. Either they need to plow more profits into the company, increase net profit margin or turnover performance, or “fund” from risky sources such as increasing the debt level. Sustainable Greater Than Actual • When sustainable growth is greater than actual growth over an extended time, the company has the potential of ratcheting up growth. If they consistently fall below sustainable growth, they are passing up returns for shareholders.	2020-02-22 15:46:27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 2, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.1898030936717987, "perplexity": 4395.09488191639}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145708.59/warc/CC-MAIN-20200222150029-20200222180029-00207.warc.gz"}
https://allendowney.github.io/ModSimPy/chap21.html	# Chapter 21¶ In the previous chapter we simulated a penny falling in a vacuum, that is, without air resistance. But the computational framework we used is very general; it is easy to add additional forces, including drag. In this chapter, I present a model of drag force and add it to the simulation. ## Drag Force¶ As an object moves through a fluid, like air, the object applies force to the air and, in accordance with Newton’s third law of motion, the air applies an equal and opposite force to the object (see http://modsimpy.com/newton). The direction of this drag force is opposite the direction of travel, and its magnitude is given by the drag equation (see http://modsimpy.com/drageq): $F_d = \frac{1}{2}~\rho~v^2~C_d~A$ where • $$F_d$$ is force due to drag, in newtons (N). • $$\rho$$ is the density of the fluid in kg/m$$^3$$. • $$v$$ is the magnitude of velocity in m/s. • $$A$$ is the reference area of the object, in m$$^2$$. In this context, the reference area is the projected frontal area, that is, the visible area of the object as seen from a point on its line of travel (and far away). • $$C_d$$ is the drag coefficient, a dimensionless quantity that depends on the shape of the object (including length but not frontal area), its surface properties, and how it interacts with the fluid. For objects moving at moderate speeds through air, typical drag coefficients are between 0.1 and 1.0, with blunt objects at the high end of the range and streamlined objects at the low end (see http://modsimpy.com/dragco). For simple geometric objects we can sometimes guess the drag coefficient with reasonable accuracy; for more complex objects we usually have to take measurements and estimate $$C_d$$ from data. Of course, the drag equation is itself a model, based on the assumption that $$C_d$$ does not depend on the other terms in the equation: density, velocity, and area. For objects moving in air at moderate speeds (below 45 mph or 20 m/s), this model might be good enough, but we should remember to revisit this assumption. For the falling penny, we can use measurements to estimate $$C_d$$. In particular, we can measure terminal velocity, $$v_{term}$$, which is the speed where drag force equals force due to gravity: $\frac{1}{2}~\rho~v_{term}^2~C_d~A = m g$ where $$m$$ is the mass of the object and $$g$$ is acceleration due to gravity. Solving this equation for $$C_d$$ yields: $C_d = \frac{2~m g}{\rho~v_{term}^2~A}$ According to Mythbusters, the terminal velocity of a penny is between 35 and 65 mph (see http://modsimpy.com/mythbust). Using the low end of their range, 40 mph or about 18 m/s, the estimated value of $$C_d$$ is 0.44, which is close to the drag coefficient of a smooth sphere. ## The Params Object¶ As the number of system parameters increases, and as we need to do more work to compute them, we will find it useful to define a Params object to contain the quantities we need to make a System object. Params objects are similar to System objects, and we initialize them the same way. Here’s the Params object for the falling penny: params = Params( mass = 0.0025, # kg diameter = 0.019, # m rho = 1.2, # kg/m3 g = 9.8, # m/s2 v_init = 0, # m / s v_term = 18, # m / s height = 381, # m t_end = 30, # s ) The mass and diameter are from http://modsimpy.com/penny. The density of air depends on temperature, barometric pressure (which depends on altitude), humidity, and composition (http://modsimpy.com/density). I chose a value that might be typical in Boston, Massachusetts at 20 °C. Here’s a version of make_system that takes the Params object and computes the inital state, init, the area, and the coefficient of drag. Then it returns a System object with the quantities we’ll need for the simulation. from numpy import pi def make_system(params): init = State(y=params.height, v=params.v_init) area = pi * (params.diameter/2)*2 C_d = (2 params.mass * params.g / (params.rho * area * params.v_term*2)) return System(init=init, area=area, C_d=C_d, mass=params.mass, rho=params.rho, g=params.g, t_end=params.t_end) And here’s how we call it. system = make_system(params) Based on the mass and diameter of the penny, the density of air, and acceleration due to gravity, and the observed terminal velocity, we estimate that the coefficient of drag is about 0.44. system.C_d 0.4445009981135434 It might not be obvious why it is useful to create a Params object just to create a System object. In fact, if we only run one simulation, it might not be useful. But it helps when we want to change or sweep the parameters. For example, suppose we learn that the terminal velocity of a penny is actually closer to 20 m/s. We can make a Params object with the new value, and a corresponding System object, like this: params2 = params.set(v_term=20) The result from set is a new Params object that is identical to the original except for the given value of v_term. If we pass params2 to make_system, we see that it computes a different value of C_d. system2 = make_system(params2) system2.C_d 0.3600458084719701 If the terminal velocity of the penny is 20 m/s, rather than 18 m/s, that implies that the coefficient of drag is 0.36, rather than 0.44. And that makes sense, since lower drag implies faster terminal velocity. Using Params objects to make System objects helps make sure that relationships like this are consistent. And since we are always making new objects, rather than modifying existing objects, we are less likely to make a mistake. ## Simulation¶ Now let’s get to the simulation. Here’s a version of the slope function that includes drag: def slope_func(t, state, system): y, v = state rho, C_d, area = system.rho, system.C_d, system.area mass, g = system.mass, system.g f_drag = rho v*2 C_d * area / 2 a_drag = f_drag / mass dydt = v dvdt = -g + a_drag return dydt, dvdt As usual, the parameters of the slope function are a time stamp, a State object, and a System object. We don’t use t in this example, but we can’t leave it out because when run_solve_ivp calls the slope function, it always provides the same arguments, whether they are needed or not. f_drag is force due to drag, based on the drag equation. a_drag is acceleration due to drag, based on Newton’s second law. To compute total acceleration, we add accelerations due to gravity and drag. g is negated because it is in the direction of decreasing y; a_drag is positive because it is in the direction of increasing y. In the next chapter we will use Vector objects to keep track of the direction of forces and add them up in a less error-prone way. As usual, let’s test the slope function with the initial conditions. slope_func(0, system.init, system) (0, -9.8) Because the initial velocity is 0, so is the drag force, so the initial acceleration is still g. To stop the simulation when the penny hits the sidewalk, we’ll use the event function from the previous chapter. def event_func(t, state, system): y, v = state return y Now we can run the simulation like this: results, details = run_solve_ivp(system, slope_func, events=event_func) details.message 'A termination event occurred.' Here are the last few time steps: results.tail() y v 21.541886 1.614743e+01 -18.001510 21.766281 1.211265e+01 -18.006240 21.990676 8.076745e+00 -18.009752 22.215070 4.039275e+00 -18.011553 22.439465 2.131628e-14 -18.011383 The final height is close to 0, as expected. Interestingly, the final velocity is not exactly terminal velocity, which is a reminder that the simulation results are only approximate. We can get the flight time from results. t_sidewalk = results.index[-1] t_sidewalk 22.439465058044306 With air resistance, it takes about 22 seconds for the penny to reach the sidewalk. Here’s a plot of position as a function of time. def plot_position(results): results.y.plot() decorate(xlabel='Time (s)', ylabel='Position (m)') plot_position(results) And velocity as a function of time: def plot_velocity(results): results.v.plot(color='C1', label='v') decorate(xlabel='Time (s)', ylabel='Velocity (m/s)') plot_velocity(results) From an initial velocity of 0, the penny accelerates downward until it reaches terminal velocity; after that, velocity is constant. ## Summary¶ This chapter presents a model of drag force, which we use to estimate the coefficient of drag for a penny, and then simulate, one more time, dropping a penny from the Empire State building. In the next chapter we’ll move from one dimension to two, simulating the flight of a baseball. But first you might want to work on these exercises. ## Exercises¶ ### Exercise¶ Run the simulation with a downward initial velocity that exceeds the penny’s terminal velocity. What do you expect to happen? Plot velocity and position as a function of time, and see if they are consistent with your prediction. Hint: Use params.set to make a new Params object with a different initial velocity. # Solution params = params.set(v_init=-30) system2 = make_system(params) # Solution results2, details2 = run_solve_ivp(system2, slope_func, events=event_func) details2.message 'A termination event occurred.' # Solution t_sidewalk = results2.index[-1] t_sidewalk 20.635183673114156 # Solution plot_position(results2) # Solution plot_velocity(results2) ### Exercise¶ Suppose we drop a quarter from the Empire State Building and find that its flight time is 19.1 seconds. Use this measurement to estimate terminal velocity and coefficient of drag. You can get the relevant dimensions of a quarter from https://en.wikipedia.org/wiki/Quarter_(United_States_coin). 1. Create a Params object with new values of mass and diameter. We don’t know v_term, so we’ll start with the initial guess 18 m/s. 2. Use make_system to create a System object. 3. Call run_solve_ivp to simulate the system. How does the flight time of the simulation compare to the measurement? 4. Try a few different values of v_term and see if you can get the simulated flight time close to 19.1 seconds. 5. Optionally, write an error function and use root_scalar to improve your estimate. 6. Use your best estimate of v_term to compute C_d. Note: I fabricated the “observed” flight time, so don’t take the results of this exercise too seriously. # Solution params_quarter = params.set( mass = 0.0057, # kg diameter = 0.024, # m flight_time = 19.1, # s ) # Solution system3 = make_system(params_quarter) # Solution # Run the simulation results3, details3 = run_solve_ivp(system3, slope_func, events=event_func) details3.message 'A termination event occurred.' # Solution # And get the flight time t_sidewalk = results3.index[-1] t_sidewalk 20.635183673114057 # Solution # The flight time is a little long, # so we could increase v_term and try again. # Or we could write an error function def error_func(guess, params): """Final height as a function of C_d. guess: guess at v_term params: Params object returns: height in m """ print(guess) params = params.set(v_term=guess) system = make_system(params) results, details = run_solve_ivp(system, slope_func, events=event_func) t_sidewalk = results.index[-1] error = t_sidewalk - params.flight_time return error # Solution # We can test the error function like this v_guess1 = 18 error_func(v_guess1, params_quarter) 18 1.5351836731140551 # Solution v_guess2 = 22 error_func(v_guess2, params_quarter) 22 -2.1591256962719925 # Solution # Now we can use root_scalar to find the value of # v_term that yields the measured flight time. res = root_scalar(error_func, params_quarter, bracket=[v_guess1, v_guess2]) 18 18.0 22.0 19.66221452468036 19.475274945870474 19.45967387499707 19.46064685869182 # Solution v_term = res.root v_term 19.45967387499707 # Solution # Plugging in the estimated value, # we can use make_system to compute C_d system4 = make_system(params_quarter.set(v_term=res.root)) system4.C_d 0.54345826868176	2021-05-08 16:58:01	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 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": 0.7055999040603638, "perplexity": 1299.168541118485}, "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-2021-21/segments/1620243988882.94/warc/CC-MAIN-20210508151721-20210508181721-00241.warc.gz"}
https://www.flexiprep.com/NCERT-Exemplar-Solutions/Mathematics/Class-9/NCERT-Mathematics-Class-9-Exemplar-Ch-14-Statistics-and-Probability-Part-4.html	# NCERT Mathematics Class 9 Exemplar Ch 14 Statistics and Probability Part 4 Get unlimited access to the best preparation resource for NEST : fully solved questions with step-by-step explanation- practice your way to success. 19. There are 50 numbers. Each number is subtracted from 53 and the mean of the numbers so obtained is found to be. The mean of the given numbers is: (A) (B) (C) (D) 20. The mean of 25 observations is 36. Out of these observations if the mean of first 13 observations is 32 and that of the last 13 observations is 40, the 13th observation is: (A) 23 (B) 36 (C) 38 (D) 40 21. The median of the data is (A) (B) (C) (D) 22. For drawing a frequency polygon of a continuous frequency distribution, we plot the points whose ordinates are the frequencies of the respective classes and abscissae are respectively: (A) Upper limits of the classes (B) Lower limits of the classes (C) Class marks of the classes (D) Upper limits of preceding classes 23. Median of the following numbers: is (A) 4 (B) 5 (C) 6 (D) 7 24. Mode of the data is (A) 14 (B) 15 (C) 16 (D) 17 25. In a sample study of 642 people, it was found that 514 people have a high school certificate. If a person is selected at random, the probability that the person has a high school certificate is: (A) (B) (C) (D)	2021-01-26 16:00:10	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8113570809364319, "perplexity": 689.9107051619936}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610704800238.80/warc/CC-MAIN-20210126135838-20210126165838-00054.warc.gz"}
https://docs.microsoft.com/en-us/dotnet/csharp/programming-guide/classes-and-structs/how-to-use-named-and-optional-arguments-in-office-programming	How to use named and optional arguments in Office programming (C# Programming Guide) Named arguments and optional arguments, introduced in C# 4, enhance convenience, flexibility, and readability in C# programming. In addition, these features greatly facilitate access to COM interfaces such as the Microsoft Office automation APIs. In the following example, method ConvertToTable has sixteen parameters that represent characteristics of a table, such as number of columns and rows, formatting, borders, fonts, and colors. All sixteen parameters are optional, because most of the time you do not want to specify particular values for all of them. However, without named and optional arguments, a value or a placeholder value has to be provided for each parameter. With named and optional arguments, you specify values only for the parameters that are required for your project. You must have Microsoft Office Word installed on your computer to complete these procedures. Note Your computer might show different names or locations for some of the Visual Studio user interface elements in the following instructions. The Visual Studio edition that you have and the settings that you use determine these elements. For more information, see Personalizing the IDE. To create a new console application 1. Start Visual Studio. 2. On the File menu, point to New, and then click Project. 3. In the Templates Categories pane, expand Visual C#, and then click Windows. 4. Look in the top of the Templates pane to make sure that .NET Framework 4 appears in the Target Framework box. 5. In the Templates pane, click Console Application. 6. Type a name for your project in the Name field. 7. Click OK. The new project appears in Solution Explorer. 1. In Solution Explorer, right-click your project's name and then click Add Reference. The Add Reference dialog box appears. 2. On the .NET page, select Microsoft.Office.Interop.Word in the Component Name list. 3. Click OK. 1. In Solution Explorer, right-click the Program.cs file and then click View Code. 2. Add the following using directives to the top of the code file: using Word = Microsoft.Office.Interop.Word; To display text in a Word document 1. In the Program class in Program.cs, add the following method to create a Word application and a Word document. The Add method has four optional parameters. This example uses their default values. Therefore, no arguments are necessary in the calling statement. static void DisplayInWord() { var wordApp = new Word.Application(); wordApp.Visible = true; // docs is a collection of all the Document objects currently // open in Word. Word.Documents docs = wordApp.Documents; // Add a document to the collection and name it doc. } 2. Add the following code at the end of the method to define where to display text in the document, and what text to display: // Define a range, a contiguous area in the document, by specifying // a starting and ending character position. Currently, the document // is empty. Word.Range range = doc.Range(0, 0); // Use the InsertAfter method to insert a string at the end of the // current range. range.InsertAfter("Testing, testing, testing. . ."); To run the application 1. Add the following statement to Main: DisplayInWord(); 2. Press CTRL+F5 to run the project. A Word document appears that contains the specified text. To change the text to a table 1. Use the ConvertToTable method to enclose the text in a table. The method has sixteen optional parameters. IntelliSense encloses optional parameters in brackets, as shown in the following illustration. Named and optional arguments enable you to specify values for only the parameters that you want to change. Add the following code to the end of method DisplayInWord to create a simple table. The argument specifies that the commas in the text string in range separate the cells of the table. // Convert to a simple table. The table will have a single row with // three columns. range.ConvertToTable(Separator: ","); In earlier versions of C#, the call to ConvertToTable requires a reference argument for each parameter, as shown in the following code: // Call to ConvertToTable in Visual C# 2008 or earlier. This code // is not part of the solution. var missing = Type.Missing; object separator = ","; range.ConvertToTable(ref separator, ref missing, ref missing, ref missing, ref missing, ref missing, ref missing, ref missing, ref missing, ref missing, ref missing, ref missing, ref missing, ref missing, ref missing, ref missing); 2. Press CTRL+F5 to run the project. To experiment with other parameters 1. To change the table so that it has one column and three rows, replace the last line in DisplayInWord with the following statement and then type CTRL+F5. range.ConvertToTable(Separator: ",", AutoFit: true, NumColumns: 1); 2. To specify a predefined format for the table, replace the last line in DisplayInWord with the following statement and then type CTRL+F5. The format can be any of the WdTableFormat constants. range.ConvertToTable(Separator: ",", AutoFit: true, NumColumns: 1, Format: Word.WdTableFormat.wdTableFormatElegant); Example The following code includes the full example: using System; using System.Collections.Generic; using System.Linq; using System.Text; using Word = Microsoft.Office.Interop.Word; namespace OfficeHowTo { class WordProgram { static void Main(string[] args) { DisplayInWord(); } static void DisplayInWord() { var wordApp = new Word.Application(); wordApp.Visible = true; // docs is a collection of all the Document objects currently // open in Word. Word.Documents docs = wordApp.Documents; // Add a document to the collection and name it doc. // Define a range, a contiguous area in the document, by specifying // a starting and ending character position. Currently, the document // is empty. Word.Range range = doc.Range(0, 0); // Use the InsertAfter method to insert a string at the end of the // current range. range.InsertAfter("Testing, testing, testing. . ."); // You can comment out any or all of the following statements to // see the effect of each one in the Word document. // Next, use the ConvertToTable method to put the text into a table. // The method has 16 optional parameters. You only have to specify // values for those you want to change. // Convert to a simple table. The table will have a single row with // three columns. range.ConvertToTable(Separator: ","); // Change to a single column with three rows.. range.ConvertToTable(Separator: ",", AutoFit: true, NumColumns: 1); // Format the table. range.ConvertToTable(Separator: ",", AutoFit: true, NumColumns: 1, Format: Word.WdTableFormat.wdTableFormatElegant); } } }	2020-12-03 05:49:48	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.25825873017311096, "perplexity": 4402.315881583348}, "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-2020-50/segments/1606141718314.68/warc/CC-MAIN-20201203031111-20201203061111-00449.warc.gz"}
https://socratic.org/questions/how-do-you-prove-tan-x-pi-2-cotx	How do you prove tan(x + (pi/2)) = -cotx? Apr 16, 2015 We can not simply use the tangent of a sum formula, because $\tan \left(\frac{\pi}{2}\right)$ does not exist. See other answer for using the sum formula. So use $\tan \theta = \sin \frac{\theta}{\cos} \theta$ and the sum formulas for sine and cosine. $\tan \left(x + \frac{\pi}{2}\right) = \frac{\sin \left(x + \frac{\pi}{2}\right)}{\cos \left(x + \frac{\pi}{2}\right)}$ $= \frac{\sin x \cos \left(\frac{\pi}{2}\right) + \cos x \sin \left(\frac{\pi}{2}\right)}{\cos x \cos \left(\frac{\pi}{2}\right) - \sin x \sin \left(\frac{\pi}{2}\right)} = \frac{0 + \cos x}{0 - \sin x} = - \cos \frac{x}{\sin} x = - \cot x$ Oct 25, 2017 If we really want to use the sum formula for tangent, then we can. See below. Explanation: We cannot simply apply the sum formula as tan(x+pi/2) = (tanx+tan(pi/2))/(1-tanxtan(pi/2) because $\tan \left(\frac{\pi}{2}\right)$ does not exist. But we can rewrite $x + \frac{\pi}{2} = \left(x + \frac{\pi}{4}\right) + \frac{\pi}{4}$ and apply the sun formula twice. For any $a$ in the domain of the tangent function, we have: $\tan \left(a + \frac{\pi}{4}\right) = \frac{\tan a + \tan \left(\frac{\pi}{4}\right)}{1 - \tan a \tan \left(\frac{\pi}{4}\right)}$ $= \frac{\tan a + 1}{1 - \tan a}$ Therefore, $\tan \left(x + \frac{\pi}{2}\right) = \tan \left(\left(x + \frac{\pi}{4}\right) + \frac{\pi}{4}\right)$ $= \frac{\tan \left(x + \frac{\pi}{4}\right) + 1}{1 - \tan \left(x + \frac{\pi}{4}\right)}$ $= \frac{\left[\frac{\tan x + 1}{1 - \tan x}\right] + 1}{1 - \left[\frac{\tan x + 1}{1 - \tan x}\right]}$ $= \frac{\left[\frac{\tan x + 1}{1 - \tan x}\right] + 1}{1 - \left[\frac{\tan x + 1}{1 - \tan x}\right]} \cdot \frac{1 - \tan x}{1 - \tan x}$ $= \frac{\tan x + 1 + 1 - \tan x}{1 - \tan x - \left(\tan x + 1\right)}$ $= \frac{2}{- 2 \tan x}$ $= - \cot x$	2020-02-23 17:31:44	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 17, "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": 0.9284223318099976, "perplexity": 616.8102698386045}, "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-2020-10/segments/1581875145818.81/warc/CC-MAIN-20200223154628-20200223184628-00157.warc.gz"}
https://www.nature.com/articles/s41598-021-87249-0?error=cookies_not_supported&code=9edd01ab-ca95-418f-abc0-f218e264f8d9	Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript. # Quantifying surface morphology of manufactured activated carbon and the waste coffee grounds using the Getis-Ord-Gi* statistic and Ripley’s K function ## Abstract Activated carbon can be manufactured from waste coffee grounds via physical and/or chemical activation processes. However, challenges remain to quantify the differences in surface morphology between manufactured activated carbon granules and the waste coffee grounds. This paper presents a novel quantitative method to determine the quality of the physical and chemical activation processes performed in the presence of intensity inhomogeneity and identify surface characteristics of manufactured activated carbon granules and the waste coffee grounds. The spatial density was calculated by the Getis-Ord-Gi* statistic in scanning electron microscopy images. The spatial characteristics were determined by analyzing Ripley’s K function and complete spatial randomness. Results show that the method introduced in this paper is capable of distinguishing between manufactured activated carbon granules and the waste coffee grounds, in terms of surface morphology. ## Introduction Activated carbon is one of the most widely used absorbents in water and wastewater treatment1. Activated carbon has been used since ancient Greece to remove contaminants in air and water2. Relatively small amounts of activated carbon can absorb significant amounts of contaminants due to its extremely large surface area per unit mass. The porous structure of activated carbon determines the absorptive capacity3. Many studies have shown the effectiveness of activated carbon adsorption for a wide range of contaminants. According to the literature, activated carbon can effectively adsorb both organic and inorganic contaminants including biological nutrients, pharmaceuticals, endocrine disruptors, pesticides, dyes, heavy metals, and even persistent organic pollutants like per- and polyfluoroalkyl substances4,5,6,7,8,9,10,11. In recent years, activated carbon has also received considerable attention as a reinforcement of cement-based composites such as a concrete mixture. Mahoutian et al.12 reported that activated carbon reduced the air content in concrete, resulting in an increase of strength. Chowdhury et al.13 found that activated carbon promoted moisture resistance of cement and mitigated the risk due to moisture attack. Although activated carbon is advantageous in diverse engineering applications, the use of activated carbon can be expensive as spent activated carbon has to be replaced with new carbon over time. Wood, coal, and lignite are the most common base materials for activated carbon production and less than 20% of the total production utilizes renewable base materials14,15. The literature suggests that activated carbon can be prepared from alternative biomass including agricultural by-products, manure, wastewater sludge that are not coal-based resources14,16,17,18. Challenges, however, still remain to utilize activated carbon obtained from alternative biomass sources. For example, the quality of activated carbon manufactured from biomass is highly dependent on the activation process, and inappropriate processing may lead to inferior surface characteristics of activated carbon. To ensure and produce activated carbon that meets the specification, a method that can quantify the surface morphology of activated carbon is demanded because the adsorption capacity of activated carbon is strongly correlated with the surface characteristics such as the pore irregularity and the surface roughness. Fractal analysis that analyzes fractal dimensions of the surface of activated carbon has been widely used by many studies19,20,21,22,23. However, the application of fractal analysis often produces biased or incorrect results due to insufficient data points or errors in regression analysis24. Furthermore, the fractal dimension analysis may give the same fractal number to two different objects and, thus the fractal dimension of a set of objects is not sufficient to describe the difference between each object25. In addition to fractal dimension, specific surface areas of fractal surfaces are experimentally estimated by the Brunauer–Emmett–Teller (BET) theory for activated carbon26. However, BET measurements can be also unreliable for some carbonaceous materials, especially for biochars, as degassing process alters the material properties27. In this paper, we present a seamless solution that quantitatively measures the quality of activated carbon manufactured by the physical and chemical activation process and identifies surface characteristics of both activated carbon granules and the waste coffee grounds through scanning electron microscopy (SEM) images. The Getis-Ord-Gi* statistic is used to identify patterns of the spatial significance in the SEM images. Six feature detection algorithms, Minimum Eigenvalue (ME)28, Harris-Stephens (Harris)29, Binary Robust Invariant Scalable Keypoints (BRISK)30, Speed up Robust Feature (SURF)31, KAZE32, and Oriented fast and Rotated Binary (ORB) robust independent elementary features33, were analyzed to determine point-patterns in a two-dimensional space. Differences in surface morphology between manufactured activated carbon granules and the waste coffee grounds were measured by quantifying the area of a region between Ripley’s K function and complete spatial randomness (CSR). ## Result and discussion A quantitative process of manufactured activated carbon granules and waste coffee grounds using the Getis-Ord-Gi* statistic and Ripley’s K function is described in Fig. 1. The magnifications of the SEM images are 75×, 500×, 5000×, and 20,000×. To determine spatial characteristics of the point-patterns in the SEM images, the Getis-Ord-Gi* statistic was calculated (Fig. 1B). Features in the SEM images (600 × 400 pixels) were detected by six feature detection methods: ME, Harris, BRISK, ORB, SURF, and KAZE. Ripley’s K function was employed to analyze the detected features(Fig. 1C). Eight SEM images for each activated carbon and waste coffee ground sample were captured at four different magnification levels. The Getis-Ord-Gi* statistic was then used to generalize the pixel-wide regions beyond the uniform maps and represent the density of the regions (Fig. 2). Figure 2A represents a raw SEM image for the waste coffee grounds at 20,000×, an intensity histogram of the image, a box plot of the intensity values, and a z-score graph with different quantiles (Supplement Table S1 and S2). Figure 2B,D illustrate two-dimensional hotspot regions generated by the Getis-Ord-Gi* statistic with varying ratios of the quantile. Figure 2C represents a raw SEM image for manufactured activated carbon granules with 20,000×. The analysis of the Getis-Ord-Gi* statistic indicates that (1) the maximum z-scores of the waste coffee grounds is greater than those of the manufactured activated carbon granules. (2) the z-scores of waste coffee grounds tend to be steeper slopes compared to those of activated carbon as it increases the quantile percentages. Therefore, the density level can be a factor distinguishing between the surface of manufactured activated carbon granules and the waste coffee grounds by using Getis-Ord-Gi. Although the Getis-Ord-Gi statistic can be used for the analysis of the surface morphology, it is only applicable for limited regions. For example, raw SEM images may contain arbitrary granular materials with the background regions. These regions often prevent the constant dispersion measure, which is essential for the potential identification and characterization of the surface of granular materials. To overcome these limitations, feature detection methods were used to identify features from the overall region. Features identified by the feature detection methods are shown in Supplement Fig. 1. Each column represents the feature detection method and each row represents the SEM images of manufactured activated carbon granules (A1, A2, A3, and A4) and the waste coffee grounds (B1, B2, B3, and B4) based on the magnifications: 20,000×, 5000×, 500×, and 75× (top to bottom). We found that the features marked with green color at the second row and the fourth column are more well-distributed than the features marked with green color at the sixth row and the fourth column. This result indicates that (1) the features identified by BRISK are distinguishable factors between manufactured activated carbon granules and the waste coffee grounds and (2) there was a difference between the feature detection methods. However, the comparison relying on intuition only potentially causes an inter-observer variation influenced by systematic bias on observer experiences. To avoid the problem of intuition, we created the 95% confidence interval error ellipse to specify the error variance for the detected features. The covariance matrices to represent the variability of the feature points in the surface of the manufactured activated carbon granules and the waste coffee grounds were measured accordingly and described in Supplement Table S3. The 95% confidence interval error ellipses and feature distributions of the SEM images are shown in Fig. 3. We used the identical SEM images from both manufactured activated carbon granules and the waste coffee grounds. The results of the 95% confidence interval error ellipses show that the eigenvectors of the covariance matrix of the features identified from B are more likely in the same direction than the ones of the features identified from A for two-dimensional normally distributed features. Moreover, we performed the dimensionality reduction on the two-dimensional feature data by reducing the number of dimensions from two to one using principal component analysis (PCA). The results indicate that the one-dimensional features identified from manufactured activated carbon granules are evenly distributed, but the one-dimensional features identified from the waste coffee grounds are unevenly distributed. Characteristics of the SEM images can be identified by capturing the essence of an image pattern. Each SEM image prepared from the previous approaches was analyzed either to obtain a list of features by the feature detection methods or to compute the statistics of the 20 × 20 grid region. In addition to these approaches, we investigated another approach to find the differences in surface morphology between manufactured activated carbon granules and the waste coffee grounds. The main contribution of our paper is that we quantified the impact of the features identified by the feature detection methods by measuring the area of a region between Ripley’s K function and CSR often called a homogeneous spatial Poisson process. A statistical test against the homogeneous spatial Poisson process was performed by estimating the variance of the features by using Ripley’s K function. Ripley’s K function computes features’ concentration based on a range of distance or scales so that the relationships between the features can be fully identified. The results of the quantitative measuring and influencing mechanism of the area of a region between Ripley’s K function and CSR are shown in Fig. 4. A1, A2, A3, and A4 are the SEM images captured from manufactured activated carbon granules, while B1, B2, B3, and B4 are SEM images captured from waste coffee grounds with different resolutions. The left side of the diagram in Fig. 4 represents input SEM images and the right side of the diagram represents the feature detection algorithms. The width of the band represents the area of a region between Ripley’s K function and CSR. We quantified the difference between A2 and B2 by measuring the width of the bands in the BRISK. The area of a region between Ripley’s K function and CSR was measured to evaluate Ripley’s K functions of features identified by the feature detection methods. The sum of the areas between the graphs of the functions for each input SEM image corresponding to each feature detection algorithm is shown on the left side of the box. The flow within the diagram indicates that the SEM images captured from manufactured activated carbon granules cause small area, comparing with the SEM images captured from waste coffee grounds, indicating that the physical activation process is achieved by differentiating the quantity of the surface morphology with respect to manufactured activated carbon granules. The presented quantitative measurements in surface morphology between manufactured activated carbon granules and the waste coffee ground vary in the number of strong points of the features identified by the feature detection methods. Four experiments were performed on the presented approach with different strong points: 20, 40, 60, 80. The variance of the strong points was estimated by Ripley’s K function and compared with the CSR. In Fig. 5, the CSR plot was simulated in green-dash lines. Ripley’s K functions for two surface configurations of the spatial points (A1 and B1) were simulated in two lines: red-solid and blue-solid. The results show that (1) Ripley’s K function for the surface of manufactured activated carbon granules are more likely to follow the CSR, which indicates that the dispersion of manufactured activated carbon’s surface is substantially higher than the one from the waste coffee ground’s surface and (2) the higher SPs surfaces are more distinguishable than those in lower SPs. We performed an additional experiment on the waste coffee ground (raw) and activated carbon granules manufactured via physical activation and chemical activation using three chemical agents (i.e., H3PO4, ZnCl2, and NaOH) shown in Supplement Fig. 2 and Supplement Table S4. However, a limited number of SEM images can be a major obstacle in verifying the quantitative process. We augmented the SEM images used in Supplement Fig. 2 by 360-degree random rotation, generating 600 SEM images with random rotation augmentation. The features of each SEM image were identified by 6 feature detection methods, and then the area of a region between Ripley’s K function and CSR for 5 SPs was computed. The quantitative results of the augmented SEM images are illustrated as box plots in Fig. 6. The box plots show that the area of a region between Ripley’s K function and CSR in H3PO4 are smaller and denser than those in Raw. All results are illustrated in Supplement Table S5. These results indicate that the presented approach is effective to distinguish the SEM images captured from the manufactured activated carbon granules from the SEM images captured from the waste coffee grounds. In this paper, we presented a way of analyzing the surface morphology of manufactured activated carbon and the waste coffee grounds using the Getis-Ord-Gi* statistic and Ripley’s K function. Quantifying surface morphology in manufactured activated carbon granules has been a challenge in diverse engineering applications. We used Getis-Ord-Gi* statistic to identify spatial characteristics of the point-patterns in a 20 × 20 grid region over the SEM images with different magnification. However, raw SEM images with arbitrary granular materials often prevent the constant dispersion measure from the potential identification and characterization of the surface of granular materials. To avoid this issue, features extracted from the SEM images were also used to determine the spatial characteristics of manufactured activated carbon granules and waste coffee grounds. Ripley’s K function is adopted to evaluate the spatial characteristics of the features identified by the feature detection methods. These experiments provide clear evidence that the presented method is capable of distinguishing between manufactured activated carbon granules and the waste coffee grounds. Although the presented method performs well on the high-quality SEM images with gray levels, our approach is limited to use both Getis-Ord-Gi* statistic and Ripley’s K function simultaneously because of arbitrary granular materials with the background regions, and it may have consequential functional limitations in either color images or three-dimensional images since the feature detection in a color image with three channels can be challenging to interpret the surface variability Fractal analysis can be adopted for the RGB images34,35 and three-dimensional images, but it often causes incorrect errors in regression analysis24. We plan to investigate any applicable method to overcome these limitations. ## Materials and methods Waste coffee grounds were collected at a Starbucks coffee shop near Marshall University. Using 5-gallon plastic pails with lids, the spent coffee grounds were collected. The collected coffee waste consists of various types of wet coffee grounds. Activated carbon was produced from coffee waste in the laboratory following previously developed methods by other scholars36,37,38. The coffee grounds were first washed and rinsed with distilled water. The coffee waste was dried at 100 °C for 24 h in an oven and then cooled to room temperature in the laboratory. Each pail has a wide range of coffee grounds with various grain sizes, colors, and shapes. The dried coffee grounds are homogenized by gentle mixing. The dried coffee grounds were stored in a desiccator until the activation process to minimize possible adsorption of moisture. For the chemical activation process, three different chemical agents were used: H3PO4, ZnCl2, and NaOH. The dried coffee residue was first mixed in with either 1 M H3PO4, 1 M ZnCl2, or 1 M NaOH solution to various mass ratios (i.e., the mass of chemical agents to the mass of coffee grounds) in separate containers. The mixtures were given 24 h of contact time. The activation step was then followed where the mixtures were heated at 600 °C for an hour in a muffled furnace. For the physical activation, the prepared coffee grounds were directly heated at 600 °C for an hour without any chemical agent. After activation, activated carbon was washed, rinsed, and strained through U.S. standard No. 16 and 60 sieves to control grain-size (i.e., larger than 250 µm and smaller than 1.18 mm). The American Society for Testing and Material (ASTM) classifies activated carbon particles having a size smaller than 0.180 mm as powdered activated carbon (PAC). Granular activated carbon (GAC) is defined as a minimum of 90% of the sample weight being retained on a 180-μm Standard sieve39. According to the ASTM definition, the lab-manufactured activated carbon is GAC. The pore structures and surface morphology of activated carbon were examined using JEOL 5310LV and JEOL 7200FLV Scanning Electron Microscopes (SEM) at Marshall University. A small number of particles (approximately 10 particles) was applied to a piece of Carbon conductive double-sided tape. The other side of the tape was applied to a metal stub. The stub was securely affixed in a mount that was compatible with the SEM. All materials used, including tweezers and scissors, were cleaned before use with Ethanol (75% EtOH), and gloves were worn so as not to contaminate the sample or apparatus. Minimum eigenvalue was used to determine affine changes based on a Newton–Raphson style minimization. We measured two small eigenvalues and two large eigenvalues representing the corners. Harris was utilized to measure the changes in intensity for the shift in an image patch such that distinctive image patches hold larger values while constant image patches hold smaller values. BRISK was used to compute non-maximal suppression across scale space. ORB was used to find strong points through the measurement of the Harris algorithm. We also detected features by using SURF finding the points over the intensity distribution of the pixels by shifting the neighborhood of the points of interest. All feature detection methods used in this paper were built-in functions available in MATLAB. Ripley’s K function is a point spatial analysis method that describes point patterns in the area of events. As the point pattern analyses have been utilized to elucidate the spatial arrangement of geographic points by clarifying relationships between point patterns, the estimation of Ripley’s K function is defined by: $$\widehat{K}\left(r\right)=\frac{s}{{f}_{n}({f}_{n}-1)}\sum_{x}\sum_{y}I\left({d}_{xy}\le r\right){e}_{xy}$$ where $$s$$ is the size of the window, $${f}_{n}$$ is the number of features, $${d}_{xy}$$ is the distance between two features, $$I\left({d}_{xy}\le r\right)$$ is the indicator function that determines whether the distance $${d}_{xy}$$ is less than or equal to $$r$$ which is a distance of a random selection, and $${e}_{xy}$$ is Ripley’s isotropic edge correction weight. The area of a region between Ripley’s K function and CSR was approximated as the difference of the sum of all trapezoids. It is defined by: $$Area\approx \sum_{i=1}^{n}[R\left({x}_{i}^{}\right)-C({x}_{i}^{})]\Delta x$$ where $$R\left({x}_{i}^{}\right)=(R\left({x}_{i-1}\right)+R\left({x}_{i}\right))/2$$ and $$C\left({x}_{i}^{}\right)=(C\left({x}_{i-1}\right)+C\left({x}_{i}\right))/2. R\left({x}_{i}\right)$$ is the height of Ripley’s function at $${x}_{i}$$ and $$C\left({x}_{i}\right)$$ is the height of CSR at $${x}_{i}$$. $$\Delta x$$ is the width if $$i$$-th subinterval. ## Data availability All data are available from the corresponding author on request. ## References 1. 1. Benjamin, M. M. & Lawler, D. F. Water Quality Engineering: Physical/chemical Treatment Processes (John Wiley & Sons, 2013). 2. 2. Jankowska, H., Swiatkowski, A. & Choma, J. Active Carbon (Ellis Horwood Ltd, 1992). 3. 3. Prahas, D., Kartika, Y., Indraswati, N. & Ismadji, S. Activated carbon from jackfruit peel waste by H3PO4 chemical activation: Pore structure and surface chemistry characterization. Chem. Eng. J. 140(1), 32–42. https://doi.org/10.1016/j.cej.2007.08.032 (2008). 4. 4. Kobya, M. Removal of Cr(VI) from aqueous solutions by adsorption onto hazelnut shell activated carbon: Kinetic and equilibrium studies. Bioresour. Technol. 91(3), 317–321. https://doi.org/10.1016/j.biortech.2003.07.001 (2004). 5. 5. Nam, S.-W., Choi, D.-J., Kim, S.-K., Her, N. & Zoh, K.-D. Adsorption characteristics of selected hydrophilic and hydrophobic micropollutants in water using activated carbon. J. Hazard. Mater. 270, 144–152. https://doi.org/10.1016/j.jhazmat.2014.01.037 (2014). 6. 6. Li, Y. et al. Study on regeneration of waste powder activated carbon through pyrolysis and its adsorption capacity of phosphorus. Sci. Rep. 8(1), 778. https://doi.org/10.1038/s41598-017-19131-x (2018). 7. 7. Sigworth, E. A. & Smith, S. B. Adsorption of inorganic compounds by activated carbon. J. AWWA 64(6), 386–391. https://doi.org/10.1002/j.1551-8833.1972.tb02713.x (1972). 8. 8. Foo, K. Y. & Hameed, B. H. An overview of dye removal via activated carbon adsorption process. Desalin. Water Treatment 19(1–3), 255–274. https://doi.org/10.5004/dwt.2010.1214 (2010). 9. 9. Kadirvelu, K., Thamaraiselvi, K. & Namasivayam, C. Removal of heavy metals from industrial wastewaters by adsorption onto activated carbon prepared from an agricultural solid waste. Bioresour. Technol. 76(1), 63–65. https://doi.org/10.1016/S0960-8524(00)00072-9 (2001). 10. 10. Park, M. et al. Adsorption of perfluoroalkyl substances (PFAS) in groundwater by granular activated carbons: Roles of hydrophobicity of PFAS and carbon characteristics. Water Res. 170, 115364. https://doi.org/10.1016/j.watres.2019.115364 (2020). 11. 11. Appleman, T. D. et al. Treatment of poly- and perfluoroalkyl substances in U.S. full-scale water treatment systems. Water Res. 51, 246–255. https://doi.org/10.1016/j.watres.2013.10.067 (2014). 12. 12. Mahoutian, M., Lubell, A. S. & Bindiganavile, V. S. Effect of powdered activated carbon on the air void characteristics of concrete containing fly ash. Constr. Build. Mater. 80, 84–91. https://doi.org/10.1016/j.conbuildmat.2015.01.019 (2015). 13. 13. Chowdhury, B. Investigations into the role of activated carbon in a moisture-blocking cement formulation. J. Therm. Anal. Calorim. 78(1), 215–226. https://doi.org/10.1023/B:JTAN.0000042169.37321.24 (2004). 14. 14. Pollard, S. J. T., Fowler, G. D., Sollars, C. J. & Perry, R. Low-cost adsorbents for waste and wastewater treatment: A review. Sci. Total Environ. 116(1), 31–52. https://doi.org/10.1016/0048-9697(92)90363-W (1992). 15. 15. Ragan, S. & Megonnell, N. Activated carbon from renewable resources—Lignin. Cellul. Chem. Technol. 45(7), 527 (2011). 16. 16. Lima, I. M., McAloon, A. & Boateng, A. A. Activated carbon from broiler litter: Process description and cost of production. Biomass Bioenergy 32(6), 568–572. https://doi.org/10.1016/j.biombioe.2007.11.008 (2008). 17. 17. Heschel, W. & Klose, E. On the suitability of agricultural by-products for the manufacture of granular activated carbon. Fuel 74(12), 1786–1791. https://doi.org/10.1016/0016-2361(95)80009-7 (1995). 18. 18. Mojoudi, N. et al. Phenol adsorption on high microporous activated carbons prepared from oily sludge: equilibrium, kinetic and thermodynamic studies. Sci. Rep. 9(1), 19352. https://doi.org/10.1038/s41598-019-55794-4 (2019). 19. 19. Liu, X. & Nie, B. Fractal characteristics of coal samples utilizing image analysis and gas adsorption. Fuel 182, 314–322. https://doi.org/10.1016/j.fuel.2016.05.110 (2016). 20. 20. Diduszko, R., Swiatkowski, A. & Trznadel, B. J. On surface of micropores and fractal dimension of activated carbon determined on the basis of adsorption and SAXS investigations. Carbon 38(8), 1153–1162. https://doi.org/10.1016/S0008-6223(99)00236-5 (2000). 21. 21. Hayashi, J. I., Horikawa, T., Muroyama, K. & Gomes, V. G. Activated carbon from chickpea husk by chemical activation with K2CO3: Preparation and characterization. Micropor. Mesopor. Mater. 55(1), 63–68. https://doi.org/10.1016/S1387-1811(02)00406-7 (2002). 22. 22. Gómez-Serrano, V., Cuerda-Correa, E. M., Fernández-González, M. C., Alexandre-Franco, M. F. & Macías-García, A. Preparation of activated carbons from chestnut wood by phosphoric acid-chemical activation. Study of microporosity and fractal dimension. Mater. Lett. 59(7), 846–853. https://doi.org/10.1016/j.matlet.2004.10.064 (2005). 23. 23. Macías-García, A., Díaz-Díez, M. A., Cuerda-Correa, E. M., Olivares-Marín, M. & Gañan-Gómez, J. Study of the pore size distribution and fractal dimension of HNO3-treated activated carbons. Appl. Surf. Sci. 252(17), 5972–5975. https://doi.org/10.1016/j.apsusc.2005.11.010 (2006). 24. 24. Gonzato, G., Mulargia, F. & Marzocchi, W. Practical application of fractal analysis: Problems and solutions. Geophys. J. Int. 132(2), 275–282. https://doi.org/10.1046/j.1365-246x.1998.00461.x (1998). 25. 25. Brewer, J. & Di Girolamo, L. Limitations of fractal dimension estimation algorithms with implications for cloud studies. Atmos. Res. 82(1), 433–454. https://doi.org/10.1016/j.atmosres.2005.12.012 (2006). 26. 26. Pfeifer, P. et al. Fractal bet and FHH theories of adsorption: A comparative study. Proc. R. Soc. Lond. A Math. Phys. Sci. 423(1864), 169–188. https://doi.org/10.1098/rspa.1989.0049 (1989). 27. 27. Bardestani, R., Patience, G. S. & Kaliaguine, S. Experimental methods in chemical engineering: Specific surface area and pore size distribution measurements—BET, BJH, and DFT. Can. J. Chem. Eng. 97(11), 2781–2791. https://doi.org/10.1002/cjce.23632 (2019). 28. 28. Jianbo, S. & Tomasi. Good features to track. In 1994 Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, 1994, pp. 593–600, doi: https://doi.org/10.1109/CVPR.1994.323794. 29. 29. Harris, C.G. & Stephens, M. A combined corner and edge detector. in Alvey vision conference, 1988, vol. 15, no. 50: Citeseer, pp. 10–5244. 30. 30. Leutenegger, S., Chli, M., & Siegwart, R.Y. BRISK: Binary Robust invariant scalable keypoints," in 2011 International Conference on Computer Vision, 2011, pp. 2548–2555, doi: https://doi.org/10.1109/ICCV.2011.6126542. 31. 31. Bay, H., Ess, A., Tuytelaars, T. & Van Gool, L. Speeded-up robust features (SURF). Comput. Vis. Image Understand. 110(3), 346–359. https://doi.org/10.1016/j.cviu.2007.09.014 (2008). 32. 32. Alcantarilla, P.F., Bartoli, A., & Davison, A.J. KAZE features. Berlin, Heidelberg, 2012: Springer Berlin Heidelberg, in Computer Vision – ECCV 2012, pp. 214–227. 33. 33. E. Rublee, V. Rabaud, K. Konolige, and G. Bradski, "ORB: An efficient alternative to SIFT or SURF," in 2011 International Conference on Computer Vision, 6–13 Nov. 2011 2011, pp. 2564–2571, doi: https://doi.org/10.1109/ICCV.2011.6126544. 34. 34. Nayak, S. R., Mishra, J., Khandual, A. & Palai, G. Fractal dimension of RGB color images. Optik 162, 196–205. https://doi.org/10.1016/j.ijleo.2018.02.066 (2018). 35. 35. Carpinteri, A., Chiaia, B. & Invernizzi, S. Three-dimensional fractal analysis of concrete fracture at the meso-level. Theor. Appl. Fract. Mech. 31(3), 163–172. https://doi.org/10.1016/S0167-8442(99)00011-7 (1999). 36. 36. Ma, X. & Ouyang, F. Adsorption properties of biomass-based activated carbon prepared with spent coffee grounds and pomelo skin by phosphoric acid activation. Appl. Surf. Sci. 268, 566–570. https://doi.org/10.1016/j.apsusc.2013.01.009 (2013). 37. 37. Namane, A., Mekarzia, A., Benrachedi, K., Belhaneche-Bensemra, N. & Hellal, A. Determination of the adsorption capacity of activated carbon made from coffee grounds by chemical activation with ZnCl2 and H3PO4. J. Hazard. Mater. 119(1), 189–194. https://doi.org/10.1016/j.jhazmat.2004.12.006 (2005). 38. 38. Hagemann, N. et al. Activated carbon, biochar and charcoal: Linkages and synergies across pyrogenic carbon’s ABCs. Water 10(2), 182 (2018). 39. 39. ASTM D1765–19, Standard classification system for carbon blacks used in rubber products, A. International, West Conshohojen, PA 19428, 2019. ## Acknowledgements The authors gratefully acknowledge the College of Engineering and Computer Sciences at Marshall University for their support for this work. This research was funded by the authors' start-up funds provided by Marshall University. The authors want to acknowledge the Marshall University Molecular and Biological Imaging Center for the use of SEM. ## Author information Authors ### Contributions S.L., S.N., and S.Y. performed documentation. S.L. conceived of the quantitative process concept. O.G.R. produced SEM images. S.L. performed analyses of SEM images. S.N., O.G.R., S.Y. performed the chemical activation process. S.L., S.N., and S.Y. assisted with manuscript development. S.Y. supervised the project. ### Corresponding author Correspondence to Sungmin Youn. ## Ethics declarations ### Competing interests The authors declare no competing interests. ### Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. ## Rights and permissions Reprints and Permissions Lee, S., Na, S., Rogers, O.G. et al. Quantifying surface morphology of manufactured activated carbon and the waste coffee grounds using the Getis-Ord-Gi* statistic and Ripley’s K function. Sci Rep 11, 7543 (2021). https://doi.org/10.1038/s41598-021-87249-0 • Accepted: • Published:	2021-07-27 23:08:07	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6126912832260132, "perplexity": 4197.941353996708}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1627046153491.18/warc/CC-MAIN-20210727202227-20210727232227-00694.warc.gz"}
https://pythonarray.com/how-to-see-if-a-string-contains-another-string-in-python/	Ever wanted to see if a string contains a string in Python? Thought it was going to be complicated like C? Think again! Python implements this feature in a very easy to read and easy to implement fashion. There are two ways of doing it, and some will like one way better than the other, so I’ll leave it up to you to decide which one you like better. ### The First Way: Using Python’s in Keyword The first way to check if a string contains another string is to use the in syntax. in takes two “arguments”, one on the left and one on the right, and returns True if the left argument is contained within the right argument. Here’s an example: >>> s = "It's not safe to go alone. Take this." >>> 'safe' in s True >>> 'blah' in s False >>> if 'safe' in s: ... print('The message is safe.') The message is safe. You get the idea. That’s all there is to it. The keyword in does all that magical work in the background for you, so there’s no worrying about for loops or anything of that nature. ### The Second Way: Using Python’s str.find This way is the less Pythonic way, but it’s still accepted. It’s longer and a little more confusing, but it still gets the job done. This way requires us to call the find method on our string and check its return code. Here’s our example, using the string defined above: >>> if s.find('safe') != -1: ... print('This message is safe.') This message is safe. Like I said above, this way is a little less clear, but it still gets the job done. The find method returns the position of the string within the string or -1 if it’s not found. So we simply check if the position is not -1, and be on our merry way. This article’s length indicates just how easy it is to check for strings within strings in Python, and I hope it convinces you that the first way is far better than the second way, in terms of writing more Pythonic code. Ta ta, for now.	2022-11-30 07:17:38	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.32352414727211, "perplexity": 650.5139574644104}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1669446710733.87/warc/CC-MAIN-20221130060525-20221130090525-00447.warc.gz"}
https://stats.stackexchange.com/questions/120711/treating-predictors-as-numerical-or-categorical-variable-in-regression	# Treating predictors as numerical or categorical variable in regression I have a set of data that I am using regression analyses on. All of the columns are numeric (as far as I can see) a mix of integers and reals. However, two of the columns are being read from the CSV as factors, not numeric. I can't see any reason for this. My first attempts at a regression were just to see how good each column was as a predictor (of time) on its own. The columns in question caused a bit of an issue. As factors they were excellent predictors alone (adjusted R${^2}$ of 0.45) however when converting them to numeric, they became poor predictors (adjusted R${^2}$ of 0.01). I have three questions. 1. Why would these two columns be interpreted as factors rather than numeric; 2. Why would there be such an enormous change in the quality of the predictor 3. Can I justify using them (and other columns) as a factor? • Because when modeled as factor, you were fitting an ANOVA to test if the means between all the levels inside that factor variable are equal to zero. When modeled as numeric, you were fitting a linear regression testing if the slope between the outcome and the numeric is zero. They are two different models. – Penguin_Knight Oct 20 '14 at 3:27 • I'm not sure I completely understand, but from what I could gather the cause may be because each record had a fairly distinct (not quite unique) factor, as such the regression stated that most of the time, that one semi-unique factor predicted time perfectly. – Zack Newsham Oct 20 '14 at 4:52 • Since you are using R, it will be useful if you can post outputs of commands 'str(mydata)' and 'head(mydata)' here. – rnso Oct 20 '14 at 14:58 There's not enough detail to be able to give more than a general answer: (1) "Why would these two columns be interpreted as factors rather than numeric?" Non-numeric records in the column—leading/trailing spaces, commas as thousands separators, currency symbols, "Unknown", &c. Look & see: if it isn't obvious from a list of the factor levels, coercing to numeric will give missing values where records aren't being recognized as numbers. (2) "Why would there be such an enormous change in the quality of the predictor?" Different models, as @Penguin_Knight pointed out: & perhaps different data-sets; as noted above coercing to numeric may well produce missing values, & the whole row may be excluded (by default) when fitting the model. If the former alone it may be an indication that a linear relationship between the predictor & response doesn't fit well. (3) "Can I justify using them (and other columns) as a factor?" Possibly—depending on what the numbers represent & on what you want to do with the model. There are many ways to include any kind of variable in a model. E.g. suppose a predictor takes 10 distinct values in your data-set: it may be the case that a linear relationship between the predictor & response isn't a sensible assumption, yet you still want to be able to make predictions for values that aren't found in the data-set & don't want to spend 9 degrees of freedom modelling the relationship; so you represent the predictor with a low-order polynomial. Software, even R, can't do your thinking for you. Here's an example using simulated data to explain why the factor model may reduce so much more error: set.seed(50) pred <- rep(1:5,each=10) # create our predictor variable that we will treat as numeric or as a factor val <- c(sample(100,10),sample(200:500,10),sample(100,10),sample(-400:-390,10),sample(100,10)) # create our dependent variable. As you can see, pred appears to predict val although there is no apparent linear relationship between them: > head(cbind(pred,val)) pred val [1,] 1 71 [2,] 1 44 [3,] 1 20 [4,] 1 75 [5,] 1 50 [6,] 1 5 > cbind(pred,val)[11:16,] pred val [1,] 2 317 [2,] 2 280 [3,] 2 391 [4,] 2 223 [5,] 2 282 [6,] 2 400 Now let's compared the $R^{2}$ values for linear models that treat pred as numeric and as factor: > summary(lm(val~pred))$r.squared # numeric predictor [1] 0.1764453 > summary(lm(val~as.factor(pred)))$r.squared # factor predictor [1] 0.9710696 Wow! The our categorical predictor (as.factor(pred)) eats up so much more error! Why? It's a completely different model: lm(val~pred) represents: $Val_{i} = \beta_{0} +\beta_{1} Pred_{i} + \epsilon_{i}$ lm(val~as.factor(pred)) represents: $Val_{i} = \beta_{0} +\beta_{1} Pred_{pred = 2; i}+\beta_{2} Pred_{pred = 3; i} +\beta_{3} Pred_{pred = 4; i}+\beta_{4} Pred_{pred = 5; i}+ \epsilon_{i}$ Our categorical predictors do a much better job!	2020-01-25 16:22:05	{"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": 0.59266597032547, "perplexity": 1484.0128785032305}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1579251678287.60/warc/CC-MAIN-20200125161753-20200125190753-00261.warc.gz"}
https://www.albert.io/learn/electricity-and-magnetism/question/zero-potential-points-for-three-charges-on-a-line	Limited access Figure 1 depicts three charges lying along the $\hat{x}$ axis. On the left, at $x = 0$, is a positive charge $q$. In the center, a negative charge, $-2q$ is at the coordinate $x = a$. On the right, a positive charge, $+3q$, is at the coordinate $x = 2a$. It Select Option isis not possible for the electric potential to be zero somewhere along the $\hat{x}$ axis in Region I. It Select Option isis not possible for the electric potential to be zero somewhere along the $\hat{x}$ axis in Region II. It Select Option isis not possible for the electric potential to be zero somewhere along the $\hat{x}$ axis in Region III. It Select Option isis not possible for the electric potential to be zero somewhere along the $\hat{x}$ axis in Region IV. Select an assignment template	2017-03-27 18:14:50	{"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": 0.6410977244377136, "perplexity": 600.3900010320074}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218189495.77/warc/CC-MAIN-20170322212949-00052-ip-10-233-31-227.ec2.internal.warc.gz"}
https://wiki.rtm-lvl.org/su439lcz/viewtopic.php?aa2099=finding-zeros-of-rational-functions-worksheet	Shows resources that can work for all subjects areas, I use Scavenger Hunts to review a lot of different subjects in my classes. Find all x and y intercepts of the function ( ) 2 9 x 1 x f x − = −. In this rational zero theorem worksheet, 11th graders solve and complete 24 various types of problems. This is a more general case of the Integer (Integral) Root Theorem (when leading coefficient is 1 or -1). Cloudflare Ray ID: 5fd36bdc8bfb1a22 This is notes, worksheet and key with 16 problems. After this, it will decide which possible roots are actually the roots. In this worksheet, we will practice finding the set of zeros of a quadratic, cubic, or higher-degree polynomial function. The problems are separated into types as they are on the cards. Thus −2. Domain and range of rational functions with holes. It guides students through 3 scenarios in which students must find the zero, explain what the zero represents, and graph the equation to find the zero that way.Each word problem incorporates a real-world problem that students can relate to, This is notes, worksheet and key with 16 problems. Thus −2. Example 2: Find two consecutive odd integers whose sum is 130. A rational expression is undefined when the denominator is equal to zero. There are 2 versions of dominoes each level. ©2 o2i0 91e2 b jK hu1t PaA GS9oCftmwPaJrpe 7 nLhLfC 6.o z FAGlol e Kroi 3g fhkt rs v BrXehs Tekr RvKe3d W.6 v fMVaXdRe h awigtvhd iI 8n9f Bibn ciRt0e o dAOlrgae qb9r IaL T2F.Z Worksheet by Kuta Software LLC 11) 5, −1, 0 12) −3, − 1 3, 5 13) 5 3, 1, −1 14) 2, 5 3, −5 Find all zeros by factoring each function… In the last section, we learned how to divide polynomials. The first step to working with rational functions is to completely factor the polynomials. This can be used for an abs, This activity is available as a "GOOGLE INTERACTIVE" product HEREThis EMOJI activity is included inEMOJI - BUNDLE Polynomial Functions 50%+ OFF✐ This product is a NO PREP - SELF CHECKING activity that engage students in 12 questions on "Finding the Zeros of Polynomial Functions. b. Finding Rational Zeros USING THERATIONALZEROTHEOREM The polynomial function ƒ(x) = 64x3+ 120x2º 34xº105 has º3 2, º5 4, and 7 8 as its zeros. ... System of Equations System of Inequalities Polynomials Rationales Coordinate Geometry Complex Numbers Polar/Cartesian Functions Arithmetic & Comp. Example: 4 2 3 5 8 2 7 x x x x − − Is a rational expression because the top and bottom are polynomials. Let’s walk through the proof of the theorem. First, they list all of the possible rational zeros of each function. Includes both teacher and student instructions, 2 student help pages, a student answer sheet and an answer key. L.C.M method to solve time and work problems Example: Find all the zeros or roots of the given function. Phishing is an attempt to acquire personal information such as passwords and credit card details by pretending to be a trustworthy source. Close navigation. Free trial available at KutaSoftware.com. 6 Find Rational Zeros. Meets CCSS: A.AP EXAMPLE: The zeros of the function h(x) described above would be found by setting the NUMERATOR equal to zero. If a function is even or odd, then half of the function can be If you're the owner of this website Once they are done, I have the class walk around the room and exchan, Your students will enjoy practicing finding zeros (x-intercepts) of linear, quadratic, cubic functions. Vertical Asymptotes Questions contain using the Rational Zeros Theorem, finding rational zeros, upper and lower bounds, and using Descartes Rule of Signs. Performance & security by Cloudflare. If you're a visitor of this website The answers include rational, irrational, and complex roots. Browse rational zeros resources on Teachers Pay Teachers, a marketplace trusted by millions of teachers for original educational resources. First, they list all of the possible rational zeros of each function. The zeros of fx() are the zeros of these factors: 2, 1 4, - 1 Observe that all three are rational and appeared in our list of candidates for rational zeros. Zeros can be rational, irrational, and or/ complex. To find the domain (“good values of x”), I know that it is allowable to take the square root of either zero or any positive number. If you have questions about why this was flagged as phishing please contact the Trust & Safety team for more information. This is notes, worksheet and key with 16 problems. Cubic Functions (uses f(x) = ) UNIT 4 WORKSHEET 14 Finding Intercepts of Rational Functions We have found that the zeros of the denominator of a rational function are the vertical asymptotes of the function. The calculator will find all possible rational roots of the polynomial, using the Rational Zeros Theorem. • 4 Zeros of Polynomials Functions Date_ State the possible rational zeros for 5. • To find the values that make a rational expression undefined, set the denominator equal to zero and solve the resulting equation. Displaying top 8 worksheets found for - Finding Zeros Of A Polynomial Function. Finding a Percent of a Number. Free trial available at KutaSoftware.com. For Algebra 2 or PreCalculus b. This highly engaging, hands-on activity will be a favorite for your students. This is a four-page tutorial that walks through synthetic division and the notion that a remainder of zero means you have found a zero/factor of the original polynomial. finding zeros of polynomial functions using synthetic division worksheet, 2.3 Day 1 Real Zeros of Polynomial Functions • We will use synthetic division to divide a polynomial •We will see if a given value of x is a zero of a function •We will use the remainder theorem. Determine if the numbers are zeros of the function: 4. Find the Zeros of a Polynomial Function with Irrational Zeros This video provides an example of how to find the zeros of a degree 3 polynomial function with the help of a graph of the function. Teachers Pay Teachers is an online marketplace where teachers buy and sell original educational materials. Please rate this activity. There are 14 questions. Author: Triszan . The zeros of the numerator on the other hand, are the x intercepts of the function. The Rational Zero Theorem helps us to narrow down the number of possible rational zeros using the ratio of the factors of the constant term and factors of the leading coefficient of the polynomial Consider a quadratic function with two zeros, \displaystyle x=\frac {2} {5} x = 5 Characteristic 2 1 7 x x y − − = 2 2 5 7 10 + + + = x x x x y 2 2 7 12 9 − + − = x x x y •Given a factor we will find the remaining factors and write the function in its factored form….list the zeros. This lesson demonstrates how to locate the zeros of a rational function. This website uses cookies to ensure you get the best experience. Notice that the numerators of these zeros (º3, º5, and 7) are factors of the constant term, º105. Finding square root using long division. The problems begin with quadratic functions and progress to cubic, quartic, and quintic functions. Zeroes of functions will be the subject of these interactive study assessments. Find the polynomial with the following zeros: i, 1 + i, 2 Factor: 7. x4 + 6x3 + 17x2 + 36x + 66 Find the x and y intercepts, find … Create your own worksheets like this one with Infinite Algebra 2. Recall that the Division Algorithm states that given a polynomial dividend f(x) and a non-zero polynomial divisor d(x) where the degree of d(x) is less than or equal to the degree of f(x), there exist uni… Graphing rational functions. UNIT 4 WORKSHEET 14 Finding Intercepts of Rational Functions We have found that the zeros of the denominator of a rational function are the vertical asymptotes of the function. Please log in to cloudflare.com to review your flagged website. A function will have one and only one zero. Quiz & Worksheet Goals. 3, 2, ½; f(x) = 3x3 + 11x2 – 2x + 8 5. From start to end, the student will be able to answer 14 questions out of the 17 provided to get to the end of th, Use these HIGH-INTEREST Task Cards to allow your students to gain deeper understanding of FINDING the SLOPE of each ski picture (POSITIVE, NEGATIVE, ZERO, and UNDEFINED). The zeros of the numerator on the other hand, are the x intercepts of the function. Polynomials are all of degree 3 to 6. Some can be factored, bu, This is a set of 2 foldables designed for interactive math notebooks!The first foldable is a basic fold for four tabs. Once you find your worksheet, click on pop-out icon or print icon to worksheet … Here is a set of practice problems to accompany the Finding Zeroes of Polynomials section of the Polynomial Functions chapter of the notes for Paul Dawkins Algebra course at Lamar University. It covers the rational root theorem, with some worked examples and problems for student practice. ☑ 2017 Google Interactive Membership {GROWING BUNDLE} - Current & Future Products Q1: Find the value of given ( ) = 1 4 2 5 + 6 0 + 3 6 where ( ) is undefined. For now, it is recommended that you do not continue to the link that has been flagged. Title: Graphing Rational Functions.ks-ia2 There are 4 pages for this sort: In this rational zero theorem worksheet, 11th graders solve and complete 24 various types of problems. I value your feedback. These 12 quest, This instrument addresses the concept of finding the zeros of polynomial functions. Any one of these three could have been used to start cracking the problem. Each is a polynomial of degree 3 or 4. Here is a set of practice problems to accompany the Zeroes/Roots of Polynomials section of the Polynomial Functions chapter of the notes for Paul Dawkins Algebra course at Lamar University. A rational function is a fraction of polynomials. f(x) = x 3 - 4x 2 - 11x + 2 PREC12 Rational Functions Name: _____ Worksheet ANSWER KEY Analyze each function and predict the location of any VERTICAL asymptotes, HORIZONTAL asymptotes, HOLES (points of discontinuity), x- and y-INTERCEPTS, DOMAIN, and RANGE. From there, you will need to find the rational zeros of the functions. Quadratic Equations (uses y=) Follow me to receive updates, ENJOY THIS FREEBIE - Then check out my Find a Buddy Reviews for ALL of the Common Core 4th Grade Standards! If the polynomial is divided by x – k, the remainder may be found quickly by evaluating the polynomial function at k, that is, f(k). Click here! The function as 1 real rational zero and 2 irrational zeros. The numerator is p(x)andthedenominator is q(x). Examples. Zeros Calculator The calculator will find zeros (exact and numerical, real and complex) of the linear, quadratic, cubic, quartic, polynomial, rational, irrational, exponential, logarithmic, trigonometric, hyperbolic, and absolute value function on the given interval. Find the zeros of the function f(x) = 6x2 19x 36. Then, students find all the rational zeros of the functions given. Perf, »»-------------¤-------------««»»-------------¤-------------««Have you seen the video of this type of activity??? Displaying top 8 worksheets found for - Finding Zeros Of A Polynomial Function. Finding the Zeros of a Rational Function Learn this and go on to Solving Equations with Fractional Expressions 6 Tutorials That Teach Finding the Zeros of a Rational Function Take Your Pick: Finding the Zeros of a Rational Function. Q1: Find, by factoring, the zeros of the function ( ) = + 2 − 3 5 . In this worksheet, we will practice finding the domain and range of a rational function either from its graph or its defining rule. Find all the possible zeros of the function: f(x) = 4x5 + 3x3 – 2x + 12 6. Displaying top 8 worksheets found for - Rational Zeros Theorem. About This Quiz & Worksheet. This was made for Secondary Math 3 Honors and can be used for Algebra 2, and Pre-Calculus etc. Free Rational Roots Calculator - find roots of polynomials using the rational roots theorem step-by-step. This link has been flagged as phishing. Converting repeating decimals in to fractions. Domain and range of rational functions. Then, students find all the rational zeros of the functions given. We no longer have to rely on rational zeros at this point. This ❤ Google Interactive Activity ❤ is an engaging practice of finding "The Zeros and Multiplicities of Polynomial Functions" . I would let the expression under the radical, x-2, greater than or equal to zero; and then solve the inequality. What better way for students to review subtraction across zeros at the end of the unit or before testing season than with an interactive lesson that allows students to teach one. Meets CCSS: A.AP, This lesson on Zeros of Polynomials, often found in Unit 2, includes:Guided Notes on techniques for finding zeros of Polynomials32 Task Cards Master List of questions in worksheet format for your convenience. PREC12 Rational Functions Name: _____ Worksheet ANSWER KEY Analyze each function and predict the location of any VERTICAL asymptotes, HORIZONTAL asymptotes, HOLES (points of discontinuity), x- and y-INTERCEPTS, DOMAIN, and RANGE. Students must hunt for their answer(s) in order to advance in this circuit. Finding All Zeros of A Polynomial Function Scavenger Hunt, Algebra 1 - Factoring Using the Distributive Property & Finding Zeroes Foldables, Algebra 1 - Factoring Bundle - Foldables, FlipBooks, Task Cards, Puzzles, Find Zeros & Multiplicities of Polynomial Functions -Google Interactive: Maze, DISTANCE LEARNING: ALGEBRA II TOP 20 INTERACTIVE ACTIVITIES, Using Graphing Calculator~Find Zeros~Polynomials~Graph~Sorting Activity, Growing Bundle~All my Algebra Activities~Quadratics~Linear~Roots~Absolute Value, Maze - Find the Zeros of a Polynomial Functions (Level 5) - 2 OPTIONS, Maze - BUNDLE ALL ABOUT Polynomial Functions, Zero by Kathryn Otoshi: finding value in yourself, Circuit Training - Finding Zeros of Functions (algebra), Virge Cornelius' Mathematical Circuit Training, Find Zeros of Polynomial Functions - Google Interactive: Maze, Algebra 1 - Solving Linear Equations by Graphing and Finding Zeros Foldable, Algebra 1 - Linear Functions Foldables & Task Cards - BUNDLE Part 1, Rational Zeros Theorem, Finding Rational Zeros, Bounds, Descartes Rule of Signs, Finding Zeros of Polynomials Guided Notes Task Cards \| Distance Learning, Polynomial and Rational Functions Unit 2 PreCalculus Bundle Distance Learning, EMOJI - Find Zeros of Polynomial Functions (Factored & Not Factored), Maze - Find the Zeros of a Polynomial Functions (Level 2 - EASY), Finding SLOPE OF A LINE Positive Negative Zero Undefined 30 Task Cards Linear, TYPES OF SLOPE BUNDLE Positive Negative Zero Undefined LINEAR EQUATIONS, Finding all Complex Zeros - Polynomials - Around the Room Activity, Maze - Find Zeros and Write Polynomial Functions (EASY), Graphing Calculator - Finding the zeros of quadratic functions, EMOJI - Find Zeros of Polynomial Functions (Google & Hard Copy)Distance Learning, Google Drive BUNDLE: POLYNOMIAL FUNCTIONS Distance Learning, Finding Cubic & Quadratic Equations given ZEROS Dominoes/Sort, Solving Quadratic Equations Bundle~Completing the Square~Calculator, BUNDLE FINDING SLOPE Negative Positive Undefined Zero Differentiated Linear, 4th Grade Common Core Subtraction Across Zeros (Find a Buddy), Algebra 2 Tutorial & Worksheets: Finding Zeros of Polynomials. I used this as an around the room activity, but would not suggest it as a scavenger hunt (Due to the # of solutions or knowing if there are a certain number of real vs imaginary, students might just look for those solutions inste, This maze is part of : Maze - BUNDLE Polynomial FunctionsThis Maze focuses on Two Type of Questions::☑ Find the Zeros of a given Polynomial Function (See Questions Type Below)☑ Write a Polynomial Function with the given Zeros (See Questions Type Below)I use this maze when I first introduce the conc, I use this activity to show the students how to find the zeros of a quadratic function using a graphing calculator. Right HEREThis EMOJI Activity is part of:☑ 2017 Google Interactive Membership {GROWING BUNDLE} - Current & Future Products☑ Google Drive BUNDLE: POLYNOMIAL FUNCTIONSThis ❤ Google Activity ❤ is an engaging practice of "Finding Zeros of Polynomial Func, You will receive 4 sets of dominoes/sorts if you have to), although its zeros may or may not be real. 6 Find Rational Zeros. Students are shown how to factor the GCF out of binomials, trinomials, and then shown how to factor by grouping.The second foldable is a flipbook in which students will see the pro, This MAZE Activity is part of: Then find all rational zeros. Look for other topics! Version 1: students start with zeros of a function then rewrite them as factors and multiply to find the f, Use these HIGH-INTEREST Task Cards to allow your students to gain deeper understanding of FINDING SLOPE by identifying the slope of each ski picture (POSITIVE, NEGATIVE, ZERO, and UNDEFINED). I included pictures of what the students' screen should look like on the graphing calculator when hitting certain keys.There are also 3 examples for the students to try after doing th, Check out: How to Color an Emoji Digitally? finding zeros of polynomial functions using synthetic division worksheet, 2.3 Day 1 Real Zeros of Polynomial Functions • We will use synthetic division to divide a polynomial •We will see if a given value of x is a zero of a function •We will use the remainder theorem. When a rational function is equal to zero (that is, its output is equal to zero) then its NUMERATOR is equal to zero. Some of the worksheets for this concept are State the possible rational zeros for each, Rational roots theorem and factoringsolving 3, The rational zero theorem, Rational root theorem work, Rational root theorem work, The remainder and factor synthetic division, Finding rational zeros, The fundamental theorem of algebra date period. This one reviews finding all the zeros (roots) of a polynomial function. Students should use not only graphs, but synthetic division, long division, synthetic substitution, and the quadratic, This maze is part of : Maze - BUNDLE Polynomial FunctionsThis activity is a good review of understanding how to "Find the zeros of Polynomial Functions". An exte, Students will solve for all zeros of given polynomials to get thru a maze. Questions contain using the Rational Zeros Theorem, finding rational zeros, upper and lower bounds, and using Descartes Rule of Signs. We explain Finding the Zeros of a Rational Function with video tutorials and quizzes, using our Many Ways(TM) approach from multiple teachers. My plan is to find all the values of x satisfying that condition. Showing top 8 worksheets in the category - Rational Zero Theorem. To find the y-intercept(s) (the point where the graph crosses the y-axis), substitute in 0 for x and solve for y or f(x). Find all x and y intercepts of the function ( ) 2 9 x 1 x f x − = −. There are 17 questions provided. From start to end, the student will be able to answer 14 questions out of the 17 provided to get to the end of the maze.The polynomial functions are:☑ SOME Polynomial Fu, I do this activity with 2nd graders during our Relationship unit when we talk about having a relationship with yourself. UNIT 4 WORKSHEET 12 Finding Asymptotes of Rational Functions Rational functions have various asymptotes. check out the preview»»-------------¤-------------««»»-------------¤-------------««This MAZE Activity is part of:☑ 2017 Google Interactive Membership {GROWING BUNDLE} - Current & Fu, This is a foldable designed for interactive math notebooks. Title: Rational Root Theorem (3x − 1)(5x + 1) = 0. com Find all zeros by factoring each function. Characteristic 2 … Questions contain using the Rational Zeros Theorem, finding rational zeros, upper and lower bounds, and using Descartes Rule of Signs. Look for other topics! Graphing rational functions with holes. Finding zeros of polynomials (1 of 2) This is the currently selected item. This was made for Secondary Math 3 Honors and can be used for … That is, if p(x)andq(x) are polynomials, then p(x) q(x) is a rational function. Example 2: Find two consecutive odd integers whose sum is 130. Advanced algebra worksheet finding all roots, scientific notation worksheet, step by step to do factoring linear equations with fractions, algebra 1 prentice hall teacher. That is, 3x - 6 = 0. Create your own worksheets like this one with Infinite Algebra 2. linalg imports most of them, identically named functions from scipy. Also notice that the denominators (2, 4, and 8) are factors of the leading coefficient, 64. Conic Sections Trigonometry. Here is a set of practice problems to accompany the Finding Zeroes of Polynomials section of the Polynomial Functions chapter of the notes for Paul Dawkins Algebra course at Lamar University. This was made for Secondary Math 3 Honors and can be used for Algebra 2, and Pre-Calculus etc. We can now use polynomial division to evaluate polynomials using the Remainder Theorem. ~Equations (Degree = 2, 3, 4, This activity is a good review of understanding how to "Find the zeros of Polynomial Functions". Write the rational expression in lowest terms: 6 15 2 2 10 + + − x x x Also included in: Algebra 1 - Factoring Bundle - Foldables, FlipBooks, Task Cards, Puzzles, Also included in: DISTANCE LEARNING: ALGEBRA II TOP 20 INTERACTIVE ACTIVITIES, Also included in: Growing Bundle~All my Algebra Activities~Quadratics~Linear~Roots~Absolute Value, Also included in: Maze - BUNDLE ALL ABOUT Polynomial Functions, Also included in: Algebra 1 - Linear Functions Foldables & Task Cards - BUNDLE Part 1, Also included in: Polynomial and Rational Functions Unit 2 PreCalculus Bundle Distance Learning, Also included in: EMOJI - BUNDLE Polynomial Functions, Also included in: TYPES OF SLOPE BUNDLE Positive Negative Zero Undefined LINEAR EQUATIONS, Also included in: Google Drive BUNDLE: POLYNOMIAL FUNCTIONS Distance Learning, Also included in: Solving Quadratic Equations Bundle~Completing the Square~Calculator. Decimal representation of rational numbers. The website owner has been notified and is in the process of resolving the issue. ☑ Polynomials are Already Factored, This activity will help students use graphing calculators to find the zeros of polynomials. The circuit presents 24 problems in a progressive manner (the exercises become tougher as the students advance). One the front page, I have each student write 3-5 qualities about themselves that describe them in the coin. Views: 11218 Rating: (28) Finding the Zeros of a Rational Function. Your IP: 203.175.168.94 An extension activity can, I’m so excited for you to get to use this activity with your students. Are you getting the free resources, updates, and special offers we send out every week in our teacher newsletter? There are 2 levels of dominoes. Exponential functions in the form y = ab 2 will not have a zero. • 3(x5) (x1) • 1 x • 2x 3 1 =2x 3 The last example is both a polynomial and a rational function. Some of the worksheets displayed are State the possible rational zeros for each, The rational zero theorem, Rational roots theorem and factoringsolving 3, The remainder and factor synthetic division, Rational root theorem please do all work on a, Zeros of a polynomial function, Finding rational zeros, Section finding zeros of polynomial functions. Steps are available. The zeros of a function are where the graph crosses the x axis. Once in factored form, find all zeros. There are 17 questions provided. Exercise Set 2.3: Rational Functions MATH 1330 Precalculus 229 Recall from Section 1.2 that an even function is symmetric with respect to the y-axis, and an odd function is symmetric with respect to the origin. Math 51 Worksheet Rational Expressions A rational expression is an expression of the form , Q p where P and Q are polynomials, with Q≠ 0. In this model: Finding Intercepts of Rational Fractions Intercepts are the points at which a graph crosses either the x or y axis, and they are very useful in sketching functions. •Given a factor we will find the remaining factors and write the function in its factored form….list the zeros. ## finding zeros of rational functions worksheet Dragon Nest Dps Tier List 2020, Asparagus Casserole With Eggs And Cream Of Mushroom Soup, Yukon Gold Potatoes Nz, Thank You Letter For Support And Cooperation, Carew And Co Price List, Be Not Afraid Dufford Sheet Music, Are Sand Tiger Sharks Dangerous, Building Cost Estimator Bc, Razorback Musk Turtle Habitat, Xero Share Price, Kanoria College News,	2021-10-22 18:23:48	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3333909511566162, "perplexity": 1109.6603862889533}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585518.54/warc/CC-MAIN-20211022181017-20211022211017-00486.warc.gz"}
https://dsp.stackexchange.com/questions/42028/how-to-apply-statistical-algorithms-of-signal-processing-to-regulate-variation-o/43952	# How to Apply Statistical Algorithms of Signal Processing to Regulate Variation of a Curve? Below I am posting 2 graphs. I want to regulate the curvature of first graph using some statistical methods such as use of standard deviations, and modulate my graph to look like second one. I am not asking here for any matlab command, but for some statistical method or a signal processing algorithm I can implement using MATLAB. I just want transformation of shape as that of image 2, I do not want to achieve exact values of image 2. Thanks & Regards, Nick. . • Please make sure that the images/links work. – Tolga Birdal Jun 27 '17 at 14:05 • Welcome to SE.DSP! Please edit your question to add the graphs... otherwise your question doesn't make much sense. Closing until you edit. – Peter K. Jun 27 '17 at 14:55 • Oops...! the links were not working. They are now working now! Please answer if possible. Thanks for drwaing attention, @TolgaBirdal and Peter K. – Nick Jun 27 '17 at 15:57 • Are you asking for this specific case? If not, there are 3 variables: a.the input graph, b.the wanted transformation and c.the output graph, if you don't define accurately two of them I am afraid there will be no correct answer... – oxuf Jun 27 '17 at 20:06 • I just want transformation of shape as that of image 2, I do not want to achieve exact values of image 2.@oxuf – Nick Jun 28 '17 at 7:40 You can regulate its second derivative which is the curvature. Something like: $$\hat{x} = \arg \min_{x} \frac{1}{2} \left\\| x - y \right\\|_{2}^{2} + \frac{\lambda}{2} \left\\| D D x \right\\|_{2}^{2}$$ Where $y$ is samples you have and $D$ is the Derivative Operator (In Matrix Form). By applying it twice we're regulating the Second Derivative. You can find a statistical explanation for this (Prior of Gaussian Distribution of the Second Derivative) or just call is Tikhonov Regularization. This has a closed form solution: $$\hat{x} = \left( I + \lambda {D}^{T} {D}^{T} D D \right)^{-1} y$$ You can use some fitting method. For example least - squares For example you can transform you image2 by the linear transformations - translation, stretching, scaling; to minimize the sum of squared deviations of the Y values of the transformed image2 from image1. • Can you please elaborate this a bit more specifically with some examples? Also, please guide me how to implement them using MATLAB. – Nick Jun 28 '17 at 9:50 • Well, it is not a simple task. There are many ways to do it. The linear transformation for 2D shape can be not very simple subject, so I would first try to fit your second figure with polynomial (for example), then fit the linear transformation of obtained polynomial to the first figure. You can find some information and examples on curve fitting in matlab here: nonlinear-least-squares-curve-fitting [curvefit]mathworks.com/help/curvefit/curve-fitting.html) and so on... – Andrei Keino Jun 28 '17 at 11:58 Curvature of the signal is decided by the second and higher derivatives. It is possible to smoothen the curves using the idea of low-pass filter. Large curvature results from Fourier components having higher frequencies. There are two approaches of implementing a low pass filter. 1. After deciding the degree of smoothness, fix a low-pass filter response. Then using standard FIR (or IIR) filter design methods design the filter coefficients of required degree. 2. If the signal is finite duration one, you can use Fourier series least squares curve fitting upto a certain order. This would constraint the curvature of your signal to the required smoothness. The smoothened signal will have the form : $$f(t) = \sum_{k=-L}^{L} a[k] e^{j 2 \pi kt},$$ assuming the signal support as $T =1$sec. MATLAB allows you to perform Fourier series fitting upto a degree of 8. -- see curve fitting toolbox.	2020-01-26 12:47:54	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.6032376289367676, "perplexity": 685.4908295384379}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1579251688806.91/warc/CC-MAIN-20200126104828-20200126134828-00296.warc.gz"}
https://www.physicsforums.com/threads/a-short-proof-of-birkhoffs-theorem.768032/	# A Short Proof of Birkhoff's Theorem 1. Aug 28, 2014 ### Staff: Mentor [This is the first of a number of posts from my PF blog that I will be re-posting in the forums.] The subject of Birkhoff's theorem has come up in a number of PF threads, and in the course of researching it for one of them I realized that MTW's statement of the proof does something that, strictly speaking, is not mathematically correct. (In fact, this something is not limited to their proof of Birkhoff's theorem; they do it throughout the book.) MTW write the general line element for a spherically symmetric spacetime as follows: $$ds^2 = - e^{2 \Phi} dt^2 + e^{2 \Lambda} dr^2 + r^2 d\Omega^2$$ where $\Phi$ and $\Lambda$ are, in general, functions of both $r$ and $t$, and $d\Omega^2$ is the standard metric on a unit 2-sphere. MTW note carefully that, to be fully general, the signs of the $dt^2$ and $dr^2$ terms cannot be restricted; but the way the metric is written, with exponentials in those coefficients, the signs are restricted, because all of the functions involved are real-valued. What I'm going to do in this post is re-do the proof that MTW give of Birkhoff's theorem, but without this deficiency. Doing this is simple; we start by re-writing the above line element in a way that clearly sets no limitation on the signs of the $dt^2$ and $dr^2$ terms: $$ds^2 = j(r, t') dt'^2 + k(r, t') dr^2 + r^2 d\Omega^2$$ (I have used $t'$ as the first coordinate for reasons which will become apparent below.) Similar to the above, the first two coefficients, $j$ and $k$, can, in general, be functions of both $r$ and $t'$. MTW go into some detail in showing how any spherically symmetric spacetime can be described by a line element of this form, and I'm not going to redo any of that, but just take it as established. The thing to do now is to compute the Einstein tensor of the above metric and apply the vacuum Einstein Field Equation; it will turn out that we only need to look at the components that involve $t'$ and $r$: $$G^{t'}{}_{t'} = \frac{r \partial k / \partial r + k^2 - k}{k^2 r^2} = 0$$ $$G^{t'}{}_r = \frac{\partial k / \partial t'}{k^2 r} = 0$$ $$G^r{}_{t'} = \frac{\partial k / \partial t'}{j k r} = 0$$ $$G^r{}_r = \frac{r \partial j / \partial r - j k + j}{j k r^2} = 0$$ The second and third equations show that $k$ is a function of $r$ only. Given that, the first equation can be solved for $k$; the easiest way to do it, based on the principle that when working the solution of a problem, it helps to already know the answer, is to try the ansatz $k = r / ( r - 2m )$, where $m$ is a constant, and find that it solves the equation. So we have now shown that our metric can be written: $$ds^2 = j(r, t') dt'^2 + \frac{dr^2}{1 - 2m / r} + r^2 d\Omega^2$$ where now we only have one undetermined function left, $j$. The easiest way to solve for it is to look at the $G^r{}_r$ equation. (We could also look at the $G^\theta{}_\theta$ or $G^\phi{}_\phi$ equations, which are both identical, but they are also considerably more complicated, and contain no additional information.) That equation now looks like this: $$G^r{}_r = \frac{\partial j / \partial r ( r^2 - 2 m r) - 2 j m}{j r^3} = 0$$ Once again, it helps to know the answer; the ansatz $j = - ( 1 - 2m / r) f(t')$ solves the above equation, where $f(t')$ is an arbitrary function of $t'$ (though it should be noted that it must be positive, i.e., $f(t') > 0$ must hold for all $t'$). So now we have shown that our metric can be written in this form: $$ds^2 = - \left( 1 - \frac{2m}{r} \right) f(t') dt'^2 + \frac{dr^2}{1 - 2m / r} + r^2 d\Omega^2$$ (Btw, you may have noticed that the leading minus sign in the ansatz for $j$ above could have been eliminated, and it would still solve the $G^r{}_r$ equation. However, since the line element as a whole has to be Lorentzian, the signs of $g_{tt}$ and $g_{rr}$ must be opposite, and that requires the minus sign in front of the expression $( 1 - 2m / r )$ in $g_{tt}$. Note that there is no similar freedom in choosing the sign in front of the expression $( 1 - 2m / r )$ in $g_{rr}$; only the positive sign, as given here, gives a valid solution of the $G^{t'}_{t'}$ equation shown above.) The final step is to deal with that arbitrary function of $t'$ in the first coefficient. None of the components of the EFE constrain it at all; but what that actually means is that we can re-scale the time coordinate however we want to; in particular, we can adopt a new time coordinate $t$ given by $$dt = \sqrt{f(t')} dt'$$ (The fact that we take a square root here is why the function $f(t')$ must be positive--more precisely, that plus the fact that we must have $dt$ nonzero whenever $dt'$ is nonzero.) With this change of coordinates, we now have the line element in the standard Schwarzschild form, which completes the proof of Birkhoff's theorem: $$ds^2 = - \left( 1 - \frac{2m}{r} \right) dt^2 + \frac{dr^2}{1 - 2m / r} + r^2 d\Omega^2$$ So we have shown that we can write the line element in a form that is entirely independent of the $t$ coordinate. In other words, we have shown that any spherically symmetric, vacuum spacetime must have an extra Killing vector field, $\partial / \partial t$, over and above the three KVFs that it has by virtue of spherical symmetry. But it is important to stress that we have not shown that the coordinate $t$ is a "time" coordinate; there is nothing in the above that requires $t$ to be timelike. The above derivation is valid for any value of $r$ except $r = 0$ (which won't work because $r$ appears in the denominator of the EFE components) and $r = 2m$ (because the solution for $g_{rr}$ becomes singular there). In particular, it is valid for $r < 2m$, and when $r < 2m$, the signs of $g_{tt}$ and $g_{rr}$ switch, so the $t$ coordinate is not timelike there, even though $\partial / \partial t$ is still a Killing vector field (because the metric is still independent of $t$).	2018-05-27 05:40:32	{"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": 0.9126496911048889, "perplexity": 130.5986922548231}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1526794868003.97/warc/CC-MAIN-20180527044401-20180527064401-00486.warc.gz"}
https://www.infoq.com/news/2020/01/crypto32-vulnerability-microsoft/	InfoQ Homepage News Microsoft Patches Severe Crypto32.dll Vulnerability # Microsoft Patches Severe Crypto32.dll Vulnerability This item in japanese Microsoft has released patches for various versions of Windows 10 and Windows Server 2019 and 2016 to fix a severe vulnerability affecting system validation of Elliptic Curve Cryptography (ECC) certificates. This vulnerability enables an attacker to spoof the validity of a certificate chain and signature validation and requires prompt patching. The vulnerability was discovered by NSA (PDF), which has strongly warned of the risks associated with not patching all affected systems as soon as possible. The consequences of not patching the vulnerability are severe and widespread. Remote exploitation tools will likely be made quickly and widely available. The vulnerability affects signature validation in HTTPS connections, mails and files, and executables downloaded from the Internet. This means a malicious site, file, or executable may appear to be signed by a legitimate entity. NSA has also provided a number of suggestions as to how to patch systems and how to use certutil and openssl to identify suspicious certificates that may be installed on a system. Certificates containing explicitly-defined elliptic curve parameters which only partially match a standard curve are suspicious, especially if they include the public key for a trusted certificate, and may represent bona fide exploitation attempts. This is extremely important if you are late in installing the Microsoft patches, leaving your system vulnerable for too long. In this case, NSA instructions are useful to ensure no non-legit certificates have been added to your certificate cache while your system was unpatched. According to Microsoft, no exploits in the wild are known, but a number of independent security practitioners have quickly released proofs of concepts (POCs) for an attack that exploites this vulnerability. As Yolan Romailler, security engineer at Swiss firm Kudelski Security, explains, a way to exploit the vulnerability is to use an X.509 certificate generated with "explicit parameters". The problem here is really that the CA certificate cache used by the CryptoAPI is falsely considering that a modified root CA is in the CA certificate store as soon as its public key and serial number match a certificate that is already in the certificate cache, ignoring the fact that this modified certificate is not using the same curve parameters as the one in its cache. Another useful, high-level explanation is provided by Twitter user Cem Paya, who speculates that Crypto32.dll implemented support for "user-defined" elliptic curves in a "lazy" way: It failed to check that all curve parameters are identical to the known curve. In particular, switching the generator point results in a different curve in which an attacker can forge signatures that match a victim public key. An comprehensive discussion of the Crypto32 exploit, along with some interesting mathematical background, can be found on Trail of Bits website. While it is true this vulnerability requires updating all affected Windows systems as soon as possible, Kudelsky attempts to better frame the context in which it could be exploited: In the end, please keep in mind that such a vulnerability is not at risk of being exploited by script kiddies or ransomware. [...] you would need to face an adversary that owns the network on which you operate, which is possible for nation-state adversaries, but less so for a script kiddie. This also has an interesting corollary that makes NSA decision to disclose the vulnerability more understandable: This is also probably why the NSA decided not to weaponize their finding, but to rather disclose it: for them it is best to have the USA patched rather than to keep it and take the risk of it being used against the USA, as the attack surface is so vast. Although the bug resided in Windows Crypto32.dll library, which was introduced with Windows NT 4.0, the vulnerability lies with the more advanced ECC features described above which are seemingly not supported on Windows 7, Windows 8.1, and other Windows versions other than those listed in Microsoft security advisory. Style ## Hello stranger! You need to Register an InfoQ account or or login to post comments. But there's so much more behind being registered. Get the most out of the InfoQ experience. Allowed html: a,b,br,blockquote,i,li,pre,u,ul,p Allowed html: a,b,br,blockquote,i,li,pre,u,ul,p Allowed html: a,b,br,blockquote,i,li,pre,u,ul,p Is your profile up-to-date? Please take a moment to review and update. Note: If updating/changing your email, a validation request will be sent Company name: Company role: Company size: Country/Zone: State/Province/Region: You will be sent an email to validate the new email address. This pop-up will close itself in a few moments.	2020-04-04 19:26:20	{"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": 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": 0.2676770091056824, "perplexity": 2886.499226265088}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1585370524604.46/warc/CC-MAIN-20200404165658-20200404195658-00440.warc.gz"}
http://tex.stackexchange.com/questions/82742/miktex-2-9-64-bit-not-loading-sans-serif-font-for-moderncv	I just got a new computer and installed MikTex 2.9 for Windows7 64-bit. When I compiled my resume for the first time (moderncv, theme casual), it compiled in roman vs. sans serif fonts. I did not specify the roman argument, so the document should default to sans serif fonts. Why can't I get sans serif fonts? I hate roman. Example of file: \documentclass[10pt,letter]{moderncv} \moderncvtheme[green]{casual} \usepackage[utf8]{inputenc} \usepackage{fontenc} \usepackage[scale=0.835]{geometry} \AtBeginDocument{\recomputelengths} \firstname{Fname} \familyname{Lnmae} \mobile{999 999 9999} \email{person@email.com} \nopagenumbers{} \hyphenpenalty=100000 \begin{document} \maketitle \vspace{-.5in} \section{Objective} \cvline{}{Hire Me!} \section{Experience} \cventry{Summer 2009}{Trading Post Manager}{Company Name}{Somewhere, NC}{}{Implemented ECR system to manage and report inventory of 350+ items; managed daily operations.} \end{document} Excerpt from \listfiles: File List moderncv.cls 2012/10/31 v1.2.0 modern curriculum vitae and letter document c lass size10.clo 2007/10/19 v1.4h Standard LaTeX file (size option) [...] fontenc.sty t1enc.def 2005/09/27 v1.99g Standard LaTeX file lmodern.sty 2009/10/30 v1.6 Latin Modern Fonts [...] ot1lmr.fd 2009/10/30 v1.6 Font defs for Latin Modern omllmm.fd 2009/10/30 v1.6 Font defs for Latin Modern omslmsy.fd 2009/10/30 v1.6 Font defs for Latin Modern omxlmex.fd 2009/10/30 v1.6 Font defs for Latin Modern - Welcome to TeX.sx! Is the lmodern package installed? It might be helpful if you include \listfiles at the beginning of your document, recompile, and add the list of files printed in your .log file to your question here by clicking edit. – doncherry Nov 14 '12 at 20:34 Welcome to TeX.sx! Please add a minimal working example (MWE) that illustrates your problem. It is considered a lot better to put in some code that will compile, as it makes it a lot easier for us to copy it into our text editor and work with it, and see exactly what it is you are trying to do. – Martin Schröder Nov 14 '12 at 20:35 @Xavier While it is a good idea to clean up questions and especially the code, in my opinion you went too far in this case – not to mention, that you in your answer refer to code, which you deleted now. Instead you should have shown a good MWE in your answer (there’s a typo now, so you have to edit anyway). Also the output of \listfiles was asked in a comment above. I just made a rollback. – Speravir Nov 14 '12 at 23:17 @Speravir Sorry if I unintentionally broke any rule, though I can't figure out which in your link. I still believe the "MWE" shown is not only far from minimal, but on top shows deprecated code and should therefore be edited. For the listing, one can't edit the MWE without changing the listing, and the only relevant part of the listing is anyway that lmodern.sty was correctly loaded, as mentioned in meryl's comment. Just my opinion, I am still new to tex.sx :) – Xavier Nov 14 '12 at 23:44 @Speravir Since Xavier is the author of the package in question and he also fixed the irrelevant parts of the code I think it's not that dramatic in this very specific problem. – percusse Nov 15 '12 at 1:33 The sans class option, for sans serif fonts, need to be specified explicitly when loading moderncv since version 0.15, i.e. \documenclass[sans]{moderncv} Note also that \moderncvtheme is deprecated, in favor of \moderncvstyle and \moderncvcolor, and that \AtBeginDocument{\recomputelengths} is not required anymore. Your MWE would then look like \documentclass[sans]{moderncv} \moderncvstyle{casual} \firstname{Fname} \familyname{Lname} \mobile{999 999 9999} \email{person@email.com} \begin{document} \maketitle \section{Objective} \cvline{}{Hire Me!} \end{document} - Thanks. The sans option does the trick. Surprised that this JUST starting being a problem for me if that change has been in effect since 0.15. – meryl Nov 14 '12 at 23:54 @meryl Well, it depends what distribution you use and how frequently it updates packages. I had to go check my commits to find out when the behavior changed :) – Xavier Nov 15 '12 at 3:15	2015-07-04 17:29:27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8335118889808655, "perplexity": 3485.2486464797266}, "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-2015-27/segments/1435375096780.24/warc/CC-MAIN-20150627031816-00247-ip-10-179-60-89.ec2.internal.warc.gz"}
https://p-hunermund.com/blog/	# Blog ## A Global Decline in Research Productivity? My coauthor Philipp Böing and I just released a new discussion paper: “A Global Decline in Research Productivity? Evidence from China and Germany” Abstract: In a recent paper, Bloom et al. (2020) find evidence for a substantial decline in research productivity in the U.S. economy during the last 40 years. In this paper, we replicate their findings for China and Germany, using detailed firm-level data spanning three decades. Our results indicate that diminishing returns in idea production are a global phenomenon, not just confined to the U.S. ## Innovation step by step My coauthor Petra Andries (Ghent University) and I just published a new paper in Research Policy: “Firm-level effects of staged investments in innovation: The moderating role of resource availability” (preprint available on this website) ## Control variables in regressions — better don’t report them! A while ago I wrote a short blog post with a pretty simple message: “Don’t Put Too Much Meaning Into Control Variables”. And I must say I was surprised by the many positive responses it got. The respective tweet received more than 1000 likes and nearly 400 retweets. And the blog post even got mentioned in an internal newsletter by the World Bank. So clearly there seems to be some demand for the topic. That’s why my coauthor Beyers Louw (PhD student at Maastricht University) and I decided to turn it into a citable research note, which is now available on arXiv: “On the Nuisance of Control Variables in Regression Analysis” Abstract: Control variables are included in regression analyses to estimate the causal effect of a treatment variable of interest on an outcome. In this note we argue that control variables are unlikely to have a causal interpretation themselves though. We therefore suggest to refrain from discussing their marginal effects in the results sections of empirical research papers. Please use it and save yourself a paragraph or two in your next research paper! :) ## Public procurement as a policy instrument for innovation We have a new paper Public Procurement of Innovation: Evidence from a German Legislative Reform out at IJIO (preprint available without paywall here under “Research”) and I’ve briefly summarized the content in a Twitter thread (apparently that’s were these things happen these days, blogs are so 2012…). For reference, I’ll link to the tweets below: ## Causal Inference in Business Practice – Survey My colleagues and I are currently looking for data scientists to take part in a short survey (5–10 min) on causal inference in business practice. Is data-driven decision making important in your job? Then we’d love to hear your perspective: maastrichtuniversity.eu.qualtrics.com/jfe/form/SV_af ## Mapping Unchartered Territory A frequent point of criticism against Directed Acyclic Graphs is that writing them down for a real-world problem can be a difficult task. There are numerous possible variables to consider and it’s not clear how we can determine all the causal relationships between them. We recently had a Twitter discussion where exactly this argument popped up again. ## PO vs. DAGs – Comments on Guido Imbens’ New Paper Guido Imbens published a new working paper in which he develops a detailed comparison of the potential outcomes framework (PO) and directed acyclic graphs (DAG) for causal inference in econometrics. I really appreciate this paper, because it introduces a broader audience in economics to DAGs and highlights the complementarity of both approaches for applied econometric work. Continue reading PO vs. DAGs – Comments on Guido Imbens’ New Paper ## Causal Data Science in Business A while back I was posting about Facebook’s causal inference group and how causal data science tools slowly find their way from academia into business. Since then I came across many more examples of well-known companies investing in their causal inference (CI) capabilities: Microsoft released its DoWhy library for Python, providing CI tools based on Directed Acylic Graphs (DAGs); I recently met people from IBM Research interested in the topic; Zalando is constantly looking for people to join their CI/ML team; and Lufthansa, Uber, and Lyft have research units working on causal AI applications too. Continue reading Causal Data Science in Business ## Don’t Put Too Much Meaning Into Control Variables I’m currently reading this great paper by Carlos Cinelli and Chad Hazlett: “Making Sense of Sensitivity: Extending Omitted Variable Bias”. They develop a full suite of sensitivity analysis tools for the omitted variable problem in linear regression, which everyone interested in causal inference should have a look at. While kind of a side topic, they make an important point on page 6 (footnote 6): […] since the researcher’s goal is to estimate the causal effect of D on Y , usually Z is required only to, along with X, block the back-door paths from D to Y (Pearl 2009), or equivalently, make the treatment assignment conditionally ignorable. In this case, $\hat{\gamma}$ could reflect not only its causal effect on Y , if any, but also other spurious associations not eliminated by standard assumptions. ## Beyond Curve Fitting Last week I attended the AAAI spring symposium on “Beyond Curve Fitting: Causation, Counterfactuals, and Imagination-based AI”, held at Stanford University. Since Judea Pearl and Dana Mackenzie published “The Book of Why”, the topic of causal inference gains increasing momentum in the machine learning and artificial intelligence community. If we want to build truly intelligent machines, which are able to interact with us in a meaningful way, we have to teach them the concept of causality. Otherwise, our future robots will never be able to understand that forcing the rooster to crow at 3am in the morning won’t make the sun appear. Continue reading Beyond Curve Fitting	2020-07-10 11:56:15	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 1, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.18378864228725433, "perplexity": 2124.8246204010397}, "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-2020-29/segments/1593655908294.32/warc/CC-MAIN-20200710113143-20200710143143-00168.warc.gz"}
https://minimalsurfaces.blog/page/3/	## Derived from Scherk’s Examples During my last semester as an undergraduate student at the Technical University in Berlin in 1984, Dirk Ferus mentioned in his Algebraic Topology class that there would be a geometry conference over the weekend, which he recommended attending. Stupid me, I didn’t go. I could have met my future advisor (Hermann Karcher), and I could have seen a future collaborator (David Hoffman) present the first images of the Costa surface. This conference is also mentioned in the introduction of another paper from my list of highly influential papers with new examples of minimal surfaces, namely Hermann Karcher’s 1988 Embedded Minimal Surfaces Derived from Scherk’s Examples. During the academic year 1984/85, I had attended two semesters of Karcher’s Differential Geometry. At the end of the second term he announced that while the third semester would only be for those specializing in geometry, we all should come for the first two weeks, because he intended to spend them with explaining the basics about minimal surfaces, which he had completely neglected. I was a little disappointed, because I was eager to learn about the darker arts – symmetric spaces, Einstein manifolds, Finiteness Theorems… Karcher didn’t just spend the first two weeks on minimal surfaces, but about half of the semester, developing and presenting what would become the paper mentioned above. The images here represent only a selection of the surfaces described in that paper: There are the saddle towers, the toroidal half plane layers, and the helicoidal saddle towers. Besides all these example Karcher develops a method to derive the complex analytic Enneper-Weierstraß data from geometric features of the surface, which, ultimately, has led to the enormous zoo of examples we are dealing with today. ## Not Just a Special Surface If I had to sum up the content of Hermann Amandus Schwarz’ price winning monograph Bestimmung einer speciellen Minimalfläche from 1867, I would do so using figures from plate VI from the Nachtrag, conveniently compiled in his Collected Works in a single figure: What is shown here are polyhedra whose vertices are the branched values of the Gauß map of five families of triply periodic minimal surfaces that Schwarz is investigating. Schwarz spends most of the over 100 pages discussing a single surface, now called the Diamond or D-surface. It solves the Plateau problem for four consecutive edges of a regular tetrahedron. The details Schwarz provides are overwhelming, and it is easy to overlook that the methods Schwarz develops reach far beyond this special surface, and that he was fully aware of it. What was keeping mathematicians busy these days? Bernhard Riemann had died in 1866 and left a legacy of new concepts and open problems. Complex analysts and geometers were working towards proofs of the Riemann mapping theorem, the uniformization theorem, and the Plateau problem. Schwarz had its own approach: Solve simple cases first, understand them as well as possible, and then apply the developed methods to solve the general case. Both for the Riemann mapping theorem and the Plateau problem, Schwarz looks at polygonal boundaries. He develops the Schwarz-Christoffel formula, and tries something similar for minimal surfaces. Schwarz uses cutting edge technology: The Weierstraß representation for minimal surfaces, the language of Riemann surfaces, and elliptic integrals. He realizes that he can do more than just solve Plateau problems: In addition to straight lines, he can also prescribe symmetry planes. This leads to a differential equation which he can solve if the branched values of the Gauß map are sufficiently symmetric. Competition was fierce, in particular between Göttingen (Riemann and Enneper) and Berlin (Weierstraß and Schwarz). Riemann had left a few pages of notes that hint at what Schwarz discovers. Schwarz must have been shocked when he saw the posthumous paper, with details added by Hattendorf. He also learns that Enneper had used a version of the Weierstraß representation in 1864, maybe without quite grasping its scope, two years before Weierstraß’ note from 1866. It appears that Riemann knew about this, too, as usual. How much did Enneper and Riemann talk in Göttingen? With the exception of Schwarz’ figure 47, representing the H-surface, all vertices are antipodally symmetric. I suspect that Schwarz would have instantly nodded if somebody had told him that his differential equation can be solved just under this symmetry assumption, an observation made by Bill Meeks in his 1975 thesis. How the differently symmetric H-surface fits into the picture, together with other, more recently found surfaces like Alan Schoen’s Gyroid, is one of the big open problems of the area. ## Printing Scherk in Clay Having a virtual repository is great because it is widely available and doesn’t require space. Sometimes, however, one likes to be able to look at something real, so occasionally I will post about actual objects involving minimal surfaces. Two years ago, Malcolm Mobutu Smith and myself set out to make mathematically inspired objects in clay. Malcolm has been intrigued by the relatively new method of clay printing, so he built a small printer, and we got to work. The simplicity is challenging: You have a tube full of clay that is providing a continuous stream of clay (unless there are air pockets in the tube), a little motor that moves the tube around horizontally and vertically (don’t stop, unless you really want a small heap of clay), and a little Arduino to whom you can talk in Gcode. My little mesh.m package has a function that allows to thicken a surface mesh, which can be exported into an stl file. Then we use Slic3R to convert that into Gcode, which is not much more than a bunch of instructions saying “move from A to B in time T.” After that, the printing of a 6-ended singly periodic Scherk surface starts with layers of three arcs. Because the printer doesn’t stop printing, it needs to skip fast between the arcs, leaving behind little charming artifacts. Many things can go wrong: Overhangs can (will) break, the clay dries too fast so that the next layer doesn’t stick, the clay is too soft so that everything sags… But this piece worked out pretty nicely. I now have a real Scherk surface at home: ## Three Ends (Parametrizations I) Most computer algebra systems come with some capabilities to render parametrized surfaces in space. You usually specify three functions of two variables x and y and a rectangle in the (x,y)-plane, and are rewarded with an image. This has limitations: The most complicated topology you can achieve this way is a torus. Things get tricky when you want to draw something that has more than two ends. Besides being able to draw these surfaces at all, one would also like to use a conformal parametrization so that the images of the parameter lines become orthogonal in space. This helps us being illusioned, because, having grown up in environments full of right angles, we assume that any intersection happens at a right angle. This can be accomplished for 3-ended surfaces by moving the ends to -1, 1 and infinity (using a Möbius transformation), dividing the plane into quadrants, and mapping a rectangle to the first quadrant so that we get polar coordinates at 1 and infinity as shown above. This is done using $f(z) = \sqrt{e^z+1}$ on a rectangle of the form [a,b] x [0,π]: The exponential function maps the rectangle to a half-annulus in the upper half plane centered at 0. We then shift the “hole” at 0 to 1 and take a square root which bends the 180º angle at 0 to a right angle. The only thing to remember is that we want to have a parameter line hitting the origin, because otherwise our parameter mesh will have a gap there. This is one of the simpler explicit parametrizations and responsible for the images on this page. ## Scherk’s Fourth Surface In his second paper about minimal surfaces from 1835, Heinrich Ferdinand Scherk summarizes his earlier findings from 1830 and gives equations for five new minimal surfaces, the first new ones since the catenoid and helicoid. Equation 7 describes the doubly periodic Scherk surface in general form (the orthogonal case is equation 6). This is the first non-trivial deformation family of minimal surfaces. Equation 9 is easily recognized as the associate family deformation of catenoid to helicoid, parametrized as screw motion invariant surfaces. These parametrizations are not conformal, and no complex analysis is involved. If only someone had realized that these surfaces share the same Gauß map, the discovery of the Enneper-Weierstraß representation could have happened decades earlier. Equation 16 is a mystery to me, I couldn’t verify that it satisfies the minimal surface equation. Equation 20, Scherk’s fourth surface, is also quite complicated, but one of the components of the implicitly given surface does satisfy the minimal surface equation. Using $t = 4\sin(x/2)^2+y^2\cos(x)\quad\text{and}\quad \rho^2 = t^2 + y^4 \sin(x)^2$ $\cosh\left( z+\sqrt{(t+\rho)/2} \csc(x/2)\right) = \frac{4 \sin(x/2)^2 + \rho}{y^2}$ To find its Enneper-Weierstraß representation and make a decent image, I looked at the level curve for x=π, which simplifies to $1+\cosh\left(\sqrt{4-y^2}\right) = \frac{8}{y^2} \ .$ This turns out to be a symmetry curve of the surface, so its normal lies in the plane x=0, and the Schwarz-Björling formula can be used to find the Enneper-Weierstraß representation: $G(z) =\frac{z-1}{z+1} \quad\text{and}\quad dh = i\frac{z}{z^4-1} \ .$ From here we can see that the surface is singly periodic with two annular and two helicoidal ends, and is also singular (at the points corresponding to 0 and infinity). Above you can see one half of the surface, with (parts of) both helicoidal ends and one of the annular ends. The singular point is where the horizontal symmetry curve in the middle meets the intersection of the two helicoidal ends, which is a straight line on the surface. Rotating about it gives a fundamental piece; below are three copies of it. For details, see the notebook under the resource below. Amusingly, there is a simpler surface with the same type of ends that I accidentally discovered a while ago. Finally, there is equation 30, giving the orthogonal case of Scherk’s singly periodic surface. Scherk does note some similarities to his doubly periodic surface. ##### Resources Mathematica Notebook for Scherk IV ## Winding Numbers In 1960, Robert Osserman proved that a complete minimal surface of finite total curvature is conformally a compact Riemann surface with finitely many points removed, and the Enneper-Weierstraß representation extends meromorphically to the punctures. One could now attach to any such surface a number of invariants: the genus g of the surface, the degree deg G of the Gauss map, the number e of ends, and for each end a winding number $\nu_j$. The latter is computed by subtracting 1 from the maximal order of the poles of the Weierstraß 1-forms at that end. Geometrically, small circles about the puncture of shrinking radius are mapped to space curve that can be rescaled so that they converge to a circle with that winding number as multiplicity. Fritz Gackstatter (1976) and independently Luquesio Jorge and William Meeks (1983) proved a useful winding number formula for oriented minimal surfaces of finite total curvature: $2 deg G = 2g-2 +\sum_{j=1}^e \nu_j+1$ For instance, the catenoid has genus 0, the degree of the Gauss map is 1, and there are two ends of winding number 2. Likewise, the the Enneper surface has genus 0, the degree of the Gauss map is 1, and there is one of winding number 3. These are, as Osserman proved, the only complete minimal surfaces with total curvature -4π. The next case of total curvature -8π was treated by F. López. Most prominently in his list is the Chen-Gackstatter surface, the only minimal torus of total curvature -8π. Besides that, there are numerous spheres. One can have (by the winding number formula) one end of winding number 5, or two ends with winding numbers 1 and 3 or 2 and 2, or three ends with winding number 1 each. You find examples for all cases somewhere on this page. Here is a question I don’t know the answer to: Can one have a complete minimal surface of finite total curvature with just one end of winding number 2? At first, this appears to contradict the winding number formula due to parity, but the surface could be non-orientable, like F. López’ amazing minimal Klein Bottle (which has a single Enneper end with winding number 3). ## Exemplum VII In 1744, Leonhard Euler published a book with the succinct title Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes. In it, he develops a general method to find curves that satisfy extremal problem, which is cow called the Calculus of Variations. In contrast to the ordinary calculus which allows to find extrema of a single function by solving an equation involving the derivative of a function, here a functional is minimized or maximized over all functions by solving an ordinary differential equation. His example VII has the title Invenire curvam, qua, inter omnes alias ejusdem longitudinis, circa axem AZ rotata, producat solidum superficies fit vel maxima vel minima. Euler’s Latin almost doesn’t require a translation into English: To find a curve, which among all others with the same length (meaning defined over the same interval) and rotated about the z-axis, produces a solid whose surface shall be maximal or minimal. Euler then proceeds, in a few lines, to apply his method to derive the differential equation for finding curves so that the corresponding surface of revolution has extremal area. Euler notes that this equation is solved by the catenary. I am not a historian, so I do not know who coined the term catenoid, nor do I know who made a first image. Euler is not concerned with two catenaries passing through the same points and thus offering two different solutions of evidently different area. Euler neither discusses nor defines the term minimal surface. This is done 1760 by Joseph Lagrange, who establishes in his note Essai d’une nouvelle methode pour determiner les maxima et les minima des formules intégrales indéfinies the minimal surface equation for a graph, observes that planar graphs satisfy his equation, and adds that “la solution générale doit être telle, que le périmètre de la surface puisse être détermine a volonté” –the general solution ought to be such that the perimeter of the surface can be prescribed arbitrarily. Lagrange gives no further examples, but his comment has triggered research that is still ongoing.	2020-02-19 01:45:24	{"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 7, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7181111574172974, "perplexity": 964.4443533601941}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1581875143963.79/warc/CC-MAIN-20200219000604-20200219030604-00486.warc.gz"}
https://gut.bmj.com/content/42/5/703	Article Text Management of occluded biliary Wallstents 1. T C K Thama, 2. D L Carr-Lockea, 3. J Vandervoorta, 4. R C K Wonga, 5. D R Lichtensteina, 6. J Van Dama, 7. F Ruymanna, 8. S Chowb, 9. J J Boscob, 10. T Qaseemc, 11. D Howellc, 12. D Pleskowd, 13. W Vannermane, 14. E D Libbyf 1. aBrigham & Women’s Hospital, Harvard Medical School, Boston, MA, USA, bLahey Medical Center, MA, USA, cMaine Medical Center, MA, USA, dDeaconess Hospital, MA, USA, eWinchester Hospital, MA, USA, fNew England Medical Center, MA, USA 1. Dr T C K Tham, Ulster Hospital, Dundonald, Belfast BT16 0RH, Northern Ireland, UK. ## Abstract Background—Wallstents (Schneider Stent, Inc., USA) used for the palliation of malignant biliary strictures, although associated with prolonged patency, can occlude. There is no consensus regarding the optimal management of Wallstent occlusion. Aims—To evaluate the efficacy of different endoscopic methods for managing biliary Wallstent occlusion. Methods—A multicentre retrospective study of patients managed for a biliary Wallstent occlusion. Results—Data were available for 38 patients with 44 Wallstent occlusions, all of which had initial endoscopic management. Twenty four patients had died and 14 were alive after a median follow up of 231 (30–1095) days following Wallstent occlusion. Occlusions were managed by insertion of another Wallstent in 19, insertion of a plastic stent in 20, and mechanical cleaning in five. Endoscopic management was successful in 43 (98%). Following management of the occlusion, bilirubin decreased from 6.0 (0.5–34.3) to 2.1 (0.2–27.7) mg/100 ml (p<0.05). No complications occurred. The median duration of second stent patency was 75 days (95% confidence interval 43 to 107) after insertion of another Wallstent, 90 days (71 to 109) after insertion of a plastic stent, and 34 days (30 to 38) after mechanical cleaning (NS). The respective median survivals were 70 days (22–118), 98 days (54–142), and 34 days (30–380) (NS). Incremental cost effective analysis showed that plastic stent insertion is the most cost effective option. Conclusion—Although all three methods are equally effective in managing an occluded Wallstent, the most cost effective method appears to be plastic stent insertion. • self expanding metal stent • biliary strictures • stent occlusion • jaundice • gastrointestinal malignancy View Full Text ## Statistics from Altmetric.com Self expanding metallic stents such as the Wallstent (Schneider Stent Inc., Minneapolis, Minnesota, USA) have been used for the management of biliary strictures to provide permanent bile drainage. The commercially available Wallstent is a tubular stainless steel uncovered super alloy mesh delivered in a constrained form on an 8 or 7.5 French gauge catheter system which, when deployed, expands to a final diameter of 24 or 30 French gauge (8 or 10 mm) and shortens to a length of 42, 68, or 80 mm. The stent is deployed using the Unistep system which allows easy retraction of the covering membrane after internal wetting of its inner hydrophilic coating. The main advantage of Wallstents over plastic stents for the palliation of malignant biliary obstruction is that they have a longer patency and despite the initial cost, have been shown to be cost effective.1 ,2 Wallstents still occlude after a median interval of nine months1 ,2 but unlike plastic stents, they cannot be extracted. Several methods have been used in the management of the occlusion such as insertion of another Wallstent or plastic stent, or mechanical cleaning. There are currently no data comparing the efficacy of the different management options and follow up of the Wallstent occlusion. Such data would be useful in deciding on optimal management. ## Methods A multicentre retrospective study of six biliary endoscopy centres in New England, USA, was undertaken to identify patients who were treated for biliary Wallstent occlusion. Detailed questionnaires were completed for each patient managed for a biliary Wallstent occlusion by the endoscopist involved. All primary Wallstents were correctly deployed across a biliary stricture to allow bile drainage. Tumour ingrowth was assumed when cholangiography showed a tight stricture within the stent, the appearance of which was similar to the original malignant stricture, and passage of a diagnostic catheter was difficult (fig 1). Tumour overgrowth was assumed when cholangiography showed a new stricture proximal or distal to the stent (fig 2). Debris or sludge occlusion was diagnosed when cholangiography showed filling defects within the lumen of the stent and further instrumentation showed passage of debris through the distal portion of the stent confirmed endoscopically (fig3). When the Wallstents became occluded, they were managed either by insertion of another Wallstent within the first (fig 4), a plastic stent within the first Wallstent (fig 5), or mechanical cleaning of the Wallstent (fig 3). Mechanical cleaning was defined as passage of an instrument (balloon, catheter, or guidewire) to allow recanalisation of the Wallstent to allow biliary drainage. Plastic stents were either the curved 10 French gauge Cotton-Leung stents (in 86%) or the straight Amsterdam stents (in 14%) (Wilson-Cook Medical Inc., Winston-Salem, North Carolina, USA). Five patients (24%) had two plastic stents inserted during the same procedure (fig5). Figure 1 Tumour ingrowth within a Wallstent. Figure 2 Tumour overgrowth proximal to a Wallstent. Figure 3 Debris within a Wallstent causing an obstruction. The debris was extracted using a balloon (mechanical cleaning). Figure 4 Second Wallstent inserted for occlusion of the first Wallstent showing good patency after deployment. Figure 5 Two plastic stents inserted for occlusion of a Wallstent. Follow up data were obtained by the physician managing the Wallstent occlusion from the patient’s medical record and/or by contacting their primary care physician. Management of the Wallstent occlusion was considered successful if there was clinical improvement with a significant fall in bilirubin following intervention. Complications of endoscopic retrograde cholangiopancreatography (ERCP) were defined according to published criteria.3 Second stent patency represented the interval between the time of treatment of the Wallstent occlusion and the time of its reocclusion or the death of the patient with jaundice and fever. Survival represented the interval between the time of treatment of the Wallstent occlusion and the patient’s death. Death without jaundice or sepsis was assumed to be due to causes other than stent occlusion. ## Discussion Most reports of the management of occluded Wallstents used in the biliary tree have either been case reports5 ,6 or a small part of a larger series evaluating other primary endpoints in the efficacy of Wallstents.2 ,7 In addition, long term results have not been published for placement of a second Wallstent or mechanical cleaning. Our retrospective study is the first to compare the different methods of treating Wallstent occlusion and also has the largest number of patients with occluded Wallstents in one series. We found no significant differences in the duration of patency or survival after managing Wallstent occlusions by insertion of another Wallstent, plastic stent, or mechanical cleaning. Thus all these methods of treatment appear to be equally effective. As median survival of patients following management of Wallstent occlusion was three months or less, the least costly method of treatment was that of a plastic stent as the insertion of another Wallstent did not seem to offer the advantage of longer patency. This is confirmed by incremental cost effectiveness analysis. As our study was retrospective and the subjects not randomly allocated, firm conclusions cannot be drawn until a prospective randomised and stratified study in larger numbers confirms our findings. However data from such a study are unlikely to be available for some time. The median duration of patency of the primary Wallstent in our study was 3.5 months which is less than the 8–10 months in previously reported studies.1 ,2 ,7 This may be due to a selection bias in our series which studied only patients whose Wallstents became occluded. The duration of patency of other Wallstents which did not occlude in the patients’ lifetime was not included. An incremental cost effectiveness analysis was performed because the three treatments for Wallstent occlusion differed only with respect to the equipment price as all appeared to be equally effective. Although ERCP related costs vary between different countries, it is apparent that the insertion of another Wallstent for an occluded Wallstent is the least cost effective option. The most cost effective option appears to be insertion of a plastic stent within the Wallstent. However in countries where ERCP costs are less than \$800, mechanical cleaning can be equally cost effective. The major cause of Wallstent occlusion was tumour ingrowth in 28 of 44 occlusions (64%), confirming previous observations.1 ,2This problem may be overcome by the development of a silicone covering8; or the emergence of newer metallic stents which do not have an open framework,9 but additional problems may ensue, including stent migration and impairment of branch duct drainage in hilar lesions as well as potential obstruction of the pancreatic and cystic duct orifices. The median patency of the second Wallstent within a Wallstent is relatively short at only 75 days which is less than that of the first Wallstent (102 days). The lack of prolonged patency following the second Wallstent insertion is one of the reasons why it has little advantage over insertion of a plastic stent. This may be because the majority of Wallstent occlusions are due to tumour ingrowth, and placement of a second Wallstent within the first Wallstent does not prevent this problem from recurring. On the other hand, a plastic stent within a Wallstent may prevent reocclusion due to tumour ingrowth although the plastic stent itself is prone to sludge encrustation.1 ,10 The positive correlation between the period of patency of the first Wallstent and the period of patency following treatment of the occlusion suggests that patients with early stent occlusion will occlude early following the second procedure. Furthermore, patients whose Wallstents remain patent for a long time may also have a longer patency following treatment of the occlusion. It could be argued that patients with early Wallstent occlusion may be selected for a subsequent cheaper procedure such as a plastic stent while patients with longer Wallstent patency might do better with another Wallstent. To determine whether such a strategy could be cost effective would require a large prospective randomised study. Although there appeared to be a trend for a lower stent patency and survival following mechanical cleaning to treat a Wallstent occlusion, this was not statistically significant. However the number of patients in the group who had mechanical cleaning is small and larger numbers may be required to confirm this. Alternative methods for treating Wallstent occlusions such as diathermy6 or hot tip laser probes11 have been described but these appear to carry an unnecessary risk of bleeding, ductal perforation, and stent fragmentation.12In contrast none of the patients in our study had any complications following treatment of their occluded Wallstents. In conclusion, although all three methods are equally effective in managing an occluded Wallstent, insertion of a plastic stent within a Wallstent appears to be the most cost effective method; in some health care environments, mechanical cleaning may be as cost effective. A prospective randomised stratified study is required to confirm this. ## Acknowledgments T C K Tham was supported as a Visiting Fellow by a grant from the Northern Ireland Council for Postgraduate Medical Education, J Vandervoort by a grant from OLV Hospital, Belgium, and both by a grant from Fujinon Photo Optical, Inc., Wayne, New Jersey and Omiya City, Japan. We are grateful to Dr Neil McDougall and David Gill of the Department of Medicine, the Queen’s University of Belfast, for assistance with graphical and statistical analysis. View Abstract ## Request Permissions If you wish to reuse any or all of this article please use the link below which will take you to the Copyright Clearance Center’s RightsLink service. You will be able to get a quick price and instant permission to reuse the content in many different ways.	2020-07-12 10:07:24	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.22109855711460114, "perplexity": 7267.264250474607}, "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-2020-29/segments/1593657134758.80/warc/CC-MAIN-20200712082512-20200712112512-00346.warc.gz"}
https://seoservicescalifornia.com/ao4tzcyd/emerald-crab-molting-process-d0cbce	# emerald crab molting process I have two EC's in my 75g and they're definitely shy. It can take up to several days to complete. Fiddler Crabs : Blue Crab Molting: As a blue crab mature, its body size increases. 1 decade ago. You might need to make a separate tank, especially for the molting process. Molting is a natural process for hermit crabs. These crabs are very easy to maintain, active, and require little attention making them a … In that time, I have seen all my shrimp molt multiple times, but as far as I know, this is the first time the Emerald has molted in that time. 12-09-2008, 07:36 PM. During the molting process, hermit crabs shed their exoskeleton, lose body parts, and become temporarily immobile, causing them to appear dead. Download Image Picture detail for : Title: Crab Molting Date: June 30, 2017 Size: 137kB Resolution: 650px x 543px Download Image. This is only according to many people I've read saying the same thing, their emerald crabs will turn white and die. boy DO they molt check out the size diferance (molt-live : left-right). While the new shell is hardening, the crab hides from predators. I just noticed him today when I was looking for him and found him with all his legs out, one claw and half his body … But have you ever watched the process? You must log in or register to reply here. every creature with an exoskeleton must molt as it grows. He has been fine for a week now, but today he has been laying on its back. Anonymous. I lost my emerald crab while he was motling. My belief is, if an emerald crab is turning white it's outer shell is aging and will eventually need to molt. Like shrimps, these emerald crabs will shed their exoskeleton as they grow leaving behind a what looks like a dead crab. He has been sitting in the same spot since yesterday barely moving but is still alive. If not youll most likely have a molt in your hand. Came home last night to find my emerald crab frozen stiff, motionless. About Molting and Brumation. The molting process of crabs is of particular interest to the CSU Crab Lab, as researchers delve into the specific processes and mechanisms that control this seemingly simple, but very complex event. Once they’ve reached sexual maturity, decorator crabs stop molting. So my emerald crab had been in the process of molting for 3 days and today when I came home I discovered him in two pieces, very obviously dead. Mithrax sculptus leaves its old shell at a visible site to dodge its predators while hiding away to allow the new shell to form completely and harden. The decorator crab recycles its living decorations during the molting process. by Fern on Tue Sep 27, 2011 12:39 pm. It makes them feel safe. WV Reef Community Member View Badges. Journey of an emerald from mines to jewelry by Ashutosh Roy The gemologists add an importance to polishing the stone, as the selection of abrasives is very crucial in polishing emerald because the abrasive may penetrate the inclusions to creep inside it. Our emerald crab molted before our very eyes! It takes weeks for a crab to prepare for molting. He most likely already has done it. His coloration has been getting lighter over the past couple of days and I wonder if he is molting? Its distinct, flat shiny green body and hairy legs easily identify the Emerald Crab. I have one emerald crab that has been sitting in a hole in a rock for most of the day for the past 3 days or so and now I can find him. Came home last night to find my emerald crab frozen stiff, motionless. This is actually the limb encased in a clear chitin sheath. I wanted to dive in deeper in this post about this crab and why this crab is a great addition to your saltwater tank. That really depends on how well they are eating. Molting begins with a secretion of hormones from the female crab, after which both female and male crabs fast. Surviving off their fat reserves, a crab will absorb as much calcium from its shell as possible — which aids in the development of the crab’s new shell. He has turned that whitish green color and has been like that since I made this thread lol. Came home last night to find my emerald crab frozen stiff, motionless. It's so cool how they just leave their old shells behind, perfectly intact. Emerald Crab molting question. This process indicates the healthy growth of the animal. Crabs. During this process the crab is highly vulnerable to predators, including the two-legged variety! Skip to main content. When your moon crab is molting, it prefers privacy. Assume that your crab is molting. In their first 2 years Dungeness crab molt as many as 6 times a year! When a female is about to molt, she releases pheromones that attract male crabs who then fight over the female. Favorite Answer. When he does move it is extremely slow. Female crabs can only mate after they molt, or shed their shell. ok my emerald wont molt. As Emerald Mythrax Crab outgrows the existing shell, it sheds the shell off, a process called molting, and a new shell replaces the old shell. I have to grab it and squeeze it to make sure its just a molt. Is this normal? This also helps to isolate your crab … More Galleries of ODFW Life Of A Dungeness Crab The first time my Emerald crab molted, I thought he had died. During this time of growing, before they reach adulthood the crab larvae go through several molting stages. In the process of backing his ass out of his old exoskeleton something got to it and killed it. In order for a hermit crab to grow it must molt. If he is unable to or refuses to molt, he will die. yes y would anybody ask that no pun intended to who created this Q/thread. Randy Pedersen ... People find empty crab shells on the beach, and these are often the molts left behind by a growing crab. It never went down, so I think its doing a surface molt. Coral Banded Shrimp $10.99$ 10.99. I have a 5.5 gallon nano tank with just two emerald crabs and elephant snail and a baby brittle star (bred from our large tank). It's about 9:15 AM and he's still not finished. I have witnessed numerous land hermit crabs during the molting process and identified four main molting positions. This must be done several times during their lifetime. 2 Answers. The crab rapidly absorbs water which causes its tissues to swell and split the old shell open across the back between the lateral spines. Answer Save. Their skeletons are on the outside, exoskeletons, and need to be shed, molted, to allow the crab to grow. Molting is very stressful for the crab. Do they just freeze The black and white breeding plumage of adult loons in summer is replaced by the gray-brown of winter. ages, from 3-4 molts in years 1-3, to a single molt every one or two years past year 6 (Baerg 1938, 1963). A soft-shell crab is nothing but a blue crab that has molted, a natural part of the growth process that occurs 20 or more times in a crab's lifetime. The under belly flap had been opened and right inside the flap it was clear but hard. The approach a hermit crab owner takes can mean the difference between life or death if the crab is molting. I blame the evil bugs but who knows. 1 0. With the emerald crab, you should notice his color becoming splotchy - more of a whitish green, rather than the solid darker green. iceni. When mine have molted, due to whatever reason, it seems that I find the body cap, with no legs attached most of the time. It may seem disturbing, but even if your crab has died during the molt, it’s safer to leave the deceased hermit crab for several months, than to attempt to move a crab during its molt. The BEST thing you can do for [a molting crab] is leave it alone! They usually only molt when they have outgrown their current shell (around every six weeks in my tank). During this process the crab is highly vulnerable to predators, including the two-legged variety! Hermit crabs are … Now one seems to be double the size of the other so I am guessing he molted. Hi guys, I have a 60gal reef tank setup and with your help in the past got through the cycling stage, last week I found one of my three emerald crabs dead, or so it appears. I'm guessing its his molt and he is somewhere else in the tank. Do they just freeze My lobster molts and it looks like the real thing. Have had a happy Sally for about 3 weeks. And yes they look like the real thing. My cleaner shrimp has done it twice and we wake up … The area where the limb is damaged will appear to grow a small nub that looks like clear gel. Hermit crabs periodically shed their exoskeleton, and this process often involves losing body parts. Now the other one is in hiding a lot and seems to have a lot of white on him instead of green. My hermit crab has been acting very lethargic lately. I came home from work yesterday and found that my emerald crab was molting. Try Prime EN Hello, Sign in Account & Lists Sign in Account & Lists Returns & Orders Try Prime Cart. Amazon.com : Emerald Crab : Pet Supplies. While molting events have not been shown to occur at a particular time during the day (Minch, 1977), the onset of a molt does seem to occur seasonally, mainly during April and … The other crab molted within the first 2 weeks of adding it to the tank. To assess the progression of each crab through the molting process, we monitored limb regenerative growth and pleopods. He needs to be left alone so he can start to regain some energy and to allow the new exo to harden. Every two days photographs of the regenerating limb were taken under a dissecting scope at 8 and \|${\rm{20}} \times$\| magnifications using a Diagnostic Instrument's SPOT Insight Model 3.2.0 digital still camera mounted on an Olympus SZ-PT dissection scope. Updated August 6, 2019 Author: Mike - FishLore Admin Social Media: The Emerald Crab (Mithrax sculptus) is a saltwater invertebrate that is often used as part of a clean up crew in marine fish tanks. They do it at night when its dark and no one is around. Hi all, my husband has a small aquarium at work (2.5 gallons) and recently got a fiddler crab for it. It is through this opening that the sperm is deposited into the female´s body and through which her eggs are laid. Do hermit crabs really molt?? It will enthusiastically feed on uneaten meaty foods and many types of nuisance algae. Emerald Crab $8.99$ 4.00. [19] This process is called molting. With the emerald crab, you should notice his color becoming splotchy - more of a whitish green, rather than the solid darker green. Molting is how hermit crabs are able to grow. My husband thought he was dead and took him out and placed him in a small plastic travel container. The crab rapidly absorbs water which causes its tissues to swell and split the old shell open across the back between the lateral spines. Just reach down there. Relevance. At this time the new exoskeleton absorbs water and become larger. I'm going to break this post down Q A style for simplicity: The number of the empty molts increased as we continued west. January 2011, Fiddler Crab – Detailed Guide: Care, Diet, and Breeding. This process indicates the healthy growth of the animal. I took out the exoskeleton and tossed it and planned on getting a new one. It will enthusiastically feed on uneaten meaty foods and many types of nuisance algae. Once the crab has shed the previous exoskeleton, the exoskeleton underneath will harden and be a nice solid color again (until it is time for the next molt). A 10-gallon aquarium would do just fine. It can also be known as the Boxing Shrimp for its propensity to hit its prey with those large claws. My other emerald is acting fine. Common Loons Molting. Some crab owners have reported crabs living to be the age of 16 or so. I was assuming the other emerald could be molting. Understanding molting can lead to successful farming of blue crabs that could produce on-demand soft-shell crab. This process typically begins at the base of the bill and spreads across the head and over the upper back. Crabs . Video shows the crab walking out of its own body and leaving behind a duplicate Emerald Crab. How often do Emerald (Mithrax) Crabs Molt? White-tailed Deer Molting. Ecdysis (molting or “busting” stage) The crab stops eating and seeks shelter in order to avoid predation. All Livestock shipments will be held for shipping until January 4th. ... Often if molting takes a few days it's a good idea to check your iodine levels, iodine aids the molting process and if levels are low the process typically takes longer. During that stage, the exoskeleton is beginning to loosen. Emerald Crab and Molting. Sally Lightfoot molting/ legs deformed Hi, I see there is a similar problem that someone has contacted you about, to one I am experiencing now. I’ve had my Emerald Crab since July. Researchers are trying to better understand how the crabs’ environment controls growth and molting. Do they often go into hiding and become very mellow like other inverts do before they molt? Although the molting process itself usually lasts only a few minutes. The Emerald Crab is known for its scavenging ability. As we proceeded with our dive, the number of crabs and carapaces quickly increased to cover the entire bottom. JavaScript is disabled. During that stage, the exoskeleton is beginning to loosen. Emerald Crabs are popular because they help keep the tank clean and have been reported to eat bubble algae (Velonia sp.). Emerald crab molting. What's wrong with my emerald crab? Molting is the process of the crab losing a layer of its exoskeleton and growing a new one, which happens about once a year (it can be twice a year or once every two years, depending on the crab and its environment). As Emerald Mythrax Crab outgrows the existing shell, it sheds the shell off, a process called molting, and a new shell replaces the old shell. Download Image. Here's a great video that shows what it looks like as a crab is crawling out of its shell. Closer inspection revealed these crabs were actually carapaces that were discarded during the molting process. You can clearly see his old molt, but it seems that he didnt develop his new shell fully. Thread starter WV Reef; Start date Nov 11, 2020; Tagged users None Nov 11, 2020 #1 WV Reef Community Member View Badges. When water evaporates, salinity concentration occurs, causing negative effects on crabs. However, crustacean aquaculture has a myriad of problems to overcome: disease, overcrowding, and optimizing conditions. Dungeness Crab Molting. Reply Like Reply. Does this mean he is about to molt? To know if your hermit crab is preparing to molt look for these signs: The hermit crab will probably become lethargic (it won’t move much). Once the crab has shed the previous exoskeleton, the exoskeleton underneath will harden and be a nice solid color again … Nov 11, 2020 #6 OP . You can move the crab to a new location and see if it changes position after a few hours to test for signs of life. During that stage, the exoskeleton is beginning to loosen. Download Image. Briefly, crabs were held suspended in the air by a clamp in a manner that allowed free motion of the swimming legs, and small wire electrodes were placed in two small holes drilled into the mesobranchial region of the dorsal carapace. I'll post a pic of him when I get home. could be molting. Yes - lobsters, crabs (hermit and emerald), all molt otherwise they can not grow larger. Emerald crab molting! The Emerald Crab is well respected for its scavenging ability. Thor amboinensis Blood Red Fire Shrimp. Ecdysis (molting or “busting” stage) The crab stops eating and seeks shelter in order to avoid predation. Also, emerald crabs do not molt often and grow slower than shrimp etc. One very important element in keeping land hermit crabs alive in captivity is to fill your crabs tank with sand deep enough minimum depth of 3 times the size of their shell and moist enough packs well that your largest crab is able to bury completely to molt successfully. A few days before molting you can notice that Arrow crabs move even slower than usual and do not eat as much. The name of Cleopatra is very closely associated with an That was about 5:30 PM. I thought mine was dead and then saw him come out of the rocks. Crabs were induced to undergo a burst swimming response, as described previously (6, 24, 30, 38). Arrow Crab and Molting. if its him hell scurry away. Like all crustaceans, Arrow crabs need to molt (shed their shells off) to grow. I just really hope he's molting and not dead. Description. I have been using iodine in an 8 week-old reef tank. We started to see some live crabs in amongst the molts. With the emerald crab, you should notice his color becoming splotchy - more of a whitish green, rather than the solid darker green. The crabs are immobile for a short time while it regains muscle control and its new exoskeleton hardens up. That's how they grow and they get there legs back!If they have lost any.They do hide a little until the new shell hardens a little also. It is an exact replica so to speak. Before its new shell hardens, the crab absorbs water and expands to a size larger than before the molt. Molting is the process of shedding old skin, feathers or skeleton to make way for new growth. As they become mature (at age 3) this slows to about once a year. When the female chooses a male, she then attaches herself to the male and rides on his back until she finally molts. If you care for hermit crabs, you'll need to be ready for this part of their life cycle. When preparing to molt, a crab’s old exoskeleton separates from the new one beneath. If you disturb your crab while it is molting, you may seriously endanger it – so be patient. Lv 7. Molting times vary from crab to crab, so if your hermit crab is still molting after the expected time, leave it be. Another transformation that takes place in the fall (as well as spring) with White-tailed Deer and other mammals is the molting of a … When Inverts molt the leave their exoskeleton out in plain view to serve as a decoy. Please support our sponsors and let them know you heard about them on AquariumAdvice.com, Aquarium Advice - Aquarium Forum Community. Unlike many other animals, Mithraculus sculptus will eat bubble algae and helps clean your aquarium of these algae. Yes - lobsters, crabs (hermit and emerald), all molt otherwise they can not grow larger. It recently stopped moving, so I put in a lot of sand so it could bury down and molt. Ok thanks. Now i've got 15 hermits in my tank and I think that's way too many but they came from a friend. One spritz of water a day over where the molter is buried is the most attention he should receive, he should not be handled in any way! I was pretty bummed. Molting is freaky. For a better experience, please enable JavaScript in your browser before proceeding. As we saw in our previous post, the Emerald Crab was rated the #1 algae eater in a saltwater aquarium. That’s why they bury themselves. not sure whats going on. Fern ; Fern Admin Posts: 2620 Join date: 2011-03-09 Age: 37 Location: Fort Myers FL. My Sally molted 5 days ago, and has not been normal since. Do they just freeze motionless like this when they molt and if so how long does the process usually take and how often do they molt? I have 2 emeralds that seemed to be the same size when I got them. The Sally Lightfoot Crab, ... Like many inverts, it will lose its exoskeleton in a process called molting as it grows. Molting is the process by which the crab discards its exoskeleton, replacing it with a temporarily soft, pliable new exoskeleton that is easy to eat. Hermit crabs require special care during the molting period, which happens at least once a year. With time, as they outgrow the existing shell (exoskeleton), they begin to shed their shells off. Blue Crab . I sprayed it with water and I picked him up and tapped his feet, but he has not moved AT ALL in the past 2 days. Yes they do. Local Delivery and Dry Good shipments remain unaffected. Often times, the emerald crab will appear back out of … The signs of fall are plentiful – skeins of migrating geese, disappearing insects, falling leaves. Once that outer shell is shed, it is common to think that the crab is dead, but it isn’t. He's still in the process, o er 24 hrs later. The crab incubation period lasts for about two weeks until the crab larvae hatch and is then released into the ocean to fend for themselves. His coloration has been getting lighter over the past couple of days and I wonder if he is molting? The crab larvae will continue to grow for the next 40 days until they reach the adult crab stage. any input is welcome World's biggest crab sheds its shell by by crawling out of its own body and leaving behind a dead looking duplicate. Loons begin a full body molt (minus their wings) in the late summer and early fall, prior to migration. His coloration has been getting lighter over the past couple of days and I wonder if he is molting? I have 90 seconds of three small crabs duking it out. All treats no tricks - Save 11% on most... Ich Treatment Destroyed my Cycle. Like all crustaceans, they need to molt to grow. A crab that is missing limbs and is nearing a molt will begin the process of regenerating its limbs. There is one way to find out. Is an emerald crab suited for a beginner? I think my emerald crab is still in the molting process, he is halfway out of his old shell, but he hasn't made much progress in the last 3 hours. Emerald Crab Molt Posted on April 21, 2012 by Amanda We were aware that shrimp’s will molt, which is the shedding of the old exoskeleton to make way for new growth, but were not aware that crabs … bbl_nk. Molting is the process where blue crabs shed their rigid outer shell (exoskeleton) to allow their bodies to increase in size. Today there was an odd, but FAINT smell of rotting fish or something musty. Molting is a key stage in the life cycle of any crab, shrimp, crayfish, lobster, etc. Many molting hermit crabs have been tossed in the trash because their owner believed them to be dead. I didn't think so since they don't have an outer shell.....which is why we have to provide shells for them as they grow. Once the crab has shed the previous exoskeleton, the exoskeleton underneath will harden and be a nice solid color again (until it is time for the next molt). Mithrax sculptus leaves its old shell at a visible site to dodge its predators while hiding away to allow the new shell to form completely and harden. Although the molting process itself usually lasts only a few minutes. This is a common question for newbies when their crab molts for the first time. Any experienced advice is appreciated. How long does this usually take? Post n°1; Emerald crab molting. Mostly peaceful towards other species, the Coral Banded Shrimp should not be housed with members of its own species unless they are a mated pair. Molting may occur once a year for larger hermit crabs or once every few months for smaller (tiny crabs). A day later he emerged from the rocks bigger and badder than ever!!!! Or death if the crab rapidly absorbs water and become larger called emerald crab molting process... Make sure it s dark and no one is in hiding a lot of sand it. White and die going to break this post about this crab is well respected its! Many inverts, it is molting style for simplicity: what 's wrong with my crab... Inverts molt the leave their old shells behind, perfectly intact environment controls growth and.. Got them encased in a saltwater aquarium usually only molt when they have outgrown current... So he can start to regain some energy and to allow the new beneath! Which causes its tissues to swell and split the old shell open across back... Save 11 % on most... Ich Treatment Destroyed my cycle, which happens at least once year... But today he emerald crab molting process turned that whitish green color and has not been normal.! Known as the Boxing shrimp for its scavenging ability never went down, so am... You may seriously endanger it – so be patient now one seems to be the same thing, emerald. Are laid Sep 27, 2011 12:39 pm o er 24 hrs later for newbies when their molts! Across the head and over the past couple of days and i its. The female´s body and hairy legs easily identify the emerald crab him instead of green!!!!!... I lost my emerald crab is molting with a secretion of hormones from the rocks depends on how they... Placed him in a process called molting as it grows helps clean your aquarium of algae! You disturb your crab while it regains muscle control and its new exoskeleton absorbs which... In amongst the molts crabs need to molt ( shed their shells off to! That could produce on-demand soft-shell crab o er 24 hrs later are eating and! Which both female and male crabs who then fight over the past couple of days and i if! And he 's still in the tank: care, Diet, and this the... New exoskeleton hardens up we monitored limb regenerative growth and molting continue to grow it must molt a! As much split the old shell open across the back between the lateral spines adding it to the tank opened. Are laid the rocks bigger and badder than ever!!!!!!!! Deer molting recycles its living decorations during the molting process: what 's wrong with emerald... Please support our sponsors and let them know you heard about them AquariumAdvice.com. Him come out of the animal week now, but it seems that he didnt develop his shell. Him instead of green to make sure it s him he ll scurry away he was motling in. Is around bigger and badder than ever!!!!!!!!!!... For smaller ( tiny crabs ) regains muscle control and its new exoskeleton hardens up stage the! Are laid continued west video that shows what it looks like as a blue crab mature, body. Is molting emerald crab molting process molting going to break this post about this crab and why this crab is for! For larger hermit crabs require special care during the molting process well they eating! Back until she finally molts its body size increases about 9:15 am and he 's still not finished left-right! 3 ) this slows to about once a year & Lists Sign in Account & Lists Sign in Account Lists! Hello, Sign in Account & Lists Sign in Account & Lists Returns & Orders Prime. Have been using iodine in an 8 week-old reef tank opening that the crab larvae will continue to it. Pedersen... People find empty crab shells on the beach, and has not been normal.!, active, and optimizing conditions is damaged will appear to grow small! Out of the bill and spreads across the back between the lateral spines of migrating geese, insects... Avoid predation for the molting process, o er 24 hrs later,. Opening that the sperm is deposited into the female´s body and through which her eggs are laid its to... Or refuses to molt, a crab ’ s old exoskeleton something got to it and it. A few minutes chooses a male, she releases pheromones that attract male crabs then. Of white on him instead of green on Tue Sep 27, 2011 12:39 pm the between. Often do emerald ( Mithrax ) crabs molt find empty crab shells on the beach, and require little making. Blue crab molting: as a decoy great addition to your saltwater tank y. Sand so it could bury down and molt laying on its back when i get.... And grow slower than shrimp etc thought he was dead and took him out and placed in. Input is welcome Also, emerald crabs do not molt often and grow slower than shrimp etc year! Shells behind, perfectly intact crab molted, i thought he had died between the lateral.. So i am guessing he molted 're definitely shy hit its prey with those large.... Up to several days to complete occur once a year that 's way too many but they from! Diet, and optimizing conditions the black and white Breeding plumage of adult loons in summer is replaced by gray-brown... But today he has been like that since i made this thread lol one... Effects on crabs reported crabs living to be the same size when i get home they reach the crab... Their current shell ( exoskeleton ) to allow the new one several molting stages inverts molt the leave old. Amboinensis Blood Red Fire shrimp owner believed them to be double the size of the animal serve... Beginning to loosen, prior to migration suited for a crab to prepare for molting week! Before they molt off ) to allow the crab is molting days and wonder!, feathers or skeleton to make sure it s just a in... She finally molts during that stage, the exoskeleton is beginning to loosen of any,... Post, the crab larvae will continue to grow a small nub that looks like clear gel it. Lobster, etc a what looks like the real thing bit new for most >! Treatment Destroyed my cycle crabs ) crabs can only mate after they molt check out the size diferance molt-live. Were actually carapaces that were discarded during the molting process itself usually lasts only few... The age of 16 or so for larger hermit crabs during the molting itself! Mature, its body size increases tank and i wonder if he molting... And die short time while it is through this opening that the sperm is deposited into the female´s body hairy. A clear chitin sheath down Q a style for simplicity: what 's wrong with my emerald since. Same spot since yesterday barely moving but is still molting after the expected time, as they outgrow existing! Is welcome Also, emerald crabs are popular because they help keep the tank clean and have been using in. Geese, disappearing insects, falling leaves hrs later water and emerald crab molting process.... Am guessing he molted its back clear chitin sheath world 's biggest crab its!, prior to migration belly flap had been opened and right inside the flap it was clear but hard your! Eventually need to be the same thing, their emerald crabs are immobile a... An exoskeleton must molt days before molting you can notice that Arrow move. Are able to grow overcrowding, and optimizing conditions we monitored limb regenerative growth and.! Like many inverts, it will enthusiastically feed on uneaten meaty foods and many types of nuisance algae 2. He ll scurry away six weeks in my 75g and they definitely... Some energy and to allow the new exo to harden Fire shrimp other molted! Hairy legs easily identify the emerald crab is known for its scavenging ability Velonia.... Time the new exo to harden Prime Cart are often the molts in this post about this and. Carapaces that were discarded during the molting process when i got them molting as grows... In Account & Lists Returns & Orders try Prime Cart, or shed their shells.. Of nuisance algae er 24 hrs later, and optimizing conditions i home. Will shed their exoskeleton as they outgrow the existing shell ( exoskeleton ) to allow the one. Produce on-demand soft-shell crab molt in your browser before proceeding this time of growing, before they molt through. Crabs: blue crab mature, its body size increases early fall, prior to migration especially the... Will shed their shell be molting 's a great addition to your saltwater tank ecdysis ( molting or busting! Life or death if the crab is known for its propensity to hit its with... During this process the crab to grow very easy to maintain, active, need... Opened and right inside the flap it was clear but hard beginning to loosen exoskeleton out plain... Yesterday and found that my emerald crab is dead, but FAINT smell of rotting fish or something.. And right inside the flap it was clear but hard assuming the other molted! He was motling video shows the crab larvae will continue to grow limb is will. Get home, before they molt to better understand how the crabs are immobile for a hermit has! Of their life cycle: 2620 Join date: 2011-03-09 age emerald crab molting process 37 Location Fort. Saltwater tank the age of 16 or so molt ( minus their wings ) in the cycle!	2021-02-28 03:00:01	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.31330910325050354, "perplexity": 5191.0333730776565}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1614178360107.7/warc/CC-MAIN-20210228024418-20210228054418-00160.warc.gz"}
https://www.physicsforums.com/threads/acceleration-of-a-stationary-mass.808452/	# Acceleration of a stationary mass? 1. Apr 14, 2015 ### houlahound so sitting in a chair gives me an acceleration of 1g. clearly I am not moving relative to the chair and Earth yet acceleration "a" is defined as the change in velocity ie; (Vfinal - Vinitial)/time in the chair Vfinal = Vinitial = 0 so by definition (and calculation) a = (Vfinal - Vinitial)/time = 0/t = 0 so why am I accelerating at 1g when I just correctly calculated my acceleration to be 0m/s/s 2. Apr 14, 2015 ### Staff: Mentor You're not accelerating at 1 g. You are feeling a force from gravity that is equal to your mass times g, and your chair is pushing you up with an equivalent force, such that the net force on you is zero, hence you are not accelerating. The fact that the force of gravity is $F = mg$ does not mean that you are being accelerated by $g$, simply that you would be if no other forces were acting on you. 3. Apr 14, 2015 ### houlahound Dr Claude given that F = ma or a = F/m, F is not zero, m is not zero so how do you get a to be zero? I thought F might be net force that cancels to zero ie gravity down reaction of chair up but F is a measurable quantity ie an accelerometer on the chair will give a non zero number. 4. Apr 14, 2015 ### DaveC426913 5. Apr 14, 2015 ### A.T. 6. Apr 14, 2015 ### Staff: Mentor Along the vertical direction: $$F_\mathrm{total} = F_\mathrm{gravity} + F_\mathrm{chair} = 0$$ The force fro the chair exactly cancels out the force of gravity, even though $F_\mathrm{gravity} = mg \neq 0$. How do you think that the accelerometer measures $F_\mathrm{gravity}$? (Hint: think about the example of the chair.) 7. Apr 14, 2015 ### houlahound i think i am getting it thanks, will work through the link (seems to be the ticket) to see the difference in defining accelerations. Geez that link goes into this in the explanation; 8. Apr 14, 2015 ### houlahound OK so; Fg = - Fc giving Fg+Fc = 0 = Ftotal = 0 I get that but it does raise the question if Ftotal = 0 why do I feel my own weight? I have no idea how an accelerometer works, I have a digital one, I doubt there is a spring and ball inside it. 9. Apr 14, 2015 ### A.T. You don't. You feel deformations in your body caused by the non-uniformly applied contact force form the chair. http://en.wikipedia.org/wiki/Accelerometer#Structure Last edited: Apr 14, 2015 10. Apr 14, 2015 ### Staff: Mentor We'll get to that question when we have finished with the other. Let's stick with old-fashioned measuring apparatus. How does a dynamometer measure force? 11. Apr 14, 2015 ### houlahound simple calibration of the spring under tension via Hooke's law restoring force, write a scale on the side in preferred units. the spring is not registering a force if it is unloaded but it still "feels" 1g force. 12. Apr 14, 2015 ### houlahound going off line, not being rude if I do not respond to further posts for awhile. will check back here later. 13. Apr 14, 2015 ### Staff: Mentor In other words, the dynamometer is measuring the force necessary for the spring to counter the force due to gravity, the equivalent of $F_\mathrm{chair}$ above. That's an arbitrary choice of the zero, since we want to measure the weight of a mass added to the spring. The spring can be slightly elongated compared to when it is at rest horizontally.	2017-10-23 10:40:32	{"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": 0.49751824140548706, "perplexity": 1496.1114230613616}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187825889.47/warc/CC-MAIN-20171023092524-20171023112524-00452.warc.gz"}
https://www.practically.com/studymaterial/blog/docs/class-9th/maths/circles/	# Circles Book 11.1. INTRODUCTION The collection of all the points in a plane which are at a fixed distance from a fixed point In the plane is called a circle. 11.2. TERMS RELATED TO A CIRCLE Centre of the Circle The fixed point is called the centre of the circle. O is the centre of the circle in the figure given below. The fixed distance from the centre and circumference of the circle is called the radius of the circle. OA = OC = r is the radius of the circle. We can draw infinite radii in a circle and all are equal in length. Chord of the circle The line segment which joins two points on the circumference of a circle is known as the chord of the circle. The chord of a circle does not pass through the centre of the circle. CD is a chord of the circle in figure. The diameter of the circle The chord, which passes through the centre of the circle, is called a diameter of the circle. We can be drawn infinite diameters in a circle and all are equal in length. In the figure, AOB is a diameter of the circle. It is denoted by d. Now, d = 2 × r $⇒$ r = $\frac{\mathrm{d}}{2}$ [Where d = diameter and r = radius of the circle] N.B. Hence, it is said that a diameter is the longest chord of a circle. A circle divides the plane on which it lies into following three parts in figure. i) Interior of the circle : The plane which exists inside of a circle or the region inside of a circle is known as the interior of the circle. ii) Circle : The geometrical figure which is surrounded by a circular line segment or a circle is a collection of all those points in a plane that are at given constant distance from a given fixed point in the plane. iii) Exterior of the circle : The plane which exists outside of a circle or the region outside of a circle is known as the exterior of the circle. Arc of a circle A continuous piece of a circle is called an arc of the circle. Minor arc The shorter (smaller) arc of a circle is called minor arc. In figure, $\stackrel{⏜}{PQ}$ is the minor arc. Major arc The longer arc of a circle is called major arc. In figure. $\stackrel{⏜}{PRQ}$ is the major arc. Semi circle If P and Q are ends of a diameter then both arcs are equal and each is called a semi circle, i.e., $\stackrel{⏜}{PXQ}$ and $\stackrel{⏜}{PYQ}$ are equal arcs having a semi-circle in figure. It is also called semicircular region. Circumference The length of the complete circle is called the circumference of the circle. It is denoted by C in figure, i.e. Circumference of the circle (C) = 2$\mathrm{\pi }$r; where $\mathrm{\pi }=\frac{22}{7}$ or 3.14 Semi Circumference Half length of the complete circle is called the semi-circumference of the circle. Both semi-circumferences of the circle are equal in length in figure. i.e. Semi circumference = $\mathrm{\pi }$.r Segment of the circle The region between a chord and either of its arcs is called a segment of the circle. Minor Segment The smaller region between a chord and smaller arc is called the minor segment of the circle, i.e. PXQ is the minor segment of the circle in figure. Major segment The bigger region between a chord and bigger arc is called the major segment of the circle, i.e., PYQ is the major segment of the circle in figure. Minor sector When a circle is divided by its two radii, the smaller region of the circle is called minor sector, e.g., OAXB is the minor sector of the circle in figure. Major Sector When a circle is divided by its two radii, the bigger region of the circle is called major sector, e.g. OAYB is the major sector of the circle in the figure above. Example 1 Prove that if the angles subtended by the chords of a circle at the centre are equal, then the chords are equal. Given : In figure, chord AB = chord CD of a circle with centre O. Construction : joined OA, OB, OC and OD respectively. To prove : $\angle \mathrm{AOB}=\angle \mathrm{COD}$ Proof : In , we have, OA = OC [Radii of a circle] OB = OD [Radii of a circle] and AB = CD [Given] [By SSS congruence rule] Hence, $\angle \mathrm{AOB}=\angle \mathrm{COD}$ [By CPCT] Example 2 Prove that the line drawn through the centre of a circle to bisect a chord is perpendicular to the chord. Given : OM $\perp$ chord AB. i.e. $\angle \mathrm{OMB}=\angle \mathrm{OMA}=90°$ To prove : AM = BM Construction : Join OA and OB respectively Proof : In , we have OA = OB [Radii of a circle] $\angle \mathrm{OMA}=\angle \mathrm{OMB}=90°$ [Given] OM = OM [common] [RHS congruence rule] Hence, AM = BM Given : AB is a chord of a circle with centre O. M is the mid-point of chord AB i.e., AM = BM in figure. Construction : Join OM, OA and OB To prove: OM $\perp$ AB Proof : In , we have AM = BM [Given] OA = OB [Radii of a circle] and OM = OM [Common] [By SSS congruence rule] [By CPCT] … (i) But $\angle \mathrm{OMA}+\angle \mathrm{OMB}=180°$ [Linear pair] $⇒\angle \mathrm{OMA}+\angle \mathrm{OMA}=180°$ [From (i)] $⇒2\angle \mathrm{OMA}=180°$ $\therefore \angle \mathrm{OMA}=90°$ … (ii) $\therefore \angle \mathrm{OMA}=\angle \mathrm{OMB}=90°$ [from (i) and (ii)] Hence, OM $\perp$ AB. Example 3 There is one and only one circle passing through three non-collinear points. Proof : Let P, Q and R be three non-collinear points. If we join PQ and QR perpendicular bisectors of PQ and QR intersect each other at O Now, join OP and it is the radius of the circle, which makes the circle PQRSP and also passes through these three non-collinear points P, Q and R. Hence, we can say that one and only one circle can be drawn through three non-collinear points. 11.3. ANGLE SUBTEND BY AN ARC The length of the perpendicular from a point to a line is the distance of the line from the point. Let AB be a line and P be a point. We know that there are infinite numbers of points on a line. If we join these points to P, we will get infinitely many line segments, PL1, PL2, PL3, PL4, …….. etc. in figure. Out of these line segments, the perpendicular from P to AB i.e. PM will be the least. Hence, this least length PM has to be the distance of AB from P. Example 3 Prove that equal chords of a circle are equidistant from the centre. Given: AB and CD are two equal chords of a circle with centre O and OE $\perp$ AB and OF $\perp$ CD in figure. To prove: OE = OF Construction : Joined OA and OC Proof : The perpendicular from the centre of a circle to a chord bisects the chord. Now, $\because$ OE $\perp$ AB, …(i) and … (ii) But AB = CD [Given] … (iii) [From (i), (ii) and (iii)] In ${△}^{s}$ OAE and OCF, we have OA = OC [Radii of a circle] AE = CF [By proof] and $\angle \mathrm{OEA}=\angle \mathrm{OFC}=90°$ [Given] [By RHS congruence rule] Hence, OE = OF [By CPCT] Example 4 To prove that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle. Given : An arc BC of a circle subtending angles BOC at the centre O and BAC at a point A on the remaining part of the circle in figure.(i), (ii) and (iii). Construction : Joined AB, AC, AO and AO is extended to D. To prove: $\angle \mathrm{BOC}=2×\angle \mathrm{BAC}$ Proof : In different three cases, BC arc is minor in figure.(i), BC arc is semicircle in figure.(ii) and BC arc is major in figure. (iii). Now, in all the cases: In $△$OAC, we have : $\angle \mathrm{DOC}=\angle \mathrm{OAC}+\angle \mathrm{ACO}$ …… (i) [Exterior angle of a triangle is equal to the sum of two interior opposite angles] Also, in $△$OAC, we have : OA = OC [Radii of a circle] … (ii) [Angles opposite to equal sides of a triangle are equal] …(iii) [From (i) and (ii)] Similarly, $\angle \mathrm{DOB}=2\angle \mathrm{OAB}$ … (iv) Now, adding (iii) and (iv), we get $\angle \mathrm{DOC}+\angle \mathrm{DOB}=2\angle \mathrm{OAC}+2\angle \mathrm{OAB}$ $⇒\angle \mathrm{BOC}=2\left(\angle \mathrm{OAC}+\angle \mathrm{OAB}\right)$ For case (iii) in figure. (iii), where BC is the major arc, (v) is replaced by reflex angle $\angle \mathrm{BOC}=2\angle \mathrm{BAC}$ Example 5 To prove that angles in the same segment of a circle are equal Given : BD is an arc of a circle with centre O and $\angle \mathrm{BAD}$ and $\angle \mathrm{BCD}$ are two angles in the same segment in figure (i) and (ii). To prove : $\angle \mathrm{BAD}=\angle \mathrm{BCD}$ Construction : Joined OB and OD Proof : We know that the angle subtended by an arc at the centre is double the angle subtended by the arc at any point in the remaining part of the circle in figure.(i) We have $\angle \mathrm{BOD}=2×\angle \mathrm{BAD}$ … (i) and $\angle \mathrm{BOD}=2×\angle \mathrm{BCD}$ .. (ii) From (i) and (ii), we get $\angle \mathrm{BAD}=\angle \mathrm{BCD}$ In figure (ii), we have Reflex $\angle \mathrm{BOD}=2×\angle \mathrm{BAD}$ …(i) and Reflex $\angle \mathrm{BOD}=2\angle \mathrm{BCD}$ ….(ii) From (i) and (ii), we get $\angle \mathrm{BAD}=\angle \mathrm{BCD}$ A quadrilateral ABCD is called a cyclic quadrilateral if all the four vertices A, B, C and D are cyclic. Example 6 AB = CB and O is the centre of the circle. Prove that BO bisects $\angle \mathrm{ABC}$ Given : In figure, AB = CB and O is the centre of the circle. To prove : BO bisects Solution : In , we have : OA = OC [Radii of a circle] AB = CB [Given] and OB = OB [Common] [By SSS congruence rule] [By CPCT] Hence, BO bisects $\angle \mathrm{ABC}$ Example 7 Two equal chords AB and CD of a circle with centre O, when produced meet at a point E, as shown in figure. Prove that BE = DE and AE = CE in figure. Given : AB and CD are two equal chords intersect at a point E in figure. To prove : BE = DE and AE = CE Construction : Joined OE OL $\perp$ AB and OM $\perp$ CD have been drawn. Solution : $\because$ AB = CD [Given] $\therefore$ OL = OM [$\because$ Equal chords are equidistant from the centre] Now, in $△S$ OLE and OME, we have OL = OM [By proof] $\angle \mathrm{OLE}=\angle \mathrm{OME}$ [Each equal to 90o] and OE = OE [Common] [By RHS congruence rule] [$\because$ BY C.P.C.T.] Now, AB = CD …(i) $⇒\frac{1}{2}\mathrm{AB}=\frac{1}{2}\mathrm{CD}⇒\mathrm{BL}=\mathrm{DM}$ … (ii) Subtracting equation (ii) from (i), we get LE – BL = ME – DM $⇒$ BE = DE Again, AB = CD and BE = DE Adding both of them, we get AB + BE = CD + DE $⇒\mathrm{AE}=\mathrm{CE}$ Hence, BE = DE and AE = CE.	2022-08-14 18:43:58	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 57, "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": 0.630205512046814, "perplexity": 1010.5895886455124}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1659882572063.65/warc/CC-MAIN-20220814173832-20220814203832-00014.warc.gz"}
http://blog.invibe.net/categories/plot.html	# 2015-02-25 Saving files in HoloViews Here, we show how to save files from figures generated with HoloViews. This was thoroughly explained in this page. # 2015-02-17 Creating an animation using Gizeh + MoviePy Gizeh (that is, Cairo for tourists) is a great interface to the Cairo drawing library. I recently wished to make a small animation of a bar moving in the visual field and crossing a simple receptive field to illustrate some simple motions that could be captured in the primary visual cortex ansd experiments that could be done there. # 2015-01-16 Rendering 3D scenes in python The above snippet shows how you can create a 3D rendered scene in a few lines of codes (from http://zulko.github.io/blog/2014/11/13/things-you-can-do-with-python-and-pov-ray/): In [1]: import vapory camera = vapory.Camera( 'location', [0, 2, -3], 'look_at', [0, 1, 2] ) light = vapory.LightSource( [2, 4, -3], 'color', [1, 1, 1] ) sphere = vapory.Sphere( [0, 1, 2], 2, vapory.Texture( vapory.Pigment( 'color', [1, 0, 1] ))) scene = vapory.Scene(camera = camera , # a Camera object objects = [light, sphere], # POV-Ray objects (items, lights) included = ["colors.inc"]) # headers that POV-Ray may need # passing 'ipython' as argument at the end of an IPython Notebook cell # will display the picture in the IPython notebook. scene.render('ipython', width=900, height=500) Out[1]: Here are more details... # 2015-01-07 The right imports in a notebook Following this post http://carreau.github.io/posts/10-No-PyLab-Thanks.ipynb.html, here is ---all in one single cell--- the bits necessary to import most useful libraries in an ipython notebook: In [1]: # import numpy and set the printed precision to something humans can read import numpy as np np.set_printoptions(precision=2, suppress=True) # set some prefs for matplotlib import matplotlib.pyplot as plt import matplotlib matplotlib.rcParams.update({'text.usetex': True}) fig_width_pt = 700. # Get this from LaTeX using \showthe\columnwidth inches_per_pt = 1.0/72.27 # Convert pt to inches fig_width = fig_width_ptinches_per_pt # width in inches FORMATS = ['pdf', 'eps'] phi = .5np.sqrt(5) + .5 # useful ratio for figures # define plots to be inserted interactively %matplotlib inline #%config InlineBackend.figure_format='retina' # high-def PNGs, quite bad when using file versioning %config InlineBackend.figure_format='svg' Below, I detail some thoughts on why it is a perfect preamble for most ipython notebooks.	2017-02-24 19:24:45	{"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": 0.2869497239589691, "perplexity": 9861.640790116815}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501171629.92/warc/CC-MAIN-20170219104611-00199-ip-10-171-10-108.ec2.internal.warc.gz"}
https://mathshistory.st-andrews.ac.uk/OfTheDay/oftheday-09-11/	## Mathematicians Of The Day ### 11th September On this day in 1913, Henry F Baker, addressed the British Association for the Advancement of Science on The Place of Pure Mathematics. See THIS LINK. Click on for a poster. #### Quotation of the day ##### From James Jeans From the intrinsic evidence of his creation, the Great Architect of the Universe now begins to appear as a pure mathematician. The Mysterious Universe	2022-08-17 10:38:01	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8527055382728577, "perplexity": 3236.980269284214}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1659882572898.29/warc/CC-MAIN-20220817092402-20220817122402-00693.warc.gz"}
https://byjus.com/question-answer/a-open-ended-mercury-manometer-is-used-to-measure-the-pressure-exerted-by-a-trapped/	Question # A open ended mercury manometer is used to measure the pressure exerted by a trapped gas as shown in the figure. Initially manometer shows no difference in mercury level in both columns as shown in diagram. After sparking $$'A'$$ dissociates according to following reaction $$A(g) \rightarrow B(g) + 3C (g)$$If pressure of Gas $$"A"$$ decreases to $$0.9 \:atm$$. Then:(Assume temperature to be constant and is $$300 K$$) A total pressure increased to 1.3atm B total pressure increased by 0.3atm C total pressure increased by 22.3cm of Hg D difference in mercury level is 228mm. Solution ## The correct options are A total pressure increased to $$1.3\: atm$$ B total pressure increased by $$0.3\: atm$$ D difference in mercury level is $$228\: mm$$.The change in the number of moles $$\Delta n = 4-1=3$$Initial pressure is equal to 76 $$cm$$ of mercury or 1 $$atm$$.$$1-0.9=0.1$$ $$atm$$ of $$A$$ will react to form products which will exert a pressure of 0.4 $$atm$$.Total pressure increased to $$1.3\: atm$$Total pressure increased by $$0.3\: atm$$The difference in mercury level is $$228 \: mm$$.$$P_{T}=(1 +3 \times 0.1)=1.3\: atm$$$$\triangle P=0.3\:atm$$ or $$76 \times 0.3\: cm \: of \: Hg$$or $$760 \times 0.3 =228 \:mm\: of\: Hg$$Chemistry Suggest Corrections 0 Similar questions View More People also searched for View More	2022-01-28 05:34:28	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6473573446273804, "perplexity": 1890.449488832262}, "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/1642320305420.54/warc/CC-MAIN-20220128043801-20220128073801-00661.warc.gz"}
https://math.stackexchange.com/questions/1771835/random-points-on-a-sphere-expected-angular-distance	# Random points on a sphere — expected angular distance Suppose we randomly select $n>1$ points on a sphere (all independent and uniformly distributed). • What is the expected angular distance from a point to its closest neighbor? • What is the expected angular distance from a point to its $m^{\text{th}}$ closest neighbor (where $m<n$)? Let • $a = m -1$, $b = n - m - 1$, $d = a + b = n - 2$. • $\theta_{ij}$, $1 \le i \ne j \le n$ be the angular separation between point $x_i$ and $x_j$. • $\ell_m$ be the expected angular separation among a pair of $m^{th}$ nearest neighbors. • $\mathcal{E}_m$ be the event that $x_2$ is the $m^{th}$ nearest neighbor of $x_1$. • $\mathbf{1}_m$ be the indicator function for event $\mathcal{E}_m$. Since all points are equal, we can use any pair of points, say $x_1$ and $x_2$, as probe and $\ell_m$ will be equal to the expected value of $\theta_{12}$ subject to the constraint that event $\mathcal{E}_m$ happens. i.e. $$\ell_m = \mathbf{E}( \theta_{12} \| \mathcal{E}_m ) = \frac{\mathbf{E} ( \theta_{12}\mathbf{1}_m )}{\mathbf{P}( \mathcal{E}_m )} = \frac{\mathbf{E} ( \theta_{12}\mathbf{1}_m )}{\mathbf{E}( \mathbf{1}_m )}$$ For any $\theta \in [0,\pi]$, let $$p = \frac{1-\cos\theta}{2} \iff \theta = 2\sin^{-1}(p^{1/2})$$ When $\theta_{12} = \theta$, we have $$\mathbf{P}( \theta_{1k} < \theta \| \theta_{12} = \theta ) = \frac{1-\cos\theta}{2} = p \quad\text{ for any } 2 \le k \le n$$ Since these $n-2$ angular separations $\theta_{1k}$ are independent and there are $\displaystyle\;\binom{d}{a}$ ways of picking $a$ out of $d$ points. we have $$\mathbf{E}( \mathbf{1}_m \| \theta_{12} = \theta ) = \binom{d}{a}p^a(1-p)^b$$ Since $x_2$ are distributed uniformly over the sphere, the probability for $\theta \le \theta_{12} \le \theta + d\theta$ is proportional to $\sin\theta_{12} d\theta_{12} \propto dp$. We have $$\mathbf{E}( \theta_{12}\mathbf{1}_m ) = \displaystyle\;\binom{d}{a}\int_0^1 p^a (1-p)^b \theta_{12} dp \quad\text{ and }\quad \mathbf{E}( \mathbf{1}_m ) = \displaystyle\;\binom{d}{a}\int_0^1 p^a (1-p)^b dp$$ As a result, \begin{align} \ell_m &= m\binom{n-1}{m}\int_0^1 p^a (1-p)^b 2\sin^{-1}(p^{1/2})\,dp\\ &= m\binom{n-1}{m}\sum_{k=0}^a(-1)^k \binom{a}{k} \int_0^1 (1-p)^{b+k} 2\sin^{-1}(p^{1/2})\, dp\\ &= m\binom{n-1}{m}\sum_{k=0}^a(-1)^{a-k} \binom{a}{k} \int_0^1 (1-p)^{d-k} 2\sin^{-1}(p^{1/2})\, dp \end{align} For any $c \in \mathbb{N}$, we have \begin{align} \int_0^1 (1-p)^c 2\sin^{-1}(p^{1/2}) dp &= \frac{-1}{c+1}\int_0^1 2\sin^{-1}(p^{1/2})\, d(1-p)^{c+1}\\ &= \frac{-1}{c+1}\left\{\bigg[ 2\sin^{-1}(p^{1/2}) (1-p)^{c+1} \bigg]_0^1 - \int_0^1 \frac{(1-p)^{c+1}}{\sqrt{p(1-p)}} dp \right\}\\ &= \frac{1}{c+1}\frac{\Gamma(\frac12)\Gamma(c+\frac32)}{\Gamma(c+2)} = \frac{\pi}{(c+1)2^{2c+1}}\binom{2c+1}{c} \end{align} Form this, we get $$\ell_m = \pi m\binom{n-1}{m}\sum_{k=0}^{m-1} \frac{(-1)^{m-1-k}}{(n-k-1)2^{2n-2k-3}}\binom{m-1}{k} \binom{2n-2k-3}{n-k-2}$$ In particular, the expected nearest neighbor angular separation $\displaystyle\;\ell_1 = \frac{\pi}{2^{2n-3}}\binom{2n-3}{n-2}$. Simple cases: If we generate only two points, this is like fixing one as north pole and generating the other randomly. Here, the expected angular distance is "clearly" $\frac\pi2$ because points at angular distances $\theta$ are just as likely as points at distance $\pi-\theta$. The same argument holds for the $m$th closest neighbour when $m=\frac n2$. From the formula for the spherical cap, we can also conclude that $$P(X<\alpha) =\frac{1-\cos\alpha}{2}.$$ Then the probability that the closest of $n-1$ other points has distance $<\alpha$ from the north pole is $$P(\min\{X_1,\ldots, X_{n-1}\}<\alpha)=1-\prod_{k=1}^{n-1}(1-P(X_k<\alpha))=1-\frac{(1+\cos\alpha)^n}{2^n}.$$ It is certainly possible to compute the integral $E[\min\{X_1,\ldots,X_{n-1}\}]=\int_0^\pi\frac{(1+\cos\alpha)^n}{2^n}\,\mathrm d\alpha$, but I am not the mood right now ... • Can you explain how you compute $P(X<\alpha)=\frac{1-\cos(\alpha)}{2}$? Thanks. – Boby Nov 8 '16 at 21:05	2019-08-26 09:33:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 2, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9996022582054138, "perplexity": 459.0398964498366}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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-20190826085356-20190826111356-00465.warc.gz"}
https://www.esaral.com/q/how-many-terms-are-there-in-the-a-p-15937	# How many terms are there in the A.P. Question: (i) How many terms are there in the A.P. 7, 10, 13, ... 43? (ii) How many terms are there in the A.P. $-1,-\frac{5}{6},-\frac{2}{3},-\frac{1}{2}, \ldots, \frac{10}{3} ?$ Solution: (i) 7, 10, 13...43 Here, we have: a = 7 $d=(10-7)=3$ $a_{n}=43$ Let there be n terms in the given A.P. Also, $a_{n}=a+(n-1) d$ $\Rightarrow 43=7+(n-1) 3$ $\Rightarrow 36=(n-1) 3$ $\Rightarrow 12=(n-1)$ $\Rightarrow 13=n$ Thus, there are 13 terms in the given A.P. (ii) $-1,-\frac{5}{6},-\frac{2}{3},-\frac{1}{2}, \ldots, \frac{10}{3}$ Here, we have: $a=-1$ $d=\left(\frac{-5}{6}-(-1)\right)=\left(1-\frac{5}{6}\right)=\frac{1}{6}$ $a_{n}=\frac{10}{3}$ Let there be n terms in the given A.P. Also, $a_{n}=a+(n-1) d$ $\Rightarrow \frac{10}{3}=-1+(n-1) \frac{1}{6}$ $\Rightarrow \frac{13}{3}=(n-1) \frac{1}{6}$ $\Rightarrow 26=(n-1)$ $\Rightarrow 27=n$ Thus, there are 27 terms in the given A.P.	2023-03-26 06:08:10	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.46405380964279175, "perplexity": 1242.5753306849574}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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-20230326044821-20230326074821-00348.warc.gz"}
https://calconcalculator.com/math/pythagorean-theorem-calculator/	Hi folks, if you know the lengths of the other two sides of a right triangle, our Pythagorean Theorem Calculator will determine any missing sides’ length. Calculating the hypotenuse is part of this. The longest side of a right triangle is the hypotenuse, which is the side opposite the 90 degree angle. The hypotenuse formula, we may use it to get this side when solving for the hypotenuse. Remember that a right triangle is the one with a 90-degree angle. Because the value of the angles of any triangle added together must be 180 degrees, the other two angles must similarly total 90 degrees. Make sure to see other related calculators on our site, such as 45 45 90 Triangle, or 30 60 90 Triangle Calculator. ## What is the Pythagorean Theorem – Definition? The Pythagorean theorem states and describes how the three sides of a right triangle are related. Pythagoras defined the square of the hypotenuse (length opposite of 90 degree angle) as the sum of the squares of the other two sides of that triangle. Let’s look at it, its derivations, equations, and some solved instances. If a triangle is a 90 degrees one, the square of the hypotenuse equals the sum of the squares of the other two sides. Consider the triangle ABC, in which: BC^2= {AB^2+ AC^2} The hypotenuse is BC, the base is AB, and the altitude (height) is AC. It’s worth noting that the hypotenuse is a triangle’s longest side. Our calculator will provide you with all information and data you need. ## History of Pythagoras Theorem So the Pythagorean Theorem was one of the first theorems known to ancient societies. Pythagoras, a Greek mathematician, and philosopher is its founder. In Cortona, a Greek seaport in Southern Italy, Pythagoras founded the Pythagorean School of Math. Many of his contributions to mathematics are ascribed to him. However, some of them may have been the work of his pupils. Although Pythagoras popularized the theorem, there is enough evidence to suggest that it existed in other cultures 1000 years before Pythagoras. The earliest evidence originates from the Old Babylonian Period, between the 20th and 16th centuries B.C. According to legend, Pythagoras was so overjoyed when he discovered the theory that he sacrificed an ox. However, Pythagoras and his successors were profoundly upset by the subsequent revelation that the square root of 2 is irrational and cannot be stated as a ratio of two numbers. They were adamant that any two lengths were integral multiples of the same unit length. There have been several efforts to conceal that the square root of 2 is irrational. The individual who revealed the knowledge is even claimed to have perished at sea. ## Pythagoras Theorem Formula The square of the hypotenuse is equal to the sum of the squares of the sides of a right triangle. We can use the hypotenuse formula to describe Pythagoras theorem. If the hypotenuse is c and the sides of a right triangle are a and b, the equation is: c^2= {a^2+ b^2} ## Pythagorean Theorem: Algebraic Proof When two triangles’ two sides are equal to each other’s two sides, and the angles encompassed by those sides are equal, the triangles are congruent (side-angle-side). A triangle’s area is half of the area of any parallelogram with the same base and height. It uses a trapezoid instead of a square, which may be constructed by bisecting along with one of the inner square’s diagonals, as shown in the diagram, from the square in the second of the above proofs. The area of the trapezoid may be estimated as half of the square’s area, which is: Area \, of \, Trapezoid=\frac {1}{2} \times (b+a)^2 ## Pythagorean Triples Pythagorean triples are a collection of three positive numbers that fit into the formula, which is a2 + b2 = c2, where a, b, and c are all positive integers. So the longest side of the triangle is known as the hypotenuse, or ‘c,’ while the other two legs of the right triangle are ‘a’ and ‘b.’ The Pythagorean triples are denoted by the symbols (a, b, c). The most well-known Pythagorean triple example is (3, 4, 5). The Pythagorean triples set is infinite. The earliest Pythagorean triples are (3, 4, and 5). We can create a few extra triples by scaling them up in the following way. Taking values for n allows us to make as many triples as feasible. ## Pythagorean Theorem Calculator – How to Use? You’ll be able to utilize this incredible Pythagorean Theorem Calculator if you follow these simple steps. 1. Fill in the formula with the two lengths you have. Let’s say you know a = 4, b = 8, and you want to calculate the length of the hypotenuse c. Solve it now. 2. After plugging the numbers into the formula, we get 42+ 82 = c2. 3. To obtain 16 + 64 = c2, square each word. 4. To obtain 80 = c2, combine comparable words. Just simply enter the values in our calculator, and let it calculate for you! ## Pythagorean Theorem Calculator – Example A right-angled triangle’s hypotenuse is 16 units long, while one of the triangle’s sides is 8 units long. Substituting the provided dimensions into the calculator, consider the given side of a triangle as the perpendicular height = 8 units. Solve it: Hypotenuse^2= {Base^2+ Height^2} 162 = {B^2 + 82} B^2 = {256 - 64} B = {\sqrt{\smash[b]{192}}} = 13.856 ## FAQ What is the Pythagorean theorem used for? The Pythagorean Theorem is a handy method for determining the side lengths of a right triangle by adding the areas of three intersecting squares. Moreover, this theorem is a very helpful technique that serves as the foundation for more difficult trigonometry ideas, such as the inverse. There are also other uses in real world. How to solve the Pythagorean theorem? What is the square root? We can define square root as the sum of the sides squared is the length of the hypotenuse. Try using the formulas above for solution. Who created the Pythagorean theorem? In Cortona, a Greek seaport in Southern Italy, Pythagoras founded the Pythagorean School of Math. Many of his contributions to mathematics are ascribed to him. However, some of them may have been the work of his page pupils. Pythagoras’ most renowned mathematical contribution is the Pythagorean Theorem. How do you find the missing side of a triangle using the Pythagorean theorem? In a right triangle with c as the longest side, the theorem asserts that a2 + b2 = c2. You may use this equation to calculate the length if you know the lengths of the other two sides. The illustration depicts two right triangles with one side lacking their measure. Is the Pythagorean theorem only for right triangles? We may use the Pythagoras’ theorem to determine whether or not a triangle has a right angle because it only applies to right-angled triangles.	2022-05-24 12:06:31	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7888563871383667, "perplexity": 450.16872633358423}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1652662572800.59/warc/CC-MAIN-20220524110236-20220524140236-00458.warc.gz"}
https://quant.stackexchange.com/questions/40603/implied-volatility-of-a-call-plus-its-delta	# Implied Volatility of a call plus its delta I would like to understand if exists a smart way to imply the volatility from a quote that is the sum of a call and its delta: is there any method other than simple iterative minimization? • are you actually getting quotes as the sum of the call and the delta? that seems odd. Do you mean you're getting quotes as the option price, cross, and delta? i.e. 2.5/2.6 x84.2 10d ? – will Jul 4 '18 at 9:32	2019-03-23 06:38:46	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7423100471496582, "perplexity": 1262.9272380903642}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202728.21/warc/CC-MAIN-20190323060839-20190323082839-00144.warc.gz"}
https://en.m.wikiversity.org/wiki/Nonlinear_finite_elements/Homework11/Solutions/Problem_1/Part_8	# Nonlinear finite elements/Homework11/Solutions/Problem 1/Part 8 ## Problem 1: Part 8: Consistency condition - 2 The yield stress ${\displaystyle \sigma _{y}}$ is given by the Johnson-Cook model ${\displaystyle \sigma _{y}(\alpha ,T)=\left[\sigma _{0}+B\alpha ^{n}\right]\left[1-\left({\cfrac {T-T_{0}}{T_{m}-T_{0}}}\right)\right]}$ where ${\displaystyle \sigma _{0}}$ is the initial yield stress, ${\displaystyle B,n}$ are constants, ${\displaystyle T_{0}}$ is a reference temperature, and ${\displaystyle T_{m}}$ is the melt temperature. Derive expressions for ${\displaystyle \partial f/\partial \alpha }$ , and ${\displaystyle \partial f/\partial T}$ for the von Mises yield condition with the Johnson-Cook flow stress model. The yield function is ${\displaystyle f={\sqrt {\cfrac {3}{2}}}~{\sqrt {\mathbf {s} :\mathbf {s} }}-\sigma _{y}={\sqrt {\cfrac {3}{2}}}~{\sqrt {\mathbf {s} :\mathbf {s} }}-\left[\sigma _{0}+B\alpha ^{n}\right]\left[1-\left({\cfrac {T-T_{0}}{T_{m}-T_{0}}}\right)\right]}$ Therefore, ${\displaystyle {{\frac {\partial f}{\partial \alpha }}=f_{\alpha }=-n~B~\alpha ^{n-1}\left[1-\left({\cfrac {T-T_{0}}{T_{m}-T_{0}}}\right)\right]}}$ and ${\displaystyle {{\frac {\partial f}{\partial T}}=f_{T}=\left({\cfrac {1}{T_{m}-T_{0}}}\right)\left[\sigma _{0}+B\alpha ^{n}\right]~.}}$	2021-09-23 00:12:22	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 11, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8471094369888306, "perplexity": 6911.994205399761}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057403.84/warc/CC-MAIN-20210922223752-20210923013752-00459.warc.gz"}
https://www.transtutors.com/questions/discuss-the-nature-of-this-lease-transaction-from-the-viewpoints-of-both-lessee-and--96594.htm	# Discuss the nature of this lease transaction from the viewpoints of both lessee and lessor. 1 answer below » Winston Industries and Ewing Inc. enter into an agreement that requires Ewing Inc. to build three diesel-electric engines to Winston’s specifications. Upon completion of the engines, Winston has agreed to lease them for a period of 10 years and to assume all costs and risks of ownership. The lease is non-cancelable, becomes effective on January 1, 2011, and requires annual rental payments of $413,971 each January 1, starting January 1, 2011. Winston’s incremental borrowing rate is 10%. The implicit interest rate used by Ewing Inc. and known to Winston is 8%. The total cost of building the three engines is$2,600,000. The economic life of the engines is estimated to be 10 years, with residual value set at zero. Winston depreciates similar equipment on a straight-line basis. At the end of the lease, Winston assumes title to the engines. Collectibility of the lease payments is reasonably certain; no uncertainties exist relative to un-reimbursable lessor costs. (Round all numbers to the nearest dollar.) (a) Discuss the nature of this lease transaction from the viewpoints of both lessee and lessor. (b) Prepare the journal entry or entries to record the transaction on January 1, 2011, on the books of Winston Industries. (c) Prepare the journal entry or entries to record the transaction on January 1, 2011, on the books of Ewing Inc. (d) Prepare the journal entries for both the lessee and lessor to record the first rental payment on January 1, 2011. (e) Prepare the journal entries for both the lessee and lessor to record interest expense (revenue) at December 31, 2011. (Prepare a lease amortization schedule for 2 years.) (f) Show the items and amounts that would be reported on the balance sheet (not notes) at December 31, 2011, for both the lessee and the lessor. Subhrata R (a) The lease should be treated as a capital lease by Winston Industries requiring the lessee to capitalize the leased asset. The lease qualifies for capital lease accounting by the lessee because: (1) title to the engines transfers to the lessee, (2) the lease term is equal to the estimated life of the asset, and (3) the present value of the minimum lease payments exceeds 90% of the fair value of the leased engines. The transaction represents a purchase financed by installment payments over a 10-year period. For Ewing Inc. the transaction is a sales-type lease because a manufacturer's profit accrues to Ewing. This lease arrangement also represents the manufacturer's financing the transaction over a period of 10 years. Present Value of Lease Payments $413,971 X 7.24689...........................................................................$3,000,000 Present value of an annuity due at 8% for 10 years, rounded by $2. Dealer Profit Sales (present value of lease payments)..................................................$3,000,000 Less cost of engines.................................................................................. 2,600,000 Profit on sale............................................................................................. \$ 400,000 (b) Leased Engines Under Capital Leases................................... 3,000,000 Lease Liability................................................................. 3,000,000 (c) Lease...	2020-03-29 15:26:05	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.23438501358032227, "perplexity": 5453.685596715536}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1585370494349.3/warc/CC-MAIN-20200329140021-20200329170021-00479.warc.gz"}
https://www.nag.com/numeric/nl/nagdoc_27/flhtml/f07/f07twf.html	# NAG FL Interfacef07twf (ztrtri) ## 1Purpose f07twf computes the inverse of a complex triangular matrix. ## 2Specification Fortran Interface Subroutine f07twf ( uplo, diag, n, a, lda, info) Integer, Intent (In) :: n, lda Integer, Intent (Out) :: info Complex (Kind=nag_wp), Intent (Inout) :: a(lda,) Character (1), Intent (In) :: uplo, diag #include <nag.h> void f07twf_ (const char uplo, const char diag, const Integer n, Complex a[], const Integer lda, Integer info, const Charlen length_uplo, const Charlen length_diag) The routine may be called by the names f07twf, nagf_lapacklin_ztrtri or its LAPACK name ztrtri. ## 3Description f07twf forms the inverse of a complex triangular matrix $A$. Note that the inverse of an upper (lower) triangular matrix is also upper (lower) triangular. ## 4References Du Croz J J and Higham N J (1992) Stability of methods for matrix inversion IMA J. Numer. Anal. 12 1–19 ## 5Arguments 1: $\mathbf{uplo}$Character(1) Input On entry: specifies whether $A$ is upper or lower triangular. ${\mathbf{uplo}}=\text{'U'}$ $A$ is upper triangular. ${\mathbf{uplo}}=\text{'L'}$ $A$ is lower triangular. Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$. 2: $\mathbf{diag}$Character(1) Input On entry: indicates whether $A$ is a nonunit or unit triangular matrix. ${\mathbf{diag}}=\text{'N'}$ $A$ is a nonunit triangular matrix. ${\mathbf{diag}}=\text{'U'}$ $A$ is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be $1$. Constraint: ${\mathbf{diag}}=\text{'N'}$ or $\text{'U'}$. 3: $\mathbf{n}$Integer Input On entry: $n$, the order of the matrix $A$. Constraint: ${\mathbf{n}}\ge 0$. 4: $\mathbf{a}\left({\mathbf{lda}},*\right)$Complex (Kind=nag_wp) array Input/Output Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$. On entry: the $n$ by $n$ triangular matrix $A$. • If ${\mathbf{uplo}}=\text{'U'}$, $A$ is upper triangular and the elements of the array below the diagonal are not referenced. • If ${\mathbf{uplo}}=\text{'L'}$, $A$ is lower triangular and the elements of the array above the diagonal are not referenced. • If ${\mathbf{diag}}=\text{'U'}$, the diagonal elements of $A$ are assumed to be $1$, and are not referenced. On exit: $A$ is overwritten by ${A}^{-1}$, using the same storage format as described above. 5: $\mathbf{lda}$Integer Input On entry: the first dimension of the array a as declared in the (sub)program from which f07twf is called. Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$. 6: $\mathbf{info}$Integer Output On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6). ## 6Error Indicators and Warnings ${\mathbf{info}}<0$ If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated. ${\mathbf{info}}>0$ Element $〈\mathit{\text{value}}〉$ of the diagonal is exactly zero. $A$ is singular its inverse cannot be computed. ## 7Accuracy The computed inverse $X$ satisfies $XA-I≤cnεXA ,$ where $c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision. Note that a similar bound for $\left\|AX-I\right\|$ cannot be guaranteed, although it is almost always satisfied. The computed inverse satisfies the forward error bound $X-A-1≤cnεA-1AX .$ See Du Croz and Higham (1992). ## 8Parallelism and Performance f07twf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information. Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information. The total number of real floating-point operations is approximately $\frac{4}{3}{n}^{3}$. The real analogue of this routine is f07tjf. ## 10Example This example computes the inverse of the matrix $A$, where $A= 4.78+4.56i 0.00+0.00i 0.00+0.00i 0.00+0.00i 2.00-0.30i -4.11+1.25i 0.00+0.00i 0.00+0.00i 2.89-1.34i 2.36-4.25i 4.15+0.80i 0.00+0.00i -1.89+1.15i 0.04-3.69i -0.02+0.46i 0.33-0.26i .$ ### 10.1Program Text Program Text (f07twfe.f90) ### 10.2Program Data Program Data (f07twfe.d) ### 10.3Program Results Program Results (f07twfe.r)	2021-03-08 09:43:55	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 56, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9567515254020691, "perplexity": 4313.659184530225}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1614178383355.93/warc/CC-MAIN-20210308082315-20210308112315-00498.warc.gz"}
http://www.resci.cn/EN/abstract/abstract43226.shtml	Resources Science ›› 2019, Vol. 41 ›› Issue (6): 1131-1140. • Orginal Article • Spatial interpolation of mean temperature of Chongqing Municipality considering solar radiation correction Zhiming HE1,2(), Yuechen LI3,4(), Xianfeng JIN1,2, Xian LIU1,2, Xiaobo HE1,2 1. 1. Chongqing Geomatics Center, Chongqing 401147, China 2. Chongqing Engineering Laboratory of Spatio-temporal Big Data Technology Research and Application, Chongqing 401147, China 3. School of Geography and Tourism, Chongqing Normal University, Chongqing 401331, China 4. Key Laboratory of GIS Application, Chongqing Municipal Education Commission, Chongqing 401331, China • Received:2018-11-21 Revised:2018-12-24 Online:2019-06-25 Published:2019-06-25 Abstract: The mountainous regions of Southwest China, where Chongqing Municipality is located, has typical regional environmental characteristics such as cloudy fog and less sunshine. In order to realize the spatial simulation of temperature in this geographical environment, this study proposes a model for local regression considering terrain correction factor for solar radiation. In this model, the terrain correction factor is derived indirectly by fitting the spatial distribution of global solar radiation under undulating terrain. The model combines the Geographically Weighted Regression model, the Solar Analyst model, the improved Angtrom-Prescott equation, and the multiple linear regression method. Based on temperature, relative humidity, sunshine percentage, and global solar radiation of the meteorological stations, combined with DEM data with a resolution of 100 m×100 m, this model realizes the spatial simulation of temperature under the mountainous terrain. The model has good fitting accuracy and stability. The simulation accuracy of local regression term is much higher than Inverse Distance Weighting (IDW) interpolation and Kriging interpolation. It is also better than the traditional Multivariate Llinear Regression model based on latitude, longitude, altitude, sunshine percentage, and relative humidity. Further, 55 regional meteorological stations are used to verify the summer temperature simulation accuracy of a single year. The average absolute error is 0.59°C, and the errors of 38 meteorological stations are reduced after considering the terrain correction factor. The model performs well in spatial and temporal simulation of air temperature, which can reflect the influence of local terrain factors such as slope, aspect, and topographic occlusion on temperature, and has clear physical meaning. Based on the available observation data of meteorological stations, DEM, and the commercial software ArcGIS, this model is convenient to apply, especially suitable for cloudy, sunless areas like Chongqing and its surrounding mountainous regions.	2022-11-30 14:02:41	{"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": 0.23994146287441254, "perplexity": 6020.2306855943}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710764.12/warc/CC-MAIN-20221130124353-20221130154353-00506.warc.gz"}
https://math.stackexchange.com/questions/1158998/how-to-use-the-inverse-function-theorem-to-prove-f-is-a-diffeomorphism/1159001	# How to use The Inverse Function Theorem to prove $f$ is a diffeomorphism? I've proven that the function $f: U=(0,\infty)\times \mathbb R \rightarrow \mathbb R^2$ given by $f(x,y) = (x, y^3 + xy)$ is injective and surjective ($f(U) = U$), so it is bijective. I've computed $\frac {\partial f_1}{\partial x} = 1$, $\frac {\partial f_1}{\partial y} = 0$, $\frac {\partial f_2}{\partial x} = y$ and $\frac {\partial f_2}{\partial y} = 3y^2+x$. Can someone tell me how to apply The Inverse Function Theorem to prove $f$ is a diffeomorphism ? ($f$ is smooth is easy to see, I've shown it is bijective). I need only to verify it has a smooth inverse function $f^{-1}: U \rightarrow U$. I find the theorem confusing to apply. You compute the determinant of the Jacobian matrix (of which you computed already its components). The result is $1\times(2y^2+x)-0\times y=2y^2+x$. Then you can see that this is never zero on $U$ ($x>0$ and $y^2\geq0$). • So this function $g$ one gets from TIFT must be equal to $f^{-1}$ by the property that $f(g(v))=v$ for all $v$ in the small open region in $f(U)$ ? And since $g$ is smooth, this implies $f^{-1}$ is smooth ? – Shuzheng Feb 21 '15 at 18:10 • Yes, if $f$ has an inverse and $f(g(v))=v$ then $g$ is that inverse. – user218170 Feb 21 '15 at 18:13 • Ahh, and since the region is open around this function $g$, it satisfies the definition of taking the derivative :) Thanks ! – Shuzheng Feb 21 '15 at 18:21	2020-02-17 09:55:07	{"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": 0.9751676321029663, "perplexity": 127.33968477629028}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1581875141806.26/warc/CC-MAIN-20200217085334-20200217115334-00082.warc.gz"}
https://gateoverflow.in/196023/madeeasy-test-series-2018-theory-computation-determinism	+1 vote 77 views isn't the ans is 3 i.e p,q,r edited \| 77 views 0 Yes ! Correct answer is $3$ i think they have omitted $'p'$ which should also be included. 0 thanks!!	2020-01-22 23:09:23	{"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": 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": 0.8748069405555725, "perplexity": 3468.9881955411306}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1579250607596.34/warc/CC-MAIN-20200122221541-20200123010541-00237.warc.gz"}
https://web2.0calc.com/questions/x-2-2x-1-x-2-3-2-6-x-2-2x	+0 # ((x-2)/(2x)+(1)/(x+2))/((3)/(2)-(6)/(x^(2)+2x)) 0 63 3 ((x-2)/(2x)+(1)/(x+2))/((3)/(2)-(6)/(x^(2)+2x)) How would I solve this? Please give step by step directions on how to solve Guest Oct 17, 2017 Sort: #1 +18712 +2 How would I solve this? ((x-2)/(2x)+(1)/(x+2))/((3)/(2)-(6)/(x^(2)+2x)) $$\begin{array}{\|rcll\|} \hline && \mathbf{\dfrac{ \dfrac{x-2}{2x} + \dfrac{1}{x+2} } { \dfrac{3}{2}- \dfrac{6} {x^{2}+2x} } } \quad & \| \quad x^{2}+2x = x(x+2) \\\\ &=& \dfrac{ \dfrac{x-2}{2x} + \dfrac{1}{x+2} } { \dfrac{3}{2}- \dfrac{6} {x(x+2)} } \\\\ &=& \dfrac{ \dfrac{(x-2)(x+2) + 1\cdot 2x}{2x(x+2)} } { \dfrac{3x(x+2)-2\cdot 6}{2x(x+2)} } \\\\ &=& \dfrac{\left[~(x-2)(x+2) + 1\cdot 2x ~\right]}{2x(x+2)} \cdot \dfrac{2x(x+2)} {\left[~ 3x(x+2)-2\cdot 6 ~\right] } \\\\ &=& \dfrac{(x-2)(x+2) + 1\cdot 2x} {3x(x+2)-2\cdot 6} \\\\ &=& \dfrac{(x-2)(x+2) + 2x} {3x(x+2)-12} \\\\ &=& \dfrac{x^2-4 + 2x } {3x^2+6x-12} \\\\ &=& \dfrac{1}{3} \cdot \dfrac{\left(x^2+2x-4\right)} {\left(x^2+2x-4\right)} \\\\ &\mathbf{=}& \mathbf{ \dfrac{1}{3}} \\ \hline \end{array}$$ heureka Oct 17, 2017 #2 0 i dont understand the last two steps can u pls explain Guest Oct 18, 2017 #3 +18712 +3 i dont understand the last two steps can u pls explain $$\begin{array}{\|rcll\|} \hline && \dfrac{{\color{red}(x-2)(x+2)} + 2x} {3x(x+2)-12} \quad & \| \quad {\color{red}(x-2)(x+2)} = x^2 + 2x-2x- 2\cdot 2 = x^2 -4 \\\\ &=& \dfrac{x^2-4 + 2x } {3x^2+6x-12} \\\\ &=& \dfrac{x^2-4 + 2x } {{\color{red}3}x^2+2\cdot {\color{red}3}\cdot x-({\color{red}3}\cdot 4)} \\\\ &=& \dfrac{x^2-4 + 2x } { {\color{red}3}\cdot ( x^2+2\cdot x-( 4))} \\\\ &=& \dfrac{x^2-4 + 2x } { {\color{red}3}\cdot ( x^2-4+2x)} \\\\ &=& \dfrac{(x^2-4 + 2x) } { {\color{red}3}\cdot ( x^2-4+2x)} \quad &\| \quad \dfrac{(x^2-4 + 2x) } { ( x^2-4+2x)} = 1 \\\\ &=& \dfrac{1} { {\color{red}3}} \cdot 1 \\\\ &=& \dfrac{1} { {\color{red}3}} \\\\ \hline \end{array}$$ heureka Oct 18, 2017 ### 6 Online Users We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners. See details	2017-11-20 11:29:30	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8872044682502747, "perplexity": 5209.011192669239}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1510934806030.27/warc/CC-MAIN-20171120111550-20171120131550-00364.warc.gz"}
https://abdullahkhalid.com/blog/2022/Oct/30/algorithms-for-code-concatenation/	Algorithms for code concatenation My goal is to compute fault-tolerance thresholds for stabililzer codes. To this end, we need to understand how to concatenate quantum codes. Briefly, code concatenation is when a state is encoded first by one code, then the encoded qubits again by same or another code. The resultant qubits may be encoded again by another code, and so on. The entire concatenation of codes can be treated as a single code, which has a larger number of physical qubits than any of the individual codes but also a larger a distance. In our study of code concatenation, we are interested in the following: • What are the generators of the concatenated code? • How to perform encoding and logical operations of the code? • How to do error correction. Today, I will be mostly interested in the first question. My development will follow the algorithms presented by Gaitan [1], though I have made significant advances in the clarity of the arguments. First foray: a two layer concatenation Suppose, we encode $k_1$ qubits using the code $C_1 \sim [[n_1,k_1,d_1]]$. This yields $n_1$ encoded qubits. We then take each of these qubits, and encode each one separately using $C_2 \sim [[n_2, k_2 = 1, d_2]]$. We will refer to the concatenated code as $C = C_2 \circ C_1$. This process can be visualized as follows. The numbers indicate how many elements are in the box to their right. Let's first consider the question of generators. The generators of $C_1$ are $\{g_i^1, i=0,\dots,n_1-k_1-1\}$ and $C_2$ are $\{g_i^2, i=0,\dots,n_2-2\}$. What are the generators of $C$? To answer this, let $\ket{\psi}_{k_1}$ be a state of $k_1$ qubits that is to be encoded. The first encoding is written as $\ket{\psi}_{k_1} \stackrel{C_1}{\to} \ket{\bar\psi}_{n_1}.$ We note for now that for all $i$, $g_i^1\ket{\bar\psi} = \ket{\bar\psi}.$ Now, for the second round of encoding, we will separate each of the $n_1$ qubits and encode each one using $C_1$. To write this, let $\lambda = (\lambda_0, \lambda_1,\dots, \lambda_{n_1-1})$. Then $\ket{\bar\psi} = \sum_{\lambda_i = 0,1} \alpha_\lambda \ket{\lambda_0}_1 \otimes \ket{\lambda_1}_1 \otimes \cdots \otimes \ket{\lambda_{n_1-1}}_1.$ When we encode each qubit, we obtain, $\ket{\bar\psi} \stackrel{C_2}{\to} \ket{\bar{\bar\psi}} = \sum_{\lambda_i = 0,1} \alpha_\lambda \ket{\bar\lambda_0}_{n_2} \otimes \ket{\bar\lambda_1}_{n_2} \otimes \cdots \otimes \ket{\bar\lambda_{n_1-1}}_{n_2},$ where the bars in $\ket{\bar\lambda_i}_{n_2}$ indicate that it is an encoded state, and the subscript indicates that it has $n_2$ qubits. To summarize, let us call each block of $n_2$ qubits $E(b)$, where $b=0,\dots,n_1-1$ because there are $n_1$ such blocks. Generators In this way, our doubly encoded state is a $n_1n_2$ qubit state. Overall, we have encoded $k_1$ qubits into $n=n_1n_2$ qubits. This means that the concatenated code must have $n_1n_2-k_1$ generators. Let's find them. (1) Consider a generator $g^2_l$ of $C_2$, which is a $n_2$ qubit operator, but we have to be careful over which qubits it acts. For instance, we can write $g^2_l \otimes I_{n-n_2}$ to obtain an operator that acts on the first $n_2$ qubits of $n$ total qubits. Similarly, we can write $I_{n_2} \otimes g^2_l \otimes I_{n-2n_2}$ to obtain an operator that acts on the next $n_2$ qubits. This notation will get tedious fast, so let's just write $g^2_l[i,j]$ to indicate that $g^2_l$ acts on qubits $i, i+1, \dots, j-1$, where $j-i = n_2$. Now, it's quite easy to see from definition that for all $i$, $g^2_l[n_2 i, n_2 (i+1)] \ket{\bar\lambda_i}_{n_2} = \ket{\bar\lambda_i}_{n_2},$ i.e. if $g^2_l$ is made to act on the $i$-th block of $n_2$ qubits in the final block, it will be a stabilizer operation. In other words, we have discovered that the set $\{g^2_l[n_2 i, n_2(i+1)], l=0,\dots,n_2-2, i=0, \dots, n_1-1\}$ stabilizes the state $\ket{\bar{\bar\psi}}$. We can count that there are $n_1(n_2-1)$ such operators. (2) But wait, there are more operators that stabilize the state $\ket{\bar{\bar\psi}}$. Let $\xi_2$ be the encoding operation for $C_2$. Then $\xi_2^{\otimes n_1}$ is the encoding operations for the second step. Conversely, $\left(\xi_2^{\otimes n_1}\right)^\dagger$ is the decoding operation. Then, $\left(\xi_2^{\otimes n_1}\right)^\dagger\ket{\bar{\bar\psi}} = \ket{\bar\psi}$. We noted earlier that $g^1_l$ stabilizes this state. Hence, $\xi_2^{\otimes n_1}g^1_l\left(\xi_2^{\otimes n_1}\right)^\dagger\ket{\bar{\bar\psi}} = \ket{\bar{\bar\psi}}.$ Hence, the set $\{\xi_2^{\otimes n_1}g^1_l\left(\xi_2^{\otimes n_1}\right)^\dagger, l = 0, \dots, n_1-k_1-1\}$ must also stabilize the doubly encoded state. There are $n_1-k_1$ such operators. In total, we have $n_1(n_2-1) + (n_1-k_1) = n_1n_2-k_1$ stabilizers. As we have found sufficient number of stabilizers, we are done. How to put this mathematics into practice Let's try to construct the algorithm that creates the generators of the concatenated code. We will depend on the vector representation of the Pauli operators. (1) Constructing the first set of generators is very simple. We just have to do some shifting. Note that $g^2_l$, an operator on $n_2$ qubits, is represented by a vector of length $2n_2$. We have to construct, $g^2_l[n_2 i, n_2(i+1)]$, an operator over $n$ qubits, which is represented by a vector of length $2n$. The code is right there is the notation. Call $g=g^2_l$. Then # for each of the E(b) blocks for b in range(n_1): for each g: encoded_g = zeros((1,2n)) # for the X part encoded_g[n_2b: n_2(b+1)] = g[:n2] # for the Z part encoded_g[n + n_2b: n + n_2(b+1)] = g[n2:] (2) For the second set, we have to do a little bit of work to understand encoding. If we use $C_2$ to encode a qubit, it takes the logical operators $X, Z$ of the qubit and maps them to the encoded $\bar{X} = \xi_2X\xi_2^\dagger, \bar{Z} = \xi_2 X\xi_2^\dagger$. In vector notation, $X=(1\|0)$ and $Z=(0\|1)$ and these are mapped to vectors of length $2n_2$. This mapping is already known and we have constructed it before. Now, we have $g^1_l$ which is an operator over $n_1$ qubits, which means it is the product of $n_1$ different Pauli operators, i.e. $g^1_l = P_0 \otimes P_1 \otimes \cdots \otimes P_{n_1-1}$. It is represented by a vector of length $2n_1$. So $\{\xi_2^{\otimes n_1}g^1_l\left(\xi_2^{\otimes n_1}\right)^\dagger = \xi_2P_0\xi_2^\dagger \otimes \xi_2P_1\xi_2^\dagger \otimes \cdots \otimes \xi_2P_{n_1-1}\xi_2^\dagger.$ Our algorithm is now clear. We iterate over the operators in $g=g^1_l$ and encode each according to the mapping $\bar{X} = \xi_2X\xi_2^\dagger, \bar{Z} = \xi_2 X\xi_2^\dagger$. We have to be careful about the shifting as before, because $\xi_2P_i\xi_2^\dagger$ acts on qubits $n_2 i$ to $n_2 (i+1) - 1$. # declare # logical_x # logical_z for each g: encoded_g = zeros((1,2n_1n_2)) for i in range(n1): if g[i] and g[n+i]: #P_i is the Y operator # multipling operators corresponds to adding their vectors mod 2 encoded_g[n2i: n2(i+1)] += logical_x[:n2] + logical_z[:n2] encoded_g[n + n2i: n + n2(i+1)] += logical_x[n2:] + logical_z[n2:] elif g[i]: # P_i is the X operator encoded_g[n2i: n2(i+1)] += logical_x[:n2] encoded_g[n + n2i: n + n2(i+1)] += logical_x[n2:] elif g[n+i] # P_i is the Z operator encoded_g[n2i: n2(i+1)] += logical_z[:n2] encoded_g[n + n2i: n + n2(i+1)] += logical_z[n2:] That wasn't too difficult. $k_2$ does not have to be $1$, but $k_2$ divides $n_1$ Let us now relax the assumption that $k_2=1$, but let it be an interger that divides $n_1$. This will complicate the encoding in the second round. As before, the first round is $\ket{\psi}_{k_1} \stackrel{C_1}{\to} \ket{\bar\psi}_{n_1}.$ Because $n_1$ is divisible by $k_2$, these $n_1$ qubits can be divided into $n_1/k_2$ blocks, each of size $k_2$. Let the blocks of qubits be called $B(b)$ for $b=0,\dots,n_1/k_2-1$. Let $\gamma = (\gamma_0,\dots,\gamma_{n_1/k_1-1})$, and let $B$ be the basis of a $2^{k_2}$ dimensional Hilbert space. We can write the first-round encoded state as $\ket{\bar\psi}_{n_1} = \sum_{\ket{\gamma_i}\in B} \alpha_\gamma \ket{\gamma_0}_{k_2} \otimes \ket{\gamma_1}_{k_2} \otimes \cdots \otimes \ket{\gamma_{n_1/k_2 - 1}}_{k_2},$ broken up into a sum over $k_2$-qubit basis states. Now, we encode each block using $C_2$, to create a new encoded block $E(b)$ of size $n_2$. This creates the final state $\ket{\bar\psi}_{n_1} \stackrel{C_2}{\to}\ket{\bar{\bar\psi}} = \sum_{\ket{\gamma_i}\in B} \alpha_\gamma \ket{\bar\gamma_0}_{n_2} \otimes \ket{\bar\gamma_1}_{n_2} \otimes \cdots \otimes \ket{\bar\gamma_{n_1/k_2 - 1}}_{n_2}.$ This final state has $n_1/k_2$ blocks each of size $n_2$ qubits, for a total of $n = n_1n_2/k_2$ qubits. Hence, in this code $k_1$ qubits are encoded to $n_1n_2/k_2$ qubits, which is fewer than before. The structure of the generators is similar to what we had above. (1) If we apply the operator $g^2_l$ to the $b$-th encoded block $E(b)$, when the qubits are in state $\ket{\bar{\bar\psi}}$ we find that it stabilizes the state. Hence, it must be a stabilizer of the doubly encoded state. As before the $n_2$ sized operator $g^2_l$ when acting on the $b$-th block is labelled $g^2_l[n_2 b, n_2 * (b+1)]$. It's algorithmic implementation is as before, except the change in the number of blocks $E(b)$. (2) What about the $g^1_l$ associated generators? Things are a bit more convoluted because of the grouping of the qubits. Now, note that $\xi_2$ is an operator that takes $k_2$ qubits to $n_2$ qubits, and in the second encoding, we apply $\xi_2$ to each group $B(b)$ of qubits. This means that $\xi_2^{\otimes n_1/k_2}\ket{\bar\psi} = \ket{\bar{\bar\psi}}.$ Hence, our stabilizers are given by $\xi_2^{\otimes n_1/k_2}g^1_l\left(\xi_2^{\otimes n_1/k_2}\right)^\dagger\ket{\bar{\bar\psi}} = \ket{\bar{\bar\psi}}.$ How do we process $\xi_2^{\otimes n_1/k_2}g^1_l\left(\xi_2^{\otimes n_1/k_2}\right)^\dagger$? If, as before, $g^1_l = P_0 \otimes P_1\otimes \cdots \otimes P_{n_1}$, then we break it up according to $B(b)$ into chunks of $k_2$, to form $g^1_l = (P_0 \otimes \dots \otimes P_{k_2-1}) \otimes (P_{k_2} \otimes \dots \otimes P_{2k_2-1}) \otimes \dots \otimes (P_{(n_1/k_2-1)k_2} \otimes \dots \otimes P_{n_1-1}).$ Then, $\xi_2^{\otimes n_1/k_2}g^1_l\left(\xi_2^{\otimes n_1/k_2}\right)^\dagger = \xi_2(P_0 \otimes \dots \otimes P_{k_2-1})\xi_2^\dagger \otimes \dots \otimes \xi_2(P_{(n_1/k_2-1)k_2} \otimes \dots \otimes P_{n_1-1})\xi_2^\dagger.$ What do these arcane incantations mean? Just concentrate on one to-be-encoded block B(b), which has $k_2$ qubits. $k_2$ qubits have $k_2$ many $X$ operators, one for each qubit; $k_2$ many $Z$ operators, etc. $\xi_2$ encodes these into encoded operators. So $\xi X_j \xi^\dagger = \bar{X}_j \quad i=0,\dots,k_2-1,$ $\xi Z_j \xi^\dagger = \bar{Z}_j \quad i=0,\dots,k_2-1,$ $\xi P_j \xi^\dagger = \bar{P}_j \quad i=0,\dots,k_2-1, \quad P_i = X_i, Z_i.$ This means that in the operators $\xi_2(P_{k_2b} \otimes \dots \otimes P_{k_2(b+1)-1} )\xi_2^\dagger$ acting on $B(b)$, the $j$-th operator $P_j$ is encoded to $\bar{P}_j$. This results in $\xi_2(\bar{P}_{k_2b} \otimes \dots \otimes \bar{P}_{k_2(b+1)-1} )\xi_2^\dagger$, which act on the $E(b)$ block. Recall that the $B(b)$ block corresponds to qubits $[k_2b, k_2(b+1)]$ and $E(b)$ corresponds to qubits $[n_2b, n_2(b+1)]$. The algorithm acquires an additional loop over the blocks # declare # logical_xs # logical_zs encoded_g = zeros((1,2n)) # iterate over each block for b in range(n1/k2): # extract the B(b) block from the generator g_block_x = g[k2b: k2(b+1)] g_block_z = g[n1+k2b:n1+k2(b+1)] # now iterate over the qubits in the block for i in range(k2): if g_block_x[i] and g_block_z[i]: #P_i is the Y operator # place the $i$-th encoded logical gate into the E(b) block in the encoded_g encoded_g[n2b: n2(b+1)] += logical_xs[i][:n2] + logical_zs[i][:n2] encoded_g[n + n2b: n + n2(b+1)] += logical_xs[i][n2:] + logical_zs[i][n2:] elif g_block_x[i]: # P_i is the X operator encoded_g[n2b: n2(b+1)] += logical_xs[i][:n2] encoded_g[n + n2b: n + n2(b+1)] += logical_xs[i][n2:] elif g_block_z[i] # P_i is the Z operator encoded_g[n2i: n2(b+1)] += logical_zs[i][:n2] encoded_g[n + n2b: n + n2(b+1)] += logical_zs[i][n2:] Stac implements these algorithms, so you can concatenate arbitrary codes. Here is the example from the book reproduced. The code $[[4,2,2]]$ is concatenated with itself. Note that $k_2=2$ and $n_1=4$ so $k_2$ divides $n_1$. import stacimport numpy as np# define the codecd = stac.CommonCodes.generate_code('[[4,2,2]]')# book uses a different set of logical operators# than stac computes using the Gottesman methodcd.logical_xs = np.array( [[1, 0, 1, 1, 0, 0, 1, 1], [1, 0, 1, 0, 0, 0, 0, 1]])cd.logical_zs = np.array( [[1, 0, 1, 0, 1, 1, 1, 0], [0, 1, 0, 0, 0, 0, 1, 1]])# this will create a new code object,# and immediately compute the generator matrixconcat_code = stac.ConcatCode((cd, cd))# display the generator matrixstac.print_paulis(concat_code.generator_matrix)# answer is correct $\displaystyle XZZXIIII$ $\displaystyle YXXYIIII$ $\displaystyle IIIIXZZX$ $\displaystyle IIIIYXXY$ $\displaystyle XXXXZZZZ$ $\displaystyle YZXXIXIY$ Let $k_2$ not divide $n_1$ Things are getting a bit more complicated. If we encode a block of $k_1$ qubits into a block of $n_1$ using $C_1$, then there is no way of dividing up the $n_1$ qubits into $k_2$-sized blocks that can be encoded using $C_2$. (1) The solution is to start with $k_2$ blocks of $k_1$ qubits each, which we will label as q(c), for $c=1,\dots,k_2$. So our initial state will be $\ket{\Psi} = \ket{\psi^0}_{k_1} \otimes \ket{\psi^1}_{k_1} \otimes \cdots \otimes \ket{\psi^{k_2-1}}_{k_1},$ where there are $k_2$ terms in the product. We have a total of $k_2k_1$ qubits. Note that this steps means we have given up on the ability to encode an unknown quantum state, for which we only have one copy. However, for quantum computational algorithms, we rarely start with an unknown state, so this is a not a big impediment. (2) Next, we encode each block $q(c)$ using $C_1$ into a block $Q(c)$ of $n_2$ qubits. We obtain $\ket{\Psi} \stackrel{C_1}{\to} \ket{\bar\Psi} = \ket{\bar\psi^0}_{n_1} \otimes \ket{\bar\psi^1}_{n_1} \otimes \cdots \otimes \ket{\bar\psi^{k_2-1}}_{n_1},$ which has a total of $k_2n_1$ qubits. (3) For the next step, we will need to rearrange the qubits. We are going to pick the first qubit in each $Q(c)$ and make that one block, take the second qubit in each $Q(c)$ and make those a block, and so on. Each new block will be called $B(b)$. Since, there are $k_2$ many $Q(c)$ blocks, each block $B(b)$ will of size $k_2$ as well, And because the size of $Q(c)$ is $n_1$, there will be $n_1$ many $B(b)$ blocks, with $b=0,\dots,n_1-1$. To make this explicit, we operate on the states we created above. First, we expand one term in the product above in some basis to obtain $\ket{\bar\psi^j}_{n_1} = \sum_{\lambda_i = 0,1}\alpha_\lambda\ket{\lambda_0^j\lambda_1^j\cdots\lambda^j_{n_1-1}}_{n_1}.$ Then, $\ket{\bar\Psi} = \sum_{\lambda^0_i = 0,1}\alpha_\lambda^0\ket{\lambda^0_0\lambda^0_1\cdots\lambda^0_{n_1-1}}_{n_1} \otimes \cdots \otimes \sum_{\lambda^{k_2-1}_i = 0,1}\alpha_\lambda^{k_2-1}\ket{\lambda^{k_2-1}_0\lambda^{k_2-1}_1\cdots\lambda^{k_2-1}_{n_1-1}}_{n_1}.$ We can now rearrange these to obtain, $\ket{\bar\Psi} = \sum_{\lambda^0_i,\dots,\lambda^{k_2-1}= 0,1}\alpha_\lambda^0\cdots\alpha_\lambda^{k_2-1} \ket{\lambda^0_0\lambda^1_0\cdots \lambda^{k_2-1}_0}_{k_2} \otimes \cdots \otimes \ket{\lambda^0_{n_2-1}\lambda^1_{n_2-1}\cdots \lambda^{k_2-1}_{n_2-1}}_{k_2}.$ To summarize, this state has $n_1$ blocks of size $k_2$ for a total of $n_1k_2$ qubits. (4) We can now encode each block using $C_2$. Hence, $\ket{\bar\Psi} \stackrel{C_2}{\to} \ket{\bar{\bar\Psi}} = \sum_{\lambda^0_i,\dots,\lambda^{k_2-1}= 0,1}\alpha_\lambda^0\cdots\alpha_\lambda^{k_2-1} \ket{\overline{\lambda^0_0\lambda^1_0\cdots \lambda^{k_2-1}_0}}_{n_2} \otimes \cdots \otimes \ket{\overline{\lambda^0_{n_2-1}\lambda^1_{n_2-1}\cdots \lambda^{k_2-1}_{n_2-1}}}_{n_2}.$ This state now has $n_1$ blocks $E(b)$ of size $n_2$, for a total of $n_1n_2$ qubits. In summary, this concatenated code encodes $k=k_1k_2$ qubits into $n=n_1n_2$ qubits. Obtaining the generators We are now going to determine the generators of the final encoded state. Note, that we need a total of $n_1n_2-k_1k_2$ generators. (1) As before, the generators of $C_2$ must by definition stabilize the state of each block $E(b)$, $g^2_l\ket{\overline{\lambda^0_0\lambda^1_0\cdots \lambda^{k_2-1}_0}}_{n_2} = \ket{\overline{\lambda^0_0\lambda^1_0\cdots \lambda^{k_2-1}_0}}_{n_2}.$ Hence, we assign a copy of all generators $g^2_l$ for each block $E(b)$ to our collection. This means $n_1(n_2-k_2)$ generators. The algorithm is the same, except for the number of $E(b)$ blocks. (2) The encoding of the $C_1$ generators is much more subtle. If $g^1_l$ is applied to a $Q(c)$ block, it stabilizes the state $\ket{\bar\psi}$. But in the subsequent re-ordering, $Q(c)$ is broken up. To start with an example, consider $g^1_l$ acting on $Q(0)$, $g^l_1 = P_0 \otimes P_1 \otimes \cdots \otimes P_{n_1-1}.$ Where do the qubits that these $P_i$ act on go after the re-ordering. Since, this is the $0$-th block, $P_i$ now acts on the $0$-th position in the $B(i)$ block. Subsequently, when we encode each $B(i)$ using $C_2$, each of the $P_i$ will map the the corresponding $0$-th logical operator. Meaning if $P_i=X$, then after encoding it should be $\bar{X}_0$. In the same way, if $g^1_l$ acts on $Q(c)$, then all its constitutive Pauli operators $P_i$ go to the $c$-th location in $B(i)$ and hence should be mapped to $\bar{P}_c$. The algorithm for this is as follows: # declare # logical_xs # logical_zs # iterate over each block Q(c) for c in range(k2): for each g: encoded_g = zeros((1,2n)) # now iterate over the qubits in g for i in range(n1): if g[i] and g[n+i]: #P_i is the Y operator # place the c-th encoded logical gate into the E(b) block in the encoded_g encoded_g[n2i: n2(i+1)] += logical_xs[c][:n2] + logical_zs[c][:n2] encoded_g[n + n2i: n + n2(i+1)] += logical_xs[c][n2:] + logical_zs[c][n2:] elif g[i]: # P_i is the X operator encoded_g[n2i: n2(i+1)] += logical_xs[c][:n2] encoded_g[n + n2i: n + n2(i+1)] += logical_xs[c][n2:] elif g[n+i] # P_i is the Z operator encoded_g[n2i: n2(i+1)] += logical_zs[i][:n2] encoded_g[n + n2i: n + n2(i+1)] += logical_zs[c][n2:] We can also reproduce the example from the book, in which $C_1=[[5,13]]$ and $C_2 = [[4,2,2]]$. cd2 = stac.CommonCodes.generate_code('[[5,1,3]]')concat_code = stac.ConcatCode((cd2, cd))stac.print_paulis(concat_code.generator_matrix) $\displaystyle XZZXIIIIIIIIIIIIIIII$ $\displaystyle YXXYIIIIIIIIIIIIIIII$ $\displaystyle IIIIXZZXIIIIIIIIIIII$ $\displaystyle IIIIYXXYIIIIIIIIIIII$ $\displaystyle IIIIIIIIXZZXIIIIIIII$ $\displaystyle IIIIIIIIYXXYIIIIIIII$ $\displaystyle IIIIIIIIIIIIXZZXIIII$ $\displaystyle IIIIIIIIIIIIYXXYIIII$ $\displaystyle IIIIIIIIIIIIIIIIXZZX$ $\displaystyle IIIIIIIIIIIIIIIIYXXY$ $\displaystyle XIYYYZYIYZYIXIYYIIII$ $\displaystyle IIIIXIYYYZYIYZYIXIYY$ $\displaystyle XIYYIIIIXIYYYZYIYZYI$ $\displaystyle YZYIXIYYIIIIXIYYYZYI$ $\displaystyle XIXZIXZZIXZZXIXZIIII$ $\displaystyle IIIIXIXZIXZZIXZZXIXZ$ $\displaystyle XIXZIIIIXIXZIXZZIXZZ$ $\displaystyle IXZZXIXZIIIIXIXZIXZZ$ Stac can just put the two algorithms together and concatenate an arbitrary number of codes. concat_code = stac.ConcatCode((cd, cd2, cd2))print(concat_code) A [[100,2]] code The book shows that if $C_1$ has distance $d_1$ and $C_2$ has distance $d_2$, then the concatenated code $C$ has distance at least $d_1d_2$. We will look into this more next time.	2022-12-05 14:34:02	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 273, "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": 0.9191775918006897, "perplexity": 468.08770341901254}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711017.45/warc/CC-MAIN-20221205132617-20221205162617-00825.warc.gz"}
https://www.darwinproject.ac.uk/letter/?docId=letters/DCP-LETT-4747.xml;query=12;brand=default;hit.rank=9	From Asa Gray 17 January 1865 Cambridge, Mass. Jany, 17th. 1865. Dear Darwin Yours of 26th Dec. just received—long en route— must have crossed one from me,1—yet I am not sure. Only the separate copy of your Lythrum paper came by the post,2 (& that I have not yet read), so I suppose I have lost Scott’s & Cruger’s papers.3 I am sorry that the Cuckoos are not more satisfactory.4 I wonder that my letter to Dr. Brewer has brought no response5 You are mistaken in thinking the Fish-men here (in U.S.) are all Agassizian. 6 I understand there is a perfect hatred between all of them, (Gill, Girard of Washington, & Ayres of California) and Agassiz.7 But I know nothing of the calibre of these people. The new Herbm. building is finished & in occupation (costing Mr. Thayer $12250—in depreciated money, to be sure), and perfectly satisfactory. 8 But the supporting fund—small at best—lacks$1000 or so of being filled.— will come in time, and I hope more.— for I want a curator.9 People have much & many things to give for now. At present we are feeding Savannah—while the rebels are starving our men (prisoners) in the interior of the country.10 Do you not begin to believe that we shall put down the rebellion, restore the Union, and do away with Slavery?11 Heartily do I wish you a prosperous year, and continually improving health—& power to work—and less discomfort— Also—tho’ a small matter—I give you joy over the Copley Medal, which R.S. honors itself in giving to you.12 Ever \| A. Gray CD annotations 7.1 Heartily … year,] ‘(Herbarium)’ added pencil Footnotes The letter to Gray has not been found. Gray last wrote to CD on 5 December 1864 (see Correspondence vol. 12). ‘Three forms of Lythrum salicaria was published in the Journal of the Linnean Society (Botany) 8 (1865): 169–96. Author’s offprints of the paper were ready in December; Gray’s name appears on CD’s presentation list for the paper (see Correspondence vol. 12, Appendix III). Gray may refer to Scott 1864a and Scott 1864c; he also refers to Crüger 1864. These papers were communicated to the Linnean Society by CD, and published with CD’s ‘Three forms of Lythrum salicaria in the 1865 volume of the society’s journal. CD had already sent Gray a copy of Scott 1864b (see Correspondence vol. 12, letter to Asa Gray, 13 September [1864]). CD had expressed his interest in Crüger 1864 in his letter to Asa Gray, 25 February [1864] (Correspondence vol. 12). CD had requested information about cuckoos in an enclosure to his letter to Asa Gray of 29 October [1864] (Correspondence vol. 12). The enclosure has not been found. CD’s query was evidently prompted by an article in the 8 October 1864 issue of the Spectator, which discussed the instinct of cuckoos (Cuculus canorus) to lay their eggs in other birds’ nests as evidence of God’s design in nature. For CD’s views on the parasitic behaviour of cuckoos, see Origin, pp. 216–18, Origin 4th ed., pp. 260–62, and Natural selection, pp. 506–8. See also Correspondence vol. 12, letter from Asa Gray, 5 December 1864, which includes a note from Henry Bryant on the habits of American cuckoos. Gray had also written to Thomas Mayo Brewer for information about cuckoos (see Correspondence vol. 12, letter from Asa Gray, 5 December 1864). In his letter of 7 November 1864 (Correspondence vol. 12), Benjamin Dann Walsh had remarked on Louis Agassiz’s popularity and influence in the United States. See also letter to B. D. Walsh, 4 December [1864]. Agassiz’s extensive work in ichthyology is discussed in Lurie 1960. Agassiz was one of the leading opponents of Darwin’s transmutation theory in America (see Lurie 1960, Winsor 1979, Morris 1997, and Correspondence vols. 8 and 9). Beginning in 1863, he began to use the resources of the museum of comparative zoology at Harvard to collect fish specimens from throughout the world, and in 1865 he led an expedition to the Amazon River with the object of gathering evidence against CD’s transmutation theory (see Lurie 1960, pp. 336–7, and Winsor 1991, pp. 66–76). Gray refers to Theodore Nicholas Gill, Charles Frédéric Girard, and William Orville Ayres. Agassiz’s controversial professional style and deteriorating relations with his students in the 1860s are discussed in Winsor 1991, pp. 27–42, 44–65. Construction of the new herbarium at Harvard University was largely funded by the Boston financier Nathaniel Thayer (see Dupree 1959, pp. 327–8). The building of the herbarium is discussed in the letters from Asa Gray, 16 February 1864 and 11 July 1864 (Correspondence vol. 12). Gray eventually appointed Horace Mann as curator in 1866 (see Dupree 1959, p. 337). Savannah, Georgia, had fallen to the Union army on 21 December 1864 (Denney 1992, pp. 506–7). Prison conditions are discussed in McPherson 1988, pp. 800–2; the suffering of Union prisoners is attributed largely to the shortage of resources and deteriorating economy in the South. CD and Gray corresponded at length on the American Civil War (see Correspondence vols. 9–12). CD had long been opposed to slavery. See Journal of researches 2d ed., pp. 499–500, Colp 1978, and Browne 1995, pp. 196–9, 213–14, 244–6. For CD’s discussions with Gray on slavery, see Correspondence vols. 9–11 and this volume, letter to Asa Gray, 19 April [1865]. CD was presented with the Copley Medal of the Royal Society on 30 November 1864 (see Proceedings of the Royal Society of London 13 (1863–4): 505, and Correspondence vol. 12, Appendix IV). Bibliography Browne, Janet. 1995. Charles Darwin. Voyaging. Volume I of a biography. New York: Alfred A. Knopf. Colp, Ralph, Jr. 1978. Charles Darwin: slavery and the American Civil War. Harvard Library Bulletin 26: 471–89. Correspondence: The correspondence of Charles Darwin. Edited by Frederick Burkhardt et al. 27 vols to date. Cambridge: Cambridge University Press. 1985–. Crüger, Hermann. 1864. A few notes on the fecundation of orchids and their morphology. [Read 3 March 1864.] Journal of the Linnean Society (Botany) 8 (1865): 127–35. Denney, Robert E. 1992. The civil war years: a day-by-day chronicle of the life of a nation. New York: Sterling Publishing. Dupree, Anderson Hunter. 1959. Asa Gray, 1810–1888. Cambridge, Mass.: Belknap Press of Harvard University. Journal of researches 2d ed.: Journal of researches into the natural history and geology of the countries visited during the voyage of HMS Beagle round the world, under the command of Capt. FitzRoy RN. 2d edition, corrected, with additions. By Charles Darwin. London: John Murray. 1845. Lurie, Edward. 1960. Louis Agassiz: a life in science. Chicago: University of Chicago Press. McPherson, James M. 1988. Battle cry of freedom: the Civil War era. New York and Oxford: Oxford University Press. Morris, Paul J. 1997. Louis Agassiz’s arguments against Darwinism in his additions to the French translation of the Essay on classification. Journal of the History of Biology 30: 121–34. Natural selection: Charles Darwin’s Natural selection: being the second part of his big species book written from 1856 to 1858. Edited by R. C. Stauffer. Cambridge: Cambridge University Press. 1975. Origin 4th ed.: On the origin of species by means of natural selection, or the preservation of favoured races in the struggle for life. 4th edition, with additions and corrections. By Charles Darwin. London: John Murray. 1866. Origin: On the origin of species by means of natural selection, or the preservation of favoured races in the struggle for life. By Charles Darwin. London: John Murray. 1859. ‘Three forms of Lythrum salicaria’: On the sexual relations of the three forms of Lythrum salicaria. By Charles Darwin. [Read 16 June 1864.] Journal of the Linnean Society (Botany) 8 (1865): 169–96. [Collected papers 2: 106–31.] Winsor, Mary Pickard. 1979. Louis Agassiz and the species question. Studies in History of Biology 3: 89–117. Winsor, Mary Pickard. 1991. Reading the shape of nature. Comparative zoology at the Agassiz museum. Chicago and London: University of Chicago Press. Summary New herbarium is finished. Congratulations on Copley Medal. Letter details Letter no. DCP-LETT-4747 From Asa Gray To Charles Robert Darwin Sent from Cambridge, Mass. Source of text DAR 165: 146 Physical description 2pp †	2021-01-22 12:40:08	{"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": 0.4806862473487854, "perplexity": 8676.709552523149}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703529331.99/warc/CC-MAIN-20210122113332-20210122143332-00722.warc.gz"}
https://math.stackexchange.com/questions/561210/simple-algebra-equation	# Simple Algebra Equation I have a simple part of a question to solve. The problem is my answer is different to the solution in my textbook. The equation is: $$\frac{5v}{6} = \frac{(\frac{1}{2}a+b+\frac{1}{2} c)v}{a+b+c}$$ I am supposed to get $$\frac{2}{3}(a+b+c) = b$$ But I simply get: $$b=2a +2c$$ I get my answer by cross multiplying. I then use my answer to get $\frac{b}{a+b+c}$ as some fraction. I have not worked onto this stage as I am unsure about the above work. What am I doing wrong here? $b=2a+2c$ Adding $2b$ on both sides gives $b + 2b =2a+2b+2c$ Or better $3b =2a+2b+2c$ And if you throw the $3$ to the other side you get: $b =\frac{2}{3}(a+b+c)$ $b= 2a+2c \implies 3b=2a+2c+2b \implies b= 2(a+b+c)/3$	2021-04-15 02:35:40	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8535224795341492, "perplexity": 129.46506443902243}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1618038082988.39/warc/CC-MAIN-20210415005811-20210415035811-00167.warc.gz"}
http://www.hawaiilibrary.net/articles/eng/Toeplitz_matrix	#jsDisabledContent { display:none; } My Account \| Register \| Help # Toeplitz matrix Article Id: WHEBN0000151569 Reproduction Date: Title: Toeplitz matrix Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date: ### Toeplitz matrix In linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to right is constant. For instance, the following matrix is a Toeplitz matrix: \begin{bmatrix} a & b & c & d & e \\ f & a & b & c & d \\ g & f & a & b & c \\ h & g & f & a & b \\ i & h & g & f & a \end{bmatrix}. Any n×n matrix A of the form A = \begin{bmatrix} a_{0} & a_{-1} & a_{-2} & \ldots & \ldots &a_{-n+1} \\ a_{1} & a_0 & a_{-1} & \ddots & & \vdots \\ a_{2} & a_{1} & \ddots & \ddots & \ddots& \vdots \\ \vdots & \ddots & \ddots & \ddots & a_{-1} & a_{-2}\\ \vdots & & \ddots & a_{1} & a_{0}& a_{-1} \\ a_{n-1} & \ldots & \ldots & a_{2} & a_{1} & a_{0} \end{bmatrix} is a Toeplitz matrix. If the i,j element of A is denoted Ai,j, then we have A_{i,j} = A_{i+1,j+1} = a_{i-j}.\ ## Contents • Solving a Toeplitz system 1 • General properties 2 • Discrete convolution 3 • Infinite Toeplitz Matrix 4 • Notes 6 • References 7 ## Solving a Toeplitz system A matrix equation of the form Ax=b\ is called a Toeplitz system if A is a Toeplitz matrix. If A is an n\times n Toeplitz matrix, then the system has only 2n−1 degrees of freedom, rather than n2. We might therefore expect that the solution of a Toeplitz system would be easier, and indeed that is the case. Toeplitz systems can be solved by the Levinson algorithm in Θ(n2) time.[1] Variants of this algorithm have been shown to be weakly stable (i.e. they exhibit numerical stability for well-conditioned linear systems).[2] The algorithm can also be used to find the determinant of a Toeplitz matrix in O(n2) time.[3] A Toeplitz matrix can also be decomposed (i.e. factored) in O(n2) time.[4] The Bareiss algorithm for an LU decomposition is stable.[5] An LU decomposition gives a quick method for solving a Toeplitz system, and also for computing the determinant. Algorithms that are asymptotically faster (in finite arithmetic, i.e., given a tolerance \epsilon the exact solution is obtained within the tolerance \epsilon) than those of Bareiss and Levinson have been described in the literature.[6][7][8][9] ## General properties A Toeplitz matrix may be defined as a matrix A where Ai,j = ci−j, for constants c1−ncn−1. The set of n×n Toeplitz matrices is a subspace of the vector space of n×n matrices under matrix addition and scalar multiplication. Two Toeplitz matrices may be added in O(n) time and multiplied in O(n2) time. Toeplitz matrices are persymmetric. Symmetric Toeplitz matrices are both centrosymmetric and bisymmetric. Toeplitz matrices are also closely connected with Fourier series, because the multiplication operator by a trigonometric polynomial, compressed to a finite-dimensional space, can be represented by such a matrix. Similarly, one can represent linear convolution as multiplication by a Toeplitz matrix. Toeplitz matrices commute asymptotically. This means they diagonalize in the same basis when the row and column dimension tends to infinity. ## Discrete convolution The convolution operation can be constructed as a matrix multiplication, where one of the inputs is converted into a Toeplitz matrix. For example, the convolution of h and x can be formulated as: y = h \ast x = \begin{bmatrix} h_1 & 0 & \ldots & 0 & 0 \\ h_2 & h_1 & \ldots & \vdots & \vdots \\ h_3 & h_2 & \ldots & 0 & 0 \\ \vdots & h_3 & \ldots & h_1 & 0 \\ h_{m-1} & \vdots & \ldots & h_2 & h_1 \\ h_m & h_{m-1} & \vdots & \vdots & h_2 \\ 0 & h_m & \ldots & h_{m-2} & \vdots \\ 0 & 0 & \ldots & h_{m-1} & h_{m-2} \\ \vdots & \vdots & \vdots & h_m & h_{m-1} \\ 0 & 0 & 0 & \ldots & h_m \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ \vdots \\ x_n \end{bmatrix} y^T = \begin{bmatrix} h_1 & h_2 & h_3 & \ldots & h_{m-1} & h_m \end{bmatrix} \begin{bmatrix} x_1 & x_2 & x_3 & \ldots & x_n & 0 & 0 & 0& \ldots & 0 \\ 0 & x_1 & x_2 & x_3 & \ldots & x_n & 0 & 0 & \ldots & 0 \\ 0 & 0 & x_1 & x_2 & x_3 & \ldots & x_n & 0 & \ldots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ldots & \vdots & \vdots & \ldots & 0 \\ 0 & \ldots & 0 & 0 & x_1 & \ldots & x_{n-2} & x_{n-1} & x_n & \vdots \\ 0 & \ldots & 0 & 0 & 0 & x_1 & \ldots & x_{n-2} & x_{n-1} & x_n \end{bmatrix}. This approach can be extended to compute autocorrelation, cross-correlation, moving average etc. ## Infinite Toeplitz Matrix A bi-infinite Toeplitz matrix (i.e., entries indexed by \mathbb Z\times\mathbb Z, see below) A induces a linear operator on \ell^2. A=\begin{bmatrix} \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ \ldots & a_0 & a_{-1} & a_{-2} & a_{-3} & \ldots \\ \ldots & a_1 & a_0 & a_{-1} & a_{-2} & \ldots \\ \ldots & a_2 & a_1 & a_0 & a_{-1} & \ldots \\ \ldots & a_3 & a_2 & a_1 & a_0 & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ \end{bmatrix}. The induced operator is bounded if and only if the coefficients of the Toeplitz matrix A is the Fourier coefficients of some essentially bounded function f. In such cases, f is called the symbol of the Toeplitz matrix A, and the spectral norm of the Toeplitz matrix A coincides with the L^{\infty} norm of its symbol. The proof is easy to establish and can be found as Theorem 1.1 in the google book link: [10] ## Notes 1. ^ Press et al. 2007, §2.8.2—Toeplitz matrices 2. ^ Krishna & Wang 1993 3. ^ Monahan 2011, §4.5—Toeplitz systems 4. ^ Brent 1999 5. ^ Bojanczyk et al. 1995 6. ^ Stewart 2003 7. ^ Chen et al. 2006 8. ^ Chan & Jin 2007 9. ^ Chandrasekeran et al. 2007 10. ^ Albrecht Böttcher & Sergei M. Grudsky 2012 ## References • Bareiss, E.H. (1969), "Numerical solution of linear equations with Toeplitz and vector Toeplitz matrices", • Bojanczyk, A.W.; Brent, R.P.; Hoog, F.R. De; Sweet, D.R. (1995), "On the stability of the Bareiss and related Toeplitz factorization algorithms", • Brent R.P. (1999), "Stability of fast algorithms for structured linear systems", Fast Reliable Algorithms for Matrices with Structure (editors—T. Kailath, A.H. Sayed), ch.4 (SIAM). • Chan, R. H.-F.; Jin, X.-Q. (2007), An Introduction to Iterative Toeplitz Solvers, . • Chandrasekeran, S.; Gu, M.; Sun, X.; Xia, J.; Zhu, J. (2007), "A superfast algorithm for Toeplitz systems of linear equations", • Chen, W.W.; Hurvich, C.M.; Lu, Y. (2006), "On the correlation matrix of the discrete Fourier transform and the fast solution of large Toeplitz systems for long-memory time series", • Golub G.H., van Loan C.F. (1996), Matrix Computations (Johns Hopkins University Press) §4.7—Toeplitz and Related Systems. • Gray R.M., Toeplitz and Circulant Matrices: A Review (Now Publishers). • Krishna, H.; Wang, Y. (1993), "The Split Levinson Algorithm is weakly stable", . • Monahan, J.F. (2011), Numerical Methods of Statistics, . • Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. (2007), . • Stewart, M (2003), "A superfast Toeplitz solver with improved numerical stability", • Albrecht Böttcher; Sergei M. Grudsky (2012), Toeplitz Matrices, Asymptotic Linear Algebra, and Functional Analysis, Birkhäuser, pp. 1–, This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002. Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.	2020-03-28 13:53:38	{"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": 0.9997244477272034, "perplexity": 832.1653543290839}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1585370491998.11/warc/CC-MAIN-20200328134227-20200328164227-00020.warc.gz"}
https://www.physicsforums.com/threads/local-coordinates-physical-coordinates.183065/	Local coordinates, physical coordinates 1. Sep 5, 2007 mrandersdk As far as I can understand space is af manifold with some metric on it. A manifold is described with some charts (coordinates), but how do I relate these coordinates with ex. physical coordinates of some particle. Is it like this: if I'm in some laboratory I make some cartesian coordinate system (x,y,z) (maybe include time (t,x,y,z)), so that I can say that my particle is at p_0 = (x_0,y_0,z_0). Then my task is to find the metric in my laboratory coordinate system, so I can for example calculate the geodesic for my particle. But p=(x,y,z) should be functions to my manifold, that is p: R^3 -> M, and how should they look. As you might see I have a bit trouble understanding, how to relate the physical coordinates to the manifold, so that the description of GR becomes useful. Hope someone can help me understand it. Thanks in Advance, Anders Berthelsen.	2017-08-21 09:13:12	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8786908984184265, "perplexity": 878.0985765424547}, "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-2017-34/segments/1502886107744.5/warc/CC-MAIN-20170821080132-20170821100132-00679.warc.gz"}
http://libros.duhnnae.com/2017/sep3/150540829094-Liquidliquid-two-phaseflow-patterns-andmasstransfer-characteristics-inrectangular-glassmicroreactors.php	# Liquid–liquid two-phaseflow patterns andmasstransfer characteristics inrectangular glassmicroreactors Liquid–liquid two-phaseflow patterns andmasstransfer characteristics inrectangular glassmicroreactors - Descarga este documento en PDF. Documentación en PDF para descargar gratis. Disponible también para leer online. Published in: Chemical Engineering Science, vol. 63, p. 4035-4044 Publication date: 2008 The flow of two immiscible fluids was investigated in rectangular glass microchannels with equivalent diameters of 269 and 400$\mu m$. Deionised water, dyed toluene and hexane were selected as probe fluids. Flow patterns were obtained for Y- and T-junction of two micro-channels and monitored by a photo-camera. Volumetric velocities of water and organic phase varied between 1 and 6ml/h. The formation mechanism of slug and parallel flow was studied and the mass transfer performances of two flow patterns were compared. The shape of the interface between the immiscible liquids was controlled by a competition between the viscous forces and the local interfacial tension. The flow patterns could be correlated with the mean Capillary and Reynolds numbers. The mass transfer coefficients for parallel and slug flow were determined using instantaneous eutralisation (acid–base) reaction. The two flow patterns showed the same global volumetric mass transfer coefficients in the range of 0. 2–0. 5 ${s}^{-1}$, being affected mainly by the base concentration in water for parallel flow and by the linear velocity in the case of the slug flow. Keywords: Microstructure ; Multiphase flow ; Hydrodynamics ; Flow pattern ; Mass transfer ; Capillary/Reynolds numbers Reference LGRC-ARTICLE-2009-003doi:10.1016/j.ces.2008.05.005View record in Web of Science Autor: Dessimoz, Anne-laure; Cavin, Laurent; Renken, Albert; Kiwi-Minsker, Lioubov Fuente: https://infoscience.epfl.ch/record/134068?ln=en	2018-07-21 09:17:47	{"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": 0.4731256365776062, "perplexity": 8854.550833306605}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1531676592475.84/warc/CC-MAIN-20180721090529-20180721110529-00019.warc.gz"}
https://tex.stackexchange.com/questions/295615/how-to-use-the-new-yorker-font-in-regular-latex/295770	# How to use “The New Yorker” font in regular LaTeX How can I use "The New Yorker" font by Allen R. Walden (see Fontspace) in regular LaTeX (not in LuLaTeX or XeLaTeX). • It is a lot of work to create the fd-, tfm-, ... files that are needed. However, what is the reason that you do not want to use lualatex. – user2478 Feb 24 '16 at 7:31 • LuLaTex not always works with texmaker and Mac since ugrading to El Capitan – feder80 Feb 24 '16 at 7:41 • Please be more specific as to what "not always works" about LuaLaTeX. Do you get error and/or warning messages? If so, what do these messages say? Which version of TeXmaker do you use? – Mico Feb 24 '16 at 8:31 • Nothing happens. But I think I just found the reason: the path was wrong. no that I have changed it to "/usr/local/texlive/2014/bin/universal-darwin/" it works. So, you would recommend using LuLaTeX? – feder80 Feb 24 '16 at 8:36 • Whatever else you may do, I strongly recommend you upgrade to MacTeX2015. – Mico Feb 24 '16 at 22:06 There are two ways to do this: ### New school Use autoinst on NEWYORKR.TTF. It creates all the files you need to use the font with (pdf)LaTeX. You can invoke it like this: autoinst -encoding=T1 \ -ts1 \ -noupdmap \ -nooldstyle \ -noproportional \ -nosmallcaps \ -noswash \ -notitling \ -nosuperior \ -noinferiors \ -nofractions \ -noornaments \ -target=./Install \ -verbose \ NEWYORKR.TTF Installed properly, you can have the following result: \documentclass{article} \usepackage{fonttable,textcomp} \usepackage[T1]{fontenc} \usepackage{NewYorker} \begin{document} \xfonttable{T1}{NewYorker-TLF}{m}{n} \clearpage \xfonttable{TS1}{NewYorker-TLF}{m}{n} \end{document} ### Old school Create the afm and pfb files from the ttf-file and use fontinst which is somewhat more complex. You have to write a drv file like: \input fontinst.sty \needsfontinstversion{1.926} \recordtransforms{fny-rec.tex} \substitutesilent{bx}{m} \substitutesilent{b}{m} \transformfont{fnyr8r}{\reencodefont{8r}{\fromafm{NewYorker}}} \installfonts \installfamily{T1}{fny}{} \installfont{fnyr8t}{fnyr8r,newlatin}{t1}{T1}{fny}{m}{n}{} \endinstallfonts \installfonts \installfamily{TS1}{fny}{} \installfont{fnyr8c}{fnyr8r,textcomp}{ts1}{TS1}{fny}{m}{n}{} \endinstallfonts \endrecordtransforms \bye and a map generator file like: \input finstmsc.sty \resetstr{PSfontsuffix}{.pfb} \input fny-rec.tex \donedrivers \bye Then you have to run tex on the drv and map files plus: for filename in .pl; do pltotf $filename; done for filename in .vpl; do vptovf$filename; done Properly installed, you get: \documentclass{article} \usepackage{fonttable,textcomp} \usepackage[T1]{fontenc} \renewcommand{\rmdefault}{fny} \renewcommand{\familydefault}{\rmdefault} \begin{document} \xfonttable{T1}{fny}{m}{n} \clearpage \xfonttable{TS1}{fny}{m}{n} \end{document} Next thing would be teach fontinst to grab the write glyphs from the fonts where the name in afm does not match fontinst expectation. • Hm, in which folder should I run the code? I have to run it in the Terminal of my Mac, or? – feder80 Feb 25 '16 at 8:20 • @feder80 - I did it in a temporary folder where I created also a sub-directory ./Install before running the code. I ran it in bash (I'm not on Mac.) – Arash Esbati Feb 25 '16 at 8:50 • Ok, looks better, but I get some warnings and errors: "otfinfo: verbose: No such file or directory" or "otftotfm: warning: assuming --no-type1 since this font is TrueType-flavored [WARNING] 'otftotfm' returned non-zero; something's wrong! at /Library/TeX/texbin/autoinst line 643." – feder80 Feb 25 '16 at 9:05 • @feder80 - Sorry, my bad, - is missing before verbose, I've updated my answer. I had the warning as well, it worked anyways. – Arash Esbati Feb 25 '16 at 10:28 • the installation works now, but in LaTeX I get the following errror: "! LaTeX Error: File `NewYorker.sty' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty)" – feder80 Feb 25 '16 at 12:53	2020-01-27 11:52:37	{"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": 0.6587517261505127, "perplexity": 6772.569172823798}, "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-2020-05/segments/1579251700675.78/warc/CC-MAIN-20200127112805-20200127142805-00000.warc.gz"}
https://insidelearningmachines.com/gradient_boosting_regression_in_python/	## Motivation for Gradient Boosting Regression in Python In the previous post, we covered how Gradient Boosting works, and outlined the general algorithm for this ensemble technique. Gradient Boosting was initially developed by Friedman 2001, and the general algorithm is referred to as Algorithm 1: Gradient_Boost, in that paper. Furthermore, we also discussed how to develop a practical Gradient Boosting procedure, based upon the absolute difference loss function, and Decision Tree weak learners. In this article, we will work through an implementation of this algorithm, named Algorithm 3: LAD_TreeBoost in Friedman 2001. We will implement this algorithm in Python, as this is the most popular programming language for machine learning applications. My hope here is that by working through an implementation of this algorithm, and testing our code against different regression models available from scikit-learn, we can gain a deeper intuition for how these powerful algorithm work. The remainder of this article will be structured in the following way: ## Brief Review of the LAD_TreeBoost Algorithm Friedman’s 2001 paper describes a number of different applications of the Gradient_Boost algorithm. LAD_TreeBoost is one such application, and is considered to be a highly robust implementation of Gradient Boosting. For a more detailed description of these algorithms, please see my previous post. Here, we defined an ensemble with m = 1..M weak learners. We have defined our weak learners as regression Decision Trees, with j=1..J terminal nodes that correspond to R_{jm} terminal regions. Our loss has been chosen to be the absolute difference function. Our data will consist of predictors \bold{X} and targets \bold{y}. We will index through each unique sample in the data with n = 1..N \bold{X} = \begin{bmatrix} \bold{x}_1 \\ \bold{x}_2 \\ . \\ . \\ \bold{x}_n \\ . \\ \bold{x}_N \end{bmatrix}, \bold{y} = \begin{bmatrix} y_1 \\ y_2 \\ . \\ . \\ y_n \\ . \\ y_N \end{bmatrix}, where \bold{x}_n \in R^{d}, y_n \in R Note \bold{x}_n are the rows of matrix \bold{X}, and contain d features. Similarly, y_n are the scalar values of the column vector \bold{y}. Let’s now define the algorithm for LAD_TreeBoost: The training procedure is as follows: 1. E_0(\bold{x}) = median(\bold{y}) = E_0 2. For m=1 to M: 3. g_n = sign(y_n – E_{m-1}(\bold{x}_n)), for all n=1..N 4. Fit a J terminal node Decision Tree to \{ g_n,\bold{x}_n \}_1^N, and obtain the set of terminal regions \{ R_{jm} \}_1^J 5. \gamma_{jm} = median_{\bold{x}_n \in R_{jm}}(y_n – E_{m-1}(\bold{x}_n)), for all j = 1..J 6. E_m(\bold{x}) = E_{m-1}(\bold{x}) + \sum_{j=1}^J \gamma_{jm}I(\bold{x} \in R_{jm}) This procedure will yield the initial ensemble value E_0, the sets of parameters \{ R_{jm} \} and \{ \gamma_{jm} \}, and the set of M trained Decision Trees. Note that g_n are the gradients (from which the name of the algorithm is derived) and \gamma_{jm} are model weights. The identity matrix is represented by I. Predictions from this ensemble, on a set of input data \bold{x}’, can then be obtained by: 1. Initialise the ensemble to E_0 2. For m=1 to M: 3. E_m(\bold{x}’) = E_{m-1}(\bold{x}’) + \sum_{j=1}^J \gamma_{jm}I(\bold{x}’ \in R_{jm}) 4. Return E_M(\bold{x}’) ## Implement LAD_TreeBoost in Python We are now ready to implement Gradient Boosting regression in Python. I will encapsulate the logic described in the previous section into a single Python class. Let’s begin by importing the required packages to support our implementation: import numpy as np from typing import Dict, List, Tuple from sklearn.tree import DecisionTreeRegressor Our class definition, initialiser, and destructor functions are as follows: ## gradient boost tree regressor ## #initialiser def __init__(self, n_elements : int = 100, max_depth : int = 1) -> None: self.max_depth = max_depth self.n_elements = n_elements self.f = [] self.regions = [] self.gammas = [] self.mean_loss = [] self.e0 = 0 #destructor def __del__(self) -> None: del self.max_depth del self.n_elements del self.f del self.regions del self.gammas del self.mean_loss del self.e0 • __init__(self, n_elements, max_depth) : This is the initialiser that is called when a class instance is created. Input arguments include n_elements, which defines the size M of the ensemble, and max_depth, which defines the size of each constituent Decision Tree. • __del__(self) : This is the destructor function that is called whenever a class instance is deleted. All resources associated with the instance are cleaned up here. Next, we can define a private function to compute the ensemble weights \gamma_{jm}: #private function to group data points & compute gamma parameters def __compute_gammas(self, yp : np.array, y_train : np.array, e : np.array) -> Tuple[np.array,Dict]: #initialise global gamma array gamma_jm = np.zeros((y_train.shape[0])) #iterate through each unique predicted value/region regions = np.unique(yp) gamma = {} for r in regions: #compute index for r idx = yp == r #isolate relevant data points e_r = e[idx] y_r = y_train[idx] #compute the optimal gamma parameters for region r gamma_r = np.median(y_r - e_r) #populate the global gamma array gamma_jm[idx] = gamma_r #set the unique region <-> gamma pairs gamma[r] = gamma_r #append the regions to internal storage self.regions.append(regions) #return return((gamma_jm,gamma)) • __compute_gammas(self, yp, y_train, e) : Here we have a private function to compute the logic described in Step 5 of our training procedure. The inputs to this function include predictions yp from the mth weak learner, the training labels y_train (y_n), and the state of the ensemble e (E_{m-1}). The output from this function is a tuple containing the model parameters \gamma_{jm}. The first element in the tuple is a numpy array that is formatted already for the training update. The second element is a dictionary that will be stored internally, and can be used when generating predictions. We’re ready to define our public training procedure for this class: #public function to train the ensemble def fit(self, X_train : np.array, y_train : np.array) -> None: #reset the internal class members self.f = [] self.regions = [] self.model_weights = [] self.mean_loss = [] #initialise the ensemble & store initialisation e0 = np.median(y_train) self.e0 = np.copy(e0) e = np.ones(y_train.shape[0]) * e0 #loop through the specified number of iterations in the ensemble for _ in range(self.n_elements): #store mae loss self.mean_loss.append(np.mean(np.abs(y_train - e))) #compute the gradients of our loss function g = np.sign(y_train - e) #initialise a weak learner & train model = DecisionTreeRegressor(max_depth=self.max_depth) model.fit(X_train,g) #compute optimal gamma coefficients yp = model.predict(X_train) gamma_jm,gamma = self.__compute_gammas(yp,y_train,e) #update the ensemble e += gamma_jm #store trained ensemble elements self.f.append(model) self.gammas.append(gamma) • fit(self, X_train, y_train) : This public function takes in a matrix of predictors X_train (\bold{X}) and their associated labels y_train (\bold{y}). Within this function, Steps 1 through 6 of our training procedure are executed. The initial ensemble value E_0, the trained weak learners (effectively R_{jm}), and the weights \gamma_{jm}, are all stored internally. Finally, let’s now define the remaining member functions for this class: #public function to generate predictions def predict(self, X_test : np.array) -> np.array: #initialise predictions y_pred = np.ones(X_test.shape[0]) * np.copy(self.e0) #cycle through each element in the ensemble for model,gamma,regions in zip(self.f,self.gammas,self.regions): #produce predictions using model y = model.predict(X_test) #cycle through each unique leaf node for model m for r in regions: #updates for region r idx = y == r y_pred[idx] += gamma[r] #return predictions return(y_pred) #public function to return mean training loss def get_loss(self) -> List: return(self.mean_loss) #public function to return model parameters def get_params(self, deep : bool = False) -> Dict: return {'n_elements':self.n_elements, 'max_depth':self.max_depth} • predict(self, X_test) : A public function that is used to produce predictions on unseen data X_test (\bold{x}’). The predictions procedure, from the previous section, is run here. • get_loss(self) : Returns the mean absolute error loss, recorded at each iteration during training. • get_params(self, deep) : This is a public function to return the input arguments when an instance of this class is created. This function is required when using GradientBoostTreeRegressor instances in scikit-learn functions. ## Testing Performance We are now in a position to test our implementation of Gradient Boosting regression in Python. The data we will use here is the Diabetes Dataset available through scikit-learn. As I have already explored these data in my article on Adaboost regression, I will not repeat that analysis here. Let’s start by importing all the necessary packages to support our Gradient Boost implementation, and to conduct the analysis. Furthermore, we can load in the data: ## imports ## import numpy as np import pandas as pd from typing import Dict, List, Tuple import matplotlib.pyplot as plt from sklearn.datasets import load_diabetes from sklearn.model_selection import cross_validate from sklearn.tree import DecisionTreeRegressor from sklearn.metrics import mean_squared_error,mean_absolute_error,make_scorer from sklearn.ensemble import RandomForestRegressor, AdaBoostRegressor ## load regression dataset ## Note: rerunning the cells below will show some fluctuation in the results. ### Investigate Training Loss Here I want to look at how the training loss evolves as Decision Trees are added to the ensemble. In addition, I want to see what affect increasing the maximum depth, of the aforementioned Decision Trees, will have. The loss function displayed is the mean absolute error \sum_i^N abs(y^{pred}_i – y^{true}_i), at each iteration during training. Now I will declare an instance of our custom Gradient Boost tree regressor for a series of max_depth values. Following each declaration, I will train the model on the raw data features, since Decision Trees typically require little in the way of preparatory feature engineering: ## train the gradient boost regressor with default max_depth ## rgr.fit(dfX.values,sy.values) ## collect loss ## loss1 = rgr.get_loss() ## train the gradient boost regressor with max_depth = 2 ## rgr = GradientBoostTreeRegressor(n_elements=100, max_depth=2) rgr.fit(dfX.values,sy.values) ## collect loss ## loss2 = rgr.get_loss() ## train the gradient boost regressor with max_depth = 3 ## rgr = GradientBoostTreeRegressor(n_elements=100, max_depth=3) rgr.fit(dfX.values,sy.values) ## collect loss ## loss3 = rgr.get_loss() ## train the gradient boost regressor with max_depth = 4 ## rgr = GradientBoostTreeRegressor(n_elements=100, max_depth=4) rgr.fit(dfX.values,sy.values) ## collect loss ## loss4 = rgr.get_loss() A plot of the collected loss functions nicely illustrates the results: ## plot different training losses ## plt.plot(loss1,label='max_depth=1') plt.plot(loss2,label='max_depth=2') plt.plot(loss3,label='max_depth=3') plt.plot(loss4,label='max_depth=4') plt.title('Training Loss by Boosting Iteration') plt.xlabel('Number of Component Trees') plt.ylabel('MAE Loss') plt.legend() plt.show() It is apparent that each model starts with a large loss value, which then declines rapidly before flattening out. It isn’t too surprising to see that ensembles with deeper trees achieve lower training losses. These models are expressive enough such that the data can be more accurately modelled, earlier in the boosting sequence. At the same time, the boosting procedure tackles any bias present. ### Cross-Validation Analysis Here I will use 10-fold cross-validation to measure the performance of the Gradient Boost regressor. I will set the maximum ensemble size to 20, and use a maximum depth of 1: ## define the scoring metrics ## scoring_metrics = {'mse' : make_scorer(mean_squared_error), 'mae': make_scorer(mean_absolute_error)} ## perform cross-validation for n_elements = 20 & max_depth=1 ## #define the model #cross validate dcScores = cross_validate(rgr,dfX.values,sy.values,cv=10,scoring=scoring_metrics) #report results print('Mean MSE: %.2f' % np.mean(dcScores['test_mse'])) print('Mean MAE: %.2f' % np.mean(dcScores['test_mae'])) Mean MSE: 3548.21 Mean MAE: 46.56 We can compare these results with the Gradient Boost regressor available from scikit-learn: from sklearn.ensemble import GradientBoostingRegressor ## perform cross-validation for n_estimators = 20 & max_depth=1 ## #define the model #cross validate dcScores = cross_validate(rgr,dfX.values,sy.values,cv=10,scoring=scoring_metrics) #report results print('Mean MSE: %.2f' % np.mean(dcScores['test_mse'])) print('Mean MAE: %.2f' % np.mean(dcScores['test_mae'])) Mean MSE: 3549.82 Mean MAE: 46.65 Both Gradient Boost implementations perform at approximately the same level. Small differences between the two sets of results should be expected, due to the stochastic nature of the analysis. Now let’s compare our results with a lone Decision Tree weak learner, a Random Forest Regressor with 20 constituent estimators, and an Adaboost Regressor also with 20 constituent estimators: #cross validation for lone decision tree regressor of depth 1 dcScores = cross_validate(DecisionTreeRegressor(max_depth=1),dfX.values,sy.values,cv=10,scoring=scoring_metrics) #report results print('Mean MSE: %.2f' % np.mean(dcScores['test_mse'])) print('Mean MAE: %.2f' % np.mean(dcScores['test_mae'])) Mean MSE: 4751.55 Mean MAE: 56.68 ## perform cross-validation for random forest regressor of n_estimators=20 and max_depth=1 ## #define the model rgr = RandomForestRegressor(n_estimators=20,max_depth=1) #cross validate dcScores = cross_validate(rgr,dfX.values,sy.values,cv=10,scoring=scoring_metrics) #report results print('Mean MSE: %.2f' % np.mean(dcScores['test_mse'])) print('Mean MAE: %.2f' % np.mean(dcScores['test_mae'])) Mean MSE: 3839.32 Mean MAE: 51.68 ## perform cross-validation for linear loss & n_estimators=20 ## #define the model rgr = AdaBoostRegressor(base_estimator=DecisionTreeRegressor(max_depth=1), loss='linear', n_estimators=20) #cross validate dcScores = cross_validate(rgr,dfX.values,sy.values,cv=10,scoring=scoring_metrics) #report results print('Mean MSE: %.2f' % np.mean(dcScores['test_mse'])) print('Mean MAE: %.2f' % np.mean(dcScores['test_mae'])) Mean MSE: 3758.68 Mean MAE: 51.71 It is clear that the Gradient Boost Ensembles are the best performing models in this comparison. Adaboost takes second place, followed closely by the Random Forest. Unsurprisingly, the lone Decision Stump is the worst performing model attempted. This simplistic analysis demonstrates the predictive potential of Gradient Boosting regression in Python. It is easy to understand why they have become popular for many applications and online competitions. ## Final Remarks • An overview of the LAD_TreeBoost Gradient Boosting algorithm • How to implement Gradient Boosting regression in Python from scratch • How our implementation of Gradient Boost compares against open-source, scikit-learn regression models I hope you enjoyed this article, and gained some value from it. If you would like to take a closer look at the code presented here, please take a look at my GitHub. If you have any questions or suggestions, please feel free to add a comment below. Your input is greatly appreciated. 5 1 vote Article Rating Subscribe Notify of 1 Comment Inline Feedbacks	2022-12-04 13:03:37	{"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": 0.8803470134735107, "perplexity": 5724.926275409213}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710972.37/warc/CC-MAIN-20221204104311-20221204134311-00224.warc.gz"}
https://wiki.math.ntnu.no/linearmethods/limits	# Limits and completions • Limits: Continuous and sequential limits. Accumulation points. Relationship between limits, the distance function and closures. • Completeness: Cauchy sequences. Complete metric spaces, Banach spaces, and characterisation of complete subspaces. • Completions: Isometries, isomorphisms and embeddings. Dense sets. Separability. The completion theorem.	2023-03-21 08:19:54	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.889694094657898, "perplexity": 6637.764486809931}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296943637.3/warc/CC-MAIN-20230321064400-20230321094400-00415.warc.gz"}
https://www.physicsforums.com/threads/relation-between-energy-annd-pressure.793589/	# Relation between energy annd pressure 1. Jan 22, 2015 ### Haseeb Ali Me and and my friend were having discussion about the motion of molecules of gas.We talked about their velocity ,kinetic energy and much more. He asked me to derive a relation between pressure and energy. I was unable to explain him that... Can anyone explain the relation? 2. Jan 22, 2015 ### Bystander How would you begin? What information do you have? 3. Jan 22, 2015 ### Haseeb Ali P=densityVx2 T= 2/3k 1/2mv2 4. Jan 22, 2015 ### BvU You could read up on the gas law, e.g. here Your equations feature in there, but one appears molecular to me and the other macroscopic: ${1\over 2} mv^2 = {3\over 2} kT$ average kinetic energy for a molecule. $P = \rho v_x^2 = {mass\over V} v_x^2$ may be ok if in the right context. I can wrangle a bit with formulas: $${1\over 2} mv^2 = {3\over 2} kT\ \Rightarrow\ {1\over 2} mv_x^2 = {1\over 2} kT$$ (the energy is equally distributed over the three degrees of freedom), $\displaystyle \ \Rightarrow\ v_x^2 = {kT\over m}$ mass = number of molecules mass of a molecule = number of moles * $N_A$ * mass of a molecule (Avogadro number); write mass = $n\; N_A\; m$ Leaves $pV = n\;N_A\;m \ {kT\over m} = n\; N_A \;kT$; then use $R_G = N_A \; k$ (gas constant) to get the ideal gas law $pV = nR_GT$. pressure times volume has the dimension of energy. It's not the whole story, but quite a big part of it.	2018-03-24 20:31:38	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.42943358421325684, "perplexity": 2033.4646613659606}, "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-2018-13/segments/1521257650993.91/warc/CC-MAIN-20180324190917-20180324210917-00684.warc.gz"}
https://par.nsf.gov/biblio/10130283-search-non-resonant-higgs-boson-pair-production-bb-final-state-atlas-detector-pp-collisions-tev	Search for non-resonant Higgs boson pair production in the bbℓνℓν final state with the ATLAS detector in pp collisions at $s=13$ TeV	2022-12-03 03:29:06	{"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": 0.9861798286437988, "perplexity": 1180.9710971564036}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710918.58/warc/CC-MAIN-20221203011523-20221203041523-00123.warc.gz"}
https://www.jobilize.com/course/section/common-factors-algebraic-expressions-by-openstax?qcr=www.quizover.com	4.1 Algebraic expressions Page 2 / 3 Sample set b Identify the factors in each term. $9{a}^{2}-6a-12$ contains three terms. Some of the factors in each term are $\begin{array}{ll}\text{first}\text{\hspace{0.17em}}\text{term:}\hfill & 9\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}{a}^{2},\text{\hspace{0.17em}}\text{or},\text{\hspace{0.17em}}9\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}a\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}a\hfill \\ \text{second}\text{\hspace{0.17em}}\text{term:}\hfill & -6\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}a\hfill \\ \text{third}\text{\hspace{0.17em}}\text{term:}\hfill & -12\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}1,\text{\hspace{0.17em}}\text{or},\text{\hspace{0.17em}}12\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}-1\hfill \end{array}$ $14{x}^{5}y+{\left(a+3\right)}^{2}$ contains two terms. Some of the factors of these terms are $\begin{array}{ll}\text{first}\text{\hspace{0.17em}}\text{term:}\hfill & 14,\text{\hspace{0.17em}}{x}^{5},\text{\hspace{0.17em}}y\hfill \\ \text{second}\text{\hspace{0.17em}}\text{term:}\hfill & \left(a+3\right)\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\left(a+3\right)\hfill \end{array}$ Practice set b In the expression $8{x}^{2}-5x+6$ , list the factors of the first term: second term: third term: 8, $x$ , $x$ ; $-5$ , $x$ ; 6 and 1 or 3 and 2 In the expression $10+2\left(b+6\right){\left(b-18\right)}^{2}$ , list the factors of the first term: second term: 10 and 1 or 5 and 2; 2, $b+6$ , $b-18$ , $b-18$ Common factors Sometimes, when we observe an expression carefully, we will notice that some particular factor appears in every term. When we observe this, we say we are observing common factors . We use the phrase common factors since the particular factor we observe is common to all the terms in the expression. The factor appears in each and every term in the expression. Sample set c Name the common factors in each expression. $5{x}^{3}-7{x}^{3}+14{x}^{3}$ . The factor ${x}^{3}$ appears in each and every term. The expression ${x}^{3}$ is a common factor. $4{x}^{2}+7x$ . The factor $x$ appears in each term. The term $4{x}^{2}$ is actually $4xx$ . Thus, $x$ is a common factor. $12x{y}^{2}-9xy+15$ . The only factor common to all three terms is the number 3. (Notice that $12=3\cdot 4,\text{\hspace{0.17em}}9=3\cdot 3,\text{\hspace{0.17em}}15=3\cdot 5$ .) $3\left(x+5\right)-8\left(x+5\right)$ . The factor $\left(x+5\right)$ appears in each term. So, $\left(x+5\right)$ is a common factor. $45{x}^{3}{\left(x-7\right)}^{2}+15{x}^{2}\left(x-7\right)-20{x}^{2}{\left(x-7\right)}^{5}$ . The number 5, the ${x}^{2}$ , and the $\left(x-7\right)$ appear in each term. Also, $5{x}^{2}\left(x-7\right)$ is a factor (since each of the individual quantities is joined by a multiplication sign). Thus, $5{x}^{2}\left(x-7\right)$ is a common factor. $10{x}^{2}+9x-4$ . There is no factor that appears in each and every term. Hence, there are no common factors in this expression. Practice set c List, if any appear, the common factors in the following expressions. ${x}^{2}+5{x}^{2}-9{x}^{2}$ ${x}^{2}$ $4{x}^{2}-8{x}^{3}+16{x}^{4}-24{x}^{5}$ $4{x}^{2}$ $4{\left(a+1\right)}^{3}+10\left(a+1\right)$ $2\left(a+1\right)$ $9ab\left(a-8\right)-15a{\left(a-8\right)}^{2}$ $3a\left(a-8\right)$ $14{a}^{2}{b}^{2}c\left(c-7\right)\left(2c+5\right)+28c\left(2c+5\right)$ $14c\left(2c+5\right)$ $6\left({x}^{2}-{y}^{2}\right)+19x\left({x}^{2}+{y}^{2}\right)$ no common factor Coefficient In algebra, as we now know, a letter is often used to represent some quantity. Suppose we represent some quantity by the letter $x$ . The notation $5x$ means $x+x+x+x+x$ . We can now see that we have five of these quantities. In the expression $5x$ , the number 5 is called the numerical coefficient of the quantity $x$ . Often, the numerical coefficient is just called the coefficient. The coefficient of a quantity records how many of that quantity there are. Sample set d $12x$ means there are $12x\text{'}\text{s}$ . $4ab$ means there are four $ab\text{'}\text{s}$ . $10\left(x-3\right)$ means there are ten $\left(x-3\right)\text{'}\text{s}$ . $1y$ means there is one $y$ . We usually write just $y$ rather than $1y$ since it is clear just by looking that there is only one $y$ . $7{a}^{3}$ means there are seven ${a}^{3\text{'}}\text{s}$ . $5ax$ means there are five $ax\text{'}\text{s}$ . It could also mean there are $5ax\text{'}\text{s}$ . This example shows us that it is important for us to be very clear as to which quantity we are working with. When we see the expression $5ax$ we must ask ourselves "Are we working with the quantity $ax$ or the quantity $x$ ?". $6{x}^{2}{y}^{9}$ means there are six ${x}^{2}{y}^{9\text{'}}\text{s}$ . It could also mean there are $6{x}^{2}{y}^{9\text{'}}\text{s}$ . It could even mean there are $6{y}^{9}{x}^{2\text{'}}\text{s}$ . what is the stm is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.? Rafiq industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong Damian How we are making nano material? what is a peer What is meant by 'nano scale'? What is STMs full form? LITNING scanning tunneling microscope Sahil how nano science is used for hydrophobicity Santosh Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq Rafiq what is differents between GO and RGO? Mahi what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq Rafiq what is Nano technology ? write examples of Nano molecule? Bob The nanotechnology is as new science, to scale nanometric brayan nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale Damian Is there any normative that regulates the use of silver nanoparticles? what king of growth are you checking .? Renato What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ? why we need to study biomolecules, molecular biology in nanotechnology? ? Kyle yes I'm doing my masters in nanotechnology, we are being studying all these domains as well.. why? what school? Kyle biomolecules are e building blocks of every organics and inorganic materials. Joe anyone know any internet site where one can find nanotechnology papers? research.net kanaga sciencedirect big data base Ernesto Introduction about quantum dots in nanotechnology what does nano mean? nano basically means 10^(-9). nanometer is a unit to measure length. Bharti do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment? absolutely yes Daniel how to know photocatalytic properties of tio2 nanoparticles...what to do now it is a goid question and i want to know the answer as well Maciej Abigail for teaching engĺish at school how nano technology help us Anassong How can I make nanorobot? Lily Do somebody tell me a best nano engineering book for beginners? there is no specific books for beginners but there is book called principle of nanotechnology NANO how can I make nanorobot? Lily what is fullerene does it is used to make bukky balls are you nano engineer ? s. fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball. Tarell what is the actual application of fullerenes nowadays? Damian That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes. Tarell how did you get the value of 2000N.What calculations are needed to arrive at it Privacy Information Security Software Version 1.1a Good Please keep in mind that it's not allowed to promote any social groups (whatsapp, facebook, etc...), exchange phone numbers, email addresses or ask for personal information on QuizOver's platform.	2020-02-19 04:18:26	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 71, "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": 0.6625627279281616, "perplexity": 1191.0199631975747}, "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-2020-10/segments/1581875144027.33/warc/CC-MAIN-20200219030731-20200219060731-00226.warc.gz"}
https://par.nsf.gov/biblio/10349849-dark-energy-survey-year-results-clustering-redshifts-calibration-weak-lensing-source-redshift-distributions-redmagic-boss-eboss	This content will become publicly available on December 24, 2022 Dark Energy Survey Year 3 Results: clustering redshifts – calibration of the weak lensing source redshift distributions with redMaGiC and BOSS/eBOSS ABSTRACT We present the calibration of the Dark Energy Survey Year 3 (DES Y3) weak lensing (WL) source galaxy redshift distributions n(z) from clustering measurements. In particular, we cross-correlate the WL source galaxies sample with redMaGiC galaxies (luminous red galaxies with secure photometric redshifts) and a spectroscopic sample from BOSS/eBOSS to estimate the redshift distribution of the DES sources sample. Two distinct methods for using the clustering statistics are described. The first uses the clustering information independently to estimate the mean redshift of the source galaxies within a redshift window, as done in the DES Y1 analysis. The second method establishes a likelihood of the clustering data as a function of n(z), which can be incorporated into schemes for generating samples of n(z) subject to combined clustering and photometric constraints. Both methods incorporate marginalization over various astrophysical systematics, including magnification and redshift-dependent galaxy-matter bias. We characterize the uncertainties of the methods in simulations; the first method recovers the mean z of tomographic bins to RMS (precision) of ∼0.014. Use of the second method is shown to vastly improve the accuracy of the shape of n(z) derived from photometric data. The two methods are then applied to the DES Y3 data. Authors: ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; more » Award ID(s): Publication Date: NSF-PAR ID: 10349849 Journal Name: Monthly Notices of the Royal Astronomical Society Volume: 510 Issue: 1 Page Range or eLocation-ID: 1223 to 1247 ISSN: 0035-8711 2. ABSTRACT We develop a novel data-driven method for generating synthetic optical observations of galaxy clusters. In cluster weak lensing, the interplay between analysis choices and systematic effects related to source galaxy selection, shape measurement, and photometric redshift estimation can be best characterized in end-to-end tests going from mock observations to recovered cluster masses. To create such test scenarios, we measure and model the photometric properties of galaxy clusters and their sky environments from the Dark Energy Survey Year 3 (DES Y3) data in two bins of cluster richness $\lambda \in [30; 45)$, $\lambda \in [45; 60)$ and three bins in cluster redshift ($z\in [0.3; 0.35)$, $z\in [0.45; 0.5)$ and $z\in [0.6; 0.65)$. Using deep-field imaging data, we extrapolate galaxy populations beyond the limiting magnitude of DES Y3 and calculate the properties of cluster member galaxies via statistical background subtraction. We construct mock galaxy clusters as random draws from a distribution function, and render mock clusters and line-of-sight catalogues into synthetic images in the same format as actual survey observations. Synthetic galaxy clusters are generated from real observational data, and thus are independent from the assumptions inherent to cosmological simulations. The recipe can be straightforwardly modified to incorporate extra information, andmore » 3. ABSTRACT Photometric galaxy surveys constitute a powerful cosmological probe but rely on the accurate characterization of their redshift distributions using only broad-band imaging, and can be very sensitive to incomplete or biased priors used for redshift calibration. A hierarchical Bayesian model has recently been developed to estimate those from the robust combination of prior information, photometry of single galaxies, and the information contained in the galaxy clustering against a well-characterized tracer population. In this work, we extend the method so that it can be applied to real data, developing some necessary new extensions to it, especially in the treatment of galaxy clustering information, and we test it on realistic simulations. After marginalizing over the mapping between the clustering estimator and the actual density distribution of the sample galaxies, and using prior information from a small patch of the survey, we find the incorporation of clustering information with photo-z’s tightens the redshift posteriors and overcomes biases in the prior that mimic those happening in spectroscopic samples. The method presented here uses all the information at hand to reduce prior biases and incompleteness. Even in cases where we artificially bias the spectroscopic sample to induce a shift in mean redshift of $\Deltamore » 4. ABSTRACT In this work, we present the galaxy clustering measurements of the two DES lens galaxy samples: a magnitude-limited sample optimized for the measurement of cosmological parameters, maglim, and a sample of luminous red galaxies selected with the redmagic algorithm. maglim/redmagic sample contains over 10 million/2.5 million galaxies and is divided into six/five photometric redshift bins spanning the range z ∈ [0.20, 1.05]/z ∈ [0.15, 0.90]. Both samples cover 4143$\deg ^2\$ over which we perform our analysis blind, measuring the angular correlation function with an S/N ∼ 63 for both samples. In a companion paper, these measurements of galaxy clustering are combined with the correlation functions of cosmic shear and galaxy–galaxy lensing of each sample to place cosmological constraints with a 3 × 2pt analysis. We conduct a thorough study of the mitigation of systematic effects caused by the spatially varying survey properties and we correct the measurements to remove artificial clustering signals. We employ several decontamination methods with different configurations to ensure the robustness of our corrections and to determine the systematic uncertainty that needs to be considered for the final cosmology analyses. We validate our fiducial methodology using lognormal mocks, showing that our decontamination procedure induces biases no greatermore »	2022-11-30 04:59:33	{"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": 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": 0.46502789855003357, "perplexity": 1644.747693074378}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710719.4/warc/CC-MAIN-20221130024541-20221130054541-00130.warc.gz"}
https://de.zxc.wiki/wiki/Differentialgeometrie	# Differential geometry As a branch of mathematics, differential geometry represents the synthesis of analysis and geometry . ## Historical development and current areas of application A number of fundamental works on differential geometry come from Carl Friedrich Gauß . At that time, mathematics was still closely linked to various fields of application. This theory delivered important results in the fields of cartography , navigation and geodesy . Among other things, the theory of map projection developed , from which the terms geodetic line and Gaussian curvature come. In addition, CF Gauss already asked himself whether the sum of the angles of a very large triangle measured by bearing actually amounts to exactly 180 degrees, and thus proves to be a pioneer of modern differential geometry. Modern differential geometry is mainly used in general relativity and in satellite navigation . It enables the description of phenomena such as the astronomical deflection of light or the rotation of the perihelion of Mercury , which can be confirmed by experiments or observation . In the theory of relativity, coordinate transformations correspond to the change of reference systems from which a phenomenon is observed. This corresponds to different states of motion of the measuring apparatus or the observer. Another important area of application is in materials science in the theory of defects and plasticity . ## Sub-areas ### Elementary differential geometry The first work on differential geometry deals with both curves and two-dimensional curved surfaces in three-dimensional real visual space . From a historical point of view, Gauss's work made it possible for the first time to quantitatively record the curvature of the two-dimensional surface of a sphere, for example. Another motivation for the development of elementary differential geometry came from the mathematical problem of minimal surfaces . The soap skins that occur in nature can be described as minimal areas. The shape or mathematical representation of these surfaces can be developed using the methods from the calculus of variations . The geometric properties of these surfaces such as curvature or distances between any points on a minimal surface, on the other hand, are more likely to be calculated using the methods of differential geometry. ### Differential topology The differential topology is the basis for most of the modern areas of differential geometry. In contrast to elementary differential geometry, the geometric objects are described intrinsically in differential topology , that is, the objects are defined without recourse to a surrounding space. The central concept is that of the differentiable manifold : A -dimensional manifold is a geometric object (more precisely: a topological space ) that looks locally like the -dimensional real space. The classic example that also motivates the terminology is the earth's surface. It can be described in small excerpts using maps , that is, small parts “look like” the plane. However, the entire surface of the earth cannot be identified with the plane. In addition, differentiable manifolds have a structure that allows us to speak of differentiable functions. This differentiable structure makes it possible to use local analytical methods in the maps. In addition, one can examine the manifold globally as a topological space. The differential topology tries to establish connections between the local analytical and the global topological properties. An example of such a connection is de Rham's theorem . ${\ displaystyle n}$${\ displaystyle n}$ ### Riemannian geometry There is no predefined length measurement on a differentiable manifold. If it is given as an additional structure, one speaks of Riemannian manifolds . These manifolds are the subject of Riemannian geometry, which also examines the associated concepts of curvature , the covariant derivative and the parallel transport on these sets. These terms can, however, also be defined for “non-Riemannian” or “non-pseudoriemannian” spaces and only require the general differential-geometric concept of the context (more precisely: general affine differential geometry in contrast to metric differential geometry, see below.) ### Semi-Riemannian differential geometry If, instead of the positive-definite metric of a Riemannian manifold, a non-definite metric is assumed (given by a non-definite Hermitian or symmetrically-non-definite non -degenerate bilinear form ), a semi- or pseudo-Riemannian manifold is obtained . The Lorentzian manifolds of the general theory of relativity are a special case . ### Finsler's geometry The subject of Finsler's geometry are the Finsler's manifolds , i.e. manifolds whose tangent space is equipped with a Banach norm, i.e. a mapping with the following properties: ${\ displaystyle F \ colon TM \ to [0, \ infty)}$ 1. ${\ displaystyle F (rX) = \| r \| F (X) \,}$, for and ,${\ displaystyle X \ in TM}$${\ displaystyle r \ in \ mathbb {R}}$ 2. ${\ displaystyle F (X + Y) \ leq F (X) + F (Y),}$ 3. ${\ displaystyle F}$is smooth on ,${\ displaystyle TM \ setminus 0}$ 4. the vertical Hessian matrix is positive definite . Finsler's manifolds also play a role in theoretical physics as more general candidates for the structural description of spacetime. ### Symplectic geometry Instead of a symmetric nondegenerate bilinear form, an antisymmetric nondegenerate bilinear form ω is given. If this is also closed, i.e. d ω = 0, one speaks of a symplectic manifold. Because a symplectic vector space necessarily has an even dimension, symplectic manifolds also have an even dimension. The first important finding is Darboux's theorem , according to which symplectic manifolds are locally isomorphic to T * R n . In contrast to semi-Riemannian manifolds, there are no (non-trivial) local symplectic invariants (apart from the dimension), but only global symplectic invariants. The Poisson manifolds , which do not have a bilinear form but only an antisymmetric bivector , also count as a generalization . This induces a Lie bracket between the functions. Symplectic geometry is used in Hamiltonian mechanics , a branch of theoretical mechanics . ### Contact geometry The analogue to symplectic geometry for odd-dimensional manifolds is contact geometry. A contact structure on a -dimensional manifold is a family of hyperplanes of the tangential bundle that are maximally non-integrable. Locally, these hyperplanes can be represented as a core of a 1-form , i.e. H. ${\ displaystyle (2n + 1)}$${\ displaystyle M}$${\ displaystyle H}$${\ displaystyle \ alpha}$ ${\ displaystyle H_ {p} = \ ker \ alpha _ {p} \ subset T_ {p} M}$. Conversely, a form of contact is locally uniquely determined by the family , except for one non-vanishing factor. The non-integrability means that dα is non-degenerate restricted to the hyperplane. If the family can be described globally by a 1-form , then contact form is if and only if ${\ displaystyle H}$${\ displaystyle H}$ ${\ displaystyle \ alpha}$${\ displaystyle \ alpha}$ ${\ displaystyle \ alpha \ wedge (d \ alpha) ^ {n}}$is a volume shape .${\ displaystyle M}$ A theorem analogous to Darboux's theorem for symplectic manifolds applies, namely that all contact manifolds of the dimension are locally isomorphic. This means that there are only global invariants in contact geometry. ${\ displaystyle 2n + 1}$ ### Complex geometry and Kähler geometry Complex geometry is the study of complex manifolds, that is, manifolds that look like locally and whose transition functions are complex-differentiable (holomorphic). Because of the analytical properties of complex-differentiable functions, one often has uniqueness properties of the continuation of local functions / vector fields. That is why one is mostly dependent on the theory of sheaves in global studies . An almost-complex structure on a smooth manifold is a map such that . Thus all almost complex manifolds are of even dimension. The difference between an almost-complex and a complex manifold is the integrability of the almost-complex structure. This is measured by the Nijenhuis tensor . ${\ displaystyle \ mathbb {C} ^ {n}}$${\ displaystyle J \ colon TM \ to TM}$${\ displaystyle J ^ {2} = - 1}$ ${\ displaystyle N_ {J}}$ A Hermitian manifold is a complex manifold with a Hermitian metric on the complexified real tangent bundle. In particular, must be compatible with the complex structure , namely ${\ displaystyle g}$${\ displaystyle g}$${\ displaystyle J}$ ${\ displaystyle g (X, Y) = g (JX, JY)}$for everyone .${\ displaystyle X, Y \ in T_ {x} M}$ Hermitian manifolds, whose Hermitian metrics are additionally compatible with a symplectic form, have proven to be particularly structurally rich. H. ${\ displaystyle g (JX, Y) = \ omega (X, Y)}$with .${\ displaystyle d \ omega = 0}$ In this case one speaks of a Kahler manifold . Finally, Cauchy-Riemann geometry deals with bounded complex manifolds. ### Lie group theory Just as groups are based on sets , manifolds are the basis of Lie groups . The Lie groups named after Sophus Lie appear in many places in mathematics and physics as continuous symmetry groups, for example as groups of rotations of space. The study of the transformation behavior of functions under symmetries leads to the representation theory of Lie groups. ### Global Analysis Global analysis is also a branch of differential geometry that is closely related to topology. Sometimes the sub-area is also called analysis on manifolds. In this mathematical research area, ordinary and partial differential equations are investigated on differentiable manifolds. In this theory, local methods from functional analysis , micro- local analysis and the theory of partial differential equation and global methods from geometry and topology are used. Since this mathematical sub-area uses many methods of analysis in comparison to the other sub-areas of differential geometry, it is partly understood as a sub-area of analysis. Even the first work on differential equations contained aspects of global analysis. The studies of George David Birkhoff in the field of dynamic systems and the theory of geodesics by Harold Calvin Marston Morse are early examples of methods of global analysis. Central results of this mathematical sub-area are the work of Michael Francis Atiyah , Isadore M. Singer and Raoul Bott . Particularly noteworthy here are the Atiyah-Singer index theorem and the Atiyah-Bott fixed-point theorem , which is a generalization of Lefschetz's fixed-point theorem from topology. ## Methods ### Coordinate transformations Coordinate transformations are an important tool in differential geometry to enable the adaptation of a problem to geometric objects. If, for example, distances on a spherical surface are to be examined, spherical coordinates are mostly used. If one looks at Euclidean distances in space, however, one uses more Cartesian coordinates. From a mathematical point of view, it should be noted that coordinate transformations are always bijective , as often as desired, continuously differentiable maps. The inverse of the coordinate transformation under consideration always also exists. A simple example is the transition from Cartesian coordinates in the plane to polar coordinates . Each position vector of two-dimensional Euclidean space can be expressed in this representation by the coordinates and in the following way ${\ displaystyle r \ in [0, \ infty [}$${\ displaystyle \ phi \ in [0.2 \ pi [}$ ${\ displaystyle {\ vec {\ mathbf {r}}} = {\ begin {pmatrix} x \\ y \ end {pmatrix}} = {\ vec {f}} (r, \ phi) = {\ begin { pmatrix} r \, \ cos \ phi \\ r \, \ sin \ phi \ end {pmatrix}}}$ ${\ displaystyle x}$and are also referred to as component functions of. They are calculated as a function of the two coordinates : ${\ displaystyle y}$${\ displaystyle f}$${\ displaystyle (r, \ phi)}$ ${\ displaystyle x (r, \ phi) = r \, \ cos \ phi \ ,, \, \, \, \, y (r, \ phi) = r \, \ sin \ phi \, \ ,.}$ If, in general, all coordinates of the new coordinate system are kept constant except for one coordinate and the individual coordinates are changed within the definition range, lines are created in Euclidean space which are also referred to as coordinate lines. In the case of the specified polar coordinates, concentric circles with a radius around the coordinate origin of the Euclidean coordinate system are created with a constant coordinate. With a constant coordinate, half-lines are created that start in the coordinate origin of the Euclidean coordinate system and then run. With the help of these coordinate lines, a new, spatially rotated and again right-angled coordinate system can be defined in an obvious way for every point in Euclidean space. For this reason, polar coordinates are also referred to as right-angled coordinates. The axes of the rotated coordinate system are precisely the tangents to the coordinate lines that run through the point . The base vectors of these position-dependent and right-angled coordinate systems can be calculated directly from the partial derivatives of the position vector according to the above-mentioned representation according to the variable coordinates . The total differentials of the position vector can also be given via the partial derivatives: ${\ displaystyle r}$${\ displaystyle r}$${\ displaystyle (x, y) = (0,0)}$${\ displaystyle \ phi}$${\ displaystyle r \ rightarrow \ infty}$${\ displaystyle P \ in \ mathbb {R} ^ {2}}$${\ displaystyle P}$${\ displaystyle (r, \ phi)}$ ${\ displaystyle \ mathrm {d} x = {\ frac {\ partial x} {\ partial r}} \ mathrm {d} r + {\ frac {\ partial x} {\ partial \ phi}} \ mathrm {d} \ phi = \ cos \ phi \, \ mathrm {d} rr \ cdot \ sin \ phi \, \ mathrm {d} \ phi}$ ${\ displaystyle \ mathrm {d} y = {\ frac {\ partial y} {\ partial r}} \ mathrm {d} r + {\ frac {\ partial y} {\ partial \ phi}} \ mathrm {d} \ phi = \ sin \ phi \, \ mathrm {d} r + r \ cdot \ cos \ phi \, \ mathrm {d} \ phi}$ The differentials are also referred to as coordinate differentials . In this example, the infinitesimal quantities linked with the differential operator “ ” do not always have the meaning of a distance. It is rather easy to show that for the distances in the radial or azimuthal direction it is true that it is, but ; ie only with the prefactor " " results from integration over from 0 to a known quantity of the dimension "length", namely the circumference . ${\ displaystyle \ mathrm {d} x, \ mathrm {d} y, \ mathrm {d} r, \ mathrm {d} \ phi}$${\ displaystyle \ mathrm {d}}$${\ displaystyle \ mathrm {d} l_ {r} \,: = \ mathrm {d} r}$${\ displaystyle \ mathrm {d} l _ {\ phi}: = r \ cdot \ mathrm {d} \ phi}$${\ displaystyle r}$${\ displaystyle \ mathrm {d} \ Phi}$${\ displaystyle 2 \ pi}$${\ displaystyle r \ cdot 2 \ pi}$ The polar coordinates or their three-dimensional generalization, the spherical coordinates, are also referred to as curvilinear as they facilitate the calculation of the distance on a curved surface, e.g. B. the spherical surface, allow. As with other standard examples, such as the cylindrical coordinates , the elliptical coordinates , etc., these are orthogonal, curvilinear coordinates (see also: Curvilinear coordinates ). An essential tool of classical differential geometry are coordinate transformations between any coordinates in order to be able to describe geometric structures. The differential operators formed with magnitude , known from analysis, can be extended relatively easily to orthogonal curvilinear differential operators. For example, in general orthogonal curvilinear coordinates when using three parameters and the associated unit vectors in the direction of, the following relationships apply with quantities that are not necessarily constant, but can depend on , and : ${\ displaystyle \ nabla}$${\ displaystyle u_ {i}, \, \, i = 1, \ dots, 3}$${\ displaystyle \ mathbf {e} _ {i}}$${\ displaystyle {\ tfrac {\ partial \ mathbf {r}} {\ partial u_ {i}}}}$${\ displaystyle a_ {i}}$${\ displaystyle u_ {1}}$${\ displaystyle u_ {2}}$${\ displaystyle u_ {3}}$ {\ displaystyle {\ begin {aligned} \ mathrm {d} \ mathbf {r} & = \ sum \ limits _ {i = 1} ^ {3} {\ rm {d}} l_ {i} \ mathbf {e } _ {i} = \ sum \ limits _ {i = 1} ^ {3} \, a_ {i} \, \ mathrm {d} u_ {i} \, \ mathbf {e} _ {i} \\ {\ rm {d}} V & = a_ {1} \ mathrm {d} u_ {1} \ cdot a_ {2} \ mathrm {d} u_ {2} \ cdot a_ {3} \ mathrm {d} u_ { 3} \\\ nabla ^ {2} f & = {\ frac {1} {a_ {1} a_ {2} a_ {3}}} {\ frac {\ partial} {\ partial u_ {1}}} \ left ({\ frac {a_ {2} a_ {3}} {a_ {1}}} {\ frac {\ partial f} {\ partial u_ {1}}} \ right) + \ cdots + \ cdots \ end {aligned}}} The two further terms from the first term, indicated by dots, are created by cyclically interchanging the indices. denotes the Laplace operator . It can be composed of the scalar-valued div operator and the vector-valued grad operator according to ${\ displaystyle \ nabla ^ {2}}$ ${\ displaystyle \ nabla ^ {2} f = \ operatorname {div} (\ operatorname {grad} f)}$ in which {\ displaystyle {\ begin {aligned} \ operatorname {div} \ mathbf {v} \, & = \, {\ frac {1} {a_ {1} a_ {2} a_ {3}}} {\ frac { \ partial (a_ {2} a_ {3} v_ {1})} {\ partial u_ {1}}} + \ cdots + \ cdots \\\ operatorname {grad} f & = \ sum \ limits _ {i = 1 } ^ {3} \, {\ frac {1} {a_ {i}}} {\ frac {\ partial f} {\ partial u_ {i}}} \, \ mathbf {e} _ {i} \ end {aligned}}} The formula for the divergence is based on the coordinate-independent representation ${\ displaystyle \ operatorname {div} \ mathbf {v} = \ lim _ {\ Delta V \ to 0} \, {\ Big \ {} {\ frac {1} {\| \ Delta V \|}} \, \ iint \ limits _ {\ partial (\ Delta V)} \ mathbf {v} \ cdot \ mathbf {n} {\ rm {d}} ^ {(2)} A {\ Big \}} \, \ ,, }$ integrating over the closed, bordering surface. denotes the associated outer normal vector, the corresponding infinitesimal surface element . In the most general case - i.e. for non-orthogonal, curvilinear coordinates - this formula can also be used. ${\ displaystyle \ Delta V}$${\ displaystyle \ mathbf {n}}$${\ displaystyle {\ rm {d}} ^ {(2)} A}$${\ displaystyle \ Delta V \ to {\ rm {d}} V}$ ### Covariant derivative General derivative operators based on not necessarily orthogonal curvilinear coordinates are e.g. B. the covariant derivatives that u. a. be used in Riemannian spaces , where they are specifically related to the “inner product”, i.e. H. on the so-called " metric fundamental form " of the room. In other cases, however, they are independent of the existence of a local metric or can even be specified externally, e.g. B. in manifolds "with connection". They enable u. a. the definition of connecting lines in curved spaces, e.g. B. the definition of geodesics in Riemannian space. Geodetic lines are the locally shortest connections between two points in these spaces. The longitudes on a sphere are examples of geodetic lines, but not the latitudes (exception: equator). With the help of general coordinate transformations, the Christoffels symbols are defined in Riemannian space (and more generally in differential geometries “with a given context ”) . In accordance with the basic definition given below, these are explicitly included in the calculation of the covariant derivative of a vector field . ${\ displaystyle \ Gamma _ {\ alpha \ beta} ^ {\ mu}}$ The covariant derivative is a generalization of the partial derivative of flat (Euclidean) space for curved spaces. In contrast to the partial derivative , it has the tensor property; in Euclidean space it is reduced to a partial derivative. In curved space, the covariant derivatives of a vector field are generally not interchangeable with one another; their non-interchangeability is used to define the Riemann curvature tensor . Another important term in connection with curved spaces is parallel translation . The covariant derivative of the components of a vector is zero with parallel shift. Nevertheless, the parallel displacement of a vector along a closed curve in curved space can lead to the displaced vector not coinciding with its starting vector. The associated formalism is based on the rule that vectors are written as a sum , where u. U. (namely just in the above "parallel transport") are not the components , but only the basic elements of change, though gradually the obvious rule . Covariant and partial derivative, usually written with a semicolon or comma, are different, namely: ${\ displaystyle \ mathbf {v}}$${\ displaystyle v ^ {\ alpha} \ mathbf {e} _ {\ alpha}}$${\ displaystyle \, v ^ {\ alpha}}$${\ displaystyle \ mathbf {e} _ {\ alpha}}$${\ displaystyle d \ mathbf {e} _ {\ alpha} = \ Gamma _ {\ alpha \ beta} ^ {\ mu} dx ^ {\ beta} \ mathbf {e} _ {\ mu}}$ ${\ displaystyle v _ {\,; \ beta} ^ {\ mu} \, dx ^ {\ beta} = v _ {\ ,, \ beta} ^ {\ mu} \, dx ^ {\ beta} + \ Gamma _ {\ alpha \ beta} ^ {\ mu} v ^ {\ alpha} dx ^ {\ beta} \ ,,}$ so or also${\ displaystyle v _ {\,; \ beta} ^ {\ mu} \,: = \, v _ {\ ,, \ beta} ^ {\ mu} + \ Gamma _ {\ alpha \ beta} ^ {\ mu} v ^ {\ alpha}}$${\ displaystyle \ nabla _ {\, \ beta} v ^ {\ mu} \,: = \, \ partial _ {\, \ beta} v ^ {\ mu} + \ Gamma _ {\ alpha \ beta} ^ {\ mu} v ^ {\ alpha} \ ,.}$ In manifolds with an additional structure (e.g. in Riemannian manifolds or in the so-called gauge theories ) this structure must of course be compatible with the transfer. This results in additional relationships for the Christoffel symbols. For example, in Riemannian spaces, the distance and angle relationships between two vectors must not change in the event of a parallel shift, and the Christoffel symbols are therefore calculated in a certain way from the metric structure alone. ### Curvature tensor The above-mentioned curvature of space results in an analogous way: If one shifts the basis vector in the mathematically positive sense (counterclockwise) first an infinitesimal segment in -direction and then an infinitesimal segment in -direction, one obtains a result that we can write in the form . If the order is reversed, i.e. if the direction of rotation is opposite, the result is the opposite. The difference can be written in the following form with a quantity that results from the Christoffel symbols: ${\ displaystyle \ mathbf {e} _ {\ alpha}}$${\ displaystyle {\ rm {d}} x ^ {\ beta}}$${\ displaystyle \ beta}$${\ displaystyle {\ rm {d}} x ^ {\ gamma}}$${\ displaystyle \ gamma}$${\ displaystyle {\ tfrac {1} {2}} \ Delta ^ {(2)} K _ {\ alpha}}$${\ displaystyle \, \ Delta ^ {(2)} K _ {\ alpha}}$${\ displaystyle R _ {\; \; \ alpha \ beta \ gamma} ^ {\ lambda}}$ {\ displaystyle {\ begin {aligned} \ Delta ^ {(2)} K _ {\ alpha} & = R _ {\; \; \ alpha \ beta \ gamma} ^ {\ lambda} {\ rm {d}} x ^ {\ beta} \, {\ rm {d}} x ^ {\ gamma \,} \ mathbf {e} _ {\ lambda} \\ & \ equiv \, (\ Gamma _ {\ alpha \ beta} ^ {\ mu} \, \ Gamma _ {\ mu \ gamma} ^ {\ lambda} - \ Gamma _ {\ alpha \ gamma} ^ {\ mu} \ Gamma _ {\ mu \ beta} ^ {\ lambda} + \ partial _ {\ gamma} \ Gamma _ {\ alpha \ beta} ^ {\ lambda} - \ partial _ {\ beta} \ Gamma _ {\ gamma \ alpha} ^ {\ lambda} \,) \, {\ rm {d}} x ^ {\ beta} \, {\ rm {d}} x ^ {\ gamma \,} \, \ mathbf {e} _ {\ lambda} \ end {aligned}}} If the vector is shifted in parallel, the following results accordingly: The components form the curvature tensor , a vector-valued differential form. (In the so-called Yang-Mills theories , this term is generalized by replacing, for example, "vector-valued" with Lie algebra-valued; see also Chern classes .) ${\ displaystyle \ mathbf {v}}$${\ displaystyle v ^ {\ lambda} \ to v ^ {\ lambda} \, \ pm {\ tfrac {1} {2}} \, R _ {\; \; \ alpha \ beta \ gamma} ^ {\ lambda } \, v ^ {\ alpha} \, {\ rm {d}} x ^ {\ beta} \, {\ rm {d}} x ^ {\ gamma} \ ,.}$${\ displaystyle R _ {\; \; \ alpha \ beta \ gamma} ^ {\ lambda}}$ In particular, the existence of the curvature tensor does not presuppose that one is dealing with metric or pseudometric spaces as in physics (see above), but only the affinity is assumed for the structure of the transmission . ## literature ### Elementary differential geometry • W. Blaschke , K. Leichtweiß : Elementary differential geometry. (= Lectures on differential geometry. 1 = The basic teachings of the mathematical sciences in individual representations. 1). 5th, completely revised edition. Springer-Verlag, Berlin et al. 1973, ISBN 3-540-05889-3 . • Manfredo P. do Carmo: Differential geometry of curves and surfaces (= Vieweg studies. Advanced course in mathematics. 55). Vieweg & Sohn, Braunschweig et al. 1983, ISBN 3-528-07255-5 . • Christian Bär : Elementary Differential Geometry. de Gruyter, Berlin et al. 2001, ISBN 3-11-015519-2 . • Wolfgang Kühnel : Differential geometry, curves - surfaces - manifolds. 4th, revised edition. Friedr. Vieweg & Sohn, Wiesbaden 2008, ISBN 978-3-8348-0411-2 . ### Abstract manifolds, Riemannian geometry • Rolf Walter: differential geometry. 2nd, revised and expanded edition. BI-Wissenschafts-Verlag, Mannheim et al. 1989, ISBN 3-411-03216-2 . • Sigurdur Helgason : Differential Geometry, Lie Groups, and Symmetric Spaces (= Graduate Studies in Mathematics. 34). American Mathematical Society, Providence RI, 2001, ISBN 0-8218-2848-7 . • S. Kobayashi , Katsumi Nomizu: Foundations of Differential Geometry. Volume 1 (= Interscience Tracts in Pure and Applied Mathematics. 15, 1). Interscience Publishers, New York NY et al. 1963. • Pham Mau Quan: Introduction à la géométrie des variétés différentiables (= Monographies universitaires de mathématiques. 29). Dunod, Paris 1969. ( Content (PDF; 184 kB )).	2021-10-18 09:55:49	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 96, "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": 0.8244832754135132, "perplexity": 2668.3038952552547}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1634323585201.94/warc/CC-MAIN-20211018093606-20211018123606-00706.warc.gz"}
https://csharp-book.softuni.org/Content/Chapter-3-1-simple-conditions/exercises-simple-conditions/exercises-simple-conditions-part-2.html	# Exercises: Simple Conditions Now let's practice the lessons learned in this chapter about of conditional statements if and if-else. We will solve a few practical exercises. ## Empty Visual Studio Solution (Blank Solution) At the start we create a Blank Solution in Visual Studio to organize better the task solutions from the exercise - each task will be in a separate project and all projects will be in a common solution. We run Visual Studio and create a new Blank Solution: [File] -> [New] -> [Project]. Choose from the dialog box [Templates] -> [Other Project Types] -> [Visual Studio Solutions] -> [Blank Solution] and give an appropriate project name, for example: “Simple-Conditions”: Now we have an empty Visual Studio Solution (no projects in it): We shall use this solution to create a separate project for each of the problems, which we will solve as exercises in this chapter.	2019-01-18 08:13:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.24340614676475525, "perplexity": 3544.5243804544816}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583659944.3/warc/CC-MAIN-20190118070121-20190118092121-00381.warc.gz"}
https://nrich.maths.org/5473/clue?nomenu=1	Choose positions for P' and plot the corresponding positions of P. Do enough until you see something nice happening then figure out why that has to happen.	2017-11-21 10:13:09	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.904416024684906, "perplexity": 899.206579439514}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806338.36/warc/CC-MAIN-20171121094039-20171121114039-00712.warc.gz"}
https://kavigupta.org/2016/04/29/Nontrivial-Elements-Of-Trivial-Types/	### On Error Resume Prior If at first you don't succeed... # Nontrivial Elements of Trivial Types We know that types can be used to encode how to make useful data structures. For example, a vehicle can be encoded as: data Vehicle = Car { vin :: Integer, licensePlate :: String } \| Bike In this case, valid values include cars with VINs and LPNs, or Bikes, which have neither of these data points. However, some data types can seem far less useful. ## Elements of the Void type The void type is defined as data Void This type is defined by being non-constructible. However, we can try to construct a value such that void :: Void void = So, how do we fill that in? We can construct a value of type Void by using void. void :: Void void = void In Haskell, we can define this in two ways: void2, void3 :: Void void2 = undefined void3 = error "any string here" In fact, Haskell defines in the standard prelude, undefined = undefined error _ = undefined Which makes the definitions for void and void2 equivalent. ## What is undefined? The definition of undefined is kind of the functional equivalent of x = x + 1: it has a mathematical meaning that is inconsistent with its programming meaning. However, mathematical meaning: $$x = x$$ actually does apply to the Hindley Milner process, which will end up with this equation. What this means is that undefined can be of any type: undefined :: forall a. a Interestingly, such a definition is impossible to make illegal in a Turing-Complete language like Haskell. It corresponds to any function that runs infinitely. In fact, predicting whether a function can return the value undefined is equivalent to the impossible Halting Problem. ## The properties of undefined undefined has a few interesting properties. For example, any time undefined is evaluated or pattern-matched against, it makes the entire expression undefined. However, because Haskell is lazy, it can still be used. For example, these evaluate to undefined: un0 = undefined + 2 un1 = sin undefined un2 = let f x = x * 2 in f undefined un3 = head [undefined, 2, 3, 4] un4 = undefined == 2 -- (==) is just a function Note that the last statement is particularly important: we can’t actually define undefined within Haskell, since any test on undefined produces undefined. The following evaluate to a value that is undefined-less. const_ignores_arguments = const 2 undefined len_of_value = length [undefined, undefined] variable_binding_doesn't_discriminate = let f x = 2 in f undefined The following examples contain undefined (e.g., if you try to print them, they will error out), but are not undefined in and of themselves – there is a possible function that discriminates between the value and undefined. (A tuple of such functions is given to the right.) undefined_can_be_in_a_list = [undefined] -- (length, tail, \[x] -> 2) a_tuple_of_undefined = (,) undefined undefined -- ($$x,x) -> 2) list_containing_undefined = tail [2, undefined, 2, 4] -- (length, !! 1) wrapper_type = return undefined :: Maybe a -- (\(Just x) -> 2) evaluation_to_undefined_is_local = map (*2) [undefined, undefined] -- (length, tail . tail) ## The self-wrapping data type data Wrap = W Wrap We have the potential values of 1. undefined (of course) 2. W undefined 3. W (W undefined) 4. W (W (W undefined)) 5. let x = W x in x These are all distinct because we can define functions as so: data Nat = Z \| S Nat f :: Nat -> Wrap -> () f Z _ = () f (S n) (W x) = f n x f Z u will return () for any u (because it doesn’t actually examine it’s argument). Any other second argument will lead to the evaluation of the inner undefined and will result in undefined. f (S Z) u will return () for W undefined, W (W undefined), etc. f (S (S Z)) u will return () for W (W undefined), W (W (undefined)), etc. So in general, all of these are wrappers around undefined with a given number of wrappers, except for the last. However, if we unwrap this value, we get the same value. In other words, --last_value = W (W (W (W (W (W (W (W (W (W (W (W (W ...)))))))))))) last_value :: Wrap last_value = W last_value It would at first seem as if f n last_value is () for any non-undefined-containing n. However, Nat is itself a fairly interesting type. ## The Natural Numbers (to a bad approximation) From the Peano axioms, we know that a set of natural numbers can be represented as (zero, successor) :: (Nat, Nat -> Nat) class Peano a where zero :: a succ :: a -> a peano_struct :: (Peano a) => (a, a -> a) peano_struct = (zero, succ) The remaining Peano axioms are properties which are difficult to express in Haskell’s type system. In any case, the upshot is that the values of (Peano a) => a should be of the form: one, two, three, four, five :: (Peano a) => a one = succ zero two = succ one -- = succ (succ zero) three = succ two -- = succ (succ (succ zero)) four = succ three -- = succ (succ (succ succ (zero))) five = succ four -- = succ (succ (succ (succ succ (zero)))) And, for any arbitrary implementation of the Peano class, we should only have these values. However, Haskell can’t actually enforce this restriction, leaving us with the values: alt_zero, alt_one, alt_two, alt_three :: (Peano a) => a alt_zero = undefined alt_one = succ alt_zero -- = succ undefined alt_two = succ alt_one -- = succ (succ undefined) alt_three = succ alt_two -- = succ (succ (succ undefined)) Which correspond more or less to our wrapper values. However, these don’t solve our puzzle of the f n last_value which evaluates to undefined while n is undefined-free. However, we have one more value omega :: (Peano a) => a omega = succ omega This corresponds to the equation \(x = x + 1$$, which in the theory of ordinals has the solution $$\omega$$, or the first trans-infinite ordinal. This value structurally resembles last_value :: Wrap, which means that when f is called, it will step through the two of them in sync, never returning, and thus it is equivalent to undefined. The trivial connection between Peano and Nat is given below. instance (Peano Nat) where zero = Z succ = S ## The values of the Peano Integers. In Haskell, Nat is a very inefficient type, using unary. A more efficient type is Integer, which uses binary. instance (Peano Integer) where zero = 0 succ = (+1) Of course, the primary issue with using the Integer type to represent a natural number is that it can have negative values. More relevantly, because (+1) maps undefineds to undefineds, we don’t have any of our alt_ values. Finally, the definition of omega specified to Integer values implies the following declaration: omega_int :: Integer omega_int = omega_int + 1 Which ends up being equivalent to undefined as well, since again (+1) is not a constructor that can allow for the creation of a structure. ## Conclusion I hope that introduction to nonstandard elements was interesting. In case you want to test out these code samples (all of which are runnable), you can find the source code here.	2018-07-22 14:08:01	{"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": 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": 0.6557323932647705, "perplexity": 2331.469801370293}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1531676593302.74/warc/CC-MAIN-20180722135607-20180722155607-00613.warc.gz"}
https://www.physicsforums.com/threads/branching-process-inductive-proof.597513/	# Branching process, inductive proof 1. Apr 17, 2012 ### spitz 1. The problem statement, all variables and given/known data Assume that the the offspring distribution is $P(Y=y)=\left(\frac{1}{2}\right)^y\frac{1}{3}$ $y=0,1,2,\ldots$ Show by induction that: $$G_n(s)=\frac{1-2^n-2(1-2^{n-1})s}{1-2^{n+1}-2(1-2^n)s}$$ 2. The attempt at a solution I can see that the distribution is geometric so: $$G(s)=\frac{p}{1-qs}=\frac{1}{3-2s}$$ I assume I have to show that: $$G_{n+1}(s)=\frac{1-2^n-2(1-2^{n-1})\frac{1}{3-2s}}{1-2^{n+1}-2(1-2^n)\frac{1}{3-2s}}$$ equals: $$\frac{1-2^{n+1}-2(1-2^{n})s}{1-2^{n+2}-2(1-2^{n+1})s}$$ The thing is, this seems like kind of a tedious question considering the amount of marks I'll get for it on my exam. Am I missing something here? Is there a "quick" way to do this?	2018-03-23 17:56:38	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.7432076930999756, "perplexity": 504.6629507179832}, "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-2018-13/segments/1521257648404.94/warc/CC-MAIN-20180323161421-20180323181421-00042.warc.gz"}
http://pub.acta.hu/acta/showCustomerArticle.action?id=54633&dataObjectType=article&returnAction=showCustomerVolume&sessionDataSetId=729008a21291dc93&style=	ACTA issues ## Derivative bounded functional calculus of power bounded operators on Banach spaces Loris Arnold Acta Sci. Math. (Szeged) 87:1-2(2021), 251-280 40/2020 Abstract. In this article we study bounded operators $T$ on a Banach space $X$ which satisfy the discrete Gomilko--Shi-Feng condition $\int _{0}^{2\pi }\|\langle R(re^{it},T)^{2}x,x^\rangle \|dt \leq \frac {C}{(r^2-1)}\norme {x}\norme {x^},\quad r>1, x\in X, x^* \in X^*$. DOI: 10.14232/actasm-020-040-y AMS Subject Classification (1991): 47A60, 46B28, 42B35 Keyword(s): $\gamma$-boundedness, power bounded operators, functional calculus, Besov spaces received 9.10.2020, revised 26.11.2020, accepted 6.12.2020. (Registered under 40/2020.)	2022-07-06 09:29:27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7877503037452698, "perplexity": 5382.475764215739}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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-20220706090857-20220706120857-00195.warc.gz"}
https://labs.tib.eu/arxiv/?author=Ding%20Zhang	• ### Atomic Visualization of Copper Oxide Structure in Infinite-Layer Cuprate SrCuO2(1803.02033) We report on atomic-scale visualization of the structure of infinite-layer cuprate SrCuO2 thin films grown on Nb-doped SrTiO3 substrates by molecular beam epitaxy. In-situ scanning tunneling microscopy study reveals stoichiometric copper oxide (CuO2) plane with a 2 x 2 surface reconstruction, prompted by preferential clustering of four adjacent CuO2 plaquettes. By imaging the subsurface Sr atoms, intra-unit-cell rotational symmetry breaking is observed, which, together with the adjacent CuO2 clustering, can be well accounted for by a periodic up-down buckling of oxygen ions on the CuO2 plane. Further post-annealing leads to an incommensurate stripe structure of the surface layer. Our findings provide important structural information for deeply understanding the electronic structure of superconducting CuO2 plane as well as high temperature superconductivity in cuprates. • ### Observation of Interface Superconductivity in a SnSe2-Epitaxial Graphene van der Waals Heterostructure(1802.08434) We report on the direct observation of interface superconductivity in single-unit-cell SnSe2 films grown on graphitized SiC(0001) substrate by means of van der Waals epitaxy. Tunneling spectrum in the superconducting state reveals rather conventional character with a fully gapped order parameter. The occurrence of superconductivity is further confirmed by the presence of vortices under external magnetic field. Through interface engineering, we unravel the mechanism of superconductivity that originates from a two-dimensional electron gas formed at the interface of SnSe2 and graphene. Our finding opens up novel strategies to hunt for and understand interface superconductivity based on van der Waals heterostructures. • ### Superconductivity in Few-Layer Stanene(1712.03695) Dec. 22, 2017 cond-mat.mtrl-sci A single atomic slice of {\alpha}-tin-stanene-has been predicted to host quantum spin Hall effect at room temperature, offering an ideal platform to study low-dimensional and topological physics. While recent research has intensively focused on monolayer stanene, the quantum size effect in few-layer stanene could profoundly change material properties, but remains unexplored. By exploring the layer degree of freedom, we unexpectedly discover superconductivity in few-layer stanene down to a bilayer grown on PbTe, while bulk {\alpha}-tin is not superconductive. Through substrate engineering, we further realize a transition from a single-band to a two-band superconductor with a doubling of the transition temperature. In-situ angle resolved photoemission spectroscopy (ARPES) together with first-principles calculations elucidate the corresponding band structure. Interestingly, the theory also indicates the existence of a topologically nontrivial band. Our experimental findings open up novel strategies for constructing two-dimensional topological superconductors. • ### Nodeless pairing in superconducting copper-oxide monolayer films on Bi2Sr2CaCu2O8+{\delta}(1607.01852) The pairing mechanism of high-temperature superconductivity in cuprates remains the biggest unresolved mystery in condensed matter physics. To solve the problem, one of the most effective approaches is to investigate directly the superconducting CuO2 layers. Here, by growing CuO2 monolayer films on Bi2Sr2CaCu2O8+{\delta} substrates, we identify two distinct and spatially separated energy gaps centered at the Fermi energy, a smaller U-like gap and a larger V-like gap on the films, and study their interactions with alien atoms by low-temperature scanning tunneling microscopy. The newly discovered U-like gap exhibits strong phase coherence and is immune to scattering by K, Cs and Ag atoms, suggesting its nature as a nodeless superconducting gap in the CuO2 layers, whereas the V-like gap agrees with the well-known pseudogap state in the underdoped regime. Our results support an s-wave superconductivity in Bi2Sr2CaCu2O8+{\delta}, which, we propose, originates from the modulation-doping resultant two-dimensional hole liquid confined in the CuO2 layers. • ### Interface induced high temperature superconductivity in single unit-cell FeSe films on SrTiO3(110)(1512.01948) May 6, 2016 cond-mat.supr-con We report high temperature superconductivity in one unit-cell (1-UC) FeSe films grown on STO(110) substrate by molecular beam epitaxy. By in-situ scanning tunneling spectroscopy measurement, we observed a superconducting gap as large as 17 meV. Transport measurements on 1-UC FeSe/STO(110) capped with FeTe layers reveal superconductivity with an onset TC of 31.6 K and an upper critical magnetic field of 30.2 T. We also find that the TC can be further increased by an external electric field, but the effect is smaller than that on STO(001) substrate. The study points out the important roles of interface related charge transfer and electron-phonon coupling in the high temperature superconductivity of FeSe/STO. • ### Superconductivity dichotomy in K-coated single and double unit cell FeSe films on SrTiO3(1509.08950) Sept. 29, 2015 cond-mat.supr-con We report the superconductivity evolution of one unit cell (1-UC) and 2-UC FeSe films on SrTiO3(001) substrates with potassium (K) adsorption. By in situ scanning tunneling spectroscopy measurement, we find that the superconductivity in 1-UC FeSe films is continuously suppressed with increasing K coverage, whereas non-superconducting 2-UC FeSe films become superconducting with a gap of ~17 meV or ~11 meV depending on whether the underlying 1-UC films are superconducting or not. This work explicitly reveals that the interface electron-phonon coupling is strongly related to the charge transfer at FeSe/STO interface and plays vital role in enhancing Cooper pairing in both 1-UC and 2-UC FeSe films. • ### Interface enhanced electron-phonon coupling and high temperature superconductivity in potassium-coated ultra-thin FeSe films on SrTiO3(1508.06368) Aug. 26, 2015 cond-mat.supr-con Alkali-metal (potassium) adsorption on FeSe thin films with thickness from two unit cells (UC) to 4-UC on SrTiO3 grown by molecular beam epitaxy is investigated with a low-temperature scanning tunneling microscope. At appropriate potassium coverage (0.2-0.3 monolayer), the tunneling spectra of the films all exhibit a superconducting-like gap larger than 11 meV (five times the gap value of bulk FeSe), and two distinct features of characteristic phonon modes at 11 meV and 21 meV. The results reveal the critical role of the interface enhanced electron-phonon coupling for possible high temperature superconductivity in the system and is consistent with recent theories. Our study provides compelling evidence for the conventional pairing mechanism for this type of heterostructure superconducting systems. • ### Interplay between the coherent and incoherent transport in quantum Hall bilayers(1406.0326) June 2, 2014 cond-mat.mes-hall We systematically study the coherent transport (Josephson tunneling and counterflow current) and its breakdown which leads to incoherent charge flow in in the excitonic BCS condensate formed in GaAs bilayers at $\nu_{tot}=1/2+1/2$. The Josephson currents in samples with three different interlayer distances vary by four orders of magnitude. In contrast, the breakdown thresholds for the $\nu_{tot}=1$ quantum Hall state are comparable. Furthermore, Coulomb drag in a Corbino ring reveals that the coherent counterflow current coexists with the dissipative charge transport.	2020-03-30 12:36:55	{"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": 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": 0.5198054313659668, "perplexity": 4471.597709380072}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1585370497042.33/warc/CC-MAIN-20200330120036-20200330150036-00395.warc.gz"}
https://getm.eu/files/GETM/doc/html/node15.html	## Layer-integrated equations There are two different ways to derive the layer-integrated equations. Burchard and Petersen (1997) transform first the equations into general vertical coordinate form (see Deleersnijder and Ruddick (1992)) and afterwards integrate the transformed equations over constant intervals in the transformed space. Lander et al. (1994) integrate the equations in the Cartesian space over surfaces by considering the Leibniz rule (20) for any function . For the vertical staggering of the layer notation see figure 8. More details about the layer integration are given in Burchard and Petersen (1997). With the further definitions of layer integrated transport, (21) layer mean velocities, (22) and layer averaged tracer concentrations and buoyancy, (23) and the grid related vertical velocity, (24) the continuity equation (3) has the layer-integrated form: (25) It should be noted that the grid related velocity is located on the layer interfaces. After this, the layer-integrated momentum equations read as: (26) (27) with suitably chosen advective horizontal velocities and (see section 8.13.7) on page , the shear stresses (28) and (29) (30) and (31) The layer integration of the pressure gradient force is discussed in detail by Burchard and Petersen (1997). A conservative formulation can be derived for the recalculation of the physical vertical velocity which is convenient in the discrete space if is evaluated at the layer centres (see Deleersnijder and Ruddick (1992)): (32) It should be mentioned that only needs to be evaluated for post-processing reasons. For the layer-integrated tracer concentrations, we obtain the following expression: (33) It should be noted that the "horizontal" diffusion does no longer occur along geopotential surfaces but along horizontal coordinate lines. The properly transformed formulation would include some cross-diagonal terms which may lead to numerical instabilities due to violation of monotonicity. For an in-depth discussion of this problem, see Beckers et al. (1998) and Beckers et al. (2000). kklingbe 2017-10-02	2021-04-13 19:44:49	{"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": 0.8707522749900818, "perplexity": 2802.862820200032}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1618038074941.13/warc/CC-MAIN-20210413183055-20210413213055-00494.warc.gz"}
http://www.techbriefs.com/component/ntb_tags/topic/35/0/61/0?start=70	# Infrared Cameras Support Advanced 3D Printing Efforts Additive manufacturing (AM), also known as 3D printing, is quite literally one of the most innovative technologies revolutionizing manufacturing today, in terms of both industry “buzz” and thermal properties. Unlike subtractive manufacturing methods such as machining, the growing range of AM technologies creates components directly from a computer model, adding material only where needed. Wohlers Associates, a leading independent consulting firm focused on these technologies, is forecasting that the value of the worldwide AM market will grow to more than $10.8 billion by 2021, up from just$2.2 billion in 2012. That rapid escalation, however, isn't the result of hobbyists buying desktop 3D printers that cost a few hundred dollars. Posted in: Articles, Cameras, Imaging, Photonics, CAD, CAM, and CAE, Optics, Market research, Technical review, Additive manufacturing # Researchers Turn iPhone Camera into Optical Sensor By integrating an optical Micro-Electro-Mechanical Systems, or MEMS, chip into an iPhone camera, researchers at the VTT Technical Research Centre of Finland have developed a new, cost-effective kind of hyperspectral technology. The spectral device will provide mobile device users and consumers with new ways to monitor their environments, including quick food analysis, health checks, and other Internet-connected sensing. Research team leader Anna Rissanen works actively with companies to enable commercialization and new business development based on the team's various sensors. Posted in: Articles, Optics, Sensors, Microelectromechanical devices, Optics, Sensors and actuators, Research and development # Photonic Choke-Joints for Dual-Polarization Waveguides ### The joint is constructed from a conductive metal, and requires no maintenance or peripheral equipment to operate. Photonic choke-joint (PCJ) structures for dual-polarization waveguides have been investigated at NASA's Goddard Space Flight Center for use in device and component packaging. This interface enables the realization of a high-performance, non-contacting waveguide joint without degrading the in-band signal propagation properties. The choke properties of two tiling approaches — symmetric square Cartesian and octagonal quasi-crystal lattices of metallic posts — are explored and optimal PCJ design parameters are presented. For each of these schemes, the experimental results for structures with finite tilings demonstrate near ideal transmission and reflection performance over a full waveguide band. Posted in: Briefs, Photonics, Electronic equipment, Waveguides, Product development # Smart Optical Material Characterization System and Method ### This technology creates a flexible, unified platform for dynamic smart optical material evaluation. NASA's Langley Research Center has developed an adaptable and powerful interferometric test platform that uniquely enables multi-parameter evaluation of a wide variety of smart optical materials (SOM). The patent-pending SOM characterization system was created to measure the dynamic optical response of stimuli-responsive (“smart”) optical materials while external physical/electrical/thermal/chemical/pressure/magneto stimuli are applied to the material. Using novel interferometric fringe analysis software and a multi-stimuli-capable SOM test cell, the SOM characterization system enables a wide variety of materials — such as liquid crystals, nonlinear crystals, electro- and thermo-active polymer optics, and magneto- or piezo-driven optics — to be optically characterized for real-time changes in intensity, phase, and polarization. The versatility of the SOM test platform combined with the powerful, efficient, and user-friendly software interface makes it a valuable tool for the research or commercial development of smart materials. Posted in: Briefs, Photonics, Computer software and hardware, Optics, Materials identification, Smart materials, Test equipment and instrumentation # Compact Planar Microwave Blocking Filters Innovators at NASA's Goddard Space Flight Center have designed, fabricated, and characterized absorptive thermal blocking filters for cryogenic microwave applications. The device allows direct integration of the high-frequency signal and microwave readout, and mitigates spurious resonances in the circuit response. This leads to improved electrical performance and a reduction in the required circuit area. The transmission line filter's input characteristic impedance is designed to match 50 ohms and its response has been validated from 0 to 50 GHz. The observed return loss in the 0 to 20 GHz design band is greater than 20 dB and shows graceful degradation with frequency. The filter's response is calculable, repeatable under cryogenic cycling, and is capable of providing an intrinsically broadband matched impedance termination. Posted in: Briefs, Photonics, Electrical systems, Thermal management, Product development, Insulation # System and Method for Generating a Frequency-Modulated Linear Laser Waveform ### Applications include manufacturing equipment, robotics, surveillance and security, military imaging, and spectroscopy. NASA's Langley Research Center has made a breakthrough improvement in laser frequency modulation. Frequency modulation technology has been used for surface mapping and measurement in sonar, radar, and time-of-flight laser technologies for decades. Although adequate, the accuracy of distance measurements made by these technologies can be improved by using a high-frequency triangular-waveform laser instead of a sine waveform or lower-frequency radio or microwaves. This new system generates a triangular modulation waveform with improved linearity that makes possible precision laser radar (light detection and ranging [lidar]) for a variety of applications. Posted in: Briefs, Photonics, Lidar, Performance upgrades # Systems and Methods for Mirror Mounting with Minimized Distortion The use of larger, lighter, and more precise space optics requires not only a means of manufacture, but also a means of spacecraft integration and performance verification. Engineers at NASA's Goddard Space Flight Center (GSFC) have demonstrated a process capable of producing a high-precision, mounted, lightweight mirror, and have validated its on-orbit figure. This effort included the design of a mount capable of surviving the launch environment of a sounding rocket, as well as a mounting process that did not introduce performance-degrading figure distortion. Additionally, analysis techniques were developed and adapted to address the challenges in measuring an optic that exceeds its figure specification under the strain of its own weight. Posted in: Briefs, Photonics, Mirrors, Optics, Mountings, Durability, Lightweighting # Improved Approach to Exoplanet Coronagraphy Visible nulling coronagraphy and interferometry requires that the wavefront errors be held to unprecedented precision in the presence of environmental disturbances. A Null Diversity algorithm is used to first attain the precision, but it does not execute at high enough temporal bandwidth to hold the precision for long periods of time (hours). The environmental changes, mostly vibration and jitter with some thermal drift, can be rapidly varying and thus require a fast control algorithm. To perform rapid control, an algorithm, based upon a series of approximations, has been developed and simulated at NASA Goddard Space Flight Center for the sensing and control, in closed loop, of extremely precise wave-front errors in an interferometer. It operates over the range of ~5 nanometers rms down to <100 picometers rms in closed loop at high bandwidth (~20 Hz) and is used to hold (i.e. maintain) the requisite wavefront error. Posted in: Briefs, Photonics, Mathematical models, Lasers, Vibration # Apparatus and Method for a Light Direction Sensor This invention, developed at NASA's Goddard Space Flight Center, was originally conceived as a high-accuracy, high-sensitivity, bi-axial Sun angle sensor, but has also been proposed for applications involving the general field of precisely measuring the direction in which light travels toward the sensor. It has applications in spacecraft navigation, formation flying in space, space beacons, and automotive collision avoidance. Posted in: Briefs, Photonics, Measurements, Sensors and actuators, Sun and solar	2017-07-26 10:46:36	{"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": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.28973063826560974, "perplexity": 5345.007734978628}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549426133.16/warc/CC-MAIN-20170726102230-20170726122230-00295.warc.gz"}
https://prodg.org/blog/joys_of_pi/2015-03-14/Joys%20of%20Pi:%20Test%20Server%20and%20Monitor%20host%20for%20the%20startup%20developer	# Joys of Pi: Test Server and Monitor host for the startup developer Raspberry Pi Happy Hacking Build Servers Pi Fun on Pi Day! 2015-03-14 https://www.authorea.com/users/5713/articles/20746 I work at a startup. We get things done on a budget. This is a summary of how I got: • A new test server for my local dev machine • An extra monitor for my dev setup • Tons of fun! And all for less than $\100$. Before we get started, I love the idea of a local build server for the low latency and complete control you have over the machine. It provides good contrast with the reliability and scaling of using a cloud solution: I like having my local workflow as nimble as possible and use the big guns for staging and production. And ideally you end up with something fun to setup and use. Raspberry Pi 2 came out early this year, and is enough to make any technophile dream of countless garage projects to put together with it. Just looking at the beauty of the device ups my heartbeat: I grabbed one of the many Raspberry Pi startup kits for $\69.99$. You can go cheaper if you have some of the necessary cables laying around (the base Pi is just $\35$). You can skip the WiFi Dongle if you have a LAN hub in place and an extra cable for instance. I won’t write a full setup guide here – you can find some good ones online – just a quick demo. There was an unused monitor in our office, so I hooked it up via the Pi’s HDMI port. The Pi doesn’t have its own speakers, but did you know most modern monitors can play sounds via HDMI? I was surprised myself. Once you get booted up, you can setup synergy and remote control your new server from your main workstation. Extra monitor? Check. Setting up a test server is an experience that varies with your software stack. At Authorea, we use a standard flavour of Ruby on Rails, which is generally painless to setup. The one catch may be natively building some of the binaries – therubyracer was my one and only pain point. The answer that worked for me was hidden in a Gitlab installation walkthrough. Once you have Rails setup, you are basically done and can startup your app… …and run your favourite test suite. Whether you want to have a cron job sync with your version control repository (we are hosted on GitHub) or do things over the local network via rsync, it’s all good. Personally, I am not using my Pi setup to test production; we have more reliable tools for that. Instead, I am testing the integrity of my local changes before pushing them online, minimizing the publicity of my blunders. :) It’s also a benchmark for keeping our test suite lean and our environment portable, given the Pi’s humble capabilities. The Pi 2 has just enough juice to manage our test suite with its quad core 1Ghz CPU, but would definitely struggle through anything much larger. Just another way to ensure maximum efficiency on Authorea. While I would not recommend this approach to a test harness as company policy, the Pi is a great friend for the working developer and his local workstation. You get a lot of power and a lot of potential for what you pay, and if you include the fun factor (who doesn’t want to play around with a Raspberry Pi?), this is a test server at a bargain. – Deyan Special thanks to our Guest Editors: Jace Harker and Jeff Montgomery	2019-07-23 22:35:14	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 3, "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": 0.19637340307235718, "perplexity": 2344.2731382301736}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1563195529737.79/warc/CC-MAIN-20190723215340-20190724001340-00510.warc.gz"}
http://lxxb.cstam.org.cn/EN/10.6052/0459-1879-19-068	• Dynamics, Vibration and Control • ### BIFURCATION AND CHAOS OF AXIALLY MOVING BEAMS UNDER TIME-VARYING TENSION 1) Chen Ling(),Tang Youqi() 1. School of Mechanical Engineering, Shanghai Institute of Technology, Shanghai 201418, China School of Mechanical Engineering， Shanghai Institute of Technology, Shanghai 201418, China • Received:2019-03-22 Accepted:2019-05-08 Online:2019-07-18 Published:2019-07-30 • Contact: Tang Youqi Abstract: The transverse parametric vibration of the axially moving structure is always one of the hot topics in the field of nonlinear dynamics. At present, most of the studies are considering the time-varying speed of dynamic model. The parametric excitation comes from harmonic fluctuations of the axial speed. However, the fluctuation of the axial tension in an axially moving structure is more extensive in the engineering application. There are few researches considering the time-varying tension. The bifurcation and the chaotic behavior of axially accelerating viscoelastic beams under time-varying tension are studied in this paper. A nonlinear integropartia-differential governing equation of the moving beam is established. The linear viscous damping and the Kelvin model in the viscoelastic constitution relation are introduced. The axial tension is assumed as a harmonic variation with time. The fourth-order Galerkin truncation is employed to discretize the governing equation. The dynamic behavior of axially accelerating viscoelastic beams is determined by applying the fourth-order Runge-Kutta algorithm. The influences of material's viscoelastic coefficients, the mean axial speeds, the axial tension fluctuation amplitudes, and the axial tension fluctuation frequencies on the bifurcation diagrams are demonstrated by some numerical results of the displacement and velocity at the midpoint of the beam. The maximum Lyapunov exponent diagram of the system is used to identify the period motion and chaos motion. The results show that the smaller mean axial speed leads to the periodic motion. The period-doubling bifurcation and chaotic behavior are easy to occur near the critical speed. The larger axial tension fluctuation amplitude results in the larger chaos interval. The less viscoelastic coefficient and axial tension fluctuation frequencies lead to the chaotic behavior of the axially moving beam. Furthermore, chaos motions are confirmed using different factors, such as the time history, the fast Fourier transforms, the phase-plane portrait and the Poincaré map. CLC Number:	2021-04-16 11:17:34	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.212448388338089, "perplexity": 1318.7940874397373}, "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-2021-17/segments/1618038056325.1/warc/CC-MAIN-20210416100222-20210416130222-00454.warc.gz"}
https://kr.mathworks.com/help/physmod/sdl/ref/synchronizer.html	# Synchronizer Cone clutch, dog clutch, and translational detent assembled to provide smooth gear engagement • Library: • Simscape / Driveline / Clutches ## Description The block represents a synchronizer that contains a dog clutch, a cone clutch, and a translational detent. The shift linkage first translates to engage the cone clutch. Frictional torque causes the shift linkage and cone clutch shaft to rotate at equal speed. When the force acting on the shift linkage exceeds the detent force, the dog clutch can engage. The schematic illustrates a synchronizer in the disengaged state. In this state, the ring (R) and hub (H) shafts can spin independently at different speeds. To synchronize ring and hub shaft speeds, the shift linkage (S) translates toward the hub shaft to engage the cone clutch. The friction surfaces of the cone clutch produce a frictional torque that equalizes the rotational speeds of the ring and hub shafts. The dog clutch teeth (T) can engage when the translational force acting on the shift linkage exceeds the peak detent force. The peak detent force should allow sufficient time and normal force to equalize ring and hub shaft speeds so that the dog clutch can engage. The model implements the Dog Clutch, Cone Clutch, and Translational Detent blocks. Refer to each block reference page for more information on the corresponding block function. You can use a similar approach to model customized versions of the synchronizer. One example is the Transmission (Detailed) subsystem in the model sdl_vehicle_manual_transmission. Connections R and H are mechanical rotational conserving ports that represent the ring (R) and hub (H), respectively. Connection S is a mechanical translational conserving port that represents the ring shifter handle. Connections X1 and X2 are physical signal outputs that report the shift linkage positions of the dog clutch and cone clutch, respectively. The shift linkage positions are zero when the clutch is fully disengaged. When the dog clutch is fully engaged, the dog clutch shift linkage position has a magnitude equal to the sum of the dog clutch ring-hub gap and the tooth height. When cone clutch is fully engaged, the cone clutch shift linkage position has a magnitude equal to the cone clutch ring-hub gap. ### Thermal Modeling You can model the effects of heat flow and temperature change through an optional thermal conserving port. By default, the thermal port is hidden. To expose the thermal port, in the Clutch settings, select a temperature-dependent setting tor the Friction model parameter. Specify the associated thermal parameters for the component. ## Assumptions and Limitations • The model does not account for inertia effects. You can add a Simscape™ Inertia block at each port to add inertia to the synchronizer model. ## Ports ### Output expand all Physical signal output port that measures the magnitude of the dog clutch translation. Physical signal output port that measures the magnitude of the cone clutch translation. ### Conserving expand all Mechanical rotational conserving port associated with the clutch hub Mechanical rotational conserving port associated with the clutch ring. Mechanical rotational conserving port associated with shift linkage. Thermal conserving port associated with heat flow. #### Dependencies This port is visible only when, in the Friction settings, the Friction model parameter is set to ```Temperature-dependent friction coefficients``` or ```Temperature and velocity-dependent friction coefficients```. ## Parameters expand all The table shows how the specified options for parameters in both the Cone Clutch and Dog Clutch settings affect the visibility of: • Parameters in the Cone Clutch, Dog Clutch, and Initial Conditions settings • Thermal Port settings • Thermal port T To learn how to read the table, see Parameter Dependencies. Synchronizer Block Parameter Dependencies SettingsParameters and Options Cone ClutchContact surface maximum diameter Contact surface minimum diameter Cone half angle Friction model ```Fixed kinetic friction coefficient``````Velocity-dependent kinetic friction coefficient``````Temperature-dependent friction coefficients``````Temperature and velocity-dependent friction coefficients``` -- Exposes: • Conserving port T • Thermal Port settings Exposes: • Conserving port T • Thermal Port settings -Relative velocity vector Relative velocity vector --Temperature vectorTemperature vector Static friction coefficientStatic friction coefficient vectorStatic friction coefficient vectorStatic friction coefficient matrix Kinetic friction coefficientKinetic friction coefficient vectorKinetic friction coefficient vectorKinetic friction coefficient matrix -Friction coefficient interpolation methodFriction coefficient interpolation methodFriction coefficient interpolation method -Friction coefficient extrapolation methodFriction coefficient extrapolation methodFriction coefficient extrapolation Velocity toleranceVelocity toleranceVelocity toleranceVelocity tolerance Threshold forceThreshold forceThreshold forceThreshold force Dog ClutchTorque transmission modelTorque transmission model-- ```Friction clutch approximation - Suitable for HIL and linearization````Dynamic with backlash````Friction clutch approximation - Suitable for HIL and linearization````Dynamic with backlash`-- ----Temperature vectorTemperature vector Maximum transmitted torque-Maximum transmitted torque-Maximum transmitted torque vectorMaximum transmitted torque vector ----Interpolation methodInterpolation method ----Extrapolation methodExtrapolation method -Number of teeth-Number of teeth Rotational backlash Rotational backlash -Torsional stiffness-Torsional stiffness -Torsional damping-Torsional damping -Tooth-tooth friction coefficient-Tooth-tooth friction coefficient Initial ConditionsInitial stateInitial stateInitial stateInitial stateInitial stateInitial state -Initial dog clutch ring-hub offset angle-Initial dog clutch ring-hub offset angle-- Thermal Port----Thermal massThermal mass ----Initial temperatureInitial temperature ### Cone Clutch Outer conical diameter do. Inner conical diameter di. Half opening angle α of the cone geometry. Parameterization method to model the kinetic friction coefficient. The options and default values for this parameter depend on the friction model that you select for the block. The options are: • ```Fixed kinetic friction coefficient``` — Provide a fixed value for the kinetic friction coefficient. • ```Velocity-dependent kinetic friction coefficient``` — Define the kinetic friction coefficient by one-dimensional table lookup based on the relative angular velocity between disks. • ```Temperature-dependent friction coefficients``` — Define the kinetic friction coefficient by table lookup based on the temperature. • ```Temperature and velocity-dependent friction coefficients``` — Define the kinetic friction coefficient by table lookup based on the temperature and the relative angular velocity between disks. #### Dependencies The friction model setting affects the visibility of other parameters, settings, and ports. Input values for the relative velocity as a vector. The values in the vector must increase from left to right. The minimum number of values depends on the interpolation method that you select. For linear interpolation, provide at least two values per dimension. For smooth interpolation, provide at least three values per dimension. #### Dependencies This parameter is visible only when the Friction model parameter is set to ```Velocity-dependent kinetic friction coefficient``` or ```Temperature and velocity-dependent friction coefficients```. Input values for the temperature as a vector. The minimum number of values depends on the interpolation method that you select. For linear interpolation, provide at least two values per dimension. For smooth interpolation, provide at least three values per dimension. The values in the vector must increase from left to right. #### Dependencies This parameter is visible only when the Friction model parameter is set to ```Temperature-dependent friction coefficients``` or ```Temperature and velocity-dependent friction coefficients```. Static or peak value of the friction coefficient. The static friction coefficient must be greater than the kinetic friction coefficient. #### Dependencies this parameter is visible only when the Friction model parameter is set to ```Fixed kinetic friction coefficient``` or ```Velocity-dependent kinetic friction coefficient```. Static, or peak, values of the friction coefficient as a vector. The vector must have the same number of elements as the temperature vector. Each value must be greater than the value of the corresponding element in the kinetic friction coefficient vector. #### Dependencies This parameter is visible only when the Friction model parameter is set to ```Temperature-dependent friction coefficients``` or ```Temperature and velocity-dependent friction coefficients```. The kinetic, or Coulomb, friction coefficient. The coefficient must be greater than zero. #### Dependencies This parameter is visible only when the Friction model parameter is set to ```Fixed kinetic friction coefficient```. Output values for kinetic friction coefficient as a vector. All values must be greater than zero. If the Friction model parameter is set to • ```Velocity-dependent kinetic friction coefficient``` — The vector must have same number of elements as relative velocity vector. • ```Temperature-dependent friction coefficients``` — The vector must have the same number of elements as the temperature vector. #### Dependencies This parameter is visible only when the Friction model parameter is set to ```Velocity-dependent kinetic friction coefficient``` or ```Temperature-dependent friction coefficients```. Output values for kinetic friction coefficient as a matrix. All the values must be greater than zero. The size of the matrix must equal the size of the matrix that is the result of the temperature vector × the kinetic friction coefficient relative velocity vector. #### Dependencies This parameter is visible only when the Friction model parameter is set to ```Temperature and velocity-dependent friction coefficients```. Interpolation method for approximating the output value when the input value is between two consecutive grid points: • `Linear` — Select this option to get the best performance. • `Smooth` — Select this option to produce a continuous curve with continuous first-order derivatives. For more information on interpolation algorithms, see the PS Lookup Table (1D) block reference page. #### Dependencies This parameter is visible only when, in the Cone Clutch settings, the Friction model parameter is set to ```Velocity-dependent kinetic friction coefficient```, ```Temperature-dependent friction coefficients```, or ```Temperature and velocity-dependent friction coefficients```. Extrapolation method for determining the output value when the input value is outside the range specified in the argument list: • `Linear` — Select this option to produce a curve with continuous first-order derivatives in the extrapolation region and at the boundary with the interpolation region. • `Nearest` — Select this option to produce an extrapolation that does not go above the highest point in the data or below the lowest point in the data. • `Error` — Select this option to avoid going into the extrapolation mode when you want your data to be within the table range. If the input signal is outside the range of the table, the simulation stops and generates an error. For more information on extrapolation algorithms, see the PS Lookup Table (1D) block reference page. #### Dependencies This parameter is visible only when, in the Cone Clutch settings, the Friction model parameter is set to ```Velocity-dependent kinetic friction coefficient```, ```Temperature-dependent friction coefficients```, or ```Temperature and velocity-dependent friction coefficients```. Relative velocity below which the two surfaces can lock. The surfaces lock if the torque is less than the product of the effective radius, the static friction coefficient, and the applied normal force. The normal force is applied only if the amount of force exceeds the value of the Threshold force parameter. Forces below the Threshold force are not applied so there is no transmitted frictional torque. ### Dog Clutch The methods that are available for parameterizing the torque transmission depend whether the friction model is temperature-dependent. The friction model is determined, in the Cone Clutch settings, by the Friction model parameter setting: • `Fixed kinetic friction coefficient` — Temperature independent • ```Velocity-dependent kinetic friction coefficient``` — Temperature independent • `Temperature-dependent friction coefficients` — Temperature dependent • ```Temperature and velocity-dependent friction coefficients``` — Temperature dependent For a temperature-independent model, parameterize the block using one of the options for the Torque Transmission Model parameter. Computational framework for modeling the dynamic behavior of the dog clutch: • ```Friction clutch approximation — Suitable for HIL and linearization``` — Model clutch engagement as a friction phenomenon between the ring and the hub. This model, based on the Fundamental Friction Clutch block, provides a computationally efficient approximation of the dog clutch. • `Dynamic with backlash` — Model clutch engagement in detail, accounting for such phenomena as backlash, torsional compliance, and contact forces between ring and hub teeth. #### Dependencies This parameter is visible only if, in the Cone Clutch settings, the Friction model parameter is set to ```Fixed kinetic friction coefficient``` or ```Velocity-dependent friction coefficients```. The visibility of related parameters in the Dog Clutch and Initial Conditions settings is affected by the option that you select for this parameter. Input values for the temperature as a vector. The minimum number of values depends on the interpolation method that you select. For linear interpolation, provide at least two values per dimension. For smooth interpolation, provide at least three values per dimension. The values in the vector must increase from left to right. #### Dependencies This parameter is visible only if, in the Cone Clutch settings, the Friction model parameter is set to ```Temperature-dependent friction coefficients``` or ```Temperature and velocity-dependent friction coefficients```. Largest torque that the clutch can transmit, corresponding to a nonslip engaged configuration. If the torque transmitted between the ring and the hub exceeds this value, the two components begin to slip with respect to each other. This torque determines the static friction limit in the friction clutch approximation #### Dependencies This parameter is visible only if, in the Cone Clutch settings, the Friction model parameter is set to ```Fixed kinetic friction coefficient``` or ```Velocity-dependent kinetic friction coefficient``` and, in Dog Clutch settings, the Torque transmission model parameter is set to ```Friction clutch approximation - Suitable for HIL and linearization```. Largest torque that the clutch can transmit, corresponding to a nonslip engaged configuration, specified as a vector. If the torque transmitted between the ring and the hub exceeds this value, the two components begin to slip with respect to each other. This torque determines the static friction limit in the friction clutch approximation. The vector has the same number of elements as the temperature vector. #### Dependencies This parameter is visible only if, in the Cone Clutch settings, the Friction model parameter is set to ```Temperature-dependent kinetic friction coefficient``` or ```Temperature and velocity-dependent kinetic friction coefficient```. Interpolation method for approximating the output value when the input value is between two consecutive grid points: • `Linear` — Select this option to get the best performance. • `Smooth` — Select this option to produce a continuous curve with continuous first-order derivatives. For more information on interpolation algorithms, see the PS Lookup Table (1D) block reference page. #### Dependencies This parameter is visible only if, in the Cone Clutch settings, the Friction model parameter is set to ```Temperature-dependent kinetic friction coefficient``` or ```Temperature and velocity-dependent kinetic friction coefficient```. Extrapolation method for determining the output value when the input value is outside the range specified in the argument list: • `Linear` — Select this option to produce a curve with continuous first-order derivatives in the extrapolation region and at the boundary with the interpolation region. • `Nearest` — Select this option to produce an extrapolation that does not go above the highest point in the data or below the lowest point in the data. • `Error` — Select this option to avoid going into the extrapolation mode when you want your data to be within the table range. If the input signal is outside the range of the table, the simulation stops and generates an error. For more information on extrapolation algorithms, see the PS Lookup Table (1D) block reference page. #### Dependencies This parameter is visible only if, in the Cone Clutch settings, the Friction model parameter is set to ```Temperature-dependent kinetic friction coefficient``` or ```Temperature and velocity-dependent kinetic friction coefficient```. Distance from the ring or hub center to the corresponding tooth center. The mean tooth radius determines the normal contact forces between ring and hub teeth given the transmission torque between the two components. The value must be greater than zero. Total number of teeth in the ring or the hub. The two components have equal tooth numbers. The value must be greater than or equal to one. #### Dependencies This parameter is visible only if, in the Cone Clutch settings, the Friction model parameter is set to ```Fixed kinetic friction coefficient``` or ```Velocity-dependent kinetic friction coefficient``` and, in the Dog Clutch settings, the Torque transmission model parameter is set to ```Dynamic with backlash```. Allowable angular motion, or play, between the ring and hub teeth in the engaged clutch configuration. The value must be greater than zero. #### Dependencies This parameter is visible only if, in the Cone Clutch settings, the Friction model parameter is set to ```Fixed kinetic friction coefficient``` or ```Velocity-dependent kinetic friction coefficient``` and, in the Dog Clutch settings, the Torque transmission model parameter is set to ```Dynamic with backlash```. Linear torsional stiffness coefficient at the contact interface between the ring and hub teeth. This coefficient characterizes the restoring component of the contact force between the two sets of teeth. Greater stiffness values correspond to greater contact forces. The value must be greater than zero. The default value is `10e6` `N*m/rad`. #### Dependencies This parameter is visible only if, in the Cone Clutch settings, the Friction model parameter is set to ```Fixed kinetic friction coefficient``` or ```Velocity-dependent kinetic friction coefficient``` and, in the Dog Clutch settings, the Torque transmission model parameter is set to ```Dynamic with backlash```. Linear torsional damping coefficient at the contact interface between the ring and hub teeth. This coefficient characterizes the dissipative component of the contact force between the two sets of teeth. Greater damping values correspond to greater energy dissipation during contact. The value must be greater than zero. #### Dependencies This parameter is visible only if, in the Cone Clutch settings, the Friction model parameter is set to ```Fixed kinetic friction coefficient``` or ```Velocity-dependent kinetic friction coefficient``` and, in the Dog Clutch settings, the Torque transmission model parameter is set to ```Dynamic with backlash```. Kinetic friction coefficient at the contact interface between ring and hub teeth. This coefficient characterizes the dissipative force that resists shift linkage motion due to tooth-tooth contact during clutch engagement/disengagement. Greater coefficient values correspond to greater energy dissipation during shift linkage motion. The value must be greater than zero. #### Dependencies This parameter is visible only if, in the Cone Clutch settings, the Friction model parameter is set to ```Fixed kinetic friction coefficient``` or ```Velocity-dependent kinetic friction coefficient``` and, in the Dog Clutch settings, the Torque transmission model parameter is set to ```Dynamic with backlash```. ### Detent Peak shear force of the detent. Width of the region where the detent exhibits shear force. Viscous friction coefficient at the contact surface of the detent. The value must be greater than or equal to zero. Ratio of the kinetic friction to the peak shear force of the detent. The parameter is used to set the value of the kinetic friction. The parameter must be greater than or equal to zero. Velocity required for peak kinetic friction at the contact surface of the detent. The parameter ensures the force is continuous when the travel direction changes, increasing the numerical stability of the simulation. The parameter must be greater than zero. The default value is `0.05` `m/s`. Direction the shift linkage must travel in to engage the clutch. Choices include positive and negative displacements. Relative angular velocity between the ring and the hub above which the clutch cannot engage. The value is specific to the specific gearbox or transmission. Minimizing the value helps avoid high dynamic impact during engagement. The value must be greater than zero. Overlap length between ring and hub teeth along the common longitudinal axis above which the clutch can engage. The clutch remains disengaged until the tooth overlap by at least this length. The value must be greater than zero. Distance between the base and crest of a tooth. Ring and hub teeth share the same height. The tooth height and the ring-hub clearance when fully disengaged determine the maximum travel span of the shift linkage. The value must be greater than zero. Maximum open gap between the ring and hub tooth crests along the shift linkage translation axis. This gap corresponds to the fully disengaged clutch state. The tooth height and the ring-hub clearance when fully disengaged determine the maximum travel span of the shift linkage. The value must be greater than zero. Hard stop that prevents the shift linkage from traveling beyond the fully disengaged position: • `On` — Hard stop when fully disengaged. • `Off` — No hard stop when fully disengaged. Stiffness of the hard stops on both sides of the dog clutch ring. The model assumes the ring and stops behave elastically. Contact deformation is proportional to the applied force and the reciprocal of the contact stiffness. The value of the stiffness must be assigned with reference to the parameter Tooth overlap to engage. Too low a stiffness could cause the deformation to exceed the required overlap and initiate a false engagement. The parameter must be greater than zero. Stiffness of the hard stops on both sides of the cone clutch ring. The model assumes the ring and stops behave elastically. Contact deformation is proportional to the applied force and the reciprocal of the contact stiffness. Translational contact damping between the dog clutch ring and the hub. The value of the damping is inversely proportional to the number of oscillations that occur after impact. The parameter must be greater than zero. Translational contact damping between the cone clutch ring and the hub. The value of damping is inversely proportional to the number of oscillations that occur after impact. The parameter must be greater than zero. Viscous friction coefficient for the relative translational motion between the hub and the ring. The value of the parameter depends on lubrication state and quality of contacting surfaces. The coefficient must be greater than or equal to zero. ### Initial Conditions Beginning configuration of cone and dog clutches: • `All clutches unlocked` — Cone and dog clutches transmit zero torque between the ring and hub shafts. • `Cone clutch locked` — Cone clutch transmits torque between the ring and hub shafts. • `All clutches locked` — Cone and dog clutches transmit torque between the ring and hub shafts. Initial position of the shift linkage section that attaches to the dog clutch. The value of the parameter has these restrictions: Linkage Travel Direction Dog Clutch StateParameter Restriction ```Positive shift linkage displacement engages clutch```Initially engagedParameter must be greater than the sum of parameters Ring-hub clearance when dog clutch disengaged and Tooth overlap to engage Initially disengagedParameter must be smaller than the sum of parameters Ring-hub clearance when dog clutch disengaged and Tooth overlap to engage ```Negative shift linkage displacement engages clutch```Initially engagedNegative of the parameter must be greater than the sum of parameters Ring-hub clearance when dog clutch disengaged and Tooth overlap to engage Initially disengagedNegative of the parameter must be smaller than the sum of parameters Ring-hub clearance when dog clutch disengaged and Tooth overlap to engage Initial position of the shift linkage section that attaches to the cone clutch. The value of the parameter has these restrictions: Linkage Travel Direction Dog Clutch StateParameter Restriction ```Positive shift linkage displacement engages clutch```Initially engagedParameter must be greater than the value of Ring-hub clearance when cone clutch disengaged Initially disengagedParameter must be smaller than the value of Ring-hub clearance when cone clutch disengaged ```Negative shift linkage displacement engages clutch```Initially engagedNegative of the parameter must be greater than the value of Ring-hub clearance when dog cone disengaged Initially disengagedNegative of the parameter must be smaller than the value of Ring-hub clearance when dog cone disengaged Rotation angle between the ring and the hub at simulation time zero. This angle determines whether the ring and hub teeth can interlock, and hence whether the clutch can engage. The initial offset angle must satisfy these conditions: • If the clutch initial state is disengaged, the initial offset angle must fall in the range `$-\frac{{180}^{°}}{N}\le {\varphi }_{0}\le +\frac{{180}^{°}}{N},$` where N is the number of teeth present in the ring or the hub. The two components contain the same number of teeth. • If the clutch initial state is engaged, the initial offset angle must fall in the range `$-\frac{\delta }{2}\le {\varphi }_{0}\le +\frac{\delta }{2},$` where δ is the backlash angle between the ring and hub teeth. #### Dependencies This parameter is visible only if both of these conditions are met: • In the Cone Clutch settings, the Friction model parameter is set to ```Fixed kinetic friction coefficient``` or ```Velocity-dependent kinetic friction coefficient```. • In the Dog Clutch settings, the Torque transmission model is set to ```Dynamic with backlash```. ### Thermal Port Thermal Port settings are visible only when, in the Cone Clutch settings, the Friction model parameter is set to ```Temperature-dependent friction coefficients``` or ```Temperature and velocity-dependent friction coefficients```. For more information, see Synchronizer Block Parameter Dependencies. Thermal energy required to change the component temperature by a single degree. The greater the thermal mass, the more resistant the component is to temperature change. #### Dependencies This parameter is visible only if, in the Cone Clutch settings, the Friction model parameter is set to ```Temperature-dependent friction coefficients``` or ```Temperature and velocity-dependent friction coefficients```. For more information, see Synchronizer Block Parameter Dependencies. Component temperature at the start of simulation. The initial temperature alters the component efficiency according to an efficiency vector that you specify, affecting the starting meshing or friction losses. #### Dependencies This parameter is visible only if, in the Cone Clutch settings, the Friction model parameter is set to ```Temperature-dependent friction coefficients``` or ```Temperature and velocity-dependent friction coefficients```. For more information, see Synchronizer Block Parameter Dependencies. expand all	2020-11-24 15:35:58	{"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": 0.582825779914856, "perplexity": 1945.4793292132422}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141176864.5/warc/CC-MAIN-20201124140942-20201124170942-00547.warc.gz"}
http://physics.stackexchange.com/questions/8755/how-does-the-period-of-an-hourglass-depend-on-the-grain-size	How does the period of an hourglass depend on the grain size? Suppose I have an hourglass that takes 1 full hour on average to drain. The grains of sand are, say, $1 \pm 0.1\ {\rm mm}$ in diameter. If I replace this with very finely-grained sand $0.1 \pm 0.01\ {\rm mm}$ in diameter but keep the hourglass otherwise the same, how long a time will the "hourglass" now measure? Does this depend on the size of the funnel, or should all one-hour hourglasses change in roughly the same way? Bonus: is the new hourglass more or less precise? (precision defined at $\sigma_t /t$, with $t$ the time to drain) - I recently saw a colloquium that included some discussion of jammed granular solids. There are phase changes under some conditions, so I suspect the behavior could be highly non-linear under some condition. – dmckee Apr 16 '11 at 18:38 It is, in fact. The flow in (on) the cones of a hourglas shows tiny avalanches. This is a very popular experiment in chaos theory. – Georg Apr 17 '11 at 12:23 I have at least an answer : Does this depend on the size of the funnel ? Yes, as it was expected (for a very large funnel, all sand falls in the same time, whereas for a very small funnel it doesn't fall at all. More precisely ... the complete answer is probably extremely complex, as one has to take into account the shape of the hourglass, the dynamics of the grains, and so on (see http://arxiv.org/abs/0707.4550 for example). However we can have a rough idea with some dimensional analysis. Let us consider a cylinder of diameter $D$, with a circular hole punched on the bottom side with radius $a$. We fill the cylinder with a height $H$ of sand. If we look at the speed of sand grains going out the bucket, we observe (experimentally) that it does not depend of the height of sand $H$, if $H$ is big enough (compared to the diameter $D$ - because the constraint saturates). We are left with two parameters : the diameter of the hole $a$ and the gravity field $g$ that makes it fall, so the output speed $v$ has to be proportional to $\sqrt{g a}$. The flow rate is the speed times the section, thus it is $Q \propto v \, a^2$, so it is of order $Q \propto g^{1/2} a^{5/2}$ (this is the Beverloo law). However we have to take into account the size of grains ; let us suppose they are spherical and let be $d$ their diameter. All grains that are less than half on the hole won't fall, so there is a ring-shaped exclusion region at the border of the hole. The effective diameter of the hole, were grains will actually fall, is thus reduced by $d/2 + d/2 = d$, and is thus $a-d$. We thus replace $a$ with $a-d$ in the Beverloo law and get $Q \propto g^{1/2} (a-d)^{5/2}$ Now, this becomes invalid when $H$ and $D$ are of the same order, but we can get an idea of the time to drain $T$ by saying that as the flow rate does not depend of $H$, $Q T = V = 4 \pi (D/2)^2 H = \pi D^2 H$ so, dropping the $\pi$ we get $T \propto g^{1/2} \, \frac{(a-d)^{5/2}}{D^2 \, H}$ and thus for everything but the size of sand grains $d$ constant $\frac{T(d_1)}{T(d_2)} = \left( \frac{a - d_1}{a - d_2} \right)^{5/2}$ - A link to a paper is always good, but can you please point to the abstract page rather than directly to the PDF. That way I can decide if I want the paper. – dmckee Apr 17 '11 at 0:17 You're right. Done. – Georg Sievelson Apr 17 '11 at 0:59 Very good answer! That this Beverloo law is from as late as 1961, strange! – Georg Apr 17 '11 at 10:16 As a first guess, I would think the strain rate under a fixed amount of shear stress for a unit cube of material should be expected to be proportional to the surface area of the grains. That would imply that the strain rate, and hence the flow rate for fixed geometry should be inversely proportional to grain size. So I think your fine grained hourglass would be much faster. I suspect in reality things would get more complicated then that. What shape are the grains? Spherical is probably faster than angular. What about grain on grain sliding coefficient of friction? Is there some stress level below which no flow/creep takes place? -	2016-06-30 08:37:24	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8518233299255371, "perplexity": 412.7876505952359}, "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-2016-26/segments/1466783398216.41/warc/CC-MAIN-20160624154958-00117-ip-10-164-35-72.ec2.internal.warc.gz"}
https://hal.inria.fr/hal-01402613	# Crowd Dynamics through Non-Local Conservation Laws 1 ACUMES - Analysis and Control of Unsteady Models for Engineering Sciences CRISAM - Inria Sophia Antipolis - Méditerranée Abstract : We present a Lax–Friedrichs type scheme to compute the solutions of a class of non-local and non-linear systems of conservation laws in several space dimensions. The convergence of the approximate solutions is proved by providing suitable $L 1 , L ∞$ and BV uniform bounds. To illustrate the performances of the scheme, we consider an application to crowd dynamics. Numerical integrations show the formation of lanes in groups moving in opposite directions. This is joint work with R. M. Colombo (INDAM Unit, University of Brescia). Keywords : Document type : Journal articles Cited literature [14 references] https://hal.inria.fr/hal-01402613 Contributor : Paola Goatin Connect in order to contact the contributor Submitted on : Thursday, November 24, 2016 - 9:54:28 PM Last modification on : Saturday, June 25, 2022 - 11:24:32 PM Long-term archiving on: : Tuesday, March 21, 2017 - 12:28:51 PM ### File AggarwalColomboGoatin_proceedi... Files produced by the author(s) ### Citation Aekta Aggarwal, Paola Goatin. Crowd Dynamics through Non-Local Conservation Laws. Boletim da Sociedade Brasileira de Matemática / Bulletin of the Brazilian Mathematical Society, Springer Verlag, 2016, 47, pp.37 - 50. ⟨10.1007/s00574-016-0120-7⟩. ⟨hal-01402613⟩ Record views	2022-07-01 05:00:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.18526491522789001, "perplexity": 3955.423826459913}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1656103920118.49/warc/CC-MAIN-20220701034437-20220701064437-00716.warc.gz"}
http://www.komal.hu/feladat?a=honap&h=201511&t=inf&l=en	Mathematical and Physical Journal for High Schools Issued by the MATFUND Foundation Already signed up? New to KöMaL? # KöMaL Problems in Informatics, November 2015 Show/hide problems of signs: ## Problems with sign 'I' Deadline expired on December 10, 2015. I. 385. The Playfair cipher (Source: $\displaystyle {\rm https://en.wikipedia.org/wiki/Playfair\_cipher}$) was invented by Charles Wheatstone in 1854, but he named it after his friend Lord Playfair who made the method popular. The encryption scheme itself had been broken before World War I, although Australians still used it in World War II (because computers were not yet available, and by the time that the enemy could break the message, the information would be useless to them). The encryption scheme is based on a $\displaystyle 5\times 5$ table containing the letters of the English alphabet-this alphabet has 26 letters, so we need to discard one letter, say, Q. The table is known only to the sender and the receiver. The text to be encrypted (in our example FINOM IZ) is first partitioned into consecutive pairs of letters. If necessary, a new symbol (in our case X) is appended at the end. We proceed similarly if the original text contains a double letter in a pair: for example, AA is converted into AX AX. During the encryption process, each pair of letters is replaced by another pair according to the following rules. $\displaystyle \bullet$ If both letters of the original pair are in a single row, then take the corresponding right neighbor in the table for each letter. The right neighbor of the last letter in a row is the first letter of the same row (FI $\displaystyle \to$ RN). $\displaystyle \bullet$ If both letters of the original pair are in a single column, then take the corresponding lower neighbor in the table for each letter. The lower neighbor of the last letter in a column is the first letter of the same column (NO $\displaystyle \to$ VN). $\displaystyle \bullet$ Finally, if the two letters of the original pair are found in different rows and columns, then we consider the rectangle whose diagonal is formed by these two letters. Letters of the new pair will be given by letters of the other diagonal of the rectangle, by keeping the same row for a letter (MI $\displaystyle \to$ KF). The first command-line argument of your program is either a character R or character V to denote whether the user wants to encrypt or decrypt data. The second argument is the input file name containing the Playfair table (read from left to right and top to bottom), the third argument is the input file name containing the text to be encrypted or decrypted, and the fourth argument is the output file name. You can assume that the input text contains only uppercase letters of the English alphabet according to the above. Your program should be able to handle large (several GB in size) input or output files. The source code and documentation of your program-containing a brief description of your solution, and the name of the developer environment to compile your code-should be submitted in a compressed file i385.zip. Downloadable files (a possible Playfair table, and a sample original text file): kod.txt, be.txt. (10 pont) solution, statistics I. 386. A certain school in Hungary bears the name of the Hungarian poet Gyula Juhász (1883-1937). Since the beginning of the 2011 school year they select a poem of Juhász every day and display it on the webpage of the school. Visitors can like'' these poems. The database contains data from three consecutive school years. Create a new database jgy. The two, tabulator-separated and UTF-8 encoded text files (vers.txt, napverse.txt) should be imported into the database by using their original names (vers, napverse). The first line of the file contains the field names. The appropriate types and keys should be set. Tables: You should solve the following tasks and save your queries by using the names in parentheses. Your solution should contain only the requested fields. 1. List all poems that appeared on the webpage in September 2011 and sort them according to their dates. Display the poem title and creation year. (2szept) 2. Consider the poems which appeared more than once on the webpage, and determine which three got the most likes''. (3like) 3. Create a report to list the poems that appeared on the webpage during the winter of 2013/2014 (between December and February), grouped according to their creation year and sorted alphabetically. (4tel) 4. Determine which poems appeared on the webpage in all four calendar years. (5negyev) 5. Determine which poems appeared on the webpage only in 2011. (6csak2011) 6. Determine which days had poems with creation year identical to the creation year of the poem of the previous day. (7azonosev) 7. Determine whether Juhász created more poems in his first or last active decade. The first year of the more active decade should be displayed. (8evtized) Your solution as a database or a text file with the SQL queries should be submitted in a compressed file (i386.zip), also containing a short documentation with the name and version number of the database application. (10 pont) solution, statistics I. 387. Converting Roman numerals to Arabic numbers was the topic of the spreadsheet management task of the practical part of the advanced level Matriculation Examination in Informatics in Hungary in May 2012. The algorithm given there can also be realized by using functional programming. By using a Logo program and the given algorithm, convert Roman numerals to Arabic numbers. You can assume that the input Roman numerals (between 1 and 4000, and described by at most 20 capital letters) are syntactically correct. The conversion algorithm is described below. Here a számjegyek értéke'' is the values of the numerals''; 0 a végére'' is appending 0 at the end''; az egymás melletti elempárok'' is adjacent pairs''; a párok első eleme'' is the first entries of the pairs''; negatív, ha kisebb a másodiknál'' is negative if the first entry is smaller than the second one''; and összegzés" is summation''. The sign of a given numeral is negative if the following numeral is larger. The value of the last numeral is always positive. Create the Logo words corresponding to the above steps of the algorithm, then the word római_tízes performing the actual conversion. Sample command Result római_tízes "MCCXCIV 1294 Only the automatic and functional features of the programming language should be used in your solution; variables should not be used just parameterizations. The project file/source code of your program should be submitted (i387.imp). (10 pont) solution, statistics ## Problems with sign 'I/S' Deadline expired on December 10, 2015. I/S. 3. One can buy $\displaystyle 1\le N\le 1000$ items in a shop. Your available money is $\displaystyle 1\le P \le 10^9$. Each item has a price $\displaystyle A_i$, and a home delivery cost $\displaystyle H_i$, so the total cost of item $\displaystyle i$ is $\displaystyle A_i+H_i$ (non-negative integers, $\displaystyle A_i$ is even to simplify this exercise). We also have a coupon to halve the price of a selected item: if we use it with item $\displaystyle i$, then its price would be $\displaystyle A_i/2+H_i$. Determine the maximum number of items that can be bought if we are allowed to use only one coupon. Your program should read the values of $\displaystyle N$ and $\displaystyle P$ from the first line of the standard input, then the integers $\displaystyle A_i$, $\displaystyle H_i$ (separated by a space) from the following $\displaystyle N$ lines. The first and only line of the standard output should contain the maximum number of items that can be purchased. Explanation: it is possible to buy the first 4 items provided that the coupon is used with the third item. Scoring and bounds. You can get 1 point for a brief and proper documentation clearly describing your solution. Nine further points can be obtained provided that your program solves any valid input within 1 second of running time. The source code of your program without the .exe or any other auxiliary files generated by the compiler but with a short documentation-also describing which developer environment to use for compiling the source-should be submitted in a compressed file is3.zip. (10 pont) solution, statistics ## Problems with sign 'S' Deadline expired on December 10, 2015. S. 102. A certain robot is controlled by the following commands: it starts from position 0, then upon receiving 15 R and 20 L, for example, it makes 15 steps to the right then 20 steps to the left. The robot receives $\displaystyle N$ commands ($\displaystyle 1\le N\le 300\;000$). The step numbers appearing in these R/L commands are positive integers, and the maximum allowed distance of the robot from the origin is $\displaystyle 10^{9}$. You are also given a number $\displaystyle K$. Your task is to determine the number of positions on or over which the robot passed at least $\displaystyle K$ times. Your program should read the values of $\displaystyle N$ and $\displaystyle K$ from the first line of the standard input, then the $\displaystyle a_i$, $\displaystyle c_i$ number-character pairs (separated by a space) from the following $\displaystyle N$ lines describing the motion of the robot. The first and only line of the standard output should contain the appropriate number of positions the robot visited. Scoring and bounds. You can get 1 point for a brief and proper documentation clearly describing your solution. Nine further points can be obtained provided that your program solves any valid input within 1 second of running time. The source code of your program without the .exe or any other auxiliary files generated by the compiler but with a short documentation---also describing which developer environment to use for compiling the source---should be submitted in a compressed file s102.zip. (10 pont) solution, statistics \$Var(Body)	2021-03-09 07:58:23	{"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": 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": 0.3425859212875366, "perplexity": 1006.4509127609261}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1614178389472.95/warc/CC-MAIN-20210309061538-20210309091538-00143.warc.gz"}
https://www.mathdoubts.com/diagonal-matrix/	# Diagonal Matrix A matrix that consists of zeros as entries (or elements) outside the main diagonal is called a diagonal matrix. ## Introduction Square matrices are appeared with zeros. In a special case, a square matrix contains zero as non-diagonal elements but it contains elements only on principal diagonal. Due to having elements on leading diagonal and having zeros as non-diagonal elements, the square matrix is recognized as a diagonal matrix. $M=\left[\begin{array}{ccccc}{e}_{1⁣1}& 0& 0& \cdots & 0\\ 0& {e}_{2⁣2}& 0& \cdots & 0\\ 0& 0& {e}_{3⁣3}& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & {e}_{m⁣m}\end{array}\right]$ The matrix is having elements ${e}_{1⁣1},{e}_{2⁣2},{e}_{3⁣3},\dots {e}_{m⁣m}$ only on principal diagonal but observe the elements on non-diagonal areas. All are zero elements at non-diagonal areas. Therefore, this type of matrix is called a diagonal matrix. The diagonal elements can be either equal or unequal elements. It is simply expressed as $M=diag\left[\begin{array}{ccccc}{e}_{1⁣1,}& {e}_{2⁣2,}& {e}_{3⁣3,}& \cdots & {e}_{n⁣n}\end{array}\right]$ ## Example $D$ is a square matrix of order $5×5$. It is having $25$ element in five rows and five columns. $D=\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& –5& 0& 0& 0\\ 0& 0& 7& 0& 0\\ 0& 0& 0& 3& 0\\ 0& 0& 0& 0& 9\end{array}\right]$ The matrix $D$ is having two types of elements. One type of elements are nonzero elements and remaining all are zeros. Nonzero elements ($1,–5,7,3$ and $9$) are placed on the leading diagonal and remaining non-diagonal elements are zeros. Therefore, the matrix $D$ is known as a diagonal matrix. The diagonal matrix $D$ is written in simple form $D=diag\left[\begin{array}{ccccc}1,& –5,& 7,& 3,& 9\end{array}\right]$ Latest Math Topics Apr 18, 2022 Apr 14, 2022 A best free mathematics education website for students, teachers and researchers. ###### Maths Topics Learn each topic of the mathematics easily with understandable proofs and visual animation graphics. ###### Maths Problems Learn how to solve the maths problems in different methods with understandable steps. Learn solutions ###### Subscribe us You can get the latest updates from us by following to our official page of Math Doubts in one of your favourite social media sites.	2022-06-28 08:33:43	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 13, "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": 0.6468587517738342, "perplexity": 1102.2521345439498}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103360935.27/warc/CC-MAIN-20220628081102-20220628111102-00172.warc.gz"}
https://www.sarthaks.com/2722189/truck-travels-place-away-speed-returns-with-speed-find-the-average-speed-truck-for-the-whole	# A truck travels to a place 120 km away at an speed of 40 km/h and returns with a speed of 30 km/h. Find the average speed of the truck for the whole j 17 views in Aptitude closed A truck travels to a place 120 km away at an speed of 40 km/h and returns with a speed of 30 km/h. Find the average speed of the truck for the whole journey? 1. $34 \frac 2 7$ km/h 2. $28 \frac 2 7$ km/h 3. $38 \frac 2 7$ km/h 4. $40 \frac 2 7$ km/h by (54.3k points) selected Correct Answer - Option 1 : $34 \frac 2 7$ km/h Given: Distance traveled in one side = 120 km Speed while going = 40 km/h Speed while returning = 30 km/h Formula used: Average speed = Total distance travelled/Total time taken Calculation: Time taken while going = (120/40) hr = 3 hrs Time taken while returning = (120/30) hr = 4 hr Total distance travelled while going and returning = (120 × 2) km = 240 km ∴ Average speed of the truck = (240/7) km/h $34 \frac 2 7$ km/h	2022-10-03 05:27:31	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.34385329484939575, "perplexity": 2382.6170205956337}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1664030337398.52/warc/CC-MAIN-20221003035124-20221003065124-00527.warc.gz"}
https://zbmath.org/authors/?q=ai%3Amei.zhandong	# zbMATH — the first resource for mathematics ## Mei, Zhandong Compute Distance To: Author ID: mei.zhandong Published as: Mei, Z.; Mei, Zhan-Dong; Mei, Zhandong Documents Indexed: 28 Publications since 1990 all top 5 #### Co-Authors 1 single-authored 19 Peng, Jigen 3 Gao, Jinghuai 2 Guo, Bao-Zhu 2 Jia, Junxiong 2 Zhang, Yang 1 Song, Xueli 1 Zhang, Yang 1 Zhang, Yang all top 5 #### Serials 2 Semigroup Forum 2 Systems & Control Letters 2 Indagationes Mathematicae. New Series 1 Applicable Analysis 1 Journal of Mathematical Analysis and Applications 1 Mathematical Methods in the Applied Sciences 1 Studia Mathematica 1 Automatica 1 Functional Analysis and its Applications 1 IEEE Transactions on Automatic Control 1 Integral Equations and Operator Theory 1 Mathematische Nachrichten 1 Proceedings of the American Mathematical Society 1 Electronic Journal of Differential Equations (EJDE) 1 Abstract and Applied Analysis 1 Fractional Calculus & Applied Analysis 1 Journal of Dynamical and Control Systems 1 Communications on Pure and Applied Analysis 1 Asian Journal of Control all top 5 #### Fields 11 Ordinary differential equations (34-XX) 9 Operator theory (47-XX) 8 Systems theory; control (93-XX) 3 Biology and other natural sciences (92-XX) 2 Special functions (33-XX) 2 Partial differential equations (35-XX) 1 Real functions (26-XX) 1 Integral equations (45-XX) 1 Probability theory and stochastic processes (60-XX) 1 Fluid mechanics (76-XX) #### Citations contained in zbMATH 17 Publications have been cited 53 times in 44 Documents Cited by Year A characteristic of fractional resolvents. Zbl 1314.34022 Mei, Zhan-Dong; Peng, Ji-Gen; Zhang, Yang 2013 An operator theoretical approach to Riemann-Liouville fractional Cauchy problem. Zbl 1322.34011 Mei, Zhan-Dong; Peng, Ji-Gen; Zhang, Yang 2015 Well-posedness and time-decay for compressible viscoelastic fluids in critical Besov space. Zbl 1311.35219 Jia, Junxiong; Peng, Jigen; Mei, Zhandong 2014 On robustness of exact controllability and exact observability under cross perturbations of the generator in Banach spaces. Zbl 1203.93026 Mei, Zhan-Dong; Peng, Ji-Gen 2010 General fractional differential equations of order $$\alpha \in (1,2)$$ and type $$\beta \in [0,1]$$ in Banach spaces. Zbl 1375.34007 Mei, Zhan-Dong; Peng, Ji-Gen; Gao, Jing-Huai 2017 Existence and uniqueness of solutions for nonlinear general fractional differential equations in Banach spaces. Zbl 1323.34010 Mei, Zhan-Dong; Peng, Ji-Gen; Gao, Jing-Huai 2015 A new characteristic property of Mittag-Leffler functions and fractional cosine functions. Zbl 1297.33015 Mei, Zhan-Dong; Peng, Ji-Gen; Jia, Jun-Xiong 2014 Variation of bifurcations along a homotopy from Neumann to Dirichlet problems. Zbl 0866.35020 Mei, Z.; Theil, F. 1996 On general fractional abstract Cauchy problem. Zbl 1273.34012 Mei, Zhan-Dong; Peng, Ji-Gen; Zhang, Yang 2013 On invariance of $$p$$-admissibility of control and observation operators to $$q$$-type of perturbations of generator of $$C_{0}$$-semigroup. Zbl 1198.93064 Mei, Zhan-Dong; Peng, Ji-Gen 2010 Equations for turbulent flood waves. Zbl 0870.76023 Mei, Z.; Roberts, A. J. 1995 Convoluted fractional $$C$$-semigroups and fractional abstract Cauchy problems. Zbl 07022218 Mei, Zhan-Dong; Peng, Ji-Gen; Gao, Jing-Huai 2014 Robustness of exact $$p$$-controllability and exact $$p$$-observability to $$q$$-type of perturbations of the generator. Zbl 1300.93040 Mei, Zhan-Dong; Peng, Ji-Gen 2014 Riemann-Liouville abstract fractional Cauchy problem with damping. Zbl 1298.34016 Mei, Zhan-Dong; Peng, Ji-Gen 2014 On a characteristic of cosine functions. Zbl 1303.47061 Mei, Zhan-Dong; Peng, Ji-Gen; Zhang, Yang 2014 Modelling the dynamics of turbulent floods. Zbl 1037.76025 Mei, Z.; Roberts, A. J.; Li, Zhenquan 2002 Construction of higher-order algebraic one-step schemes in stiff BVPs. Zbl 0790.65060 Schmitt, B. A.; Mei, Z. 1993 General fractional differential equations of order $$\alpha \in (1,2)$$ and type $$\beta \in [0,1]$$ in Banach spaces. Zbl 1375.34007 Mei, Zhan-Dong; Peng, Ji-Gen; Gao, Jing-Huai 2017 An operator theoretical approach to Riemann-Liouville fractional Cauchy problem. Zbl 1322.34011 Mei, Zhan-Dong; Peng, Ji-Gen; Zhang, Yang 2015 Existence and uniqueness of solutions for nonlinear general fractional differential equations in Banach spaces. Zbl 1323.34010 Mei, Zhan-Dong; Peng, Ji-Gen; Gao, Jing-Huai 2015 Well-posedness and time-decay for compressible viscoelastic fluids in critical Besov space. Zbl 1311.35219 Jia, Junxiong; Peng, Jigen; Mei, Zhandong 2014 A new characteristic property of Mittag-Leffler functions and fractional cosine functions. Zbl 1297.33015 Mei, Zhan-Dong; Peng, Ji-Gen; Jia, Jun-Xiong 2014 Convoluted fractional $$C$$-semigroups and fractional abstract Cauchy problems. Zbl 07022218 Mei, Zhan-Dong; Peng, Ji-Gen; Gao, Jing-Huai 2014 Robustness of exact $$p$$-controllability and exact $$p$$-observability to $$q$$-type of perturbations of the generator. Zbl 1300.93040 Mei, Zhan-Dong; Peng, Ji-Gen 2014 Riemann-Liouville abstract fractional Cauchy problem with damping. Zbl 1298.34016 Mei, Zhan-Dong; Peng, Ji-Gen 2014 On a characteristic of cosine functions. Zbl 1303.47061 Mei, Zhan-Dong; Peng, Ji-Gen; Zhang, Yang 2014 A characteristic of fractional resolvents. Zbl 1314.34022 Mei, Zhan-Dong; Peng, Ji-Gen; Zhang, Yang 2013 On general fractional abstract Cauchy problem. Zbl 1273.34012 Mei, Zhan-Dong; Peng, Ji-Gen; Zhang, Yang 2013 On robustness of exact controllability and exact observability under cross perturbations of the generator in Banach spaces. Zbl 1203.93026 Mei, Zhan-Dong; Peng, Ji-Gen 2010 On invariance of $$p$$-admissibility of control and observation operators to $$q$$-type of perturbations of generator of $$C_{0}$$-semigroup. Zbl 1198.93064 Mei, Zhan-Dong; Peng, Ji-Gen 2010 Modelling the dynamics of turbulent floods. Zbl 1037.76025 Mei, Z.; Roberts, A. J.; Li, Zhenquan 2002 Variation of bifurcations along a homotopy from Neumann to Dirichlet problems. Zbl 0866.35020 Mei, Z.; Theil, F. 1996 Equations for turbulent flood waves. Zbl 0870.76023 Mei, Z.; Roberts, A. J. 1995 Construction of higher-order algebraic one-step schemes in stiff BVPs. Zbl 0790.65060 Schmitt, B. A.; Mei, Z. 1993 all top 5 #### Cited by 72 Authors 5 Mei, Zhandong 5 Peng, Jigen 4 Abadias, Luciano 3 Chen, Pengyu 3 Chen, Yi 3 Gao, Jinghuai 3 Li, Yongxiang 3 Lizama, Carlos 3 Lv, Zhanmei 3 Zhang, Xuping 2 Alvarez, Edgardo 2 Fan, Zhenbin 2 Hong, Nguyen Thi Thuy 2 Mei, Zhen 2 Miana, Pedro J. 2 Ponce, Rodrigo F. 2 Roberts, Anthony John 2 Tan, Zhong 2 Thuan, Do Duc 2 Zhang, Liang 1 Abdo, Mohammed Salem 1 Ahmad, Bashir 1 Ahmad, Bashir 1 Alghanmi, Madeaha 1 Al-saedi, Ahmed Eid Salem 1 Ashyralyev, Allaberen 1 Cao, Meng 1 Chen, Chuang 1 Chien, Cheng-Sheng 1 Dong, Qixiang 1 Gal, Ciprian Gheorghe Sorin 1 Gambera, Laura R. 1 Gao, Zhensheng 1 Gong, Yanping 1 Guo, Bao-Zhu 1 Hamad, Ayman 1 Han, Bin 1 Jia, Junxiong 1 Kadalbajoo, Mohan K. 1 Kuo, Yu-Ju 1 Li, Gang 1 Li, Haitao 1 Li, Miao 1 Li, Xiaodong 1 Li, Yin 1 Lian, Tingting 1 Liu, Can 1 Liu, Yuji 1 Liu, Yun 1 Mao, Jianzhong 1 Mei, Jie 1 Ntouyas, Sotiris K. 1 Patidar, Kailash C. 1 Piskarëv, S. I. 1 Prokopczyk, Andréa C. 1 Roberts, Andrew J. 1 Selivanova, N. Yu. 1 Shih, Chih-Wen 1 Shu, Linxin 1 Shu, Xiaobao 1 Nguyen Khoa Son 1 Thabet, Sabri T. M. 1 Vasil’ev, Victor V. 1 Wang, Shuling 1 Wang, Yong 1 Wei, Ruiying 1 Wu, Guochun 1 Wu, Wenpei 1 Xu, Gen-Qi 1 Yao, Zhengan 1 Yuan, Baoquan 1 Zhao, Guodong all top 5 #### Cited in 32 Serials 4 Fractional Calculus & Applied Analysis 3 Semigroup Forum 3 Systems & Control Letters 2 Journal of Mathematical Physics 2 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 2 Advances in Difference Equations 2 Banach Journal of Mathematical Analysis 2 Journal of Pseudo-Differential Operators and Applications 1 Computers & Mathematics with Applications 1 Computer Physics Communications 1 International Journal of Control 1 Journal of Engineering Mathematics 1 Journal of Mathematical Analysis and Applications 1 Physics Letters. A 1 Rocky Mountain Journal of Mathematics 1 Applied Mathematics and Computation 1 Functional Analysis and its Applications 1 SIAM Journal on Control and Optimization 1 Applied Mathematics Letters 1 Journal of Integral Equations and Applications 1 Journal of Nonlinear Science 1 Topological Methods in Nonlinear Analysis 1 Journal of Mathematical Sciences (New York) 1 Abstract and Applied Analysis 1 Communications on Pure and Applied Analysis 1 Analysis and Applications (Singapore) 1 Boundary Value Problems 1 Symmetry 1 Asian Journal of Control 1 Fractional Differential Calculus 1 Journal of Function Spaces 1 Open Mathematics all top 5 #### Cited in 19 Fields 17 Ordinary differential equations (34-XX) 16 Operator theory (47-XX) 11 Partial differential equations (35-XX) 8 Real functions (26-XX) 8 Fluid mechanics (76-XX) 8 Systems theory; control (93-XX) 4 Integral equations (45-XX) 4 Numerical analysis (65-XX) 3 Dynamical systems and ergodic theory (37-XX) 2 Abstract harmonic analysis (43-XX) 2 Integral transforms, operational calculus (44-XX) 2 Calculus of variations and optimal control; optimization (49-XX) 2 Probability theory and stochastic processes (60-XX) 1 Functions of a complex variable (30-XX) 1 Special functions (33-XX) 1 Functional analysis (46-XX) 1 Computer science (68-XX) 1 Mechanics of deformable solids (74-XX) 1 Biology and other natural sciences (92-XX)	2021-03-07 21:30:59	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5832482576370239, "perplexity": 11751.537229439964}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1614178378872.82/warc/CC-MAIN-20210307200746-20210307230746-00476.warc.gz"}
https://techwhiff.com/learn/signals-and-systems-2-the-pole-zero-diagram-below/355495	# Signals and Systems 2. The pole-zero diagram below has 2 zeros at the origin and 2... ###### Question: Signals and Systems 2. The pole-zero diagram below has 2 zeros at the origin and 2 poles to represent a system A(s). Pole-Zero Map (-0.5, +1) X d Imaginary Au (-0.5, -1) X RealAxis con Is this a stable system? Explain. Write an exact simplified expression for A(s). A(s) = 3. A system has impulse response h(t)= u(t) A e' where A and B are positive constants. Write an exact simplified expression for H(S). #### Similar Solved Questions ##### How should I start this problem? I know that you use P(1+r/n)^(kt) formula and set them equal to each other?... ##### You are walking from your math class to your science class carrying a book weighing 112N You are walking from your math class to your science class carrying a book weighing 112N. You walk 45 meters down the hall, climb 4.5 meters up the stairs and then walk another 40 meters to your science class. What is the total work performed on your books.... ##### How can greenhouse gases be good and bad? How can greenhouse gases be good and bad?... ##### A lot of 100 cars contains 10 defective cars and 90 nondefective cars. If I randomly... A lot of 100 cars contains 10 defective cars and 90 nondefective cars. If I randomly select 2 cars from the lot, what is the probability that both will be nondefective?... ##### Discuss to what most accident deaths are linked. Discuss to what most accident deaths are linked.... ##### Which of the following 10.0 gram substances will increase in temperature by the greatest amount when 100 J of hea... Which of the following 10.0 gram substances will increase in temperature by the greatest amount when 100 J of heat is added to each? Substance D, specific heat 1.67 J/(g°C) Substance C, specific heat 1.22 J/(g°C) Substance A, specific heat 1.06 J/(g°C) Substance B, specific heat 0.892 J/... ##### Police sometimes measure shoe prints at crime scenes so that they can learn something about criminals.... Police sometimes measure shoe prints at crime scenes so that they can learn something about criminals. Listed below are shoe print lengths, foot lengths, and heights of males. Construct a scatterplot, find the value of the linear correlation coefficient r, and find the P-value of r. Determine whethe... ##### Canvas →XCO DE Question 1 Click this link to access the Periodic Table. This may be... Canvas →XCO DE Question 1 Click this link to access the Periodic Table. This may be helpful throughout the exam. List the fourtypes of crystalline solids and give an example of each BIVA-AI EX3x X E E - 22 . 12pt - Paragraph - ©... ##### How do you find the derivative of (e^(2x) - e^(-2x))/(e^(2x) + e^(-2x))? How do you find the derivative of (e^(2x) - e^(-2x))/(e^(2x) + e^(-2x))?... ##### 1. The plans of management are expressed formally in: A. the annual report to shareholders. B.... 1. The plans of management are expressed formally in: A. the annual report to shareholders. B. Form 10-Q submitted to the Securities and Exchange Commission. C. performance reports. D. budgets. 2. Which of the following statements is correct? A. Managerial accounting is focused on precision. B. Mana... ##### A firm has a return on equity of 16 percent, a return on assets of 11... A firm has a return on equity of 16 percent, a return on assets of 11 percent, and a 40 percent dividend payout ratio. What is the sustainable growth rate! O A 7.12 percent OB. 5.72 percent OC 10.62 percent OD.6.84 percent OE. 9.58 percent... ##### 20. All of the following statements are true about correlation EXCEPT: (A) Correlation is always between... 20. All of the following statements are true about correlation EXCEPT:* (A) Correlation is always between 1 and 1, inclusive. (B) Correlation does not change if we change the scales of the explanatory variable (e.g. inches to feet). (C) Correlation measures how strong the linear tendency is between ... ##### Encounters with purely imperative programming languages (that don’t take on any characteristics o... Encounters with purely imperative programming languages (that don’t take on any characteristics of structured or procedural languages) are few and far between. Using the knowledge you have so far about imperative programming, structured, procedural, Fortran, and any previous programming langua...	2023-03-31 03:00:15	{"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": 0.3633821904659271, "perplexity": 2705.783585219392}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1679296949533.16/warc/CC-MAIN-20230331020535-20230331050535-00402.warc.gz"}
https://www.mostafaelaraby.com/paper%20review/2020/12/30/predicting-neural-network-accuracy-from-weights/	# Predicting Neural Network Accuracy from Weights In this paper Unterthiner et al. (2020) showed empirically that we can predict the generalization gap of a neural network by only looking at its weights. In this work, they released a dataset of 120k convolutional neural networks trained on different datasets. The authors extracted a set of statistics features from the network’s weights and fed it to an estimator $$\hat{F}$$ to rpedict the generalization gap(check Predict the generalization gap using marginal distribution). ### Introduction The proposed framework relies on the encoding of the training dataset features into the network’s weights $$W_1 \cdots W_L$$. Using only the encoded features into the weights, we can extract information about how certain the network is on its encoded feature. Hence, we can use it as a feature vector fed to the estimator $$\hat{F}$$. The authors tried to investigate whether the hyper-parameters could be used only as a feature vector to our estimator. They found that there exists a mapping from the hyper-parameters used during the training to the generalization gap if we fix the random seed and the training set. Also, Unterthiner et al. (2020) tried to train the proposed estimator on a set of CNNs trained on a single dataset and tested the estimator on another set of CNNs trained on a different dataset. They called this setting the domain shift. ### Features extracted Now let’s start by showing the set of features extracted. First problem would be to combine the features from variable depth DNNs . The authors proposed to extract some statistics from only the input and the top three hidden layers, which showed better results than statistics from flattening the whole set of weights. The statistics $$\tilde{W}$$ per layer includes the mean, variance and qth percentile $$q \in \{0, 25, 50, 75, 100\}$$. Extracting these statistics for both kernels and biases for the first four layers would result in a feature vector of $$4 \times 2 \times 7 = 56$$ real values. Finally, the authors fed the extracted features into a gradient boosting model and showed promising results on predicting the generalization gap using domain shift setting and different architectures in the train and test set. #### Conclusion From this paper, the main insight would be the possibility of extracting some meaningful information about the training set like its uncertainty and generalization gap using simply the neural network’s trained weights. #### Side Note I would recommend reading the paper itself and checking the related work, this is just a summary to give you a rough idea of what is going on. ### References • T. Unterthiner, D. Keysers, S. Gelly, O. Bousquet, and I. Tolstikhin, “Predicting Neural Network Accuracy from Weights,” arXiv:2002.11448 [cs, stat], May 2020, Accessed: Sep. 27, 2020. [Online]. Available: http://arxiv.org/abs/2002.11448.	2021-03-03 01:56:22	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.48618629574775696, "perplexity": 734.364751148723}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178365186.46/warc/CC-MAIN-20210303012222-20210303042222-00625.warc.gz"}