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https://www.physicsforums.com/threads/the-area-of-a-square.630758/	[ "# The area of a square\n\n## Main Question or Discussion Point\n\nI was just wondering the other day about the concept of area....Area to me is the space occupied in 2d by a bounded figure..... I wanted to find out WHY the area of a square is s^2 or why area of a rectangle is lxb...Consider the dimensions of a rectangle 7x5. The area can be expressed as 5 strips of length 7 i.e. 7+7+7+7+7=35, but now the strips are each 1 unit wide and hence the formula works (width of 5 units is divided into 5 parts of 1 unit), what if the strips are made smaller and smaller such that the strips are infinitesimally small then the formula doesn't make any sense because the width of 5 units is being divided into infinite parts and hence the area is coming out to be zero, thus making me dumbfounded as to the approach adopted by me earlier.\n\nThus, why is the area of a rectangle or square- lxb or sxs???\n\nwhat if the strips are made smaller and smaller such that the strips are infinitesimally small then the formula doesn't make any sense\nObviously the formula will not make sense if you compute it completely differently.\nOf course, it's entirely possible to add together all of those infinitesimally small strips, which will give you the right answer; you just have to use calculus.\n\nThe approach you described (using strips of elements unit 1 wide) seems perfectly intuitive to me and clearly describes why the area of the square is computed as it is, so I'm not sure what else you want. You could use calculus and integrate over a square to the find the area, I suppose, which would give you the formula s2. Let's do that (if you don't have any experience with calculus, just watch how a bunch of incomprehensible stuff happens that gives you the right answer):\n\nWe want to compute the area of a square with side length a. We can do this by integrating the function f(x) = a over the interval 0 < x < a...\n\n$$\\int_0^a\\! a \\, \\mathrm{d} x \\: = \\: a \\cdot a - a \\cdot 0\\: = \\: a^2$$\n\nWhat we've done here is exactly what you described; we took each of those slices and shrunk them down until they were infinitesimally small, and then added them up. This gives us the right answer.\n\nLast edited:\nHi.\n\nHm.\n\nIf width of single strip is identically zero, then there is nothing that could be done. However, if strip width is identically zero, then width is not infinitesimal. So Your reasoning got astray at the moment You passed from \"infinitesimal\" to \"identically zero\". You might have not noticed this transition. If infinitesimal, then not zero. Yes, as close to zero as You like. Never zero though. Use limits. Or integral, of course. I guess question was of speculative nature.\n\nThis reminds me of Zeno paradox.\n\nCheers.\n\nthanks i understood...i guess i have to learn calculus to understand the realm of mathematics which deals with infinitesimally small quantites...how is this book by michael spivak? Everybody told me it's great. What do you suggest?\n\nHi.\n\nI don't know about Spivak's Calculus. I do know about a ton of other books I've read on calculus, though. There are 2 types of calculus books. First type are books for students of mathematics and theoretical physics. They have to know it rigorously because it's in the curriculum and profs lecture it that way. Second type of calculus books are books written for engineers. Those written for engineers go straight for the head: they aim at calculating things. Engineer books are not concerned with purely theoretical aspects of calculus. So, one might go for second type books first, and when differential details become clear, one is advised to take a byte at the real thing. Otherwise, if not informed on the subject at all, one is easily lost in all the details of theoretical math. And calculus is not easy one way or another. It's a huge area and one never gets to master it entirely. Ever. Finally, in my opinion, reading only one book on calculus is not enough. The beast is too huge for only one weapon.\n\nCheers.\n\nThanks...! I will look into these things.\n\nSpivak's a pretty serious Calculus textbook. I've heard it's one of the best, but from what I've gathered unless you're very confident about your mathematic skills (which you may well be), I'd start with a simpler book.\n\nIn Euclidean geometry, the area of a square with side $a$ is postulated to be $a^2$ (hence, the name squared). To justify this claim, imagine you increase the side n times. How many small squares fill the large square?\n\nThen, to prove the formula for the area of a rectangle:\nThe side of the outer square is $a + b$, and the side of the inner square is $a - b$ (assuming $a > b$). Then, the area of the outer square is the sum of the areas of the inner square and the four identical rectangles:\n$$(a + b)^2 = (a - b)^2 + 4 A$$\n$$A = \\frac{(a + b)^2 - (a - b)^2}{2}$$\n$$A = \\frac{(a^2 + 2 a b + b^2) - (a^2 - 2 a b + b^2)}{4}$$\n$$A = a \\, b$$\n\nWhich is that simpler book for calculs? I don't think my math skills are that great. I would surely start with that book where everything is simple enough to completely grasp a particular idea and a concept. Please suggest some simpler book....!!\n\nAre you asking why a square with a side length of say, 2cm, has an area of 4cm2?\nIf so, you can think of it as taking the 2cm along one side and multiplying it by how many of those there are. one side is 2cm wide, and stretches over 2cm. If you are asking how you get 4cm2 from multiplying 2cm by 2cm, you can make a small dimensional analysis.\n\nsubstitute cm into \"d\" you then have ${(2d)^2}={2^2}{d^2}=4{d^2}$\n\nplug our dimension back into d's place, you get:\n\n$${2cm{\\cdot}2cm}=4{cm^2}$$\n\nAnother example is acceleration. In algebriac terms acceleration is:\n\n$\\frac{{v_f}-{v_i}}{t}=a$ where:\n\nvf=final velocity\nvi=initial velocity\nt=time\na=acceleration.\n\nthere are fundemental units of quantity; here is the wiki page for the SI Units http://en.wikipedia.org/wiki/International_System_of_Units#Units_and_prefixes\nWe take these fundemental units to make others, such as these: http://en.wikipedia.org/wiki/List_of_physical_quantities\n\nnotice the velocity is $\\frac{d}{t}$, time is (fundemntal)=t, and acceleration is $\\frac{d}{t^2}$.\n\nYou may ask, \"why is the time squared for acceleration?\" treat the dimensions algebriaclly.\n$$\\frac{(\\frac{d}{t})}{t}=\\frac{d}{tt}=\\frac{d}{t^2}$$\nThe quantities' dimensions treat each other algebriaclly." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9465698,"math_prob":0.98757464,"size":940,"snap":"2020-24-2020-29","text_gpt3_token_len":246,"char_repetition_ratio":0.13461539,"word_repetition_ratio":0.09944751,"special_character_ratio":0.26489362,"punctuation_ratio":0.124423966,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9990351,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-05-29T01:59:19Z\",\"WARC-Record-ID\":\"<urn:uuid:94537b29-db79-4598-9458-b24dae99c3e4>\",\"Content-Length\":\"100987\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a5e5ebe2-5d68-48a3-9f22-74edb73150b3>\",\"WARC-Concurrent-To\":\"<urn:uuid:3e900405-06ec-4aa2-9366-8ca0e5642b46>\",\"WARC-IP-Address\":\"23.111.143.85\",\"WARC-Target-URI\":\"https://www.physicsforums.com/threads/the-area-of-a-square.630758/\",\"WARC-Payload-Digest\":\"sha1:5HJF32OYTVJCV3AGTU2V7YXCBQCUQU33\",\"WARC-Block-Digest\":\"sha1:NIYF23T6FYXEW4NN4T4BFDMWISRPI5YY\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-24/CC-MAIN-2020-24_segments_1590347401004.26_warc_CC-MAIN-20200528232803-20200529022803-00197.warc.gz\"}"}
http://www.celestrak.com/columns/v02n03/	[ "", null, "## Orbital Coordinate Systems, Part III\n\nBy Dr. T.S. Kelso", null, "January/February 1996", null, "", null, "January/February 1996\n\n## Orbital Coordinate Systems, Part III\n\nBy Dr. T.S. Kelso\n\nLast time, we worked through the process of calculating the ECI (Earth-Centered Inertial) coordinates of an observer's position on the Earth's surface, starting with the observer's latitude and longitude. Then, we used those coordinates to calculate look angles (azimuth and elevation) from the observer's position to an orbiting satellite. The most difficult part of that process was in calculating the sidereal time, a quantity necessary to determines the Earth's orientation in inertial space.\n\nIn the process of performing those calculations, however, we made one simplifying assumption: that the Earth is a sphere. Unfortunately, this assumption is not a good one. Ignoring the fact that the Earth's shape can more accurately be described as an oblate spheroid (a flattened sphere) can have a significant effect in certain types of satellite tracking applications. In this column, we will examine the implications of our initial assumption by modifying our calculations to allow for the Earth's flattening at the poles and then tackle the related problem of determining the sub-point of an orbiting satellite. Let's start by looking at a cross-section of the Earth and defining some terms.", null, "Figure 1. Cross-Section of Oblate Earth\n\nFigure 1 is an exaggerated view of the cross-section of the Earth. For an observer on the Earth's surface, we can define a couple of terms fairly easily. The first is the local zenith. The local zenith direction is just a fancy way of saying \"straight up.\" It is the direction away from a point on the Earth's surface perpendicular (at a right angle to) the local horizon. On a sphere, this direction is always directly away from the Earth's center. However, on an oblate spheroid, this is not the case since a line from the center of the Earth to the observer's position would not point to the local zenith (except on the equator and at the poles).\n\nSince the local zenith direction depends upon the local horizon, let's take some time to better define it, as well. The local horizon is a plane which is tangent (touching at a point) to the Earth's surface at the observer's position. For our purposes, we will consider the local horizon to be the plane tangent to the reference spheroid. The term reference spheroid is used to define the oblate spheroid which 'best' defines the shape of the Earth. How 'best' is defined is a complicated process and depends upon whether the fit of the reference spheroid is regional or global. We will use the reference spheroid defined in WGS-72 (World Geodetic System, 1972) for our standard.\n\nIn WGS-72, the Earth's equatorial radius, a, is defined to be 6,378.135 km. The Earth's polar radius, b, is related to the equatorial radius by something called the flattening, f, where\n\nb = a(1 - f)\n\nThe flattening term, as defined in WGS-72, is only 1/298.26—a very small deviation from a perfect sphere. Using this value, the Earth's polar radius would be 6,356.751 km—only 22 kilometers difference from the equatorial radius.\n\nThe first real significance of using an oblate spheroid instead of a sphere to define the Earth's shape comes in determining the observer's latitude. On a sphere, latitude is defined as the angle between the line going from the center of the Earth to the observer and the Earth's equatorial plane. However, on an oblate spheroid, geodetic latitude is the angle between the local zenith direction and the Earth's equatorial plane. This angle, φ, is the latitude used on maps; the angle formed by the observer's position, the Earth's center, and the equatorial plane is more properly referred to as the geocentric latitude, φ'.\n\nThe impact of this change is that in order to calculate the observer's ECI position, we must determine the geocentric latitude from the geodetic latitude. Knowing the geocentric latitude, φ', we can then calculate the geocentric radius, ρ, and from that calculate the z coordinate (ρ sin φ') and the projection in the equatorial plane (ρ cos φ'). Let's start by developing the relationship between φ and φ' since we'll usually be given φ.\n\nFrom the basic definition of an ellipse,", null, "where\n\nR' = ρ cos(φ')\n\nand\n\nz' = ρ sin(φ').\n\nNow,\n\ntan(φ') =", null, "and\n\ntan(φ) =", null, "(that is, the normal to the tangent of the spheroid). Differentiating the equation of the ellipse,", null, "and rearranging terms,", null, "which can be written as,\n\ntan(φ') =", null, "tan(φ) = (1-f)2 tan(φ).\n\nSo, knowing the geodetic latitude and the flattening, we can now determine the geocentric latitude. Now, let's see how much of a difference results from using an oblate spheroid. Figure 2 plots the difference between geodetic and geocentric latitude as a function of geodetic latitude.", null, "Figure 2. Geocentric vs. Geodetic Latitude\n\nThat's it? All that work and the maximum error is less than two-tenths of a degree? It would hardly seem worth the effort to perform the calculation. But let's explore a little further.\n\nAlthough the development is too complicated to present here, it can be shown that\n\nρ sin(φ') = z' = a S sin(φ)\n\nand\n\nρ cos(φ') = R' = a C cos(φ)\n\nwhere", null, "", null, ".\n\nOur ECI coordinates, are now\n\nx' = a C cos(φ) cos(θ)\n\ny' = a C cos(φ) sin(θ)\n\nz' = a S sin(φ).\n\nUsing the example of calculating the ECI coordinates of 40° N (geodetic) latitude, 75° W longitude on 1995 October 01 at 9h UTC,\n\nx' = 1703.295 km, y' = 4586.650 km, z' = 4077.984 km.\n\nAlthough close to our calculations assuming a spherical Earth, we find this simplification resulted in a position error of 22.8 km.\n\nWhat we really want to know, however, is just how big an error will result when generating look angles to a satellite from an observer's position on the Earth's surface if we assume a spherical Earth. From Figure 2, we would expect to have the largest errors for observers around 45° N latitude, so let's use a location near Minneapolis at 45° N latitude and 93° W longitude for our example. On a pass of the Mir space station over Minneapolis on 1995 November 18, Mir passed almost directly overhead. At 12h 46m UTC, its ECI position was calculated to be: x = -4400.594 km, y = 1932.870 km, z = 4760.712 km. Calculating the look angles for both a spherical and oblate Earth yields the results shown in Table 1.\n\nTable 1. Look Angles for Spherical vs. Oblate Earth\n\n Spherical Earth Oblate Earth Azimuth 118.80° 100.36° Elevation 80.24° 81.52°\n\nThe pointing error produced by assuming a spherical Earth is 3.17 degrees. For most applications, this error might not be significant. However, in applications involving tracking with high-gain, typically narrow-beamwidth, antennas, an error of 3 degrees can result in a loss of communications.\n\nSo, now that we've completed the calculation of a satellite look angle for an oblate Earth, let's look at how to calculate the sub-point of a satellite in Earth orbit. We'll begin by examining the calculations for a spherical Earth first before looking at the case for an oblate Earth.\n\nFirst, let's be sure we understand what we're looking for. The satellite sub-point is that point on the Earth's surface directly below the satellite. For the case of a spherical Earth, this point is the intersection of the line from the center of the Earth to the satellite and the Earth's surface, as shown in Figure 3.", null, "Figure 3. Calculating Satellite Sub-Point—Spherical Earth\n\nGiven the ECI position of the satellite to be [x, y, z], the latitude is", null, "and the (East) longitude is", null, "where θg is the Greenwich Mean Sidereal Time (GMST). The altitude of the satellite would be", null, "where Re is the Earth's circular radius.\n\nAs seen in Figure 4, the calculation for an oblate Earth is somewhat more complicated. The first thing we notice is that our definition of satellite sub-point requires some refinement. The point on the Earth's surface directly below the satellite is not on a line joining the satellite and the center of the Earth. Instead, it is that point on the Earth's surface where the satellite would appear at the zenith.", null, "Figure 4. Calculating Satellite Sub-Point—Oblate Earth\n\nCalculating the longitude of the satellite's sub-point doesn't change. However, to calculate the geodetic latitude of the satellite sub-point, we'll want to begin by approximating φ with φ' (as calculated above) and letting", null, "(for computational efficiency). Then, we'll want to loop through the following calculations", null, "", null, "", null, "until", null, "is within the desired tolerance. To compute the altitude of the satellite above the sub-point,", null, ".\n\nUsing our example of Mir passing over Minneapolis on 1995 November 18 at 12h 46m UTC yields a sub-point at 44.91° N (geodetic) latitude, 92.31° W longitude, and 397.507 km altitude. And while we cannot solve for the sub-point directly, the number of iterations required is typically quite small. For this example, the value of", null, "after the first iteration is 0.180537 degrees, after the second iteration it's 0.000574 degrees, and after the third iteration it's 0.000002 degrees.\n\nAdmittedly, some of the differences we've found may seem small, but that will depend upon your tracking requirements. And, since they are not that much more difficult to calculate, there is little reason not to use them. As always, if you have questions or comments on this column, feel free to send me e-mail at [email protected] or write care of Satellite Times. Until next time, keep looking up!" ]	[ null, "http://www.celestrak.com/images/stlogo.gif", null, "http://www.celestrak.com/columns/v02n03/v02n03.gif", null, "http://www.celestrak.com/images/stlogo.gif", null, "http://www.celestrak.com/columns/v02n03/v02n03.gif", null, "http://www.celestrak.com/columns/v02n03/fig-1.gif", null, "http://www.celestrak.com/columns/v02n03/eq-01.gif", null, "http://www.celestrak.com/columns/v02n03/eq-02.gif", null, "http://www.celestrak.com/columns/v02n03/eq-03.gif", null, "http://www.celestrak.com/columns/v02n03/eq-04.gif", null, "http://www.celestrak.com/columns/v02n03/eq-05.gif", null, "http://www.celestrak.com/columns/v02n03/eq-06.gif", null, "http://www.celestrak.com/columns/v02n03/fig-2.gif", null, "http://www.celestrak.com/columns/v02n03/eq-07.gif", null, "http://www.celestrak.com/columns/v02n03/eq-08.gif", null, "http://www.celestrak.com/columns/v02n03/fig-3.gif", null, "http://www.celestrak.com/columns/v02n03/eq-09.gif", null, "http://www.celestrak.com/columns/v02n03/eq-10.gif", null, "http://www.celestrak.com/columns/v02n03/eq-11.gif", null, "http://www.celestrak.com/columns/v02n03/fig-4.gif", null, "http://www.celestrak.com/columns/v02n03/eq-12.gif", null, "http://www.celestrak.com/columns/v02n03/eq-13.gif", null, "http://www.celestrak.com/columns/v02n03/eq-14.gif", null, "http://www.celestrak.com/columns/v02n03/eq-15.gif", null, "http://www.celestrak.com/columns/v02n03/eq-16.gif", null, "http://www.celestrak.com/columns/v02n03/eq-17.gif", null, "http://www.celestrak.com/columns/v02n03/eq-16.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.88859415,"math_prob":0.97961223,"size":9347,"snap":"2021-31-2021-39","text_gpt3_token_len":2243,"char_repetition_ratio":0.15230654,"word_repetition_ratio":0.031786397,"special_character_ratio":0.23494169,"punctuation_ratio":0.11594992,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9969256,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52],"im_url_duplicate_count":[null,null,null,5,null,null,null,5,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,6,null,3,null,6,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-09-27T09:14:41Z\",\"WARC-Record-ID\":\"<urn:uuid:6f450eca-e368-47e0-890b-6b6ac0b869ce>\",\"Content-Length\":\"19410\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:967fa7d1-ba79-4994-a978-51cffab7b288>\",\"WARC-Concurrent-To\":\"<urn:uuid:b2836f0d-3f16-4b08-bc4d-4d7d3016510b>\",\"WARC-IP-Address\":\"104.168.149.178\",\"WARC-Target-URI\":\"http://www.celestrak.com/columns/v02n03/\",\"WARC-Payload-Digest\":\"sha1:33RE2DOR7SLWSVKI44Q2WADPBBDKIXYA\",\"WARC-Block-Digest\":\"sha1:FV2NOCI6WSHVRKM45QXITKIAZX22T6MI\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780058415.93_warc_CC-MAIN-20210927090448-20210927120448-00544.warc.gz\"}"}
https://www.nuget.org/packages/cs-pca/	[ "#", null, "cs-pca 1.0.1\n\n`Install-Package cs-pca -Version 1.0.1`\n`dotnet add package cs-pca --version 1.0.1`\n`<PackageReference Include=\"cs-pca\" Version=\"1.0.1\" />`\nFor projects that support PackageReference, copy this XML node into the project file to reference the package.\n`paket add cs-pca --version 1.0.1`\n`#r \"nuget: cs-pca, 1.0.1\"`\n#r directive can be used in F# Interactive, C# scripting and .NET Interactive. Copy this into the interactive tool or source code of the script to reference the package.\n```// Install cs-pca as a Cake Addin\n\n// Install cs-pca as a Cake Tool\n#tool nuget:?package=cs-pca&version=1.0.1```\n\n## cs-pca\n\nPrincipal Component Analysis implemented in C#\n\n## Install\n\n``````Install-Package cs-pca\n``````\n\n## Usage\n\nThe sample codes below shows how to use the library to reduce the number of dimension or reconstruct the original data from the reduced data:\n\n``````List<double[]> source = GetNormalizedData();\nList<double[]> Z; // PCA output\ndouble variance_retained;\nK = 5; // dimension of the Z (note that Z will have K+1 dimensions where the first dimension will be ignored)\nPCA.PCADimReducer.CompressData(source, K, out Z, out variance_retained);\n\n// To reconstruct some compressed data point from Z\nList<double[]> compressed_data_point = GetCompressedDataPoints(); // K+1 dimension data points\nList<double[]> uncompressed_data_point = ReconstructData(compressed_data_point, Z);\n``````\n\nThis package has no dependencies.\n\n### NuGet packages\n\nThis package is not used by any NuGet packages.\n\n### GitHub repositories\n\nThis package is not used by any popular GitHub repositories." ]	[ null, "https://api.nuget.org/v3-flatcontainer/cs-pca/1.0.1/icon", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.63431925,"math_prob":0.4126395,"size":1868,"snap":"2021-43-2021-49","text_gpt3_token_len":482,"char_repetition_ratio":0.125,"word_repetition_ratio":0.1328125,"special_character_ratio":0.24571735,"punctuation_ratio":0.15041783,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96113616,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-20T11:08:37Z\",\"WARC-Record-ID\":\"<urn:uuid:d8182ad1-7bcf-4bcf-b5c7-ca33caae594c>\",\"Content-Length\":\"41409\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:726fb0b3-bd94-40c0-a12e-bf33ea7c511b>\",\"WARC-Concurrent-To\":\"<urn:uuid:2f46d2e6-f9d7-43ff-91a6-dd8bd2259738>\",\"WARC-IP-Address\":\"52.240.159.111\",\"WARC-Target-URI\":\"https://www.nuget.org/packages/cs-pca/\",\"WARC-Payload-Digest\":\"sha1:BHE7P6O6B4IM5UKBFCUKWNPFWWQVT64G\",\"WARC-Block-Digest\":\"sha1:WLMOAX2MXLJO6QGVTAWMNQ7NL6RJ7E5K\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585305.53_warc_CC-MAIN-20211020090145-20211020120145-00349.warc.gz\"}"}
https://gemma.feri.um.si/projects/industrial-projects/robust-trapezoidation-and-triangulation-algorithm-for-simple-polygons/	[ "## Robust Trapezoidation and Triangulation algorithm for simple polygons\n\nThe first industrial project of GeMMA was aimed to provide two robust dynamic link libraries for the German branch of the renowned engineering software producer. We have developed a novel polygon trapezoidation algorithm and adapted an existing polygon triangulation algorithm. Both of them are based on a sweep-line paradigm. They can handle simple (non-self-intersecting) polygons with or without holes. The latter may be separately trapezoidated/ triangulated on a request. After months of intensive testing in the AUTODESK’s testing department, our solutions were accepted for incorporation in a worldwide used commercial software product, while GeMMA was honoured with the AUTODESK’s certificate of quality. Besides the feasibility, correctness and speed, we also had to achieve high level of numerical robustness by avoiding all arithmetical operations that could produce inexact results: floating point divisions, triangular functions and square root calculations.\n\nThe proposed polygon trapezoidation algorithm represents an original approach. As the sweepline glides over the plane, a set of so-called open trapezoids is generated and maintained. An actual trapezoid is cut from the open one by adding a horizontal side in accordance with the context of a processed vertex. In this manner, a simple polygon is partitioned into a set of trapezoids with horizontal parallel sides. These may be limited either with original or with newly added vertices. Some trapezoids may also be degenerated into triangles. In comparison to the fastest Seidel’s method at the time, our algorithm revealed 30 to 100% faster performance, although both methods require O(n log n) time with respect to the number of vertices n.\n\nAnother problem considered in the project was the polygon triangulation. One could use the trapezoidation and then partition each trapezoid into two triangles, but our costomers preferred the solution without additional vertices that would typically appear during the trapezoidation. We have utilized quite traditional two-step approach which firstly decomposes the input polygon into the so-called y-monotone pieces as proposed by Garey, Johnson, Preparata and Tarjan (1978), and then separately triangulates each separate y-monotone polygon with the method introduced by Lee and Preparata (1977). To achieve the required speed, we have slightly adapted the algorithm by utilizing some redundant data structures, particularly the so-called neighbour tree for each vertex.\n\nThe theoretical analysis has shown that the first step runs in O(n log NMS), where NMS is the number of monotone pieces, while the triangulation itself requires linear O(n) time. Since the sweep-line algorithm must be pre-processed by sorting and since a popular quicksort performs best in random case, but appears slow in intuitively the simplest cases (corresponding to polygons with low NMS), this project initiated our increased interest in developing adaptive sorting algorithms. Our original solutions include vertex sort, smart quicksort and finally (still unpublished) smart merge sort." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9317705,"math_prob":0.9056323,"size":3152,"snap":"2020-34-2020-40","text_gpt3_token_len":627,"char_repetition_ratio":0.12229987,"word_repetition_ratio":0.0,"special_character_ratio":0.17385787,"punctuation_ratio":0.0806142,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96065205,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-09-25T22:13:00Z\",\"WARC-Record-ID\":\"<urn:uuid:90a20fbb-db57-4944-bbc3-cad99efbe323>\",\"Content-Length\":\"6731\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f3053081-b405-4dfb-82d8-84d77aaa8a1b>\",\"WARC-Concurrent-To\":\"<urn:uuid:4996ccc2-b4d5-4044-a8d9-50866b82e639>\",\"WARC-IP-Address\":\"164.8.9.62\",\"WARC-Target-URI\":\"https://gemma.feri.um.si/projects/industrial-projects/robust-trapezoidation-and-triangulation-algorithm-for-simple-polygons/\",\"WARC-Payload-Digest\":\"sha1:4HNDV65LXKELVK4MI5S7L2UBFT2Y7BBL\",\"WARC-Block-Digest\":\"sha1:7G5ZFRQH76R2RRW5KRGGQK5UOYSBXPQF\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600400228998.45_warc_CC-MAIN-20200925213517-20200926003517-00395.warc.gz\"}"}
https://quant.stackexchange.com/questions/16280/black-litterman-how-to-choose-the-uncertainty-in-the-views-omega-for-smooth/16281	[ "# Black-Litterman, how to choose the uncertainty in the views $\\Omega$ for smooth transitions from prior to posterior\n\nIn Black-Litterman we get a new vector of expected returns of the form: \\begin{align} \\Pi_{BL} = \\Pi + \\underbrace{\\tau \\Sigma P^T[P\\tau\\Sigma P^T+\\Omega]^{-1}}_{\\text{correction}}[Q-P\\Pi] \\end{align} where $P$ is the pick matrix and we mix the prior $\\Pi$ with the expected value of the views $Q$. $\\Sigma$ is the historical covariance matrix and $\\Omega$ is the covariance matrix of the views.\n\nLet us assume that $P$ is just the identity matrix and look at the choice $\\Omega = \\tau\\Sigma$, then we see that $$\\Pi_{BL} = \\frac12 \\Pi + \\frac12 Q,$$ thus we have a 50:50 mix and the covariance of the matrix does not affect the posterior at all - it is just a trivial mixture. This is against my intuition. Furthermore optimal weights using this $\\Pi_{BL}$ will differ relatively much from optimal weights of the prior (of course depending on $Q$).\n\nIf we assume $\\Omega = \\text{diag}(\\tau \\Sigma)$ then I can not find a closed form for $\\Pi_{BL}$ but appearantly the posterior is more compatible with the prior and the optimal weights are more similar than in the other setting.\n\nMy question: how can I choose $\\Omega$ best in order to get results that do not deviate too much from my prior? I know that in the literature there are theories (e.g. here The Black-Litterman Model In Detail) but I can't see through. What is used in practice?\n\nIn practice, $\\Omega$ (the covariance of the investor views) often 'inherits' the market covariance $\\Sigma$. A convenient choice is\n\n$\\Omega = \\left( 1/c -1 \\right) P \\Sigma P^T$\n\nwhere $c$ is a confidence parameter: the case $c \\rightarrow 1$ corresponds to a strongly peaked distribution of views (the investor views dominate the market), while $c \\rightarrow 0$ gives an infinitely disperse distribution where investor views have no influence. Tuning $c$ allows you to deviate smoothly from the prior $\\Pi$.\n\nThis choice for $\\Omega$ is proposed in Attilio Meucci's Risk and Asset Allocation, chapter 9.2.\n\nEdit: In the example you give ($P$ is the identity matrix and $\\Omega = \\tau \\Sigma$), the investor provides views on each asset with the same uncertainty as the market. In that case, the posterior return $\\Pi_{BL}$ is just the average of market prior $\\Pi$ and investor expectation $Q$. This seems plausible by symmetry: if you switch market and investor, $\\Pi_{BL}$ stays the same.\n\n• But you have to agree that setting $1/c-1 = \\tau$ leads to what I write above ... I will play with your $c$ factor - thanks for your answer. – Ric Jan 20 '15 at 15:49\n• Or where does $\\tau$ enter? do we have 2 factors: $(1/c-1) \\tau$? Then one would see things quite clearly in the correction term above ... – Ric Jan 20 '15 at 16:15\n• yes. I would deviate from this choice only if you can assign different confidences to your individual views (which is a fairly common situation in practice). – Felix Jan 20 '15 at 16:24\n• You might also refer to Equations 21-23 in papers.ssrn.com/sol3/papers.cfm?abstract_id=1213325 – John Jan 20 '15 at 16:28\n• no, it is $(1/c-1)$. $\\tau \\Sigma$ is the covariance of the posterior, assumed to be a normal distribution in the Black Litterman framework. – Felix Jan 20 '15 at 16:41\n\nWhen I implemented a BL model, I chose to do the omega optimization using the technique Idzorek proposed here:\n\nhttps://corporate.morningstar.com/ib/documents/MethodologyDocuments/IBBAssociates/BlackLitterman.pdf\n\nIt's a numerical procedure though.\n\n• Thanks for the link to the full publication - I have already read summaries of it ... – Ric Jan 26 '15 at 7:35" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.895965,"math_prob":0.98715985,"size":1340,"snap":"2021-21-2021-25","text_gpt3_token_len":349,"char_repetition_ratio":0.11302395,"word_repetition_ratio":0.0,"special_character_ratio":0.27014926,"punctuation_ratio":0.061068702,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9974595,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-05-17T20:44:35Z\",\"WARC-Record-ID\":\"<urn:uuid:0531f1f9-1e90-474e-897b-5a00fd617680>\",\"Content-Length\":\"180334\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:01a80c72-d3ea-4609-afcd-df3f173d2191>\",\"WARC-Concurrent-To\":\"<urn:uuid:9cda0f04-594b-4b8b-b7fd-4d0dcb52e407>\",\"WARC-IP-Address\":\"151.101.65.69\",\"WARC-Target-URI\":\"https://quant.stackexchange.com/questions/16280/black-litterman-how-to-choose-the-uncertainty-in-the-views-omega-for-smooth/16281\",\"WARC-Payload-Digest\":\"sha1:7OVJX3NIFZGEOQ3J6OKFJ3SE4SJ3JDLE\",\"WARC-Block-Digest\":\"sha1:XGP2KZI6NDIBOL7333CXOOLUHF5WPQYJ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-21/CC-MAIN-2021-21_segments_1620243992440.69_warc_CC-MAIN-20210517180757-20210517210757-00185.warc.gz\"}"}
https://isabelle.in.tum.de/repos/isabelle/rev/d47eabd80e59?revcount=30	[ "author hoelzl Mon, 14 Mar 2011 14:37:41 +0100 changeset 41975 d47eabd80e59 parent 41974 6e691abef08f child 41976 3fdbc7d5b525\nsimplified definition of open_extreal\n```--- a/src/HOL/Library/Extended_Reals.thy\tMon Mar 14 14:37:40 2011 +0100\n+++ b/src/HOL/Library/Extended_Reals.thy\tMon Mar 14 14:37:41 2011 +0100\n@@ -1076,128 +1076,89 @@\n\nsubsubsection \"Topological space\"\n\n+lemma\n+ shows extreal_max[simp]: \"extreal (max x y) = max (extreal x) (extreal y)\"\n+ and extreal_min[simp]: \"extreal (min x y) = min (extreal x) (extreal y)\"\n+ by (simp_all add: min_def max_def)\n+\ninstantiation extreal :: topological_space\nbegin\n\n-definition \"open A \\<longleftrightarrow>\n- (\\<exists>T. open T \\<and> extreal ` T = A - {\\<infinity>, -\\<infinity>})\n+definition \"open A \\<longleftrightarrow> open (extreal -` A)\n\\<and> (\\<infinity> \\<in> A \\<longrightarrow> (\\<exists>x. {extreal x <..} \\<subseteq> A))\n\\<and> (-\\<infinity> \\<in> A \\<longrightarrow> (\\<exists>x. {..<extreal x} \\<subseteq> A))\"\n\n-lemma open_PInfty: \"open A ==> \\<infinity> : A ==> (EX x. {extreal x<..} <= A)\"\n+lemma open_PInfty: \"open A \\<Longrightarrow> \\<infinity> \\<in> A \\<Longrightarrow> (\\<exists>x. {extreal x<..} \\<subseteq> A)\"\nunfolding open_extreal_def by auto\n\n-lemma open_MInfty: \"open A ==> (-\\<infinity>) : A ==> (EX x. {..<extreal x} <= A)\"\n+lemma open_MInfty: \"open A \\<Longrightarrow> -\\<infinity> \\<in> A \\<Longrightarrow> (\\<exists>x. {..<extreal x} \\<subseteq> A)\"\nunfolding open_extreal_def by auto\n\n-lemma open_PInfty2: assumes \"open A\" \"\\<infinity> : A\" obtains x where \"{extreal x<..} <= A\"\n+lemma open_PInfty2: assumes \"open A\" \"\\<infinity> \\<in> A\" obtains x where \"{extreal x<..} \\<subseteq> A\"\nusing open_PInfty[OF assms] by auto\n\n-lemma open_MInfty2: assumes \"open A\" \"(-\\<infinity>) : A\" obtains x where \"{..<extreal x} <= A\"\n+lemma open_MInfty2: assumes \"open A\" \"-\\<infinity> \\<in> A\" obtains x where \"{..<extreal x} \\<subseteq> A\"\nusing open_MInfty[OF assms] by auto\n\n-lemma extreal_openE: assumes \"open A\" obtains A' x y where\n- \"open A'\" \"extreal ` A' = A - {\\<infinity>, (-\\<infinity>)}\"\n- \"\\<infinity> : A ==> {extreal x<..} <= A\"\n- \"(-\\<infinity>) : A ==> {..<extreal y} <= A\"\n+lemma extreal_openE: assumes \"open A\" obtains x y where\n+ \"open (extreal -` A)\"\n+ \"\\<infinity> \\<in> A \\<Longrightarrow> {extreal x<..} \\<subseteq> A\"\n+ \"-\\<infinity> \\<in> A \\<Longrightarrow> {..<extreal y} \\<subseteq> A\"\nusing assms open_extreal_def by auto\n\ninstance\nproof\nlet ?U = \"UNIV::extreal set\"\nshow \"open ?U\" unfolding open_extreal_def\n- by (auto intro!: exI[of _ \"UNIV\"] exI[of _ 0])\n+ by (auto intro!: exI[of _ 0])\nnext\nfix S T::\"extreal set\" assume \"open S\" and \"open T\"\n- from `open S`[THEN extreal_openE] guess S' xS yS . note S' = this\n- from `open T`[THEN extreal_openE] guess T' xT yT . note T' = this\n-\n- have \"extreal ` (S' Int T') = (extreal ` S') Int (extreal ` T')\" by auto\n- also have \"... = S Int T - {\\<infinity>, (-\\<infinity>)}\" using S' T' by auto\n- finally have \"extreal ` (S' Int T') = S Int T - {\\<infinity>, (-\\<infinity>)}\" by auto\n- moreover have \"open (S' Int T')\" using S' T' by auto\n- moreover\n- { assume a: \"\\<infinity> : S Int T\"\n- hence \"EX x. {extreal x<..} <= S Int T\"\n- apply(rule_tac x=\"max xS xT\" in exI)\n- proof-\n- { fix x assume : \"extreal (max xS xT) < x\"\n- hence \"x : S Int T\" apply (cases x, auto simp: max_def split: split_if_asm)\n- using a S' T' by auto\n- } thus \"{extreal (max xS xT)<..} <= S Int T\" by auto\n- qed }\n- moreover\n- { assume a: \"(-\\<infinity>) : S Int T\"\n- hence \"EX x. {..<extreal x} <= S Int T\"\n- apply(rule_tac x=\"min yS yT\" in exI)\n- proof-\n- { fix x assume : \"extreal (min yS yT) > x\"\n- hence \"x<extreal yS & x<extreal yT\" by (cases x) auto\n- hence \"x : S Int T\" using a S' T' by auto\n- } thus \"{..<extreal (min yS yT)} <= S Int T\" by auto\n- qed }\n- ultimately show \"open (S Int T)\" unfolding open_extreal_def by auto\n+ from `open S`[THEN extreal_openE] guess xS yS .\n+ moreover from `open T`[THEN extreal_openE] guess xT yT .\n+ ultimately have\n+ \"open (extreal -` (S \\<inter> T))\"\n+ \"\\<infinity> \\<in> S \\<inter> T \\<Longrightarrow> {extreal (max xS xT) <..} \\<subseteq> S \\<inter> T\"\n+ \"-\\<infinity> \\<in> S \\<inter> T \\<Longrightarrow> {..< extreal (min yS yT)} \\<subseteq> S \\<inter> T\"\n+ by auto\n+ then show \"open (S Int T)\" unfolding open_extreal_def by blast\nnext\n- fix K assume openK: \"ALL S : K. open (S:: extreal set)\"\n- hence \"ALL S:K. EX T. open T & extreal ` T = S - {\\<infinity>, (-\\<infinity>)}\" by (auto simp: open_extreal_def)\n- from bchoice[OF this] guess T .. note T = this[rule_format]\n-\n- show \"open (Union K)\" unfolding open_extreal_def\n- proof (safe intro!: exI[of _ \"Union (T ` K)\"])\n- fix x S assume \"x : T S\" \"S : K\"\n- with T[OF `S : K`] show \"extreal x : Union K\" by auto\n- next\n- fix x S assume x: \"x ~: extreal ` (Union (T ` K))\" \"S : K\" \"x : S\" \"x ~= \\<infinity>\"\n- hence \"x ~: extreal ` (T S)\"\n- by (auto simp: image_UN UN_simps[symmetric] simp del: UN_simps)\n- thus \"x=(-\\<infinity>)\" using T[OF `S : K`] `x : S` `x ~= \\<infinity>` by auto\n- next\n- fix S assume \"\\<infinity> : S\" \"S : K\"\n- from openK[rule_format, OF `S : K`, THEN extreal_openE] guess S' x .\n- from this(3) `\\<infinity> : S`\n- show \"EX x. {extreal x<..} <= Union K\"\n- by (auto intro!: exI[of _ x] bexI[OF _ `S : K`])\n- next\n- fix S assume \"(-\\<infinity>) : S\" \"S : K\"\n- from openK[rule_format, OF `S : K`, THEN extreal_openE] guess S' x y .\n- from this(4) `(-\\<infinity>) : S`\n- show \"EX y. {..<extreal y} <= Union K\"\n- by (auto intro!: exI[of _ y] bexI[OF _ `S : K`])\n- next\n- from T[THEN conjunct1] show \"open (Union (T`K))\" by auto\n- qed auto\n+ fix K :: \"extreal set set\" assume \"\\<forall>S\\<in>K. open S\"\n+ then have : \"\\<forall>S. \\<exists>x y. S \\<in> K \\<longrightarrow> open (extreal -` S) \\<and>\n+ (\\<infinity> \\<in> S \\<longrightarrow> {extreal x <..} \\<subseteq> S) \\<and> (-\\<infinity> \\<in> S \\<longrightarrow> {..< extreal y} \\<subseteq> S)\"\n+ by (auto simp: open_extreal_def)\n+ then show \"open (Union K)\" unfolding open_extreal_def\n+ proof (intro conjI impI)\n+ show \"open (extreal -` \\<Union>K)\"\n+ using [unfolded choice_iff] by (auto simp: vimage_Union)\n+ qed ((metis UnionE Union_upper subset_trans )+)\nqed\nend\n\n-lemma open_extreal_lessThan[simp]:\n- \"open {..< a :: extreal}\"\n-proof (cases a)\n- case (real x)\n- then show ?thesis unfolding open_extreal_def\n- proof (safe intro!: exI[of _ \"{..< x}\"])\n- fix y assume \"y < extreal x\"\n- moreover assume \"y ~: (extreal ` {..<x})\"\n- ultimately have \"y ~= extreal (real y)\" using real by (cases y) auto\n- thus \"y = (-\\<infinity>)\" apply (cases y) using `y < extreal x` by auto\n- qed auto\n-qed (auto simp: open_extreal_def)\n-\n-lemma open_extreal_greaterThan[simp]:\n+lemma continuous_on_extreal[intro, simp]: \"continuous_on A extreal\"\n+ unfolding continuous_on_topological open_extreal_def by auto\n+\n+lemma continuous_at_extreal[intro, simp]: \"continuous (at x) extreal\"\n+ using continuous_on_eq_continuous_at[of UNIV] by auto\n+\n+lemma continuous_within_extreal[intro, simp]: \"x \\<in> A \\<Longrightarrow> continuous (at x within A) extreal\"\n+ using continuous_on_eq_continuous_within[of A] by auto\n+\n+lemma open_extreal_lessThan[intro, simp]: \"open {..< a :: extreal}\"\n+proof -\n+ have \"\\<And>x. extreal -` {..<extreal x} = {..< x}\"\n+ \"extreal -` {..< \\<infinity>} = UNIV\" \"extreal -` {..< -\\<infinity>} = {}\" by auto\n+ then show ?thesis by (cases a) (auto simp: open_extreal_def)\n+qed\n+\n+lemma open_extreal_greaterThan[intro, simp]:\n\"open {a :: extreal <..}\"\n-proof (cases a)\n- case (real x)\n- then show ?thesis unfolding open_extreal_def\n- proof (safe intro!: exI[of _ \"{x<..}\"])\n- fix y assume \"extreal x < y\"\n- moreover assume \"y ~: (extreal ` {x<..})\"\n- moreover assume \"y ~= \\<infinity>\"\n- ultimately have \"y ~= extreal (real y)\" using real by (cases y) auto\n- hence False apply (cases y) using `extreal x < y` `y ~= \\<infinity>` by auto\n- thus \"y = (-\\<infinity>)\" by auto\n- qed auto\n-qed (auto simp: open_extreal_def)\n-\n-lemma extreal_open_greaterThanLessThan[simp]: \"open {a::extreal <..< b}\"\n+proof -\n+ have \"\\<And>x. extreal -` {extreal x<..} = {x<..}\"\n+ \"extreal -` {\\<infinity><..} = {}\" \"extreal -` {-\\<infinity><..} = UNIV\" by auto\n+ then show ?thesis by (cases a) (auto simp: open_extreal_def)\n+qed\n+\n+lemma extreal_open_greaterThanLessThan[intro, simp]: \"open {a::extreal <..< b}\"\nunfolding greaterThanLessThan_def by auto\n\nlemma closed_extreal_atLeast[simp, intro]: \"closed {a :: extreal ..}\"\n@@ -1227,19 +1188,17 @@\nobtains e where \"e>0\" \"{x-e <..< x+e} \\<subseteq> S\"\nproof-\nobtain m where m_def: \"x = extreal m\" using assms by (cases x) auto\n- obtain A where \"open A\" and A_def: \"extreal ` A = S - {\\<infinity>,(-\\<infinity>)}\"\n- using assms by (auto elim!: extreal_openE)\n- hence \"m : A\" using m_def assms by auto\n- from this obtain e where e_def: \"e>0 & ball m e <= A\"\n- using open_contains_ball[of A] `open A` by auto\n- moreover have \"ball m e = {m-e <..< m+e}\" unfolding ball_def dist_norm by auto\n- ultimately have : \"{m-e <..< m+e} <= A\" using e_def by auto\n- { fix y assume y_def: \"y:{x-extreal e <..< x+extreal e}\"\n- from this obtain z where z_def: \"y = extreal z\" by (cases y) auto\n- hence \"z:A\" using y_def m_def * by auto\n- hence \"y:S\" using z_def A_def by auto\n- } hence \"{x-extreal e <..< x+extreal e} <= S\" by auto\n- thus thesis apply- apply(rule that[of \"extreal e\"]) using e_def by auto\n+ from `open S` have \"open (extreal -` S)\" by (rule extreal_openE)\n+ then obtain e where \"0 < e\" and e: \"ball m e \\<subseteq> extreal -` S\"\n+ using `x \\<in> S` unfolding open_contains_ball m_def by force\n+ show thesis\n+ proof (intro that subsetI)\n+ show \"0 < extreal e\" using `0 < e` by auto\n+ fix y assume \"y \\<in> {x - extreal e<..<x + extreal e}\"\n+ then obtain t where \"y = extreal t\" \"t \\<in> ball m e\"\n+ unfolding m_def by (cases y) (auto simp: ball_def dist_real_def)\n+ then show \"y \\<in> S\" using e by auto\n+ qed\nqed\n\nlemma extreal_open_cont_interval2:\n@@ -1266,41 +1225,36 @@\nfixes S :: \"extreal set\"\nassumes \"open S\"\nshows \"open (uminus ` S)\"\n-proof-\n- obtain T x y where T_def: \"open T & extreal ` T = S - {\\<infinity>, (-\\<infinity>)} &\n- (\\<infinity> : S --> {extreal x<..} <= S) & ((-\\<infinity>) : S --> {..<extreal y} <= S)\"\n- using assms extreal_openE[of S] by metis\n- have \"extreal ` uminus ` T = uminus ` extreal ` T\" apply auto\n- by (metis imageI extreal_uminus_uminus uminus_extreal.simps)\n- also have \"...=uminus ` (S - {\\<infinity>, (-\\<infinity>)})\" using T_def by auto\n- finally have \"extreal ` uminus ` T = uminus ` S - {\\<infinity>, (-\\<infinity>)}\" by (auto simp: extreal_uminus_reorder)\n- moreover have \"open (uminus ` T)\" using T_def open_negations[of T] by auto\n- ultimately have \"EX T. open T & extreal ` T = uminus ` S - {\\<infinity>, (-\\<infinity>)}\" by auto\n- moreover\n- { assume \"\\<infinity>: uminus ` S\"\n- hence \"(-\\<infinity>) : S\" by (metis image_iff extreal_uminus_uminus)\n- hence \"uminus ` {..<extreal y} <= uminus ` S\" using T_def by (intro image_mono) auto\n- hence \"EX x. {extreal x<..} <= uminus ` S\" using extreal_uminus_lessThan by auto\n- } moreover\n- { assume \"(-\\<infinity>): uminus ` S\"\n- hence \"\\<infinity> : S\" by (metis image_iff extreal_uminus_uminus)\n- hence \"uminus ` {extreal x<..} <= uminus ` S\" using T_def by (intro image_mono) auto\n- hence \"EX y. {..<extreal y} <= uminus ` S\" using extreal_uminus_greaterThan by auto\n- }\n- ultimately show ?thesis unfolding open_extreal_def by auto\n+ unfolding open_extreal_def\n+proof (intro conjI impI)\n+ obtain x y where S: \"open (extreal -` S)\"\n+ \"\\<infinity> \\<in> S \\<Longrightarrow> {extreal x<..} \\<subseteq> S\" \"-\\<infinity> \\<in> S \\<Longrightarrow> {..< extreal y} \\<subseteq> S\"\n+ using `open S` unfolding open_extreal_def by auto\n+ have \"extreal -` uminus ` S = uminus ` (extreal -` S)\"\n+ proof safe\n+ fix x y assume \"extreal x = - y\" \"y \\<in> S\"\n+ then show \"x \\<in> uminus ` extreal -` S\" by (cases y) auto\n+ next\n+ fix x assume \"extreal x \\<in> S\"\n+ then show \"- x \\<in> extreal -` uminus ` S\"\n+ by (auto intro: image_eqI[of _ _ \"extreal x\"])\n+ qed\n+ then show \"open (extreal -` uminus ` S)\"\n+ using S by (auto intro: open_negations)\n+ { assume \"\\<infinity> \\<in> uminus ` S\"\n+ then have \"-\\<infinity> \\<in> S\" by (metis image_iff extreal_uminus_uminus)\n+ then have \"uminus ` {..<extreal y} \\<subseteq> uminus ` S\" using S by (intro image_mono) auto\n+ then show \"\\<exists>x. {extreal x<..} \\<subseteq> uminus ` S\" using extreal_uminus_lessThan by auto }\n+ { assume \"-\\<infinity> \\<in> uminus ` S\"\n+ then have \"\\<infinity> : S\" by (metis image_iff extreal_uminus_uminus)\n+ then have \"uminus ` {extreal x<..} <= uminus ` S\" using S by (intro image_mono) auto\n+ then show \"\\<exists>y. {..<extreal y} <= uminus ` S\" using extreal_uminus_greaterThan by auto }\nqed\n\nlemma extreal_uminus_complement:\nfixes S :: \"extreal set\"\n- shows \"(uminus ` (- S)) = (- uminus ` S)\"\n-proof-\n-{ fix x\n- have \"x:uminus ` (- S) <-> -x:(- S)\" by (metis image_iff extreal_uminus_uminus)\n- also have \"... <-> x:(- uminus ` S)\"\n- by (metis ComplI Compl_iff image_eqI extreal_uminus_uminus extreal_minus_minus_image)\n- finally have \"x:uminus ` (- S) <-> x:(- uminus ` S)\" by auto\n-} thus ?thesis by auto\n-qed\n+ shows \"uminus ` (- S) = - uminus ` S\"\n+ by (auto intro!: bij_image_Compl_eq surjI[of _ uminus] simp: bij_betw_def)\n\nlemma extreal_closed_uminus:\nfixes S :: \"extreal set\"\n@@ -1309,7 +1263,6 @@\nusing assms unfolding closed_def\nusing extreal_open_uminus[of \"- S\"] extreal_uminus_complement by auto\n\n-\nlemma not_open_extreal_singleton:\n\"~(open {a :: extreal})\"\nproof(rule ccontr)\n@@ -1491,22 +1444,17 @@\nqed\nqed\n\n-lemma open_extreal: assumes \"open S\" shows \"open (extreal ` S)\"\n- unfolding open_extreal_def apply(rule,rule,rule,rule assms) by auto\n-\n-lemma open_real_of_extreal:\n- fixes S :: \"extreal set\" assumes \"open S\"\n- shows \"open (real ` (S - {\\<infinity>, -\\<infinity>}))\"\n-proof -\n- from `open S` obtain T where T: \"open T\" \"S - {\\<infinity>, -\\<infinity>} = extreal ` T\"\n- unfolding open_extreal_def by auto\n- show ?thesis using T by (simp add: image_image)\n-qed\n+lemma inj_extreal[simp]: \"inj_on extreal A\"\n+ unfolding inj_on_def by auto\n+\n+lemma open_extreal: \"open S \\<Longrightarrow> open (extreal ` S)\"\n+ by (auto simp: inj_vimage_image_eq open_extreal_def)\n+\n+lemma open_extreal_vimage: \"open S \\<Longrightarrow> open (extreal -` S)\"\n+ unfolding open_extreal_def by auto\n\nsubsubsection {* Convergent sequences }\n\n-lemma inj_extreal[simp, intro]: \"inj_on extreal A\" by (auto intro: inj_onI)\n-\nlemma lim_extreal[simp]:\n\"((\\<lambda>n. extreal (f n)) ---> extreal x) net \\<longleftrightarrow> (f ---> x) net\" (is \"?l = ?r\")\nproof (intro iffI topological_tendstoI)\n@@ -1516,12 +1464,9 @@\nby (simp add: inj_image_mem_iff)\nnext\nfix S assume \"?r\" \"open S\" \"extreal x \\<in> S\"\n- have : \"\\<And>x. x \\<in> real ` (S - {\\<infinity>, - \\<infinity>}) \\<longleftrightarrow> extreal x \\<in> S\"\n- apply (safe intro!: rev_image_eqI)\n- by (case_tac xa) auto\nshow \"eventually (\\<lambda>x. extreal (f x) \\<in> S) net\"\n- using `?r`[THEN topological_tendstoD, OF open_real_of_extreal, OF `open S`]\n- using `extreal x \\<in> S` by (simp add: )\n+ using `?r`[THEN topological_tendstoD, OF open_extreal_vimage, OF `open S`]\n+ using `extreal x \\<in> S` by auto\nqed\n\nlemma lim_real_of_extreal[simp]:\n@@ -1744,21 +1689,18 @@\nobtain r where r[simp]: \"m = extreal r\" using m by (cases m) auto\nobtain p where p[simp]: \"t = extreal p\" using t by (cases t) auto\nhave \"r \\<noteq> 0\" \"0 < r\" and m': \"m \\<noteq> \\<infinity>\" \"m \\<noteq> -\\<infinity>\" \"m \\<noteq> 0\" using m by auto\n- from `open S`[THEN extreal_openE] guess T l u . note T = this\n+ from `open S`[THEN extreal_openE] guess l u . note T = this\nlet ?f = \"(\\<lambda>x. m x + t)\"\nshow ?thesis unfolding open_extreal_def\nproof (intro conjI impI exI subsetI)\n- show \"open ((\\<lambda>x. rx + p)`T)\"\n- using open_affinity[OF `open T` `r \\<noteq> 0`] by (auto simp: ac_simps)\n- have affine_infy: \"?f ` {\\<infinity>, - \\<infinity>} = {\\<infinity>, -\\<infinity>}\"\n- using `r \\<noteq> 0` by auto\n- have \"extreal ` (\\<lambda>x. r x + p) ` T = ?f ` (extreal ` T)\"\n- by (simp add: image_image)\n- also have \"\\<dots> = ?f ` (S - {\\<infinity>, -\\<infinity>})\"\n- using T(2) by simp\n- also have \"\\<dots> = ?f ` S - {\\<infinity>, -\\<infinity>}\"\n- using extreal_inj_affinity[OF m' t] by (simp only: image_set_diff affine_infy)\n- finally show \"extreal ` (\\<lambda>x. r * x + p) ` T = ?f ` S - {\\<infinity>, -\\<infinity>}\" .\n+ have \"extreal -` ?f ` S = (\\<lambda>x. r * x + p) ` (extreal -` S)\"\n+ proof safe\n+ fix x y assume \"extreal y = m * x + t\" \"x \\<in> S\"\n+ then show \"y \\<in> (\\<lambda>x. r * x + p) ` extreal -` S\"\n+ using `r \\<noteq> 0` by (cases x) (auto intro!: image_eqI[of _ _ \"real x\"] split: split_if_asm)\n+ qed force\n+ then show \"open (extreal -` ?f ` S)\"\n+ using open_affinity[OF T(1) `r \\<noteq> 0`] by (auto simp: ac_simps)\nnext\nassume \"\\<infinity> \\<in> ?f`S\" with `0 < r` have \"\\<infinity> \\<in> S\" by auto\nfix x assume \"x \\<in> {extreal (r * l + p)<..}\"\n@@ -1769,7 +1711,7 @@\nusing m t by (cases rule: extreal3_cases[of m x t]) auto\nhave \"extreal l < (x - t)/m\"\nusing m t by (simp add: extreal_less_divide_pos extreal_less_minus)\n- then show \"(x - t)/m \\<in> S\" using T(3)[OF `\\<infinity> \\<in> S`] by auto\n+ then show \"(x - t)/m \\<in> S\" using T(2)[OF `\\<infinity> \\<in> S`] by auto\nqed\nnext\nassume \"-\\<infinity> \\<in> ?f`S\" with `0 < r` have \"-\\<infinity> \\<in> S\" by auto\n@@ -1781,7 +1723,7 @@\nusing m t by (cases rule: extreal3_cases[of m x t]) auto\nhave \"(x - t)/m < extreal u\"\nusing m t by (simp add: extreal_divide_less_pos extreal_minus_less)\n- then show \"(x - t)/m \\<in> S\" using T(4)[OF `-\\<infinity> \\<in> S`] by auto\n+ then show \"(x - t)/m \\<in> S\" using T(3)[OF `-\\<infinity> \\<in> S`] by auto\nqed\nqed\nqed\n@@ -1864,12 +1806,9 @@\nproof (rule topological_tendstoI, unfold eventually_sequentially)\nobtain rx where rx_def: \"x=extreal rx\" using assms by (cases x) auto\nfix S assume \"open S\" \"x : S\"\n- then obtain A where \"open A\" and A_eq: \"extreal ` A = S - {\\<infinity>,(-\\<infinity>)}\"\n- by (auto elim!: extreal_openE)\n- then have \"x : extreal ` A\" using `x : S` assms by auto\n- then have \"rx : A\" using rx_def by auto\n- then obtain r where \"0 < r\" and dist: \"!!y. dist y (real x) < r ==> y : A\"\n- using `open A` unfolding open_real_def rx_def by auto\n+ then have \"open (extreal -` S)\" unfolding open_extreal_def by auto\n+ with `x \\<in> S` obtain r where \"0 < r\" and dist: \"!!y. dist y rx < r ==> extreal y \\<in> S\"\n+ unfolding open_real_def rx_def by auto\nthen obtain n where\nupper: \"!!N. n <= N ==> u N < x + extreal r\" and\nlower: \"!!N. n <= N ==> x < u N + extreal r\" using assms(3)[of \"extreal r\"] by auto\n@@ -1881,13 +1820,11 @@\nfrom this obtain ra where ra_def: \"(u N) = extreal ra\" by (cases \"u N\") auto\nhence \"rx < ra + r\" and \"ra < rx + r\"\nusing rx_def assms `0 < r` lower[OF `n <= N`] upper[OF `n <= N`] by auto\n- hence \"dist (real (u N)) (real x) < r\"\n+ hence \"dist (real (u N)) rx < r\"\nusing rx_def ra_def\nby (auto simp: dist_real_def abs_diff_less_iff field_simps)\n- from dist[OF this]\n- have \"u N : extreal ` A\" using `u N ~: {\\<infinity>,(-\\<infinity>)}`\n+ from dist[OF this] show \"u N : S\" using `u N ~: {\\<infinity>,(-\\<infinity>)}`\nby (auto intro!: image_eqI[of _ _ \"real (u N)\"] simp: extreal_real)\n- thus \"u N : S\" using A_eq by simp\nqed\nqed\n\n@@ -2933,21 +2870,6 @@\nfrom this show ?thesis using continuous_imp_tendsto by auto\nqed\n\n-\n-lemma continuous_at_extreal:\n-fixes x0 :: real\n-shows \"continuous (at x0) extreal\"\n-proof-\n-{ fix T assume T_def: \"open T & extreal x0 : T\"\n- from this obtain S where S_def: \"open S & extreal ` S = T - {\\<infinity>, (-\\<infinity>)}\"\n- using extreal_openE[of T] by metis\n- moreover hence \"x0 : S\" using T_def by auto\n- moreover have \"ALL y:S. extreal y : T\" using S_def by auto\n- ultimately have \"EX S. x0 : S & open S & (ALL y:S. extreal y : T)\" by auto\n-} from this show ?thesis unfolding continuous_at_open by blast\n-qed\n-\n-\nlemma continuous_at_of_extreal:\nfixes x0 :: extreal\nassumes \"x0 ~: {\\<infinity>, (-\\<infinity>)}\"\n@@ -2995,9 +2917,6 @@\nusing continuous_at_iff_extreal assms by (auto simp add: continuous_on_eq_continuous_at)\n\n-lemma continuous_on_extreal: \"continuous_on UNIV extreal\"\n- using continuous_at_extreal continuous_on_eq_continuous_at by auto\n-\nlemma open_image_extreal: \"open(UNIV-{\\<infinity>,(-\\<infinity>)})\"\nby (metis range_extreal open_extreal open_UNIV)\n```" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6558791,"math_prob":0.91898483,"size":20123,"snap":"2021-21-2021-25","text_gpt3_token_len":6871,"char_repetition_ratio":0.23356031,"word_repetition_ratio":0.20636286,"special_character_ratio":0.38602594,"punctuation_ratio":0.15107296,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9993873,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-22T11:24:53Z\",\"WARC-Record-ID\":\"<urn:uuid:9ef88938-5d87-485f-b260-ce7743fcf2dc>\",\"Content-Length\":\"58185\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:90311652-bef7-4cf1-b1c9-cbab82fbd652>\",\"WARC-Concurrent-To\":\"<urn:uuid:930fc348-9147-4029-9144-5c0df01ee6c9>\",\"WARC-IP-Address\":\"131.159.46.82\",\"WARC-Target-URI\":\"https://isabelle.in.tum.de/repos/isabelle/rev/d47eabd80e59?revcount=30\",\"WARC-Payload-Digest\":\"sha1:KPXZ24IF6BDSRSTPBATQHS5ZSVUFYKMF\",\"WARC-Block-Digest\":\"sha1:YPKKBTJVIPYXSCOKFMZW733IL5KFXZA4\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623488517048.78_warc_CC-MAIN-20210622093910-20210622123910-00615.warc.gz\"}"}
https://link.springer.com/article/10.1007/s11047-014-9436-7	[ "# Topology driven modeling: the IS metaphor\n\n## Abstract\n\nIn order to define a new method for analyzing the immune system within the realm of Big Data, we bear on the metaphor provided by an extension of Parisi’s model, based on a mean field approach. The novelty is the multilinearity of the couplings in the configurational variables. This peculiarity allows us to compare the partition function $$Z$$ with a particular functor of topological field theory—the generating function of the Betti numbers of the state manifold of the system—which contains the same global information of the system configurations and of the data set representing them. The comparison between the Betti numbers of the model and the real Betti numbers obtained from the topological analysis of phenomenological data, is expected to discover hidden n-ary relations among idiotypes and anti-idiotypes. The data topological analysis will select global features, reducible neither to a mere subgraph nor to a metric or vector space. How the immune system reacts, how it evolves, how it responds to stimuli is the result of an interaction that took place among many entities constrained in specific configurations which are relational. Within this metaphor, the proposed method turns out to be a global topological application of the S[B] paradigm for modeling complex systems.\n\n## Introduction\n\nThe objective pursued in this note is to frame the research on the immune system as part of data science. Such research is naturally complex and articulated and our contribution intends to be here along the lines of seeing it as a viable candidate for topological data analytics and an example of the S[B] paradigm for modeling complex systems. We recall that data science is the practice to deriving valuable insights from data by challenging all the issues related to the processing of very large data sets, while Big Data is jargon to indicate such a large collection of data (for example, exabytes) characterized by high-dimensionality, redundancy, and noise. The analysis of Big Data requires handling high-dimensional vectors capable of weaning out the unimportant, redundant coordinates. The notion of data space, its geometry and topology are the most natural tools to handle the unprecedentedly large, high-dimensional, complex sets of data (Carlsson 2009; Edelsbrunner and Harer 2010); basic ingredient of the new data-driven complexity science (TOPDRIM 2012; Merelli and Rasetti 2013).\n\nTopology, the branch of mathematics dealing with qualitative geometric information such as connectivity, classification of loops and higher dimensional manifolds, studies properties of geometric objects (shapes) in a way which is less sensitive to metrics than geometric methods: it ignores the value of distance function and replaces it with the notion of connective nearness: proximity. All these features make topology ideal for analysing the space of data.\n\nStarting from the notion of a mean field proposed by Parisi in his simple model for idiotypic network (Parisi 1990), we propose a more sophisticated version that is multilinear in the configurational variables (the antibody concentrations) instead of being constant or at most linear. Multi-linearity allows us to recognize in the partition function $$Z$$ of the model, that embodies all the statistical properties of the system at equilibrium, features similar to those of a particular functor of a topological field theory. The latter contains indeed the same global information about the topological properties (specifically its global invariants) of the system configuration space and can be identified with the generating function of Betti numbers, namely the Poincaré polynomial of data space (Atiyah and Bott 1983). Once the homology of the space of data has been constructed, and its generating cycles have been defined, the related two sets of Betti numbers can be compared. In this way, self-consistent information is obtained, regarding $$2{\\hbox {-}}ary$$, $$3{\\hbox {-}}ary,\\, \\dots n{\\hbox {-}}ary$$ relations among antibodies. Comparison between the Betti numbers of the model and the real Betti numbers, obtained by constructing the topology of phenomenological immune system space of data, will unveil the hidden relations between idiotypes and anti-idiotypes; in particular, those relations where components interact indistinctly and therefore can not be reduced to a mere subgraph, but rather they bear on a new concept of interaction, scale-free and metric-free. The analysis of Betti numbers on phenomenological data can be dealt with techniques based on persistent homology (Carlsson 2009; Petri et al. 2013).\n\nThe challenge we are facing is to unveil whether in natural, multi-level complex systems, $$n$$-body interactions can drive the emergence of novel qualia in these systems. In physics, the interactions between material objects in real space are binary. This means that mutual forces and motions are produced by two-body interactions, the building blocks of any many-particle system. Thus at the atomic or molecular level description of matter (living or not) the total force acting on any given particle is the result of the composition of binary interactions. However, how can we discover if $$n$$-body interactions do exist? What we are proposing here is to use the IS metaphor, i.e. a complex system whose adaptivity is driven by data, as a global topological application of the S[B] paradigm. S[B] allows us to entangle in a unique model the computational component with the coordination. In particular, B accounts for the computation while S describes the global computation context (Merelli et al. 2013). The adaptation phase occurs when a machine can no longer compute in a given state of the system, thus the system changes state, i.e. the global context of computation. In the IS metaphor the computation context can be identified by the global invariants while the computation with the model of interactions, a sort of interactive machine. Each time we discover new global invariants, a new context of computation arises and with it a new IS model must be generated; we call this step the adaptation phase.\n\nIn the following, after giving a brief description of the antigen-free immune system and recalling Parisi’s mean field model, we formally define the new topological field model, and, finally, discuss the S[B] paradigm. An appendix is provided with a general introduction to the fundamental tool of persistent homology and Betti numbers.\n\n## The antigen-free immune system\n\nCells and molecules of the immune system not only recognize foreign substances; they react and regulate each other, so that the immune system can be seen as a network of interacting cells and antibodies. This perspective is known as the idiotypic or immune network theory (Jerne 1974). It refers to the immune system as a complex process that takes place at the cellular level for protecting organisms from infectious agents (the antigens), which are antibody generators. In the scheme proposed by Jerne, it is the antigen that provokes an immune response and each antibody is represented as a large Y-shaped protein. The immune system uses this protein to identify and neutralize foreign objects. The antibody can recognize and bind a specific part of the antigen; resorting to this binding mechanism it can block the attack. Moreover, in Jerne’s network theory, antibodies are capable of being recognized by other antibodies; whenever this happens the former is suppressed and its concentration is reduced while the latter is stimulated and its concentration increases (see Fig. 1).\n\nThe mechanism whereby the production of a given antibody elicits or suppresses the production of other antibodies that, in turn, elicit or suppress the production of other antibodies like a concatenation of events, hints to a strict analogy of the immune system function with memory in the brain. It recalls the way in which a firing neuron may induce or inhibit the firing of other neurons, and so forth. On the assumption that a functional network of antibodies is possible, several models have been constructed, among which Parisi’s model. The latter studies the persistence of immune memory in the absence of any driving effect of external antigens and it offers a robust, though simple, theoretical framework without providing detailed description of the system (Parisi 1990).\n\nThe model we propose is a preliminary test of data field theory; it aims at a deeper understanding of the functional properties that the global and persistent topological properties of an antibodies data space can imply. In particular, it targets at discovering the existence of $$n$$-ary relations among antibodies and determining how the ensuing configurations influence the immune system reaction to the presence of antigens. The extraction of global qualitative information from an antibodies data space (e.g. concentrations), should lead to the discovery of those characteristics that are shared in a group of immunoglobulin receptor molecules. This means not only discovering a single idiotype, but the capacity of being active in the presence of $$n$$ others. We want to prove that topological data analysis, through persistent homology and its Betti numbers, allows us to determine the effective $$n$$-antibody configurations. Note that the models proposed in literature to describe the relationship between structure and function in biological networks are all based on the concept that any relation can be reduced to a set of binary relation (Hart et al. 2009): we argue that this is not necessarily the case. We start thinking of models as relationships, i.e. facts in logical space of forms. Forms that can be directly classified by Betti numbers, extracted by calculating the Betti numbers through the persistent homology of the space of data and used in the frame of a conceptual model able to bear on those topological features.\n\n### Parisi mean field model for IS\n\nThe simplest and most efficient network of the immune system is represented by a model that can be easily formulated in the absence of antigens. Although it is well known that the number of specific lymphocytes plays a crucial role, the variables of the network model are limited to the antibody concentrations.\n\nThe mean-field idiotypic network model of antigen-free Immune System, proposed by G. Parisi and inspired by an earlier Hopfield’s model conceived to represent the brain and many other similar models (Hopfield 1982; Hoffmann 1975, 2010; Farmer et al. 1986; Varela et al. 1988), describes essentially an iterated cascade of events, in which the production of a given antibody provokes, or possibly inhibits, the production of other antibodies, which in turn induce, or possibly impede, the production of other antibodies, which in turn give rise to or prevent the production of other antibodies, etc..\n\nIn the Parisi model, the concentration $$c_i(t)$$ of antibody $$i$$ is assumed to have, in absence of external antigens, only two values, conventionally $$0$$ or $$1$$ (in the presence of antigen concentrations $$c_i$$ might become $$\\gg 1$$); $$t$$ is time. The immune system state at time $$t$$ is determined by the values of all $$c_i$$’s for all possible antibodies $$(i = 1, \\dots , N)$$. The dynamical process is typically described by a discretized time (the time step $$\\tau$$ being the time needed to implement the immune response). The dynamical variable $$h_i$$ (the mean field) represents the total stimulatory/inhibitory (depending on its sign) effect of the whole network on the $$i$$-th antibody. $$h_i$$ is positive when the excitatory effect of the other antibodies is greater than the suppressive effect and then $$c_i$$ is one. Otherwise $$h_i$$ is negative and $$c_i$$ is zero. The mean-field is expressed typically as\n\n\\begin{aligned} \\displaystyle {h_i (t)= S + {{\\mathop {\\mathop {\\sum }\\limits _{k=1}}\\limits _{k \\ne i}^{N}}} J_{ik} c_k (t)}, \\,\\,\\, where \\,\\,\\, c_i (t) = \\Theta [h_i (t-\\tau )] \\end{aligned}\n(1)\n\n$$\\Theta (x)$$ denotes the Heaviside function that is zero for negative $$x$$ and 1 for positive $$x$$, while $$J_{ik}$$ ($$J_{ii} = 0, J_{ki} = J_{ik}$$) represents the influence of antibody $$k$$ on antibody $$i$$. If $$J_{ik}$$ is positive, antibody $$k$$ triggers the production of antibody $$i$$, whereas if $$J_{ik}$$ is negative, antibody $$k$$ suppresses the production of antibody $$i$$. $$\\left\| J_{ik} \\right\|$$ is a measure of control efficiency that the antibody $$k$$ exercises on antibody $$i$$. The $$J_{ik}$$ are distributed in the interval $$[-1, +1]$$. $$S$$ is the threshold parameter; it regulates the dynamics when the couplings $$J_{ik}$$ are all very small; otherwise $$S$$ is equal to zero. At equilibrium, when the concentrations of antibodies are time independent, the Eq. (1) simplifies to\n\n\\begin{aligned} \\displaystyle {h_i=S+ {{\\mathop {\\mathop {\\sum }\\limits _{k=1}}\\limits _{k \\ne i}^{N}}} J_{ik} c_k}\\, , \\, c_i= \\Theta (h_i )\\in \\{ 0 , 1 \\} \\; . \\end{aligned}\n(2)\n\nThis idiotypic network model has the advantage of being simple and easy to analyze. The phenomenon of dependence of the immunity/tolerance pathway on the amount of antigens suggests that the concentration of any given antibody is crucial to determine the effects on the other antibodies. The assumption of two levels of concentration (0 or 1) bypasses the problem of the choice of a pathway.\n\nHowever, this model is elementary in view of testing the perspectives of a data field theory. We need to increase its complexity in order to reach a description of the system sufficiently detailed to catch the global features of its data space. We generalize the mean field in such a way that it crucially depends on those topological features of the space of antibody concentrations that will be reflected in the topological properties of the system space of data. We construct a model sensitive to global features, designed to benefit of the advantage of lending itself to a kind of reverse engineering of the process of field construction. In the model, the antibodies with positive $$c_i \\, (=1)$$ are actually produced by the system while those absent $$(c_i=0)$$ are suppressed. Suppression due to clonal abortion is neglected.\n\n## The topological field model for antigen-free immune system\n\nIn this section, we generalize the way how Parisi’s linear model represents immunological memory by a linear mean field. The antibodies of the idiotypic cascade are denoted by $$Ab_i$$; during the production of $$Ab_1$$, ignited directly by the antigen, the environment of lymphocytes is modified by $$Ab_2$$: the life-span of the $$Ab_1$$-producing cells and the population of helper cells specific for $$Ab_1$$ increase. The symmetry of the couplings $$(J_{ik}=J_{ki})$$ implies that $$Ab_3$$ should be rather similar to $$Ab_1$$, the internal image of $$Ab_2$$ should persist after it disappeared, its presence induces the survival of memory cells directed against the antigen. The process continues by iteration. In the extended model, we assume the production of $$Ab_i$$ is conditioned to different extents and also by the simultaneous presence of a subset of $$2, 3, ... , N$$, antibodies.\n\nA weakness of this representation is that the possible equilibrium configurations of the network are fixed, whereas we want the network to be capable of learning which antibodies should be produced without assuming that only a fraction of all antibodies have physiological relevance. Therefore, whilst we maintain the global cost function\n\n\\begin{aligned} \\displaystyle { E= \\sum _{i=1}^N h_i c_i }, \\, \\,\\, \\; c_i= \\Theta (h_i )\\in [0,1] \\; , \\end{aligned}\n(3)\n\nwe consider in the space of antibodies $$\\mathcal {A}$$, the points of which are labelled by $$i=1 \\dots ,N$$, the graph $$\\mathcal {G}$$ generated by the $$J_{ik} \\ne 0$$ (for simplicity we assume here that $$J_{ik}\\in [ -1, +1 ]$$ when $$J_{ik} \\ne 0$$). We next extend $$\\mathcal {G}$$ to the simplicial complex $$\\mathcal {C}$$, obtained from $$\\mathcal {G}$$ by completion, constructing the simplicial complex $$\\mathcal {C}$$ which has $$\\mathcal {G}$$ as 1-skeleton (scaffold), see Fig. 5. Each $$n$$-cycle in $$\\mathcal {C}$$ cannot be seen as composition of two-body interactions, but represents a true $$n$$-body interaction; in other words, any relationship expressed in the cycle is unique in its configuration. We denote by $$C^{(n)} ([l_1, \\dots l_{(n+1)} ]$$) the cycles of $$\\mathcal {C}$$, and by $$\\delta _{k,i}$$ the presence or the absence of $$i$$ in the cycle ($$\\delta _{k,i}=1$$ if $$k=i$$, $$\\delta _{k,i}=0$$ if $$k\\ne i$$) and we generalize then the standard linear form for the mean field $$h_i$$ to the form:\n\n\\begin{aligned} h_i=S+ \\sum _{k = 1}^N \\, {\\mathop {\\mathop {\\sum }\\limits _{C^{(n)} ([ \\ell _1, \\dots \\ell _{(n+1)} ])}}\\limits _{1 \\le n \\le N - 1}} J_{\\ell _1 \\dots \\ell _k \\dots \\ell _{n+1}} \\prod _{j=1}^n c_{\\ell _j} \\, \\delta _{k, i} \\; \\end{aligned}\n(4)\n\nIn the partition function\n\n\\begin{aligned} \\displaystyle {Z(x) \\doteq \\sum _{\\left\\{ c_\\ell \\right\\} } e^{-x E\\left( \\left\\{ c_\\ell \\right\\} \\right) }} \\; \\; x \\in \\mathbb {R}, \\end{aligned}\n(5)\n\nthe sum runs over the set of all possible valuations $$c_\\ell = 0 , 1 \\; , \\; \\forall \\ell$$, subdivides the set of states in classes of equivalence, giving different statistical weights—depending on a parameter $$x \\in {\\mathbb {R}}\\; , \\; x >0$$—to those states which are invariant with respect to a given set of transformations. A phase transition, if any, would allow us to pass from one class of equivalence to the other when the state symmetry is (partially or fully) broken. This turns the model into a theoretical framework where, given a parameter—for example the average specific antibody concentration—we can predict when and if a configuration may break into another, giving rise to a different immunity type, i.e. change the adaptive immunity. In terms of formal language theory, going from one configuration to another belonging to a different class of equivalence has the following meaning: if we associate to the space of data a group of possible transformations preserving its topology (e.g., its mapping class group), and the related regular language, the general semantics thus naturally generated describes the set of all transformations and hence of all ‘phases’ in the form of relations.\n\nWe consider then the functor partition function, $$Z(x)$$. We might of course access more information (patterns) by considering higher ($$k$$-th) order correlation functions,\n\n\\begin{aligned} \\Gamma _k (x) \\doteq \\frac{1}{Z(x)} \\sum _{\\left\\{ c_\\ell \\right\\} } c_{\\ell _1} \\dots c_{\\ell _k} e^{-x E\\left( \\left\\{ c_\\ell \\right\\} \\right) } \\; , \\end{aligned}\n(6)\n\nfor any given set of points $${\\ell _1 \\dots \\ell _k } \\in \\mathcal {A}$$. We can represent with strings of $$N$$ dichotomic variables the set of $$\\{c_\\ell \\}$$, $$2^{N-1}$$ possible configurations.\n\nA crucial assumption we add to the model is that the coupling constants $$J_{\\ell _1 \\dots \\ell _k \\dots \\ell _{n+1}}$$ are taken to be proportional to a linear combination (with negative coefficients) of the simplex $$n$$-volume $$V^{(n)}$$, the simplex corresponding to the cell defined by the set $$\\bigl \\{ \\ell _1 , \\dots , \\ell _n \\bigr \\}$$ in the cells of cycle $$C^{(n)} ([\\ell _1, \\dots , \\ell _{(n+1)} ])$$, with the volume of the cell boundary of dimension $$n-2$$, weighted by the curvature at that boundary. The latter measures the ease with which the $$n$$-body interaction is favored by the manifold bending. The ensuing action is expected to measure reasonably well the probability that the $$n$$-body process described by that coupling takes place.\n\nWhen the model with such interaction form is dealt with as a statistical field theory it turns out to be fully isomorphic with a Euclidean topological field theory describing a totally different physical system: gravity coupled with matter in a simplicial complex setting, consistent with general relativity. We think back to the standard example of the Ising model, which also has variables in $${\\mathbb {Z}}_2$$ (Parisi 1998) and recall that a statistical field theory is any model in statistical mechanics where the degrees of freedom comprise a field; i.e. the microstates of the system are expressed through field configurations. The features of the ensuing theory are quite general and far reaching. The topology of the associated moduli space depends only on the manifold genus $$g$$, on the dimension $$n$$ of the (vector) bundle over it used to define the field, and on the dimension $$\\delta ({\\mathrm{mod}} \\, n)$$ of the associated determinant bundle. Such space is a projective variety, smooth only if $$(\\delta ,n) = 1$$. The recursive determination of the Betti numbers in this case is given by the Harder and Narasimhan and Atiyah and Bott recursions (Harder and Narasimhan 1975; Atiyah and Bott 1983). The former explicitly counts points of the moduli space, the latter resorts to an infinite-dimensional Morse theory with the field action functional as Morse function. These recursions lead to a closed formula for the Poincaré polynomial, i.e. for the Betti numbers of the moduli space. These implicit methods were successively made explicit (Desale and Ramanan 1975).\n\nWhat is intriguing is that our field theory turns out to be isomorphic to $${\\mathbb {Z}}_2$$ (quantum) gravity, dealt with in nonperturbative fashion by standard Regge calculus (Regge 1961).\n\nLet us recall here that the construction of a consistent theory of quantum gravity in the continuum is a problem in theoretical physics that has so far defied all attempts of a rigorous formulation and resolution. The only effective approach to try and obtain a non-trivial quantum theory proceeded via discretization of space-time and of the Einstein action, i.e., by replacing the space-time continuum by a combinatorial simplicial complex and deriving the action from simple physical principles.\n\nQuantum Regge calculus, based on the well-explored classical discretization of the Einstein action due to Regge, and the essentially equivalent method of dynamical triangulations are the tools that proved most successful. Regge’s method consists in approximating Einstein’s continuum theory by a simplicial discretization of the space-time (in gravity a four-dimensional Lorentz manifold) resorting to local building blocks (simplices) and then constructing the gravitational action as the sum of a term depending on the (hyper)volumes of the different simplicial complexes and another reflecting the space-time curvature. The metric tensor associated with each simplex is expressed as a function of the squared edge lengths, which are the dynamical variables of this model. Summing over all interpolating geometries (state sum) generated by the simplicial complex construction in the embedding higher-dimensional ones (filtration), allows us to derive both the Einstein action and the equilibrium configurations simply by means of counting procedure (entropy estimate).\n\nThe $${\\mathbb {Z}}_2$$ version of the model is one in which representations of $$SU(2)$$ labeling the edges in quantum Regge calculus are reduced to $${\\mathbb {Z}}_2$$. The power of the method resides in the property that the infinite degrees of freedom of Riemannian manifolds are reduced by discretization; and the theory can deal with PL spaces, described by a finite number of parameters. Moreover, for the manifolds approximated by a simplicial complex (or by dynamically triangulated random surfaces), the local coordination numbers are automatically included among the dynamical variables, leaving the quadratic link lengths $$q_\\ell$$, globally constrained by triangle inequalities, as true degrees of freedom.\n\nMore precisely, the model adopted here for the immune system is isomorphic to the $${\\mathbb {Z}}_2$$ Regge model, where the quadratic link lengths $$q_\\ell$$ of the simplicial complexes are restricted to take on only two values: $$q_\\ell = 1 + {\\mathfrak {l}} \\sigma _\\ell$$, where $$\\sigma _\\ell = \\pm 1 = 2 c_{\\ell } - 1$$. Such model has been exactly solved (in the case of quantum gravity) via the matrix model approach (Ambjørn et al. 1985) and with the help of conformal field theory (Knizhnik et al. 1988). A crucial ingredient is the choice of functional integration measure, whose behavior, with respect to diffeomorphisms, is fundamental. The very definition of diffeomorphism is a heavy constraint in constructing the PL space exactly invariant under the action of the full diffeomorphism group (Menotti 1998), and only the recent construction of a simplicial version of the mapping class group made it viable (Merelli and Rasetti 2013).\n\nAs Regge regularization leads to the usual Liouville field theory in the continuum limit based on a description of PL manifolds with deficit angles, not edge lengths, we may assume that also in our case the correct measure has to be nonlocal. Starting point for the $${\\mathbb {Z}}_2$$ Regge model is a discrete description of general relativity in which space-time is represented by a piecewise flat, simplicial manifold (Regge skeleton). The procedure works for any space-time dimension $$d$$, metrics of arbitrary signature, and action\n\n\\begin{aligned} A ( \\mathbf{{q}} ) = x \\left( \\sum _{s^d} V^{(d)} \\left( s^d \\right) - \\zeta \\sum _{s^{d-2}} {\\mathfrak {d}} ( s^{d-2} ) \\, V^{(d-2)} \\left( s^{d-2} \\right) \\right) \\; \\end{aligned}\n(7)\n\nwith the quadratic edge lengths $$\\left\\{ q_\\ell \\right\\}$$ (more precisely, the $$\\sigma _{\\ell }$$’s) describing the dynamics of the complex. $$x$$ and $$\\zeta$$ denote free constants (in the discrete time picture, with uniform time step $$\\tau$$, energy functional and action are merely proportional). The first sum runs over all $$d$$-simplices $$s^d$$ of the simplicial complex, while $$V ( s^d )$$ is the $$d$$-volume of $$s^d$$. The second term represents the curvature of the simplicial complex, concentrated along the $$(d - 2)$$-simplices, leading to deficit angles $${\\mathfrak {d}} ( s^{d-2} )$$. The physical meaning of the terms entering action $$A$$ is what makes it acceptable for a consistent description of the immune system with higher order (‘many body’) interactions: the lower the volumes and the higher the curvature, the lower is the action $$(x, \\zeta > 0)$$.\n\nAt equilibrium, i.e. in the absence of an explicit time-dependence of the expectation values of the variables, the partition function for our antigen-free IS model is nothing but the field propagator of the theory, expressed via path integral\n\n\\begin{aligned} \\displaystyle {Z = \\int \\mathcal{{D}} \\, [\\mathbf{{q}}] \\, \\mathrm{{e}}^{- A ( \\mathbf{{q}} )}} \\end{aligned}\n(8)\n\nFunctional integration should extend over all metrics on all possible topologies, hence the path-integral approach, typically suffers from a nonuniqueness of the integration measure and a need for a nonlocal measure is advocated. The standard ‘simplicial’ measure\n\n\\begin{aligned} \\displaystyle { \\int \\mathcal{{D}} \\, [\\mathbf{{q}}] = \\prod _\\ell \\, \\int \\frac{\\mathrm{{d}} q_\\ell }{q_\\ell ^\\alpha } \\, \\mathcal{{F}} ( \\mathbf{{q}} )}, \\,\\,\\, \\mathrm{{where}} \\,\\,\\, \\alpha \\in {\\mathbb {R}} \\end{aligned}\n(9)\n\nallows exploring a family of measures, as $$\\mathcal{{F}} ( \\mathbf{{q}} )$$ can be designed to constrain integration to those configurations which do not violate triangular inequalities, and moreover can be chosen so as to remove non realistic simplices. The characteristic partition function of the model becomes then\n\n\\begin{aligned} Z = \\left[ \\prod _\\ell ^\\mathcal{{N}} \\int\\limits _0^\\infty {\\mathrm{d}} q_\\ell \\, q_\\ell ^{- \\alpha } \\right] \\, \\mathcal{{F}} ( \\mathbf{{q}} ) \\, \\mathrm{{e}}^{- \\sum _s A_s ( \\mathbf{{q}} )} \\; , \\end{aligned}\n\nwhere $$\\mathcal{{N}}$$ is the number of links and $$A_s$$ is the contribution to the action of simplex $$s$$.\n\nIt is worth recalling that in (Desale and Ramanan 1975) arithmetic techniques and the Weil conjecture were used, and a crucial ingredient was the property that the volume of a particular locally symmetric space attached to $$SL_n$$ with respect to the canonical measure—an invariant known as the Tamagawa number of $$SL_n$$—equals 1. The simplicial volume is a homotopy invariant of oriented, closed, connected manifolds defined in terms of the singular chain complex with real coefficients. Such invariant measures the efficiency of representing the fundamental space class using singular simplices. Since the fundamental class is nothing but a generalized triangulation of the manifold, the simplicial volume can be interpreted as well both as a measure for the complexity of the manifold and as a homotopy invariant approximation of the Riemannian volume. $$Z(x)$$ provides then the generating function (Poincaré polynomial) of the Betti numbers of $$\\mathcal {A}$$.\n\nThe final step is to compare the Betti numbers obtained empirically from the data against such generating function, thus determining [simply through the solution of a system of (non-linear) algebraic equations] the set of non-zero $$J_{\\ell _1 \\dots \\ell _k \\dots \\ell _{n+1}}$$. This fully determines which antibody influences which, including ‘many-body’ influences, i.e. when and if it may happen that a given set of (two or more than two) antibodies play a role only when simultaneously active.\n\nA short discussion of Regge calculus, meant to introduce in simple way, accessible also to readers not familiar with the notion of geometry over discrete spaces (simplicial complexes), and some of the notions actually used in the derivation can be found in Battaglia and Rasetti (2003), where some of the preliminary ideas of the scheme are described, successively developed in extended way for present and other applications. As for the work in $$\\mathbb {Z}_2$$ quantum gravity which our generalized model of immune system is isomorphic to, a more articulated and complete set of references is available in Giulini (2007) and Bittner et al. (1999).\n\n## A global topological application of the S[B] paradigm\n\nIn this section we introduce the $$S[B]$$ paradigm for modeling complex adaptive systems and we discuss the IS metaphor as a global topological application of the adaptation phase; the aim is to contribute to understand the adaptability feature that, as addressed in the paper of Stepney et al. (2005), still remains ‘poorly understood’.\n\nIn the $$S[B]$$ paradigm a complex system consists of two components, the computation level from where its behavior $$B$$ emerges, the interactive machine, and the context of the computation, its global structure $$S$$. Both levels are distinct but entangled in a unique computational model that evolves by learning and adapting. The computational model associated to the $$S[B]$$ plays a crucial role in the characterization of the adaption phase, it can be represented by any mathematical model of computation, provided that it allows to express the dependency between different levels of abstraction.\n\nFigure 2 shows a simple adaptive system represented by finite state machines, which is the most general among other models, such as complex automata, higher dimensional automata, hypernetworks, recurrent neural network, multiagent, etc. On the left hand side, the two components are entangled in such a way that the emergent behaviour $$B$$ is subject to the global constraints while the global structure $$S$$ is affected by the emergent behavior. On the right, an $$S[B]$$ system is depicted as a light oval $$S$$ that embeds a dark round $$B$$, showing the adaptation phase that takes place whenever the computation can no longer evolve in the current context (the $$S[B]$$ on the lower right corner). The adaptation phase allows $$S$$ to relax the set of constraints so as to permit further computations—in the figure the black arrow drawn between the two $$S$$ components, represents the change of the global context, and the dashed arrow between the dark rounds represents the unfolding of the computation. The evolution of such a model relies on the ability of the system to adapt its computation to global requirements.\n\nA full yet concise description of the formal definition of $$S[B]$$ on a finite state machine that encapsulates both the computation $$(B)$$ and its controller $$(S)$$ follows. In this framework, both $$B$$ and $$S$$ are classically described as a finite state machine of the form $$B=(Q, q_0, \\rightarrow _B)$$ ($$Q$$ set of $$B$$ states, $$q_0$$ initial $$B$$ state and $$\\rightarrow _B$$ transition relation) and $$S = (R, r_0, \\mathcal {O}, \\rightarrow _S, L)$$ where $$R$$ is a set of $$S$$ states, $$r_0$$ is the initial $$S$$ state, $$\\mathcal {O}$$ is an observation function of $$B$$ states, $$\\rightarrow _S$$ is a transition relation and $$L$$ is a state labeling function. The function $$L$$ labels each $$S$$ state with a formula representing a set of constraints over an observation of the $$B$$ states. Therefore, a $$S$$ state $$r$$ can be directly mapped to the set of $$B$$ states satisfying $$L(r)$$. Through this hierarchy, $$S$$ can be viewed as a second-order structure $$(R \\subseteq 2^Q, r_0,\\rightarrow _S \\subseteq 2^Q \\times 2^Q, L)$$ where each $$S$$ state $$r$$ is identified with its corresponding set of $$B$$ states. An $$S[B]$$ system is the combination of an interactive machine $$B=(Q, q_0, \\rightarrow _B)$$ and a coordinator $$S=(R, r_0,\\mathcal {O}, \\rightarrow _S, L)$$ such that for all $$q \\in Q$$, $$\\mathcal {O}(q) \\ne \\perp$$. In any $$S[B]$$ system the initial $$B$$ state must satisfy the constraints of the initial $$S$$ state, i.e. $$q_0 \\models L(r_0)$$.\n\nDuring adaptation phase the $$B$$ machine is no longer regulated by the $$S$$ controller, except for a condition, called transition invariant, that must be fulfilled by the system undergoing adaptation. The complete and formal definition of the $$S[B]$$ based on finite state machine, its semantics and the adaptability checking can be found in Merelli et al. (2013).\n\nIt is quite evident that the model described above can be applied when the system requirements are known a priori and the adaptation phase reduces to dynamic selection of possible states with respect to environmental changes. To overcome this limit and allow the definition of a model that can change the set of global constraints and consequently the set of computations at run-time, we adopt the IS metaphor to characterize the adaptation phase of an $$S[B]$$ model. The global context is defined as a function of the topological invariants extracted from the analysis of the space of data: the Betti numbers. In the model proposed in previous section the Betti numbers and the $$J_{\\{\\ell \\}}$$ interaction matrix faithfully represent the relations hidden in the current space of data. Thus, the adaptation phase of an $$S[B]$$ system is indeed represented as the interplay capabilities of the immune system to identify, classify and learn the new relationship emerging among the actors of the system. Figure 3 graphically mimics the adaptability checking performed by an $$S[B]$$ system; it starts on the upper left corner of the figure with the actual model $$S[B]$$ that, when necessary, may be adapted to a new context provided by the topological analysis of the space of data (set of observations of real system). The changes in the context is determined by comparing the Betti numbers of the space of data with the Betti numbers of the actual model. If there is no new knowledge, the model remains $$S[B]$$ otherwise it adapts to the new context by learning the knowledge provided by the Betti numbers, updating its computation with new set of relations $$J_{\\{\\ell \\}}$$ and becoming $$S'[B']$$. This learning process reminds us of what in literature is called recurrent neural network, a process based on active exploration of an unknown environment and the generation of a finite state automata model of the environment.\n\nSummarizing, inspired by the IS metaphor we present a computational model as an higher order relational model which deals with multilinear n-body interactions, the interactions characteristic of the immune response. In such case, the model adapts when it no longer fits the space of observed data, and the construction of the topological field model allows us to determine the values of the $$J_{{\\{\\ell \\}}}$$ matrix, hence, e.g., the classes of antibodies that are in relation in the current immune response. We call this step a recursive construction of a relational model that learns new antibody relations as immune response to the presence of an antigene.\n\nAs future work, we aim to apply the proposed approach to real-world IS phenomena treated bot in silico and in vivo experiments and compare the results with other similar models.\n\n## Concluding remarks\n\nWe have defined a new topology-based method suitable to provide a benchmarking application of the S[B] paradigm. The method relies on a multi-linear model of immune system inspired by the topology of space of data. Starting from the notion of an Ising model in a mean field, given by Parisi and others in their seminal work, we proposed a more sophisticated version that is multilinear in the configurational variables (the antibody concentrations) instead of constant or at most linear. This work is not intended to be the study of the dynamics of the immune network in view of establishing the equilibrium among antibodies, but, instead, it has a prospective interest and strategic aim at defining a new approach for the analysis of the immune system as a metaphor of a real-life system represented in terms of Big Data.\n\n## References\n\n• Ambjørn J, Durhuus B, Fröhlich J (1985) Diseases of triangulated random surface models, and possible cures. Nucl Phys B257:433–449\n\n• Atiyah MF, Bott R (1983) The Yang–Mills equations over Riemann surfaces. Philos Trans R Soc Lond Ser A 308(1505):523–615\n\n• Battaglia D, Rasetti M (2003) Quantumlike diffusion over discrete sets. Phys Lett A 313:8–15\n\n• Bittner E, Hauke A, Markum H, Riedler J, Holm C, Janke W (1999) $$Z_2$$-Regge versus standard Regge calculus in two dimensions. Phys Rev D 59(124018):1–9\n\n• Carlsson G (2009) Topology of data. Bull New Ser AMS 46(2):255–308\n\n• Desale UV, Ramanan S (1975) Poincaré polynomials of the variety of stable bundles. Math Ann 216(3):233–244\n\n• Edelsbrunner H, Harer J (2010) Computational topology, an introduction. American Mathematical Society, Providence\n\n• Farmer J, Packard N, Perelson A (1986) The immune system, adaptation, and machine learning. Physica D22:187–204\n\n• Giulini D (2007) Mapping class groups of 3-manifolds. In: Fauser B, Tolksdorf J, Zeidler E (eds) Quantum gravity: mathematical models and experimental bounds. Birkhuser Verlag, Basel, pp 161–201\n\n• Harder G, Narasimhan MS (1974/75) On the cohomology groups of moduli spaces of vector bundles on curves. Math Ann 212:215–248\n\n• Hart E, Bersini H, Santos F (2009) Structure versus function: a topological perspective on immune networks. Natural Computing 9138(8):603–624\n\n• Hoffmann GW (1975) A network theory of the immune system. Eur J Immunol 5:638–647\n\n• Hoffmann GW (2010) An improved version of the symmetrical immune network theory. www.arXiv.org/abs/1004.5107\n\n• Hopfield JJ (1982) Neural networks and physical systems with emergent collective computational abilities. Proc Natl Acad Sci USA 79(8):2554–2558\n\n• Jerne N (1974) Towards a network theory of the immune system. Ann Immunol Inst Pasteur 125C:373–389\n\n• Knizhnik V, Polyakov A, Zamolodchikov A (1988) Fractal structure of 2d-quantum gravity. Mod Phys Lett A 3(8):819–826\n\n• Menotti P (1998) Group theoretical derivation of Liouville action for Regge surfaces. Nucl Phys B 523(3):611–619\n\n• Merelli E, Paoletti N, Tesei L (2013) Adaptability checking in multi-level complex systems. http://arxiv.org/abs/1404.0698\n\n• Merelli E, Rasetti M (2013) Non-locality, topology, formal languages: new global tools to handle large data sets. Procedia Comput Sci 18:90–99\n\n• Parisi G (1998) Statistical field theory. Westview Press, Boulder\n\n• Parisi G (1990) A simple model for the immune network. Proc Natl Acad Sci USA 87:429–433\n\n• Petri G, Scolamiero M, Donato I, Vaccarino F (2013) Topological strata of weighted complex networks. PLoS One 8(6):e66506\n\n• Regge T (1961) General relativity without coordinates. Nuovo Cimento 19:558–571\n\n• Stepney S, Smith RE, Timmis J, Tyrrell AM, Neal MJ, Hone ANW (2005) Conceptual frameworks for artificial immune systems. Int J Unconv Comput 1(3):315–338\n\n• TOPDRIM. Topology driven methods of complex systems project, Future Emerging Technologies (FET) programme within Seventh Framework Programme (FP7), www.topdrim.eu\n\n• Varela F, Coutinho A, Dupire B, Vaz N (1988) Cognitive networks: immune, neural and otherwise. In: Perelson A (ed) Theoretical immunology: part two, SFI studies in science of complexity, vol 2. Addison Wesley, Reading, pp 359–371\n\n## Acknowledgments\n\nWe thank the organizers of the ICARIS Workshop for offering the opportunity to present new ideas related to the TOPDRIM project in such a stimulating and pleasant environment. We acknowledge the financial support of the Future and Emerging Technologies (FET) programme within the Seventh Framework Programme (FP7) for Research of the European Commission, under the FET-Proactive grant agreement TOPDRIM, number FP7-ICT-318121.\n\n## Author information\n\nAuthors\n\n### Corresponding author\n\nCorrespondence to Emanuela Merelli.\n\n## Appendix\n\n### Persistent homology and Betti numbers\n\nIn this appendix we describe a general approach that allows to extract global topological information from a space of data. It is based on three basic steps: (i) The interpretation of the huge collection of $$points$$ that constitutes the space of data; this is achieved by resorting to a family of simplicial complexes (Fig. 4), parametrized by some suitably chosen ‘proximity parameter’ (Fig. 5). This operation converts the data set into a global topological object. In order to fully exploit the advantages of topology, the choice of such parameter should be metric independent. In our context it measures the expression of a possible $$relation$$. (ii) The reduction of noise, affecting the data space, as the result of the parametrized persistent homology. (iii) The encoding of the data set persistent homology in the form of a parameterized version of topological invariants, in particular Betti numbers, i.e. the invariant dimensions of the homology groups. These three steps provide an exhaustive knowledge of the global features of the space of data, even though such a space is neither a metric space nor a vector space, as other approaches require (Carlsson 2009).\n\nIn order to better comprehend the scheme, it is necessary to recall that homology is a mathematical tool that ‘measures’ the shape of an object (typically a manifold). The result of this measure is an algebraic object, a succession of groups. Informally, these groups encode the number and the type of ‘holes’ in the manifold. A basic set of invariants of a topological space $$X$$ is just its collection of homology groups, $$H_i(X)$$. Computing such groups is certainly non-trivial, even though efficient algorithmic techniques are known to do it systematically. Important ingredients of these techniques, and outcomes as well of the computation, are just Betti numbers; the $$i$$-th Betti number, $$b_i=b_i(X)$$, denotes the rank of $$H_i(X)$$. It is worth remarking that Betti numbers often have an intuitive meaning, for example, $$b_0$$ is simply the number of connected components of the space considered. While oriented 2-dimensional manifolds are completely classified by $$b_1=2g$$, where $$g$$ is the genus (i.e. number of ‘holes’) of the manifold; $$b_j$$ with $$j \\ge 2$$ classifies the features (number of higher-dimensional holes) of higher-dimensional manifolds. What makes them convenient is the fact that in several cases knowing the Betti numbers is the same as knowing the full space homology. Sometimes to know the homology groups it is sufficient to know the corresponding Betti numbers, typically much simpler to compute. In the absence of torsion, if we want to distinguish two topological objects via their homology, their Betti numbers may already do it.\n\nData can be represented as a collection, unordered sequence, of points in a $$n$$-dimensional space $$E_n$$, the space of data. The conventional way to convert a collection of points within a space such as $$E_n$$ into a global object, is to use the point cloud as the vertex set of a combinatorial graph, $$\\mathcal {G}$$. The edges of the graph are exclusively determined by a given notion of proximity, specified by some weight parameter $$\\delta$$. The parameter $$\\delta$$ should not fix a ‘distance’, that would imply fixing some sort of metric, but rather provide information about ‘dependence’, i.e. correlation or, even better, relation. If dependence had to do with distance, it should be a non-metric notion, rather a chemical distance or ontological distance just to mention an example. A graph of this sort, while capturing pretty well connectivity data, essentially ignores a wealth of higher order features beyond clustering. Such features can instead be accurately discerned by thinking of the graph as the scaffold (1-skeleton) of a different, higher-dimensional, richer (more complex) discrete object, generated by completing the graph $$\\mathcal {G}$$ to a simplicial complex, $$\\mathcal {C}$$. The latter is a piecewise-linear space built from simple linear constituents (simplices) identified combinatorially along their faces. The decisions as how this is done, implies a choice of how to fill in the higher dimensional simplices of the proximity graph. Such choice is not unique, and different options lead to different global representations. Two among the most natural and common ones, equally effective to our purpose, but with different characteristic features, are: (i) the Čech simplicial complex, where $$k$$-simplices are all unordered $$(k+1)$$-tuples of points of the space $$E_n$$, whose closed $$\\frac{1}{2} \\delta$$-ball neighborhoods have a non-empty mutual intersection; (ii) the Rips complex, an abstract simplicial complex whose $$k$$-simplices are the collection of unordered $$(k+1)$$-tuples of points pairwise within distance $$\\delta$$. The Rips complex is maximal among all simplicial complexes with the given 1-skeleton (the graph), and the combinatorics of the 1-skeleton completely determines the complex. The Rips complex can thus be stored as a graph and reconstructed out of it. For a Čech complex, on the contrary, one needs to store the entire boundary operator, and the construction is more complex; however, this complex contains a larger amount of information about the topological structure of the data space.\n\nAlgebraic topology provides a mature set of tools for counting and collating holes and other topological pattern features, both spaces and maps between spaces, for simplicial complexes. It is therefore able to reveal patterns and structures not easily identifiable otherwise. As persistent homology is generated recursively, corresponding to an increasing sequence of values of $$\\delta$$. Complexes grow with $$\\delta$$. This leads us to naturally identifying the chain maps with a sequence of successive inclusions. Persistent homology is nothing but the image of the homomorphism thus induced. The available algorithms for computing persistent homology groups focus typically on this notion of filtered simplicial complex. Most invariants in algebraic topology are quite difficult to compute efficiently. Fortunately, homology is exceptional under this respect because the invariants arise as quotients of finite-dimensional spaces.\n\nTopological information is contained in persistence homology, that can be determined and presented as a sort of parameterized version of the set of Betti numbers. Its role is just that of providing summaries of information over domains of parameter values, so as to better understand relationships among the geometric objects constructed from data. The emerging geometric/topological relationships involve continuous maps between different objects, and therefore become manifestations of functoriality, i.e, imply the notion that invariants can be extended not just to the objects studied, but also to the maps between such objects. Functoriality is central in algebraic topology because the functoriality of homological invariants is what permits one to compute them from local information. We recall the K$$\\ddot{u}$$nneth theorem that allows to consider the Poincaré polynomial of the space $$X$$ as the generating function of the Betti numbers of $$X$$." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.86234266,"math_prob":0.9949964,"size":15740,"snap":"2022-05-2022-21","text_gpt3_token_len":4011,"char_repetition_ratio":0.12722419,"word_repetition_ratio":0.0066197766,"special_character_ratio":0.26404065,"punctuation_ratio":0.089234106,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9982657,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-05-22T05:46:37Z\",\"WARC-Record-ID\":\"<urn:uuid:61391ef4-e4a2-4d23-9140-f4bd492dd686>\",\"Content-Length\":\"201592\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7574af1b-3db8-4edf-925d-fa32bee334da>\",\"WARC-Concurrent-To\":\"<urn:uuid:89aa25a3-8788-4caf-a511-2067ed37fb5e>\",\"WARC-IP-Address\":\"146.75.36.95\",\"WARC-Target-URI\":\"https://link.springer.com/article/10.1007/s11047-014-9436-7\",\"WARC-Payload-Digest\":\"sha1:USLFLKNBKY4RREEVFBUXRJD3GWFHFDRF\",\"WARC-Block-Digest\":\"sha1:BGM5GO3JEOBM66BKUDNHYRHPHH7QV2OH\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-21/CC-MAIN-2022-21_segments_1652662543797.61_warc_CC-MAIN-20220522032543-20220522062543-00766.warc.gz\"}"}
https://percentage-calculator.net/what-is-x-percent-of-y/what-is-30-percent-of-1700.php	[ "# What is 30 percent of 1700 (30% of 1700)?\n\nAnswer: 30 percent of 1700 is 510\n\n## Fastest method for calculating 30 percent of 1700 (30% of 1700)\n\nAssume the unknown value is 'Y'\n\nY = 30 / 100\n\nY = 30 / 100 x 1700\n\nY = 510\n\nAnswer: 30 percent of 1700 is 510\n\nIf you want to use a calculator, simply enter 30÷100x1700 and you will get your answer which is 510\n\nYou may also be interested in:\n\nHere is a calculator to solve percentage calculations such as what is 30% of 1700. You can solve this type of calculation with your own values by entering them into the calculator's fields, and click 'Calculate' to get the result and explanation.\n\nWhat is% of\n?\n\n## Have time and want to learn the details?\n\nLet's solve the equation for Y by first rewriting it as: 100% / 1700 = 30% / Y\n\nDrop the percentage marks to simplify your calculations: 100 / 1700 = 30 / Y\n\nMultiply both sides by Y to transfer it on the left side of the equation: Y ( 100 / 1700 ) = 30\n\nTo isolate Y, multiply both sides by 1700 / 100, we will have: Y = 30 ( 1700 / 100 )\n\nComputing the right side, we get: Y = 510\n\nThis leaves us with our final answer: 30% of 1700 is 510" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.91571975,"math_prob":0.99871683,"size":950,"snap":"2020-34-2020-40","text_gpt3_token_len":286,"char_repetition_ratio":0.1448203,"word_repetition_ratio":0.030927835,"special_character_ratio":0.3694737,"punctuation_ratio":0.08163265,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9994054,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-08-09T03:19:56Z\",\"WARC-Record-ID\":\"<urn:uuid:5ff29ec5-e0df-45ef-86de-ad4b1baa909f>\",\"Content-Length\":\"58830\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:641f9f06-6aab-40e2-8f99-2e7019a8baa8>\",\"WARC-Concurrent-To\":\"<urn:uuid:d258c343-05e0-4d0f-a289-009ea80ac8bc>\",\"WARC-IP-Address\":\"68.66.224.6\",\"WARC-Target-URI\":\"https://percentage-calculator.net/what-is-x-percent-of-y/what-is-30-percent-of-1700.php\",\"WARC-Payload-Digest\":\"sha1:K5ZAGW3SSU76BYKCQ3TPBLUVKA2ERS22\",\"WARC-Block-Digest\":\"sha1:K7UAML25HJJLLGOXXDJRTH2ZBIXAPOUD\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-34/CC-MAIN-2020-34_segments_1596439738380.22_warc_CC-MAIN-20200809013812-20200809043812-00282.warc.gz\"}"}
https://slideplayer.com/slide/7819321/	[ "", null, "# + The Practice of Statistics, 4 th edition – For AP* STARNES, YATES, MOORE Chapter 8: Estimating with Confidence Section 8.2 Estimating a Population Proportion.\n\n## Presentation on theme: \"+ The Practice of Statistics, 4 th edition – For AP* STARNES, YATES, MOORE Chapter 8: Estimating with Confidence Section 8.2 Estimating a Population Proportion.\"— Presentation transcript:\n\n+ The Practice of Statistics, 4 th edition – For AP* STARNES, YATES, MOORE Chapter 8: Estimating with Confidence Section 8.2 Estimating a Population Proportion\n\n+ Chapter 8 Estimating with Confidence 8.1Confidence Intervals: The Basics 8.2Estimating a Population Proportion 8.3Estimating a Population Mean\n\n+ Section 8.2 Estimating a Population Proportion After this section, you should be able to… CONSTRUCT and INTERPRET a confidence interval for a population proportion DETERMINE the sample size required to obtain a level C confidence interval for a population proportion with a specified margin of error DESCRIBE how the margin of error of a confidence interval changes with the sample size and the level of confidence C Learning Objectives\n\n+ Estimating a Population Proportion Activity: The Beads Your teacher has a container full of different colored beads. Your goal is to estimate the actual proportion of red beads in the container. Form teams of 3 or 4 students. Determine how to use a cup to get a simple random sample of beadsfrom the container. Each team is to collect one SRS of beads. Determine a point estimate for the unknown population proportion. Find a 90% confidence interval for the parameter p. Consider any conditions that are required for the methods you use. Compare your results with the other teams in the class.\n\n+ Conditions for Estimating p Suppose one SRS of beads resulted in 107 red beads and 144 beads of another color. The point estimate for the unknown proportion p of red beads in the population would be Estimating a Population Proportion How can we use this information to find a confidence interval for p?\n\n+ Conditions for Estimating p Check the conditions for estimating p from our sample. Estimating a Population Proportion Random: The class took an SRS of 251 beads from the container. Normal: Both np and n(1 – p) must be greater than 10. Since we don’t know p, we check that The counts of successes (red beads) and failures (non-red) are both ≥ 10. Independent: Since the class sampled without replacement, they need to check the 10% condition. At least 10(251) = 2510 beads need to be in the population. The teacher reveals there are 3000 beads in the container, so the condition is satisfied. Since all three conditions are met, it is safe to construct a confidence interval.\n\n+ Constructing a Confidence Interval for p We can use the general formula from Section 8.1 to construct a confidence interval for an unknown population proportion p : Estimating a Population Proportion Definition: When the standard deviation of a statistic is estimated from data, the results is called the standard error of the statistic.\n\n+ Finding a Critical Value How do we find the critical value for our confidence interval? Estimating a Population Proportion If the Normal condition is met, we can use a Normal curve. To find a level C confidence interval, we need to catch the central area C under the standard Normal curve. For example, to find a 95% confidence interval, we use a critical value of 2 based on the 68-95-99.7 rule. Using Table A or a calculator, we can get a more accurate critical value. Note, the critical value z* is actually 1.96 for a 95% confidence level.\n\n+ Finding a Critical Value Use Table A to find the critical value z* for an 80% confidence interval. Assume that the Normal condition is met. Estimating a Population Proportion Since we want to capture the central 80% of the standard Normal distribution, we leave out 20%, or 10% in each tail. Search Table A to find the point z* with area 0.1 to its left. So, the critical value z* for an 80% confidence interval is z* = 1.28. The closest entry is z = – 1.28. z.07.08.09 – 1.3.0853.0838.0823 – 1.2.1020.1003.0985 – 1.1.1210.1190.1170\n\n+ One-Sample z Interval for a Population Proportion Once we find the critical value z, our confidence interval for the population proportion p is Estimating a Population Proportion Choose an SRS of size n from a large population that contains an unknown proportion p of successes. An approximate level C confidence interval for p is where z is the critical value for the standard Normal curve with area C between – z* and z. Use this interval only when the numbers of successes and failures in the sample are both at least 10 and the population is at least 10 times as large as the sample. One-Sample z Interval for a Population Proportion\n\n+ Calculate and interpret a 90% confidence interval for the proportion of red beads in the container. Your teacher claims 50% of the beads are red.Use your interval to comment on this claim. Estimating a Population Proportion z.03.04.05 – 1.7.0418.0409.0401 – 1.6.0516.0505.0495 – 1.5.0630.0618.0606 For a 90% confidence level, z = 1.645 We checked the conditions earlier. sample proportion = 107/251 = 0.426 statistic ± (critical value) (standard deviation of the statistic) We are 90% confident that the interval from 0.375 to 0.477 captures the actual proportion of red beads in the container. Since this interval gives a range of plausible values for p and since 0.5 is not contained in the interval, we have reason to doubt the claim.\n\n+ The Four-Step Process We can use the familiar four-step process whenever a problem asks us to construct and interpret a confidence interval. Estimating a Population Proportion State: What parameter do you want to estimate, and at what confidence level? Plan: Identify the appropriate inference method. Check conditions. Do: If the conditions are met, perform calculations. Conclude: Interpret your interval in the context of the problem. Confidence Intervals: A Four-Step Process\n\n+ Choosing the Sample Size In planning a study, we may want to choose a sample size that allows us to estimate a population proportion within a given margin of error. Estimating a Population Proportion The margin of error (ME) in the confidence interval for p is z* is the standard Normal critical value for the level of confidence we want. To determine the sample size n that will yield a level C confidence interval for a population proportion p with a maximum margin of error ME, solve the following inequality for n: Sample Size for Desired Margin of Error\n\n+ Example: Customer Satisfaction Read the example on page 493. Determine the sample size needed to estimate p within 0.03 with 95% confidence. Estimating a Population Proportion The critical value for 95% confidence is z* = 1.96. Since the company president wants a margin of error of no more than 0.03, we need to solve the equation Multiply both sides by square root n and divide both sides by 0.03. Square both sides. Substitute 0.5 for the sample proportion to find the largest ME possible. We round up to 1068 respondents to ensure the margin of error is no more than 0.03 at 95% confidence.\n\n+ Section 8.2 Estimating a Population Proportion In this section, we learned that… Summary\n\n+ Section 8.2 Estimating a Population Proportion In this section, we learned that… When constructing a confidence interval, follow the familiar four-step process: STATE: What parameter do you want to estimate, and at what confidence level? PLAN: Identify the appropriate inference method. Check conditions. DO: If the conditions are met, perform calculations. CONCLUDE: Interpret your interval in the context of the problem. The sample size needed to obtain a confidence interval with approximate margin of error ME for a population proportion involves solving Summary\n\n+ Looking Ahead… We’ll learn how to estimate a population mean. We’ll learn about The one-sample z interval for a population mean when σ is known The t distributions when σ is unknown Constructing a confidence interval for µ Using t procedures wisely In the next Section…\n\n+\n\n+ Homework Chapter 8 #33-36, 38, 40, 42\n\nDownload ppt \"+ The Practice of Statistics, 4 th edition – For AP* STARNES, YATES, MOORE Chapter 8: Estimating with Confidence Section 8.2 Estimating a Population Proportion.\"\n\nSimilar presentations" ]	[ null, "https://slideplayer.com/static/blue_design/img/slide-loader4.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.81369376,"math_prob":0.95363694,"size":7972,"snap":"2021-21-2021-25","text_gpt3_token_len":1837,"char_repetition_ratio":0.20820783,"word_repetition_ratio":0.089686096,"special_character_ratio":0.239714,"punctuation_ratio":0.11290322,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99594134,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-05-09T17:34:51Z\",\"WARC-Record-ID\":\"<urn:uuid:e787b44f-52ac-4ae5-ab60-406437cbb234>\",\"Content-Length\":\"172043\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e2f00a5f-e343-419c-b6db-d0eda3e0a95a>\",\"WARC-Concurrent-To\":\"<urn:uuid:aec9a2a9-493e-4034-a2b2-eca94ad9d816>\",\"WARC-IP-Address\":\"138.201.54.25\",\"WARC-Target-URI\":\"https://slideplayer.com/slide/7819321/\",\"WARC-Payload-Digest\":\"sha1:2ACS7DMVN3DKDMKYIVS2OXHBDRUDIXTW\",\"WARC-Block-Digest\":\"sha1:GXQULUWF24SLKOVS6HMM7CLHSOTW6QSN\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-21/CC-MAIN-2021-21_segments_1620243989006.71_warc_CC-MAIN-20210509153220-20210509183220-00478.warc.gz\"}"}
https://research.tue.nl/en/publications/on-the-calculation-of-nearest-neighbors-in-activity-coefficient-m	[ "# On the calculation of nearest neighbors in activity coefficient models\n\n2 Citations (Scopus)\n\n### Abstract\n\nGuggenheim proposed a theoretical expression for the combinatorial entropy of mixing of unequal sized and linear and branched molecules to improve the Flory-Huggins model. Later the combinatorial activity coefficient equation, which was derived from Guggenheim's model, was applied in the UNIQUAC, UNIFAC, and COSMOSAC models. Here we derive from Guggenheim's entropy theory a new function for the number of nearest neighbors of a compound in a multicomponent mixture for which the knowledge of the coordination number and a reference area are not needed. This new relation requires only the mole, volume and surface fraction of the compounds in the mixture. The benefit of the new relation is that both the combinatorial and the residual term in the aforementioned models can be made lattice-independent. We demonstrate that the proposed relation simplifies the Staverman-Guggenheim combinatorial model and can be applied with success to the UNIQUAC and COSMOSPACE model in the description of vapor-liquid phase equilibria and excess enthalpy. We also show that the new expression for the number of nearest neighbors should replace the relative surface area and the number of surface patches in the residual part of the UNIQUAC and the COSMOSPACE model, respectively. As a result a more rigorous version of the UNIQUAC and the COSMOSPACE model is obtained. This could serve as a better basis for predictive models like UNIFAC, COSMO-RS and COSMOSAC.\n\nOriginal language English 10-23 14 Fluid Phase Equilibria 465 https://doi.org/10.1016/j.fluid.2018.02.024 Published - 15 Apr 2018\n\n### Fingerprint\n\nActivity coefficients\ncoefficients\nEntropy\nentropy\ncoordination number\nPhase equilibria\nEnthalpy\nliquid phases\nenthalpy\nVapors\nvapors\nMolecules\nLiquids\n\n### Keywords\n\n• COSMO-RS\n• COSMOSAC\n• COSMOSPACE\n• GEQUAC\n• Lattice theory\n• UNIFAC\n• UNIQUAC\n\n### Cite this\n\n@article{b03feace49a14fa19e4644a6956c145c,\ntitle = \"On the calculation of nearest neighbors in activity coefficient models\",\nabstract = \"Guggenheim proposed a theoretical expression for the combinatorial entropy of mixing of unequal sized and linear and branched molecules to improve the Flory-Huggins model. Later the combinatorial activity coefficient equation, which was derived from Guggenheim's model, was applied in the UNIQUAC, UNIFAC, and COSMOSAC models. Here we derive from Guggenheim's entropy theory a new function for the number of nearest neighbors of a compound in a multicomponent mixture for which the knowledge of the coordination number and a reference area are not needed. This new relation requires only the mole, volume and surface fraction of the compounds in the mixture. The benefit of the new relation is that both the combinatorial and the residual term in the aforementioned models can be made lattice-independent. We demonstrate that the proposed relation simplifies the Staverman-Guggenheim combinatorial model and can be applied with success to the UNIQUAC and COSMOSPACE model in the description of vapor-liquid phase equilibria and excess enthalpy. We also show that the new expression for the number of nearest neighbors should replace the relative surface area and the number of surface patches in the residual part of the UNIQUAC and the COSMOSPACE model, respectively. As a result a more rigorous version of the UNIQUAC and the COSMOSPACE model is obtained. This could serve as a better basis for predictive models like UNIFAC, COSMO-RS and COSMOSAC.\",\nkeywords = \"COSMO-RS, COSMOSAC, COSMOSPACE, GEQUAC, Lattice theory, UNIFAC, UNIQUAC\",\nauthor = \"G.J.P. Krooshof and R. Tuinier and {de With}, G.\",\nyear = \"2018\",\nmonth = \"4\",\nday = \"15\",\ndoi = \"10.1016/j.fluid.2018.02.024\",\nlanguage = \"English\",\nvolume = \"465\",\npages = \"10--23\",\njournal = \"Fluid Phase Equilibria\",\nissn = \"0378-3812\",\npublisher = \"Elsevier\",\n\n}\n\nIn: Fluid Phase Equilibria, Vol. 465, 15.04.2018, p. 10-23.\n\nTY - JOUR\n\nT1 - On the calculation of nearest neighbors in activity coefficient models\n\nAU - Krooshof, G.J.P.\n\nAU - Tuinier, R.\n\nAU - de With, G.\n\nPY - 2018/4/15\n\nY1 - 2018/4/15\n\nN2 - Guggenheim proposed a theoretical expression for the combinatorial entropy of mixing of unequal sized and linear and branched molecules to improve the Flory-Huggins model. Later the combinatorial activity coefficient equation, which was derived from Guggenheim's model, was applied in the UNIQUAC, UNIFAC, and COSMOSAC models. Here we derive from Guggenheim's entropy theory a new function for the number of nearest neighbors of a compound in a multicomponent mixture for which the knowledge of the coordination number and a reference area are not needed. This new relation requires only the mole, volume and surface fraction of the compounds in the mixture. The benefit of the new relation is that both the combinatorial and the residual term in the aforementioned models can be made lattice-independent. We demonstrate that the proposed relation simplifies the Staverman-Guggenheim combinatorial model and can be applied with success to the UNIQUAC and COSMOSPACE model in the description of vapor-liquid phase equilibria and excess enthalpy. We also show that the new expression for the number of nearest neighbors should replace the relative surface area and the number of surface patches in the residual part of the UNIQUAC and the COSMOSPACE model, respectively. As a result a more rigorous version of the UNIQUAC and the COSMOSPACE model is obtained. This could serve as a better basis for predictive models like UNIFAC, COSMO-RS and COSMOSAC.\n\nAB - Guggenheim proposed a theoretical expression for the combinatorial entropy of mixing of unequal sized and linear and branched molecules to improve the Flory-Huggins model. Later the combinatorial activity coefficient equation, which was derived from Guggenheim's model, was applied in the UNIQUAC, UNIFAC, and COSMOSAC models. Here we derive from Guggenheim's entropy theory a new function for the number of nearest neighbors of a compound in a multicomponent mixture for which the knowledge of the coordination number and a reference area are not needed. This new relation requires only the mole, volume and surface fraction of the compounds in the mixture. The benefit of the new relation is that both the combinatorial and the residual term in the aforementioned models can be made lattice-independent. We demonstrate that the proposed relation simplifies the Staverman-Guggenheim combinatorial model and can be applied with success to the UNIQUAC and COSMOSPACE model in the description of vapor-liquid phase equilibria and excess enthalpy. We also show that the new expression for the number of nearest neighbors should replace the relative surface area and the number of surface patches in the residual part of the UNIQUAC and the COSMOSPACE model, respectively. As a result a more rigorous version of the UNIQUAC and the COSMOSPACE model is obtained. This could serve as a better basis for predictive models like UNIFAC, COSMO-RS and COSMOSAC.\n\nKW - COSMO-RS\n\nKW - COSMOSAC\n\nKW - COSMOSPACE\n\nKW - GEQUAC\n\nKW - Lattice theory\n\nKW - UNIFAC\n\nKW - UNIQUAC\n\nUR - http://www.scopus.com/inward/record.url?scp=85043368492&partnerID=8YFLogxK\n\nU2 - 10.1016/j.fluid.2018.02.024\n\nDO - 10.1016/j.fluid.2018.02.024\n\nM3 - Article\n\nAN - SCOPUS:85043368492\n\nVL - 465\n\nSP - 10\n\nEP - 23\n\nJO - Fluid Phase Equilibria\n\nJF - Fluid Phase Equilibria\n\nSN - 0378-3812\n\nER -" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.86601776,"math_prob":0.8821751,"size":5463,"snap":"2019-51-2020-05","text_gpt3_token_len":1348,"char_repetition_ratio":0.13134274,"word_repetition_ratio":0.79151946,"special_character_ratio":0.2121545,"punctuation_ratio":0.09677419,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96714056,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-12-12T00:38:06Z\",\"WARC-Record-ID\":\"<urn:uuid:e55411df-0d98-4ae9-8311-1336199bec42>\",\"Content-Length\":\"46869\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:aa775c1c-930f-4a9f-a84c-05a2321d068c>\",\"WARC-Concurrent-To\":\"<urn:uuid:f36d9d5f-c901-4306-81be-cb7306f57d71>\",\"WARC-IP-Address\":\"52.51.22.49\",\"WARC-Target-URI\":\"https://research.tue.nl/en/publications/on-the-calculation-of-nearest-neighbors-in-activity-coefficient-m\",\"WARC-Payload-Digest\":\"sha1:4NYS4ZHX4I7U2GOB3YX6W6BA7TKL54IO\",\"WARC-Block-Digest\":\"sha1:7D5R5TH4IH5QMOSR22CT4K7ASO3HCIFM\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-51/CC-MAIN-2019-51_segments_1575540534443.68_warc_CC-MAIN-20191212000437-20191212024437-00438.warc.gz\"}"}
http://www.kpubs.org/article/articleMain.kpubs?articleANo=E1JSE6_2014_v6n2_206	[ "Numerical simulations of two-dimensional floating breakwaters in regular waves using fixed cartesian grid\nNumerical simulations of two-dimensional floating breakwaters in regular waves using fixed cartesian grid\nInternational Journal of Naval Architecture and Ocean Engineering. 2014. Jun, 6(2): 206-218", null, "This is an Open-Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.\n• Published : June 30, 2014", null, "PDF", null, "e-PUB", null, "PubReader", null, "PPT", null, "Export by style\nArticle\nAuthor\nMetrics\nCited by\nTagCloud\nKwang-Leol, Jeong\nDepartment of Naval Architecture and Ocean Engineering, Graduate School of Inha University, Incheon, Korea\nYoung-Gill, Lee\nDepartment of Naval Architecture and Ocean Engineering, Inha University, Incheon, Korea\n\nAbstract\nThe wave attenuation by floating breakwaters in high amplitude waves, which can lead to wave overtopping and breaking, is examined by numerical simulations. The governing equations, the Navier-Stokes equations and the continuity equation, are calculated in a fixed Cartesian grid system. The body boundaries are defined by the line segment connecting the points where the grid line and body surface meet. No-slip and divergence free conditions are satisfied at the body boundary cell. The nonlinear waves near the moving body is defined using the modified markerdensity method. To verify the present numerical method, vortex induced vibration on an elastically mounted cylinder and free roll decay are numerically simulated and the results are compared with those reported in the literature. Using the present numerical method, the wave attenuations by three kinds of floating breakwaters are simulated numerically in a regular wave to compare the performance .\nKeywords\nINTRODUCTION\nAlthough bottom fixed breakwaters have an advantage in calming harbors, there are some problems with water pollution in the harbors due to the blockage of currents, moreover, the cost of construction increases considerably in deep sea or soft ground conditions. Floating breakwaters can be an alternative, but they have poorer performance than bottom fixed breakwaters. In addition, disaster can occur if the mooring lines are damaged ( Lee and Song, 2005 ). To increase the performance of a breakwater, the effects of nonlinear phenomena, such as overtopping and breaking waves need to be analyzed. To secure the safety of mooring lines, the maximum tensions on the mooring lines have to be estimated considering the second order wave drift force on a floating breakwater.\nThe motion of a floating body has been analyzed mostly based on potential theory. Most studies of floating breakwaters were also performed based on potential theory ( Loukogeorgaki and Angelides, 2005 ; Lee and Cho, 2003 ; Lee and Song, 2005 ; Song and Kim, 2005 ). The potential theories use linearized governing equations and free surface boundary conditions. The theories, however, show satisfactory results when linear assumptions are suitable for the purpose only. If the flows include highly nonlinear phenomena such as wave breaking and wave overtopping, the potential theories cannot produce meaningful results. Because such nonlinear phenomena are not negligible estimating the performance and evaluating the safety of mooring lines, a method that can analyze nonlinear phenomena is necessary. Experiments are an alternative method, and there have been performed many researchers ( Ruol et al., 2008 ; Chun and Cho, 2011 ). Computational Fluid Dynamics (CFD) can be another alternative method. Koftis and Prinos (2005) examined flow near a floating breakwater using the CFD. In Koftis and Prinos (2005) , the motions of the breakwater were not considered. The motions should be considered in the analysis to obtain more rational results. A grid technique for a moving body and a free surface model for a nonlinear wave are important. The grid system is essential in most numerical simulations for practical purposes even though there are meshless methods using particle such as Najafi-Jilani and Rezaie-Mazyak (2011) and Jung et al. (2008) . The grid systems have been categorized in two groups; a body fitted grid system and a non-body fitted Cartesian grid system ( Mittal and Iaccarino, 2005 ). The body fitted grid system provides a more accurate result particularly near the body boundary. Many research using body fitted grid have been conducted for floating body in waves with grid deformation, overset grid or grid regeneration ( Guo et al., 2012 ; Sadat-Hosseini et al., 2013 ; Hadzic et al., 2005 ; Simonsen et al., 2013 ). In body fitted grid system, grid generation near a complex body requires considerable time and effort. Moreover, grid regeneration and interpolation between overset grids for moving a body take huge time. On the other hand, the Cartesian grid system has advantages on easy grid generation and fast calculation. In the Cartesian grid system, the grid lines do not consist with the body surface. Therefore, a special treatment is required for a body boundary treatment. These methods are called the Immersed Boundary Method (IBM). The greatest advantage of the Cartesian grid is the easy treatment of moving bodies in a fixed grid system. A range of methods for a moving body in a Cartesian grid system have been developed. On the other hand, few methods consider the problems including the free surface. Normally, zero gradient boundary conditions are imposed for pressure and velocity on the body boundary. These boundary conditions are difficult to be satisfied if a body boundary cell contains the free surface, due to the drastic differences in the densities and viscous coefficients. Lin (2007) proposes the Partial Cell Treatment (PCT) method to simulate a moving body on the free surface. The PCT method defines the body boundary using the volume ratio in a cell between the body and fluid. Therefore, the PCT method cannot define the body surface as sharply as the Ghost Cell Immersed Boundary Method (GCIBM) and Cartesian Cut Cell (CCM) ( Hu and Kashiwagi, 2009 ). In the present study, a body boundary treatment technique using no-slip and divergence free conditions is suggested for the sharp definition of the body boundaries.\nThe differences in the densities and viscous coefficients of water and air are sources of solution instability. The most popular method for free surface modeling is the Volume of Fluids (VOF) method ( Hirt and Nichols, 1981 ). In the VOF method, the free surface is treated as a transient zone, where the density and viscosity varies continuously in space. The densities and viscous coefficients of the cells near the free surface are defined using the volume fractional function. The transient zone is a source of the reduction of the accuracy of the solution. Park et al. (1999) proposed the marker-density method, which does not use transient zone. To obtain the sufficient stability, the velocities of air in the free surface cells are extrapolated from the water velocities. Lee et al. (2012) suggested the modified marker-density method which calculates the velocities of air in the free surface cells with the governing equations. In this study, the modified marker-density method is implemented for the nonlinear free surface definition.\nTo verify developed numerical method, vortex induced vibrations on the elastically mounted circular cylinder are numerically simulated and the results are compared with the results of other researches. Moreover, a free roll decay simulation is performed and the results are compared with the data in the published literature. Finally, three kinds of breakwaters in regular wave are simulated using the present numerical method.\nNUMERICAL METHOD\n- Governing equations and discretization\nThe filtered Navier-Stokes equations and the continuity equation are employed as governing equations. Subgrid-scale (SGS) turbulence model is implemented to consider the effects of the turbulence flow smaller than a grid. f i in Eq. (1) denotes the gravitational acceleration and r ij is SGS Reynolds stress.", null, "PPT Slide\nLager Image", null, "PPT Slide\nLager Image\nThe governing equations are descretized with Foreword Time, Centred Space (FTCS) except for convection terms. The convection terms are descretized using the Kawamura-Kuwahara scheme ( Kawamura and Kuwahara, 1984 ) and first order upwind scheme considering the number of fluid cells in space and the Adams-Bashforth scheme in time. The descretized governing equation is shown in Eqs. (3) and (4).", null, "PPT Slide\nLager Image", null, "PPT Slide\nLager Image\nwhere", null, "PPT Slide\nLager Image\nrepresents a discretized spatial derivation and u * denotes the tentative velocity in Eq. (5).", null, "PPT Slide\nLager Image\nBy substituting Eq. (3) into Eq. (4), the pressure Poisson equation Eq. (6) is obtained and the pressure Poisson equation is solved using the Successive Over Relaxation method. More details of spatial descretization can be found in Lee et al. (2012) .", null, "PPT Slide\nLager Image\n- Body boundary conditions\nThe governing equations are solved in a staggered Cartesian grid. Because the grid line does not consist with the body surface, the body boundaries are defined by line segments connecting the points where the grid line and body surface meet. The grids containing the line segment are defined as body boundary cell. The no-slip condition is imposed on the body surface and the zero-divergence condition is imposed in body boundary cell. The velocity near a body is assumed to follow a quadratic polynomial w ( x ) in Fig. 1. Inducing the velocity profile, '0' is imposed on the body surface for the no-slip boundary condition.\nTo satisfy the divergence free condition, the divergence of a body boundary cell is calculated by joining the fluxes passing through the grid lines together. The flux Q (in Fig. 1 ) through the grid line of the cut cell H is calculated by integrating the quadratic polynomial. Using the divergence, new pressures are calculated in Eq. (7), and the velocities are computed from them using Eq. (3). The superscripts in Eq. (7) mean the inner iteration steps. Until the pressures and velocities are converged, the iterative calculation continues to satisfy the zero-divergence condition.", null, "PPT Slide\nLager Image\nwhere D is the divergence of body boundary cells, and ω is the relaxation factor.", null, "PPT Slide\nLager Image\nSchematic sketch of a velocity profile near a fixed body boundary.", null, "PPT Slide\nLager Image\nSchematic sketch of velocity profiles near a moving body.\nWhen a body moves, velocity profiles are induced with the velocities of the body surface ( w b1 in Fig. 2 ) instead of the zero value in Fig. 1 . The velocity profiles for the flux calculation change abruptly when the body surface passes the velocity definition point in a certain time interval because the velocity definition points used for the velocity profiles change. This causes spurious pressure oscillations. To prevent the abrupt change in the volume flux through the grid face, the volume flux is determined with the weighted average of the two fluxes according to Eq. (8). Q 1 and Q 2 are obtained by integrating the velocity profile w 1 ( x ) and w 2 ( x ) , respectively. The velocity profile w 1 ( x ) is induced with w b1 , w i 1 , k , and w i 2 , k , and the velocity profile w 2 ( x ) is induced with w b1 , w i, k , and w i 1 , k .", null, "PPT Slide\nLager Image\nEq. (9) must be satisfied to conserve the mass in a body boundary cell. A B is the area occupied by the body in a cell. The second term in Eq. (9) is calculated by adding all fluxes together through grid faces. Instead of calculating A B , the amount of the body passing through the face is added to the volume flux as shown in Eq. (10) for an easy calculation of the first term of Eq. (9). Eq. (10) cannot substitute for Eq. (9) sufficiently if the body shape change sharply like a corner.", null, "PPT Slide\nLager Image", null, "PPT Slide\nLager Image\nIf a fluid cell neighboring the body boundary undergoes a change to become a body boundary cell, the pressure changes abruptly because the pressure calculation undergoes a different process according to the change in the geometrical surrounding. In this case, the pressure in the fluid cell neighboring the body boundary cell can be determined using the same process that is applied to the body boundary cell described in previous paragraphs.\n- Free surface boundary conditions\nEqs. (11) and (12) show the dynamic and kinematic boundary conditions of the free surface. Eq. (11) means that the air pressure is the same as that of water on the free surface and the surface tension and the viscous stress on free surface are ignored. Eq. (12) means that the velocities of a fluid particle and the free surface normal to the free surface are the same on that location of the free surface.", null, "PPT Slide\nLager Image", null, "PPT Slide\nLager Image\nIn the above,", null, "PPT Slide\nLager Image\nand", null, "PPT Slide\nLager Image\nmean the velocity vectors of the fluid particle and free surface respectively, and", null, "PPT Slide\nLager Image\nstands for the normal vector of the free surface.\nThe transport equation of the Marker-density is used to define the position of the free surface, as shown in Eq. (13) instead of Eq. (12). The value of the Marker-density is an artificial density obtained numerically from the water density ( ρ water ) and air density ( ρ air ). A cell is considered to be a water cell if the Marker-densities of a cell and its neighboring cells are larger than the average of the air and water densities. On the other hand, the cell is considered an air cell if the Marker-densities of a cell and its neighboring cells are less than the average of the air and water densities. Other fluid cells are treated as free surface cells. The location of the free surface is determined where the Marker-density is the average of the air and water densities.\nThe method for the free-surface definition is similar to the VOF method. In the VOF method, the volume fraction is used to determine the density and viscosity of the free surface cell. The velocity and pressure of free surface cells are calculated using the weighted average density and viscosity by a volume fraction. In the Marker-density method, on the other hand, the air and water regions are treated as separate regions. Therefore, the Marker-density value is used only to define the free surface position. In this study, the pressure and velocity distributions are calculated from the density and the viscosity of air or water, not from the Marker-density.", null, "PPT Slide\nLager Image\nIn the traditional Marker-density method, the pressure on the free surface is extrapolated equally from the nearest air cell including gravitational acceleration. Also the air velocity of free surface cell is extrapolated with Lagrangian manner to obtain the stability of the solutions ( Park et al., 1999 ). In the present method, on the other hand, the air velocity of free surface cell is calculated with governing equations. For the stability of the solution, continuous pressure gradient condition (Eq. (14)) is additionally imposed. Fig. 3 and Eq. (15) show how to calculate the pressure on the free surface.", null, "PPT Slide\nLager Image", null, "PPT Slide\nLager Image\nSchematic drawing of the pressure calculation on free surface.", null, "PPT Slide\nLager Image\nThe velocities and pressures of the free surface cells are calculated using a simultaneous iteration method. The velocities of the free surface cells are calculated according to Eq. (16).", null, "PPT Slide\nLager Image\nand", null, "PPT Slide\nLager Image\nin Eq. (16) are the tentative velocities in Eq. (5). The divergences of free surface cells are calculated using the velocities of the free surface. The pressure is updated or calculated at each inner iteration step with the new divergence according to Eq. (7). Until the pressures and velocities converge, they are calculated iteratively.", null, "PPT Slide\nLager Image\nThe process to obtain the pressure and velocity of body boundary and free surface cells are identical. Therefore, additional treatment is not necessary to get the stability of the body boundary cell including the free surface.\n- In/out flow boundary conditions\nVelocity distribution of the Stokes's 2nd order wave theory is imposed on the inflow boundary to generate regular waves. Neumann boundary condition is imposed for the pressure and wave elevation on the inflow boundary. To prevent unintended wave reflection or pressure oscillation on the outflow boundary, artificial damping zone is imposed near the outflow boundary. As shown in Eq. (17), the velocity of z- direction are artificially damped 1% every time step.", null, "PPT Slide\nLager Image\nVERIFICATIONS\n- Elastically mounted circular cylinder\nVortex shedding on the circular cylinder suspended by an elastic spring is simulated numerically at Re = 100. The SGS turbulence model is not employed in these calculations due to the low Reynolds number. The amplitude and period of the cylinder motion are affected by the mass of cylinder ( m ) and spring constant ( k ). The mass ratio ( m * = 4 m /πρ D 2 L ) is 5 and the non-dimensional velocities (", null, "PPT Slide\nLager Image\n) ranged from 0.5 to 1.5.", null, "PPT Slide\nLager Image\nAmplitude of the oscillation of circular cylinder due to vortex shedding by non-dimensional velocity.\nFig. 4 shows the amplitude of oscillating cylinder at various non-dimensional velocities. The present calculation results agree with the other research data of Bahmani and Akbari (2011) and Shiels et al. (2001) . A 'Lock-in' phenomenon, in which the period of vortex shedding coincides with the period of the mass spring system are observed from 0.8 to 1.1 of the nondimensional velocity. Fig. 5 shows the time histories of lift force coefficient of the cylinder in U R = 0.5 and U R =1.1 conditions. Even though, there are several spurious pressure oscillations, the effects are small enough to ignore. Fig. 6 shows the pressure contours, stream lines and velocity vectors near the moving cylinder. In the figure the dashed lines indicate the initial position of the cylinder. The velocity vectors in the cylinder mean the velocity of the moving cylinder. Fig. 6 shows the spatially continuous variation of velocity and pressure. The present numerical simulation method is applicable to the simulation of translational body motion by fluid force.", null, "PPT Slide\nLager Image\nLift force coefficient histories in Ur = 0.5 and Ur = 1.1 conditions.", null, "PPT Slide\nLager Image\nPressure contours, stream lines and velocity vectors in UR = 0.8 at t/T = 140.1sec.\n- Free roll decay test\nUsing a rectangular floating body, a free roll decay test is performed to check the applicability to the problem including the free surface. The breadth of the body is 0.3 m and the draft is 0.05 m . The translational movements are restrained and the rotational movement is set free. The moment of inertia is equal to 0.262 kg/m2 . The SGS turbulence model is implemented in the calculation. The initial roll angle is set 15°. Fig. 7 shows the calculation domain. Three grid size cases (0.004 m , 0.002 m , and 0.001 m ) are employed to check the grid dependency and the time interval is set to 5/10000 s .", null, "PPT Slide\nLager Image\nSchematic sketch of computation domain for the free roll decay test.\nFig. 8 presents the time histories of the roll moment and roll angle. In the case of a grid size of 0.004 m , there are spiky variations in the moment history. On the other hand, the roll moment variations are smooth in the case of a grid size of 0.002 m and 0.001 m . In the case of a grid size of 0.004 m , the roll angle is different from other cases. Between the cases of grid sizes of 0.002 m and grid size 0.001 m , there are small differences that could be ignored.", null, "PPT Slide\nLager Image\nTime histories of the roll moments and angle during free decay.\nThe results of the present simulation are compared with the experimental data reported in Jung et al. (2007) in Fig. 9 . The agreement is sufficient to apply the present method to simulate the rotational motion of a floating body when the roll angle is smaller than 6°. Fig. 10 shows the vorticity distribution near the floating body. A number of vortexes exist near the body and affect the motion of the body.", null, "PPT Slide\nLager Image\nComparison of the roll angle extinction with experimental data.", null, "PPT Slide\nLager Image\nVorticity contours and free surface shape at 2.0s.\nFLOATING BREAKWATERS\n- Calculation conditions\nThree breakwater shapes are investigated under the same displacement. The shape of the submerged area of CASE 1 is a square shape with 0.15 m edges. The ratio between the breadth and draft of CASE 2 is equal to 2. CASE 3 is designed by giving a slope to the side walls of CASE 1. Fig. 11 shows the shape of each case. Table 1 lists the principal dimensions of all the cases.", null, "PPT Slide\nLager Image\nSection shapes of the floating breakwaters.\nPrincipal dimensions of floating breakwaters.", null, "PPT Slide\nLager Image\nPrincipal dimensions of floating breakwaters.\nTo investigate the effects of wave overtopping, CASE 1-1, which is designed by reducing the freeboard of CASE 1, is additionally simulated. To limit the drift due to waves, it is assumed that all the breakwaters are moored by linear spring in the x- direction. The spring constant is set to 2 N / m . The length of incident wave is 0.75 m and the height is 0.030 m .", null, "PPT Slide\nLager Image\nSchematic sketch of computation domain for floating breakwaters.\nFig. 12 shows the schematic sketch for the calculation domain. The calculations cannot continue after the waves reflected by the breakwater reaches the inflow boundary. Therefore, the inflow boundary is located sufficiently far from the floating breakwater. A damping zone is placed near the end of the computational domain to avoid unintended reflections by the outflow boundary. The minimum grid sizes in the x- and z- directions are 0.002 m . The time interval is equal to 4/10,000 s .\n- Simulation results\nFig. 13 shows the pressure distributions around each breakwater from 21.0 s to 22.6 s . The pressure distributions around CASE 1 are similar to those of CASE 1-1 in overall time, except for those on the weather side at 21.1 s and 21.2 s . The pressure on weather side of CASE 1 is much higher than that of CASE 1-1. Such a pressure difference, caused by the difference of the free board height, causes the difference of sway or drift forces. The amplitude of sway and drift of CASE 1 is a little larger than CASE 1-1 as shown Figs. 14 and 15. There is not large difference in the pressure distribution on the bottom of CASE 1 and CASE 1-1 as shown in Fig. 13 . However, the heave amplitude of CASE 1-1 is smaller than that of CASE 1 as shown in Fig. 15 . The difference of heave is caused by the overtopped water of CASE 1-1; nevertheless the amount of the overtopped water is not large.", null, "PPT Slide\nLager Image\nComparison of pressure distributions around the floating breakwaters.\nThe pressure variation on the lee side affects the height of transmitted waves. The pressure on the lee side of CASE 3 is much higher than those of other cases as shown in Fig. 13 . Therefore the transmitted wave of CASE 3 is largest among the cases as shown in Fig. 16 . The transmitted waves are measured at x = 0.5 m . On the contrary, the transmitted wave height of CASE 2 is smallest among the cases due to the same reason. In CASE 1, CASE 1-1 and CASE 2, the break water moves to right side due to the wave loads. Especially, the horizontal displacement of CASE 2 is larger than any other as shown in Fig. 14 . In CASE 3, on the contrary, the breakwater does not move to right side because of the large pressure on the lee side as shown in Fig. 14 . Therefore the tension on the mooring line of CASE 3 would be smaller than any other. In contrast the tension on the mooring line of CASE 2 would be largest due to large horizontal displacement.", null, "PPT Slide\nLager Image\nTime histories of the motions of each breakwater.", null, "PPT Slide\nLager Image\nComparison of the motions of breakwaters in frequency domain.", null, "PPT Slide\nLager Image\nComparison of transmitted waves of each the floating breakwater.\nCONCLUSIONS\nIn the present study, the numerical simulation method developed by Lee and Jeong (2013) was verified for the simulation of two-dimensional floating body and applied to two-dimensional breakwater. To verify the present numerical method, 'lock-in' phenomena on the circular cylinder are simulated and the results are compared to existing research data and the results shows that the present numerical method is applicable to the simulating a translational movement of a body due to fluid force. A few spurious pressure oscillations are occurred but these are ignorable. Free roll decay also simulated. Because the present method cannot resolve the boundary layer accurately in high Reynolds number, there are some differences when the roll angle is larger than 6°. However, when the roll angle is smaller than 6°, the agreements are good enough to apply the present method to the simulation for floating breakwaters.\nThe attenuation performance of a floating breakwater is investigated using the present method. Wave overtopping due to small free board reduces the horizontal displacement of floating breakwater since the pressure on the weather side is reduced. The overtopped water also reduces the heave motion. The high pressure on the lee side of trapezoidal shape (CASE 3) drasticcally reduces the horizontal displacement of floating breakwater; however, the high pressure increases the height of transmitted wave. On the other hand, the small pressure on lee side of the rectangular shape (CASE 2) increases the horizontal displacement but the height of transmitted wave is reduced. Rectangular type floating breakwater has the advantage in calming harbors and trapezoidal shape floating breakwater has the advantage in the small tension on mooring lines.\nFor more accurate simulation, a method to resolve the boundary layer near the wall has to be developed. Moreover, the present numerical method has to be extended to three-dimension to consider oblique incident waves and the wave deflection.\nAcknowledgements\nThis study was supported by INHA University.\nReferences" ]	[ null, "http://www.kpubs.org/resources/images/licenceby.png", null, "http://www.kpubs.org/resources/images/pc/down_pdf.png", null, "http://www.kpubs.org/resources/images/pc/down_epub.png", null, "http://www.kpubs.org/resources/images/pc/down_pub.png", null, "http://www.kpubs.org/resources/images/pc/down_ppt.png", null, "http://www.kpubs.org/resources/images/pc/down_export.png", null, "http://www.kpubs.org/S_Tmp/J_Data/E1JSE6/2014/v6n2/E1JSE6_2014_v6n2_206_e901.jpg", null, "http://www.kpubs.org/S_Tmp/J_Data/E1JSE6/2014/v6n2/E1JSE6_2014_v6n2_206_e902.jpg", null, "http://www.kpubs.org/S_Tmp/J_Data/E1JSE6/2014/v6n2/E1JSE6_2014_v6n2_206_e903.jpg", null, "http://www.kpubs.org/S_Tmp/J_Data/E1JSE6/2014/v6n2/E1JSE6_2014_v6n2_206_e904.jpg", null, 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https://pinoybix.org/2015/02/mcqs-in-physics-part14.html	[ "# MCQ in Physics Part 14 \| ECE Board Exam\n\n(Last Updated On: February 22, 2020)", null, "This is the Multiples Choice Questions Part 14 of the Series in Physics as one of the General Engineering and Applied Sciences (GEAS) topic. In Preparation for the ECE Board Exam make sure to expose yourself and familiarize in each and every questions compiled here taken from various sources including past Board Questions in General Engineering and Applied Sciences (GEAS), Physics Books, Journals and other Physics References.\n\n### Online Questions and Answers in Physics Series\n\nFollowing is the list of multiple choice questions in this brand new series:\n\nCollege Physics MCQs\nPART 1: MCQs from Number 1 – 50 Answer key: PART I\nPART 2: MCQs from Number 51 – 100 Answer key: PART II\nPART 3: MCQs from Number 101 – 150 Answer key: PART III\nPART 4: MCQs from Number 151 – 200 Answer key: PART IV\nPART 5: MCQs from Number 201 – 250 Answer key: PART V\nPART 6: MCQs from Number 251 – 300 Answer key: PART VI\nPART 7: MCQs from Number 301 – 350 Answer key: PART VII\nPART 8: MCQs from Number 351 – 400 Answer key: PART VIII\nPART 9: MCQs from Number 401 – 450 Answer key: PART IX\nPART 14: MCQs from Number 651 – 700 Answer key: PART XIV\n\n### Continue Practice Exam Test Questions Part XIV of the Series\n\nChoose the letter of the best answer in each questions.\n\nNewton’s First Law\n\n651. A light hangs from two cables. One cable has a tension of 39.72 lb. and is at an angle of 43.4° with respect to the ceiling. What is the weight of the lamp if the other cable makes an angle of 17.1° with respect to the ceiling?\n\n• a. 37.2 lb.\n• b. 35.8 lb.\n• c. 36.8 lb.\n• d. 36.17 lb.\n\n652. A 46.07 N light hangs from two cables at angles 54.9° and 61.4° with respect to the ceiling. What is the tension in the first cable?\n\n• a. 24.6 N\n• b. 25 N\n• c. 23.9 N\n• d. 26.4 N\n\n653. A light hangs from two cables. One cable has a tension of 28.75 N and is at an angle of 58.1° with respect to the ceiling. What is the tension in the other cable if it makes an angle of 9.4° with respect to the ceiling?\n\n• a. 15.9 N\n• b. 16.1 N\n• c. 15.4 N\n• d. 14.9 N\n\nNewton’s Second Law\n\n654. A 8.3 kg mass and a 17.1 kg mass are tied to a light string and hung over a frictionless pulley. What is their acceleration?\n\n• a. 33.95 m/s2\n• c. 4.395 m/s2\n• b. 4 m/s2\n• d. 3.395 m/s2\n\n655. An unknown mass and a 9.9 kg mass are tied to a light string and hung over a frictionless pulley. If the tension in the string is 14.5 N, what is the unknown mass?\n\n• a. 1 kg\n• b. 0.8 kg\n• c. 1.8 kg\n• d. 0.5 kg\n\n656. A lady pulls a cart with a force of 1837 N. Neglecting friction, if the cart changes from resting to a speed of 1.3 m/s in a distance of0.03289 m, what is the total mass of the cart?\n\n• a. 71.5 kg\n• b. 75.1 kg\n• c. 70.5 kg\n• d. 17.5 kg\n\n657. A 3.66 lb. book is resting on a 19.41 lb. table. What is the normal force from the floor on each table leg?\n\n• a. 5 lbs.\n• b. 5.7675 lbs.\n• c. 4.9 lbs.\n• d. 6.7675 lbs.\n\n658. A box sits on a ramp inclined at 21.7° to horizontal. If the normal force on the box from the ramp is 20.94 N, what is the mass of the box?\n\n• a. 3.2 kg\n• b. 2.9 kg\n• c. 2.3 kg\n• d. 3.9 kg\n\n659. A 7.9 kg box sits on a ramp. If the normal force on the box from the ramp is 41.82 N, what is the angle the ramp makes with the (horizontal) ground?\n\n• a. 75.3°\n• c. 57.3°\n• b. 57.5°\n• d. 75.5°\n\n660. A man sees a 44.5 kg cart about to bump into a wall at 1.7 m/s. If the cart is 0.04203 m from the wall when he grabs it, how much force must he apply to stop it before it hits?\n\n• a. 1530 N\n• c. 1350 N\n• b. 1250 N\n• d. 1520 N\n\n661. What is the minimum force required to start a 4.2 kg box moving across the floor if the coefficient of static friction between the box and the floor is 0.6?\n\n• a. 23.761 N\n• b. 25.469 N\n• c. 24.696 N\n• d. 26.496 N\n\n662. What is the kinetic energy of a 70 kg man running along at 6.36 m/s?\n\n• a. 1315 J\n• b. 1515 J\n• c. 1215 J\n• d. 1415 J\n\n663. What is the speed of a 53.6 kg woman running with a kinetic energy of 1617 J?\n\n• a. 7.77 m/s\n• b. 7.57 m/s\n• c. 7.67 m/s\n• d. 7.87 m/s\n\n664. What is the gravitational potential energy of a149.1 kg man at a height of h = 74.21 m above the ground? (consider h = 0 to be the reference where Ug = 0)\n\n• a. 100,000 J\n• b. 107,300 J\n• c. 108,400 J\n• d. 110,580 J\n\n665. What is the height where a 121.2 kg woman would have a gravitational potential energy of 10610 J? (consider h = 0 to be the reference where the Ug = 0)\n\n• a. 9.54m\n• b. 7.57m/s\n• c. 8.94 m\n• d. 7.87m/s\n\n666. What is the change in gravitational potential energy for a 68.9 kg man walking up stairs from a height of 63.07 m to 107.69 m?\n\n• a. -30133 J\n• b. 301230 J\n• c. 30320 J\n• d. 30130 J\n\n667. What is the change in gravitational potential energy for a 132.5 kg woman walking down a hill from a height of 102.86 m to 70.38 m?\n\n• a. -4123 J\n• b. -42175 J\n• c. -5 J\n• d. 4321 J\n\n668. How much work is done by gravity when a 82.3 kg. diver jumps from a height of 5.23 m into the water?\n\n• a. 4321 J\n• b. 4218 J\n• c. 4871 J\n• d. 4334 J\n\n669. How much work must be done to move a 34.6 kg box 3.66 m across the floor if the coefficient of kinetic friction between the box and the floor is 0.3.\n\n• a. 372.3 J\n• b. -321.9 J\n• c. -372.3 J\n• d. 3.234 J\n\n670. What is the coefficient of kinetic friction between a 16 kg box and the floor if it takes 140.4 J of work to move it a distance of 3.2 m?\n\n• a. 0.28\n• b. -0.25\n• c. 0.281\n• d. 0.21\n\n671. What is the length of a 73.7 m wide rectangular lab if its mass is 1.01 kg and the moment of intertia about an axis through the center and perpendicular to the large flat face if its mass is 713.6 kgm2?\n\n• a. 55.2 m\n• b. 53.5 m\n• c. 52.5 m\n• d. 54.5 m\n\n672. What is the mass of a hollow cylinder of radius3.38 m if it has a moment of inertia of 33.930468 kgm2 about the central axis or rotation?\n\n• a. 3.07 kg\n• b. 2.50 kg\n• c. 2.97 kg\n• d. 1.95 kg\n\n673. A 0.324 kg ball is stuck 0.54 m from the center of a disk spinning at 5.55 rad/s. What is its angular momentum?\n\n• a. 0.5243 Js\n• b. 0.6321 Js\n• c. 1.021 Js\n• d. 1.1524 Js\n\n674. A 40.2 kg child is sitting on the edge of a 165.3 kg merry-go-round of radius 2.1 m while it is spinning at a rate of 3.229 rpm. If the child moves to the center, how fast will it be spinning? (Hint: use conservation of angular momentum)\n\n• a. 4.5 rpm,\n• b. 4.4 rpm\n• c. 4.2 rpm\n• d. 4.8 rpm\n\n675. An empty metal can rolling down a hill gets to the bottom with a speed of 1.06 m/s. What would have been the speed if the can was full? (Assume the ends of the hollow can don’t significantly affect its moment of inertia and the walls are so thin that the full can may be considered as a solid cylinder of the same radius)\n\n• a. 1.333 m/s\n• b. 1.423 m/s\n• c. 1.223 m/s\n• d. 1.323 m/s\n\n676. A light hangs from two cables. One cable has a tension of 23.83 lb. and is at an angle of 26.2° with respect to the ceiling. What is the weight of the lamp if the other cable makes an angle of 48.7° with respect to the ceiling?\n\n• a. 34.86 lb.\n• b. 33.9 lb.\n• c. 35.8 lb.\n• d. 36 lb.\n\n677. A light hangs from two cables. One cable has a tension of 25.55 N and is at an angle of 7.5° with respect to the ceiling. What is the tension in the other cable if it makes an angle of 20.2° with respect to the ceiling?\n\n• a. 28 N\n• b. 26 N\n• c. 27 N\n• d. 29 N\n\n678. A man is pulling a cart (total 26.7 kg) with a force of 1612 N. Neglecting friction, how much time does it take to get the cart from rest up to 1.5 m/s?\n\n• a. 0.04583s\n• b. 0.01252s\n• c. 0.03567s\n• d. 0.02484 s\n\n679. A lady is pulling a cart (total 55.7 kg) with a force of 395 N. Neglecting friction, what is the acceleration of the cart?\n\n• a. 7.1 m/s2\n• b. 7.092 m/s2\n• c. 7.091 m/s2\n• d. 7.093 m/s2\n\n680. A lady pulls a cart with a force of 1454 N. Neglecting friction, if the cart changes from resting to a speed of 1.7 m/s in a distance of 0.02872 m, what is the total mass of the cart?\n\n• a. 28.90 kg\n• b. 28.91 kg\n• c. 27.90 kg\n• d. 29.00 kg\n\n681. A man sees a 44.5 kg cart about to bump into a wall at 1.7 m/s. If the cart is 0.04203 m from the wall when he grabs it, how much force must he apply to stop it before it hits?\n\n• a. 1530 N\n• b. 1730 N\n• c.1630 N\n• d.1830 N\n\n682. A 5.4 kg mass and a 6.2 kg mass are tied to a light string and hung over a frictionless pulley. What is the tension in the string?\n\n• a. 56.57 N\n• b. 55.569 N\n• c. 57.1 N\n• d. 56.569 N\n\n683. A 4.6 kg mass and an 8.5 kg mass are tied to a light string and hung over a frictionless pulley. What is their acceleration?\n\n• a. 2.92 m/s2\n• b. 2.916 m/s2\n• c. 2.917 m/s2\n• d. 3 m/s2\n\n684. An unknown mass and a 13.2 kg mass are tied to a light string and hung over a frictionless pulley. If the tension in the string is 61.3114 N, what is the unknown mass?\n\n• a. 4.12 kg\n• b. 4.1 kg\n• c. 4.0 kg\n• d. 4.2kg\n\n685. A 3.1 lb. book is resting on a 73.76 lb. table. What is the normal force of the book on the table?\n\n• a. -3.2 lbs.\n• b. -3.1 lbs.\n• c. 3.2 lbs.\n• d. 3.1 lbs.\n\n686. A 5.2 kg box sits on a ramp inclined at 42.4° to horizontal. What is the normal force on the box from the ramp?\n\n• a. 37.64N\n• b. 37.53N\n• c. 37.54 N\n• d. 37.63 N\n\n687. A 6.7 kg box sits on a ramp inclined at 37.4° to horizontal. What is the normal force on the ramp from the box?\n\n• a. 52.17 N\n• b. 52.16 N\n• c. -52.16 N\n• d. -52.17 N\n\n688. A box sits on a ramp inclined at 21.7° to horizontal. If the normal force on the box from the ramp is 20.94 N, what is the mass of the box?\n\n• a. 2.3 kg\n• b. 2.4 kg\n• c. 2.2 kg\n• d. 2.1 kg\n\n689. A 7.9 kg box sits on a ramp. If the normal force on the box from the ramp is 41.82 N, what is the angle the ramp makes with the (horizontal) ground?\n\n• a. 57.3°\n• b. 56.3°\n• c. 57.4°\n• d. 55.3°\n\n690. What is the minimum force required to start a 11.5 kg box moving across the floor if the coefficient of static friction between the box and the floor is 0.64?\n\n• a. 72.13N\n• b. 72.1 N\n• c. 72.13 N\n• d. 72.128 N\n\n691. What is the mass of a box which requires a minimum pushing force of 74.088 N to start moving across a floor with a coefficient of static friction between the box and the floor of 0.6?\n\n• a. 12.2 kg\n• b. 12.6 kg\n• c. 13 kg\n• d. 12.5 kg\n\n692. If a minimum force of 79.4682 N is required to push on a 15.9 kg box to begin moving it across the floor, what is the coefficient of static friction between the box and the floor?\n\n• a. 0.51\n• b. 0.52\n• c. 0.53\n• d. 0.54\n\n693. A box is sliding down a ramp with an acceleration of 1.621 m/s2. If the ramp is at an angle of 25.1° relative to the ground, what is the coefficient of kinetic friction between the box and the ramp?\n\n• a. 0.2857\n• b. 0.3000\n• c. 0.2856\n• d. 0.2867\n\n694. What is the kinetic energy of a 70 kg man running along at 6.36 m/s?\n\n• a. 1416 J\n• b. 1417 J\n• c. 1415 J\n• d. 1418 J\n\n695. What is the change in gravitational potential energy for a 68.9 kg man walking up stairs from a height of 63.07 m to 107.69 m?\n\n• a. 30130 J\n• b. 30132J\n• c. 30131 J\n• d. 30133 J\n\n696. What is the mass of a diver whose gravitational potential energy changes by -160,500 J when diving into water from a height of 130.61 m?\n\n• a. 125.2 kg\n• b. 125.3 kg\n• c. 125.5 kg\n• d. 125.4 kg\n\n697. How much work is done by gravity when a 82.3 kg diver jumps from a height of 5.23 m into the water?\n\n• a. 4220 J\n• b. 4218 J\n• c. 4229 J\n• d. 4219 J\n\n698. What height above the water does a 133.9 kg diver jumps need to jump from for gravity to do 6062 J of work on him/her?\n\n• a. 4.61 m\n• b. 4.60 m\n• c. 4.62 m\n• d. 4.63 m\n\n699. What is the mass of a diver if gravity does 8100 J of work on him/her when jumping into the water from a height of 6.55 m? What is the mass of a diver if gravity does 8100 J of work on him/her when jumping into the water from a height of 6.55 m?\n\n• a. 126.2 kg\n• b. 126.1 kg\n• c. 126.4 kg\n• d. 126.3 kg\n\n700. How much work must be done to move a 34.6 kg box 3.66 m across the floor if the coefficient of kinetic friction between the box and the floor is 0.3?\n\n• a. 372.2 J\n• b. 372.3 J\n• c. 372.4 J\n• d. 372.5 J\n\n### Complete List of MCQs in General Engineering and Applied Science per topic\n\nHelp Me Makes a Difference!\n\n P inoyBIX educates thousands of reviewers/students a day in preparation for their board examinations. Also provides professionals with materials for their lectures and practice exams. Help me go forward with the same spirit. “Will you make a small \\$5 gift today?” Option 1 : \\$1 USD Option 2 : \\$3 USD Option 3 : \\$5 USD Option 4 : \\$10 USD Option 5 : \\$25 USD Option 6 : \\$50 USD Option 7 : \\$100 USD Option 8 : Other Amount", null, "© 2014 PinoyBIX Engineering. © 2019 All Rights Reserved \| How to Donate? \|", null, "", null, "#### GEAS Solution\n\nDynamics problem Economics problem Physics problem Statics problem Strength problem Thermodynamics problem\n\n#### Questions and Answers in GEAS\n\nEngineering Economics Engineering Laws and Ethics Engineering Management Engineering Materials Engineering Mechanics General Chemistry Physics Strength of Materials Thermodynamics\nConsider Simple Act of Caring!: LIKE MY FB PAGE\n\nOur app is now available on Google Play, Pinoybix Elex", null, "" ]	[ null, "https://pinoybix.org/wp-content/uploads/2015/02/geas_mcqs_in_physics.png", null, "https://www.paypalobjects.com/en_US/i/scr/pixel.gif", null, "http://www.blogarama.com/images/button_sm_1.gif", null, "https://images.dmca.com/Badges/dmca-badge-w100-5x1-09.png", null, "https://play.google.com/intl/en_us/badges/static/images/badges/en_badge_web_generic.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8416425,"math_prob":0.96845067,"size":11305,"snap":"2020-10-2020-16","text_gpt3_token_len":4159,"char_repetition_ratio":0.16228652,"word_repetition_ratio":0.30462268,"special_character_ratio":0.42158338,"punctuation_ratio":0.19585745,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9976251,"pos_list":[0,1,2,3,4,5,6,7,8,9,10],"im_url_duplicate_count":[null,1,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-02-29T10:25:41Z\",\"WARC-Record-ID\":\"<urn:uuid:880f3e36-aadc-462b-9f8d-2e54cb7c690a>\",\"Content-Length\":\"121862\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d888e9e7-99d9-4630-a74d-4bd4c0098544>\",\"WARC-Concurrent-To\":\"<urn:uuid:7bef6fca-f632-4b8b-bc1b-d1a31da8b758>\",\"WARC-IP-Address\":\"104.24.98.66\",\"WARC-Target-URI\":\"https://pinoybix.org/2015/02/mcqs-in-physics-part14.html\",\"WARC-Payload-Digest\":\"sha1:KGT5RCVMSCEUPXIMFNTXSXLT7FYGY4JU\",\"WARC-Block-Digest\":\"sha1:JHB6YW3QIGDP52Z46MLPCYLTXOB7MDZB\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-10/CC-MAIN-2020-10_segments_1581875148850.96_warc_CC-MAIN-20200229083813-20200229113813-00019.warc.gz\"}"}
https://mindstat.medium.com/random-is-not-actually-random-79c22c3c62f8?source=post_page-----4b3d1209f9b1--------------------------------	[ "# Random is NOT actually random\n\nIf we take a random page from any book or a paper or whatever you like, and select all the numbers from it, almost 30% of all numbers will begin with 1, 17% begin with 2 , 11% begin with 3, and the percentage goes down with the number till it reach 9. this is called the benford’s law\n\nIt is said that if the first digits of the numbers you selected does not comply with this law, then the numbers were not random.\n\nThis phenomenon was first observed by Simon Newcomb and later formulated by Frank Benford. This phenomenon can be used for almost everything, where all the 9 numbers have equal probability of appearing as the first digit(eg: length of river, height of a tower.. etc.).\n\nThe most common use of this phenomenon is in the accounting and taxes. where they use it to asses fraud or irregularities in the books.\n\nThis is how it works, let’s take the general ledger and take the first digit of all the revenue and expenditure and consider benford’s law, if it does not comply with the benford’s law, then there is a chance of irregularities in the books, and better review it again, this is also used by the tax legislators but with a more complex equation.\n\nThe benford’s law is being used in a lot of areas including the elections which is held a lot of disagreements from the experts. they say there are lot of factors other than fraud that would lead to the change in the curve. but still other possibilities of the uses of the phenomenon are being explored.\n\nThis will completely change the way we asses randomness, so whatever data we consider there is a high chance that the fist digit obey the benford’s law. this is happening because the as we go to the bigger numbers we will have to pass the smaller first digits, that means the further we go the more chance are given for the 1s and 2s.\n\nMathematically the benford’s law is based on the logarithm with base-10, which shows that the probability that the leading digit of a number will be n can be calculated as log(1+1/n). there are more complex and advanced studies and explanations based on the same law, but this is the general idea. The graph that represent the benford’s law is shown below" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.959214,"math_prob":0.97989076,"size":2657,"snap":"2021-31-2021-39","text_gpt3_token_len":596,"char_repetition_ratio":0.13494158,"word_repetition_ratio":0.0,"special_character_ratio":0.21678585,"punctuation_ratio":0.083636366,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9848108,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-07-29T08:15:42Z\",\"WARC-Record-ID\":\"<urn:uuid:6ea370fe-8ba8-4336-8dd1-86d8cdca5257>\",\"Content-Length\":\"116869\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:cc373625-d91c-4c28-a27d-53300408cf9c>\",\"WARC-Concurrent-To\":\"<urn:uuid:fccf5657-dbad-49da-a055-3903d503b2b2>\",\"WARC-IP-Address\":\"162.159.153.4\",\"WARC-Target-URI\":\"https://mindstat.medium.com/random-is-not-actually-random-79c22c3c62f8?source=post_page-----4b3d1209f9b1--------------------------------\",\"WARC-Payload-Digest\":\"sha1:WC3HDGSVS5PYT3XDUNYMOPQZDZMTD576\",\"WARC-Block-Digest\":\"sha1:MIEVPJCBCB7S6FRMGPTDMKTRZ6NSKUPJ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-31/CC-MAIN-2021-31_segments_1627046153854.42_warc_CC-MAIN-20210729074313-20210729104313-00503.warc.gz\"}"}
https://www.teacherspayteachers.com/Product/Writing-and-Solving-Linear-Equations-Bundle-4184593?utm_source=taylorjsmathmaterials.com&utm_campaign=Resource%20Guide%208th	[ "", null, "# Writing and Solving Linear Equations Bundle", null, "", null, "", null, "", null, "7th - 10th\nSubjects\nStandards\nResource Type\nFormats Included\n• Zip\n•", null, "Activity\nPages\n38 pages\n\\$11.47\nBundle\nList Price:\n\\$12.75\nYou Save:\n\\$1.28\n\\$11.47\nBundle\nList Price:\n\\$12.75\nYou Save:\n\\$1.28\nEasel Activities Included\nSome resources in this bundle include ready-to-use interactive activities that students can complete on any device. Easel by TpT is free to use! Learn more.\n\n### Description\n\nStudents practice writing and solving equations in this 9 pack bundle!\n\nIf you only need one of the lessons, each is sold individually!\n\nWhat is covered:\n\n- It reviews the following vocabulary and key terms: coefficient, term, linear, nonlinear, equation, and types of angles\n\nWorksheet Breakdown:\n\n1. Writing Equations Using Symbols Worksheet: Students practice writing equations from words\n\n2. Linear and Nonlinear Expressions Worksheet: Students write an expression from words and then determine whether it is linear or nonlinear\n\n3. Linear Equations Worksheet: Students determine if given values are solutions to equations\n\n4. Solving a Linear Equation Worksheet: Students solve one-step and two-step equations\n\n5. Writing and Solving Linear Equations with Geometry Worksheet: Students practice writing and solving equations based on geometry word problems\n\n6. Solutions of a Linear Equation Worksheet: Students practice solving equations where distributive property is needed; no solution problems are included\n\n7. Classifications of Solutions Worksheet: Students simplify equations and determine if they have one, no, or infinite solutions\n\n8. Linear Equations as Proportions Worksheet: Students solve equations by multiplying each numerator with the other sides denominator\n\n9. Application of Linear Equations Worksheet: Students write and solve equations from words\n\nPossible uses:\n\n- End-of-topic cumulative review guide\n\n- Extra practice for struggling students\n\n- Homework\n\n- Test prep\n\n*********************************************************************************************************\n\nYou might be interested in:\n\n- these FREEBIES\n\n- the unit on Congruence\n\n- the unit on Linear Equations\n\n- the unit on Percents & Proportional Relationships\n\nor my store for other material to supplement the 7th & 8th grade curriculum!\n\nFollow My Store to receive email updates on new items, product launches, and sales!\n\nIf you have any requests and/or updates for my work, please message me!\n\nTotal Pages\n38 pages\nN/A\nTeaching Duration\nN/A\nReport this resource to TpT\nReported resources will be reviewed by our team. Report this resource to let us know if this resource violates TpT’s content guidelines.\n\n### Standards\n\nto see state-specific standards (only available in the US).\nSolve linear equations in one variable.\n\n### Questions & Answers\n\nTeachers Pay Teachers is an online marketplace where teachers buy and sell original educational materials." ]	[ null, "https://www.facebook.com/tr", null, "https://ecdn.teacherspayteachers.com/thumbitem/Writing-and-Solving-Linear-Equations-Bundle-4184593-1659455444/original-4184593-1.jpg", null, "https://ecdn.teacherspayteachers.com/thumbitem/Writing-and-Solving-Linear-Equations-Bundle-4184593-1659455444/original-4184593-2.jpg", null, "https://ecdn.teacherspayteachers.com/thumbitem/Writing-and-Solving-Linear-Equations-Bundle-4184593-1659455444/original-4184593-3.jpg", null, "https://ecdn.teacherspayteachers.com/thumbitem/Writing-and-Solving-Linear-Equations-Bundle-4184593-1659455444/original-4184593-4.jpg", null, "https://static1.teacherspayteachers.com/tpt-frontend/releases/production/current/tpt-easel-icon.atx6cuo0bp.svg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.84038615,"math_prob":0.76164305,"size":2095,"snap":"2022-40-2023-06","text_gpt3_token_len":397,"char_repetition_ratio":0.23529412,"word_repetition_ratio":0.027303753,"special_character_ratio":0.22863962,"punctuation_ratio":0.12386707,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9758982,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12],"im_url_duplicate_count":[null,null,null,1,null,1,null,1,null,1,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-09-28T18:58:33Z\",\"WARC-Record-ID\":\"<urn:uuid:5d37afe6-4210-4a68-9b95-f5a088ea65c8>\",\"Content-Length\":\"266039\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:65686364-3b8e-4b84-b902-b69b46ee9e03>\",\"WARC-Concurrent-To\":\"<urn:uuid:0a41aa13-6fcb-4ecf-b85f-0750568518b0>\",\"WARC-IP-Address\":\"23.197.178.13\",\"WARC-Target-URI\":\"https://www.teacherspayteachers.com/Product/Writing-and-Solving-Linear-Equations-Bundle-4184593?utm_source=taylorjsmathmaterials.com&utm_campaign=Resource%20Guide%208th\",\"WARC-Payload-Digest\":\"sha1:JQOUOJ2RZ23QPQRNJESM64VBHRH67QVH\",\"WARC-Block-Digest\":\"sha1:D6VQFHVP6XCK5XLFVVQM2ZYWOMS5WTRW\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030335276.85_warc_CC-MAIN-20220928180732-20220928210732-00025.warc.gz\"}"}
https://cloud.originlab.com/doc/COM/Classes/Application/FindMatrixSheet	[ "# 2.1.15 FindMatrixSheet\n\n## Description\n\nFind a matrix sheet by name.\n\n## Syntax\n\nVB: Function FindMatrixSheet(Name As ByVal String ) As MatrixSheet C++: MatrixSheet FindMatrixSheet(LPCSTR Name ) C#: MatrixSheet FindMatrixSheet(string Name ) \n\n## Parameters\n\nName\nThe range string of the Origin matrix sheet to be found. The range string contains the workbook name between square brackets followed by the sheet name:\n[<bookName>]<sheetName>\nThe following special notations are also supported:\n1. Empty string -- this means the active sheet from the active book\n2. Book name only -- like \"Book1\", will get the active sheet from named book\n3. Sheet name only with ! at the end -- like \"Sheet2!\", will get the named sheet from the active book\n\n## Return\n\nIf the named matrix sheet is found then an Origin MatrixSheet is returned.\n\n## Examples\n\n### VBA\n\nPrivate Sub SetMatrixSheetDimensions(sheetName As String, numRows As Integer, numCols As Integer)\nDim app As Origin.ApplicationSI\nSet app = New Origin.ApplicationSI\n\nDim msheet As Origin.MatrixSheet\nSet msheet = app.FindMatrixSheet(sheetName)\nIf msheet Is Nothing Then\nMsgBox (\"Failed to find matrix sheet\")\nExit Sub\nEnd If\n\n' Put current dimensions in Excel sheet cells A1 and B1\nRange(\"A1\") = msheet.Rows ' put number of rows in Excel cell A1\nRange(\"B1\") = msheet.Cols ' put number of columns in Excel cell B1\n\nmsheet.Rows = numRows\nmsheet.Cols = numCols\n\n' Put new dimensions in Excel sheet cells A2 and B2\nRange(\"A2\") = msheet.Rows ' put number of rows in Excel cell A1\nRange(\"B2\") = msheet.Cols ' put number of columns in Excel cell B1\nEnd Sub\n\n### C#\n\nusing Origin; // allow using MatrixSheet without having to write Origin.MatrixSheet\n\nstatic void SetMatrixSheetDimensions(string strSheetName, int nRows, int nCols)\n{\nApplicationSI app = new Origin.ApplicationSI();\n\nMatrixSheet msheet = app.FindMatrixSheet(strSheetName);\nif( msheet == null )\n{\nreturn;\n}\nConsole.WriteLine(\"Current dimensions of \" + msheet.Name + \" are \" + msheet.Rows + \" rows by \" + msheet.Cols + \" columns.\");\n\nmsheet.Rows = nRows;\nmsheet.Cols = nCols;\nConsole.WriteLine(\"New dimensions of \" + msheet.Name + \" are \" + msheet.Rows + \" rows by \" + msheet.Cols + \" columns.\");\n\n}\n\n### Python\n\nimport OriginExt as O\napp = O.Application(); app.Visible = app.MAINWND_SHOW\npageName = app.CreatePage(app.OPT_MATRIX)\nlayer = app.FindMatrixSheet(pageName)\nprint(layer.Name)\n\n8.0SR2" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5114903,"math_prob":0.94976044,"size":1906,"snap":"2021-43-2021-49","text_gpt3_token_len":492,"char_repetition_ratio":0.17770767,"word_repetition_ratio":0.23443224,"special_character_ratio":0.24973767,"punctuation_ratio":0.16969697,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9911066,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-12-06T09:22:44Z\",\"WARC-Record-ID\":\"<urn:uuid:bb57c5a4-33aa-477b-adcc-9cd42c83f5eb>\",\"Content-Length\":\"179220\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a86b19b0-155d-4836-a104-1fa2c10a0fe4>\",\"WARC-Concurrent-To\":\"<urn:uuid:27a20778-ef73-4d35-a31b-0788ca4714ac>\",\"WARC-IP-Address\":\"13.249.32.96\",\"WARC-Target-URI\":\"https://cloud.originlab.com/doc/COM/Classes/Application/FindMatrixSheet\",\"WARC-Payload-Digest\":\"sha1:RDMWLB66CKTZEQS56MFXWEB7VKSCUIXW\",\"WARC-Block-Digest\":\"sha1:B642EW4AZD74D7YGGJMTKB3HWDS7JWBD\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964363290.59_warc_CC-MAIN-20211206072825-20211206102825-00250.warc.gz\"}"}
https://blog.changkun.de/archives/2016/04/194/	[ "$P(A丨B)=\\frac{P(AB)}{P(B)}$\n\n$P(B)=P(A)P(B\|A)+P(\\bar A)P(B \| \\bar A) =ab+(1-a)(1-b) = 2ab+1-a-b$\n\n1. Scientific Regress： http://www.firstthings.com/article/2016/05/scientific-regress" ]	[ null ]	{"ft_lang_label":"__label__zh","ft_lang_prob":0.9646047,"math_prob":0.987168,"size":725,"snap":"2020-45-2020-50","text_gpt3_token_len":631,"char_repetition_ratio":0.073509015,"word_repetition_ratio":0.0,"special_character_ratio":0.32413793,"punctuation_ratio":0.045454547,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9991184,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-11-23T16:00:59Z\",\"WARC-Record-ID\":\"<urn:uuid:8fedfab7-644f-4158-8254-b48d39da7e06>\",\"Content-Length\":\"21332\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:1e7a5f90-31d7-4df2-ae89-a286c7eee8f8>\",\"WARC-Concurrent-To\":\"<urn:uuid:52782ac6-d7f8-49e7-af01-84a5abbda313>\",\"WARC-IP-Address\":\"139.59.204.66\",\"WARC-Target-URI\":\"https://blog.changkun.de/archives/2016/04/194/\",\"WARC-Payload-Digest\":\"sha1:UN65EOKNYPTZZAGKH2DAOT6VTEIWZABN\",\"WARC-Block-Digest\":\"sha1:Z7K2YXXRMM3BHTS52B3H4AYZD4WHEK3W\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-50/CC-MAIN-2020-50_segments_1606141163411.0_warc_CC-MAIN-20201123153826-20201123183826-00405.warc.gz\"}"}
https://mathoverflow.net/questions/324842/what-is-a-hypergraph-minor	[ "# What is a hypergraph minor?\n\nIs there a theory of hypergraph minors? I could only find some attempts to define them at papers/theses, whose main topic was something else. What would be a useful definition? Does the hypergraph version of the Robertson–Seymour theorem hold?\n\n• Arguably one of the baby steps of graph minor theory is the Wagner theorem on planar graphs. Already this is highly non-trivial for hypergraphs. Recent work of Carmesin has provided a finite list of forbidden minors (for some definition of minor) for the embeddability of simply-connected locally 3-connected 2-complexes in R^3, but there are infinite antichains when these hypotheses are lifted. Related notions of minor towards embedabillity have also been introduced by Nevo and Wagner – Arnaud Mar 8 at 14:47\n\nLet $$H$$ and $$H′$$ be hypergraphs. Then $$H$$ is a minor of $$H′$$ if $$H$$ can be obtained from $$H′$$ by a sequence of operations of the following kinds:\n• addition of ahyperedge $$e$$ such that the set $$e$$ induces a clique in the underlying graph, and" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9385511,"math_prob":0.9867391,"size":2002,"snap":"2019-13-2019-22","text_gpt3_token_len":491,"char_repetition_ratio":0.12112112,"word_repetition_ratio":0.0,"special_character_ratio":0.22877122,"punctuation_ratio":0.08672087,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9981258,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-03-26T11:03:31Z\",\"WARC-Record-ID\":\"<urn:uuid:31dc99ef-d84d-435d-a9f1-ffcde65c2e70>\",\"Content-Length\":\"115877\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:13fbbae9-78ff-49f2-86a1-876f5b845081>\",\"WARC-Concurrent-To\":\"<urn:uuid:7439a931-73a8-41c7-b9aa-e4527d42e337>\",\"WARC-IP-Address\":\"151.101.65.69\",\"WARC-Target-URI\":\"https://mathoverflow.net/questions/324842/what-is-a-hypergraph-minor\",\"WARC-Payload-Digest\":\"sha1:STSIIL2GNAXL7VDOCDO3FJS6CLKK3OFE\",\"WARC-Block-Digest\":\"sha1:VYB4DETPWVOG7Y2IJK66RRP3JBZTMEJC\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-13/CC-MAIN-2019-13_segments_1552912204969.39_warc_CC-MAIN-20190326095131-20190326121131-00502.warc.gz\"}"}
http://macphilly.com/page/is-14-an-even-number-39591933.html	[ "# Is 14 an even number", null, "This means that if the integer is divided by 2, it yields no remainderso it has 0 parity. Is 14 a palindrome? If the number is larger than 10, if it ends in 0 it is also an even number. Is 14 a perfect square? You can help Wikipedia by adding to it. How is 14 spelled out in other languages or countries? For example: 0N, 2N, 4N, 6N, What are the factor combinations of the number 14? What is the place value chart for the number 14?\n\n• Is 14 an Even Number\n• Even and Odd Numbers Between 1 and Even and Odd Numbers Examples\n• Even and Odd Numbers\n• Is 14 an even or odd number\n• [SOLVED] Is 14 an Even number\n• What is Even Number Definition, Facts & Example\n\n• You can divide. Related Links: Is 14 an odd number? What is the prime factorization of 14? What are the factors of 14? Is 14 a perfect square?", null, "What are the multiples of 14? The number five can be divided into two groups of two and one group of one. Even numbers always end with a digit of 0, 2, 4, 6 or 8.\n\n## Is 14 an Even Number\n\n2, 4, 6, 8, 10, 12, 14, 16,\nThis short article about mathematics can be made longer. From Wikipedia, the free encyclopedia. What are the factor combinations of the number 14? What are the prime factors of the number 14? How is 14 written in roman numerals?\n\n## Even and Odd Numbers Between 1 and Even and Odd Numbers Examples", null, "Bholi si surat ringtone for iphone\nHidden category: Math stubs.\n\nHow is 14 spelled out in other languages or countries? What is the total number of prime factors of the number 14?\n\nVideo: Is 14 an even number Even Number Song\n\nIf you divide by 2 and there is a remainder left, then the number is odd. They are even because they can be divided by 2 evenly.\n\n### Even and Odd Numbers\n\nIf the number is larger than 10, if it ends in 0 it is also an even number. Factoring Questions What are the factors or divisors of the number 14?\n\nLet's try that calculation 14 ÷ 2 = 7. Notice the answer is a whole number and doesn't have a remainder? That means the number is even, as it can cleanly be. To find out if 14 is an even number, we divided 14 by two. When we did that, we found that the answer is a whole number.\n\n## Is 14 an even or odd number\n\nIf you divide any number, such as Here are a couple of methods you can use to figure out if 14 is an even or odd number: Divide By Two Method You can divide 14 by two and if the result is an.\nIs 14 an even number?\n\nIf the number is larger than 10, if it ends in 0 it is also an even number. In other projects Wikimedia Commons. Any number multiplied with an even number will result in an even number. What is log 10 14?\n\n### [SOLVED] Is 14 an Even number\n\nHow is 14 formatted in other languages or countries? For example: 0N, 2N, 4N, 6N,", null, "Is 14 an even number Is 14 a Fibonacci number? See Terms of Use for details. They are even because they can be divided by 2 evenly.", null, "That means the number is even, as it can cleanly be divided by 2, with no remaining remainder. Is 14 an odd number?\nDefinition of Even Number explained with real life illustrated examples.\n\n### What is Even Number Definition, Facts & Example\n\nAlso learn the For example: 2, 4, 6, 8, 10, 12, 14, 16 are sequential even numbers. All the even and odd numbers between 1 and are discussed here. What are the even numbers from 1 to ?\n\nVideo: Is 14 an even number Even and Odd Numbers Song for Kids - Odds and Evens for Grades 2 & 3\n\nThe even 12 14 16 18 20 22 24 26 28 An even number is an integer which is \"evenly divisible\" by two. 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50,\nAny number multiplied with an even number will result in an even number.\n\nWhat is the negative version of the number 14? Views Read Change Change source View history. Even numbers are either positive or negative,as even number are types of integers.\n\nZero is an even number because zero divided by two equals zero, which despite not being a natural numberis an integer.", null, "Potassium persulfate initiator mechanism definition Is 14 a Fibonacci number? What are the factor combinations of the number 14? From Wikipedia, the free encyclopedia. What is the total number of prime factors of the number 14? How is 14 spelled out in other languages or countries?\n\n## 1 thoughts on “Is 14 an even number”\n\n1.", null, "Nizahn:\n\nCalculation Questions What are the answers to common fractions of the number 14?" ]	[ null, "http://macphilly.com/media/is-14-an-even-number-6.jpg", null, "http://macphilly.com/media/is-14-an-even-number-4.jpg", null, "http://macphilly.com/media/is-14-an-even-number.png", null, "http://macphilly.com/media/is-14-an-even-number-2.png", null, "http://macphilly.com/media/is-14-an-even-number-5.jpg", null, "http://macphilly.com/media/is-14-an-even-number-3.jpg", null, "https://1.gravatar.com/avatar/1cb1c39857f5eef49897f849251861a9", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9193964,"math_prob":0.99207664,"size":3196,"snap":"2020-34-2020-40","text_gpt3_token_len":894,"char_repetition_ratio":0.19235589,"word_repetition_ratio":0.2015873,"special_character_ratio":0.2963079,"punctuation_ratio":0.15725806,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9989466,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"im_url_duplicate_count":[null,1,null,1,null,1,null,1,null,1,null,1,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-01T00:26:56Z\",\"WARC-Record-ID\":\"<urn:uuid:6df3656c-b265-43ff-bc8f-be0535022b24>\",\"Content-Length\":\"18837\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f0cfc5e4-5396-4bc0-a3df-2bdc4887d4c8>\",\"WARC-Concurrent-To\":\"<urn:uuid:37135a44-3a94-4b93-a864-3ccf7cf3e5fe>\",\"WARC-IP-Address\":\"104.28.21.43\",\"WARC-Target-URI\":\"http://macphilly.com/page/is-14-an-even-number-39591933.html\",\"WARC-Payload-Digest\":\"sha1:QBG6XD2OVYW6HIBZNQU23ODSMQIEUB34\",\"WARC-Block-Digest\":\"sha1:SO4SLCYIGUOYAGBZQWIJ5SD364UBCNIF\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600402130531.89_warc_CC-MAIN-20200930235415-20201001025415-00231.warc.gz\"}"}
https://www.eleccircuit.com/simple-voltage-regulator-using-2n3055/	[ "# Simple Voltage regulator using 2N3055\n\nYou want to use a DC regulator or learn about voltage regulators using 2N3055. Why use this transistor? Normally, it can be used with loads that require a current not exceeding 2A and a voltage of not more than 30V.\n\nThis is enough for general work. It’s a transistor that people like to use for a long time. Therefore easy to find And very cheap. There is a lot of circuits using 2N3055.\n\nNow we recommend you 2 circuit diagram. Both circuits use a Zener diode and transistor.\n\n## 12V DC regulator circuit using 2N3055\n\nHere are 12V 1A linear regulator using a transistor and Zener diode. It is a series voltage regulator because the load current passes through the series transistor.\n\nAs the circuit diagram below, The input terminal wants an unregulated DC supply,15V to 20V. Then, the regulated voltage will come out to the load.\n\n12V 1A linear voltage regulator using 2n3055 transistor and Zener diode\n\nTo begin with, An electrical current flows through the resistor-R1 to limits current to the Zener diode. So it provides the reference voltage.\nIn the same, the base voltage of transistor-Q1 is also a constant.\n\nWhen ZD1 is 12V, the base voltage is also the 12 volts.\n\nIf we set the transistor in this form. The output voltage is the same as the Zener diode voltage. And we always call this that emitter follower. In practice, the voltage of output is lower than ZD1. Because when a transistor is working. It needs to has a base-emitter voltage.\n\n• VBE = Base-Emitter voltage\n• VZD = Zener diode voltage\n• Vout = Output voltage\n\nVout = VZD – VBE\nVBe = 0.6V\nVout = 12V – 0.6V = 11.4V\n\nThis voltage is still suitable for many loads using the 12V supply such as a receiver radios.\n\nSince it is a power supply that regulates a certain power output.\n\nIn the circuit, the transistor has a proper gain and changing of VBE help it.\n\n• When a load use more current. In general, the output voltage is low down. But the base-emitter voltage rises up, transistor Q1 works more. So it keeps the output voltage to be a constant level.\n• Then, if load use less current. The output voltage increase. But the output is still a fixed voltage. Because the voltage of Base-emitter less, transistor Q1 works less too.\n\nThe advantage of this circuit, we can use a tiny current to Zener diode and base of a transistor. Thus, it has a much more stabilized output.\n\nThe function of Others components\n\n• C1 is a smoothing capacitor at an input.\n• C2 keeps the reference voltage to be stable better.\n• C3 is a 0.047uF decoupler capacitor to filter out the transient noise.\n• R1 increases the stability of the load circuit\n• Do you know what is transient noise?\nThe power supply has a stray magnetic field. The circuit will induct them into the transient noise. The transistor-2N3055 can power load current up to 2A. But it is so hot. It so needs a proper heatsink.\n\n## Power-loss in a series regulator circuit\n\nGood power supply circuit design. It should reduce the loss of energy in the circuit to a minimum. Of course, energy will be expressed by heat.\n\nIn this series pass transistor regulator. The transistor-Q1 works look like a resistor. When we consider the power loss. It must dissipate or reduce it.\n\nDo you see an image? It is simple. Let me explain to you.\n\nLook at three cases below:\n\nIn these 3 examples, A, B, and C. The outputs are 15V, 12V and 5V. At 1A the current.\n\nDo you know which transistor has the most heat loss? Or…\nWhich transistor will heat up the most?\nYes, C example. Why?\nBecause the reason is simple.\n\nThe transistor of C is dropping the most voltage. It is effectively a dropper resistor and must dissipate heat according to ohms law.\n\nHere is show each case:\n\n• In the case of A:\nThe voltage across the transistor is 20V -15V = 5V.\nIt needs to dissipate wattage is 5V x 1A = 5W.\n• In the case of B:\nthe voltage across the transistor is 20V -12V = 7V.\nIt needs to dissipate wattage is 7V x 1A = 7W.\n\nBut…\n\n• In the case of C, the wattage is 15 watts — A much increase.\n\n### Short-circuited case\n\nIf a power supply is short-circuited. The whole input voltage will be dropped across the power transistor. And it will result in enormous heating problems.\n\nSo, for this reason, we should keep it cold with an effective heat sink.\n\n## 38V Power Supply Using 2N3055\n\nMy friend is learning about CNC, he wants a 38V Regulated Power Supply for servo motor. We have many ways to use it, but what is best for him. This circuit is one of the right choices. Because he has all the equipment. No need to buy a new one.\n\n### How this circuit works\n\nWe use the Simple Zener diode voltage regulator as main ideas, and two transistors to increase current to the load at 1A-2A.\n\nThis regulated power supply included a transformer-T1, a bridge-D1…D4, and the 38V DC filtering voltage regulator circuits, which consists of C1, C2, R1, R2, R3, Q1, and Q2.\n\nWhen 230VA or 120VAC (USA) is provided, step-down transformer-T1 changes power line AC to about 30VAC. The full-wave rectifier bridge, D1 through D4 to rectifies the AC into pulsating DC, then filtered by C1.\n\nThe capacitor C1, C3 acts as the storage capacitor or filters the noise and spikes off the AC. The 40V Zener Diode-ZD1 keeps the voltage constant across the base of BD139 NPN transistor-Q1 and Q2-2N3055 as the Darlington form, The electrolytic capacitor-C2 is used for the smoothed Zener voltage. This makes the 38V constant voltage and high power across resistor R3 and the (+) and (-) output terminals.\n\nWhen the output is connected with the low resistance load, the power transistor-Q2 will get very hot, so we always use a heat sink on it.\n\n#### Parts will you need\n\nSemiconductors:\n\n• D1-D1: 1N4002, 100V 1A Diodes\n• ZD1: 40V 1w Zener Diode\n• Q1: BD139, 80V 1.5A NPN Transistor\n• Q2: 2N3055 or TIP3055 100V, 15A, NPN transistor\n\nResistors (All 0.25 watt,5% metal/carbon film, Unless stated otherwise)\n\n• R1, R3: 3.9K\n• R2: 470 ohms\n\nElectrolytic Capacitors\n\n• C1: 470µF 50V\n• C2: 47µF 50V\n• C3: 100µF 50V\n\nT1: 230V or 120V AC primary to 30V,2A secondary transformer\n\nSW1: Power ON-OFF Switch\nF1: 0.5A Fuse\n\nNote:\nYou can use the Bridge Diode 2A-4A 200V to replace D1-D4. The transformer is used 2A min for the 1-2A load. This circuit has\n\nSharing is caring!\n\n## GET UPDATE VIA EMAIL\n\nI always try to make Electronics Learning Easy.\n\nJLCPCB - Only \\$2 for PCB Protytpe(Any Color)\n\nWith 600,000+ Customers Worldwide, 10,000+ PCB Orders Per Day\n\nUp to \\$20 shipping discount on first order now: https://jlcpcb.com/quote" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8553692,"math_prob":0.9565939,"size":6448,"snap":"2019-51-2020-05","text_gpt3_token_len":1767,"char_repetition_ratio":0.15782122,"word_repetition_ratio":0.0068906117,"special_character_ratio":0.26147643,"punctuation_ratio":0.12917595,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95923024,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-12-08T05:23:59Z\",\"WARC-Record-ID\":\"<urn:uuid:0832294a-bdbb-4965-80c6-184a60f928f0>\",\"Content-Length\":\"103797\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:03cf08a6-2835-42c1-8976-c023296f6412>\",\"WARC-Concurrent-To\":\"<urn:uuid:328053a1-4883-471b-91b3-0c7585226370>\",\"WARC-IP-Address\":\"104.27.154.229\",\"WARC-Target-URI\":\"https://www.eleccircuit.com/simple-voltage-regulator-using-2n3055/\",\"WARC-Payload-Digest\":\"sha1:4LTUMFE5X6DDF4NYQXRSBFVI5PHRHAXE\",\"WARC-Block-Digest\":\"sha1:DHS4JEVZSGGEAQKCPXI2QCZBBILHN3FA\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-51/CC-MAIN-2019-51_segments_1575540506459.47_warc_CC-MAIN-20191208044407-20191208072407-00214.warc.gz\"}"}
https://www.physicsforums.com/threads/converge-absolutely-or-conditionally-or-diverges.153883/	[ "# Converge absolutely or conditionally, or diverges?\n\n## Homework Statement\n\nDetermine if the series converges absolutely, converges conditionally, or diverges.\n\nequation is here: http://img409.imageshack.us/img409/7353/untitledly5.jpg [Broken]\n\n## Homework Equations\n\nmaybe alternating series, or harmonic series?\n\n## The Attempt at a Solution\n\nnot real familiar with tan with series.\nhaven't tried much, need supporting work for the answer.\nneed help.\n\nLast edited by a moderator:\n\nRelated Calculus and Beyond Homework Help News on Phys.org\nAs n-> infinity, tan(1/n) -> tan(0) -> 0\nDoes this help?\n\nquasar987\nHomework Helper\nGold Member\nActually, this says nothing at all about the series. The implication is one way only: \"Sum a_n converges ==> a_n-->0\" but \"a_n-->0 ==> nothing\".\n\nActually the series satisfies all the criteria corresponding to the convergence of an alternating series. Remains to see if it converges absolutely. I.e. does\n\n$$\\sum_{n=1}^{\\infty}\\tan(n^{-1})<\\infty$$\n\n??\n\nno it doesn't converge absolutely because it continues on to infinity.\n\nhowever, i do ask, how do you know to test it to be less than infinity? in other words, the convergence for a alternating series passes. but what other series convergence did not pass?\n\nso ultimately, this will converge conditionally.\n\nfor my work, i could prove this by showing the alternating series? and then showing that it also continues on to infinity?\n\nthanks again for all the help so far.\n\nquasar987\nHomework Helper\nGold Member\nWhat do you mean by \"continues on to infinity\" ?\n\nmjsd\nHomework Helper\nrcmango, i think you mean using the Leibniz test (for alternating series)\nthere are three conditions, check all to prove.\n\nHallsofIvy" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.81222224,"math_prob":0.5688212,"size":433,"snap":"2020-10-2020-16","text_gpt3_token_len":109,"char_repetition_ratio":0.10955711,"word_repetition_ratio":0.0,"special_character_ratio":0.24711317,"punctuation_ratio":0.1923077,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.951321,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-04-01T21:29:49Z\",\"WARC-Record-ID\":\"<urn:uuid:ba5d48a3-1abf-46e2-aeea-01eec5a658a8>\",\"Content-Length\":\"87764\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7ecf7d6f-9555-4367-a8dc-70441afb915b>\",\"WARC-Concurrent-To\":\"<urn:uuid:51200360-5c66-48e1-996e-c65d290a8224>\",\"WARC-IP-Address\":\"23.111.143.85\",\"WARC-Target-URI\":\"https://www.physicsforums.com/threads/converge-absolutely-or-conditionally-or-diverges.153883/\",\"WARC-Payload-Digest\":\"sha1:3KADNMLG27UEPMZBFVZRJPW2LXRUEPP4\",\"WARC-Block-Digest\":\"sha1:PMIIX4XZAAGQOUO7ZSAWXNIILEP5CE53\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-16/CC-MAIN-2020-16_segments_1585370506121.24_warc_CC-MAIN-20200401192839-20200401222839-00469.warc.gz\"}"}
https://www.gradesaver.com/textbooks/math/geometry/geometry-common-core-15th-edition/chapter-8-right-triangles-and-trigonometry-mid-chapter-quiz-page-515/23	[ "## Geometry: Common Core (15th Edition)\n\nThe angle with a tangent of $1$ would be an angle that measures $45^{\\circ}$. This is logical because the tangent ratio is $\\frac{opposite}{adjacent}$. In a $45^{\\circ}-45^{\\circ}-90^{\\circ}$ triangle, the two legs (which correspond to opposite and adjacent sides of each of the angles, not including the right angle) would be congruent, so the ratio of their lengths is $1$." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.92710125,"math_prob":0.9988152,"size":403,"snap":"2020-34-2020-40","text_gpt3_token_len":107,"char_repetition_ratio":0.13533835,"word_repetition_ratio":0.0,"special_character_ratio":0.2878412,"punctuation_ratio":0.077922076,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9996871,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-09-27T06:15:32Z\",\"WARC-Record-ID\":\"<urn:uuid:4ceefde9-330b-46e0-b43a-7eade581b8be>\",\"Content-Length\":\"83791\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:99e2f29f-f435-46b0-a207-74c5cdd822e0>\",\"WARC-Concurrent-To\":\"<urn:uuid:3f17ab7c-df78-4c04-8a0a-2daa7851565a>\",\"WARC-IP-Address\":\"54.87.252.231\",\"WARC-Target-URI\":\"https://www.gradesaver.com/textbooks/math/geometry/geometry-common-core-15th-edition/chapter-8-right-triangles-and-trigonometry-mid-chapter-quiz-page-515/23\",\"WARC-Payload-Digest\":\"sha1:OPIVBHISUJ6WNHBYEOV7JLOQ6ZCGTDPW\",\"WARC-Block-Digest\":\"sha1:DIGMTEQ6HI2IDFP62KWQOH4QRAUDHLI7\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600400265461.58_warc_CC-MAIN-20200927054550-20200927084550-00292.warc.gz\"}"}
https://forum.math.toronto.edu/index.php?PHPSESSID=2ts9icbcibl8sc9m99801fiik1&topic=2314.0;wap2	[ "APM346--2020S > Quiz 2\n\nQuiz2 TUT5101\n\n(1/1)\n\nJingjing Cui:\n$$2u_{t}+t^2u_{x}=0\\\\ \\frac{dt}{2}=\\frac{dx}{t^2}=\\frac{du}{0}\\\\ \\int\\frac{1}{2}t^2dt=\\int1dx\\\\ \\frac{1}{6}t^3+A=x\\\\ A=x-\\frac{1}{6}t^3\\\\$$\nBecause c=0, so\n$$u(t,x)=g(A)=g(x-\\frac{1}{6}t^3)$$\n\nThe initial condition given in the question: u(x,0)=f(x)\nThe characteristics curves ($A=x-\\frac{1}{6}t^3$) will always intersect t=0 (x-axis) at a unique point, no matter what value A takes. Thus, the solution always exist.\n\nVictor Ivrii:\nIn the tsecond/third lines should be $dt$, ..., not $\\partial t$,..." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5581202,"math_prob":0.9999782,"size":606,"snap":"2023-40-2023-50","text_gpt3_token_len":251,"char_repetition_ratio":0.1345515,"word_repetition_ratio":0.0,"special_character_ratio":0.41914192,"punctuation_ratio":0.13013698,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99995804,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-10-01T12:52:13Z\",\"WARC-Record-ID\":\"<urn:uuid:490262dd-6607-4584-8162-70cdd64fa19f>\",\"Content-Length\":\"2907\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b724f1b1-3da9-40eb-9ab1-8f2f555d8d5e>\",\"WARC-Concurrent-To\":\"<urn:uuid:0709f2b6-eb4e-4e75-b73d-1e04dc4f2fe9>\",\"WARC-IP-Address\":\"142.150.233.214\",\"WARC-Target-URI\":\"https://forum.math.toronto.edu/index.php?PHPSESSID=2ts9icbcibl8sc9m99801fiik1&topic=2314.0;wap2\",\"WARC-Payload-Digest\":\"sha1:3B2FUIWYW7K4L667UZ3U2L5MI53EN5SA\",\"WARC-Block-Digest\":\"sha1:4GTTMVUKKVQNIQO5SH4NMLKNKIT2VD7G\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233510888.64_warc_CC-MAIN-20231001105617-20231001135617-00274.warc.gz\"}"}
https://support.sas.com/documentation/onlinedoc/or/ex_code/132/dcmpe08.html	[ "## Vehicle Routing Problem (dcmpe08)\n\n```\n/**************************************************************/\n/ /\n/ S A S S A M P L E L I B R A R Y /\n/ /\n/ NAME: dcmpe08 /\n/ TITLE: Vehicle Routing Problem (dcmpe08) /\n/ PRODUCT: OR /\n/ SYSTEM: ALL /\n/ KEYS: OR /\n/ PROCS: OPTMODEL, SGPLOT /\n/ DATA: /\n/ /\n/ SUPPORT: UPDATE: /\n/ REF: /\n/ MISC: Example 8 from the Decomposition Algorithm /\n/ chapter of Mathematical Programming. /\n/ /\n/*************************************************************/\n\n/ number of vehicles available /\n%let num_vehicles = 8;\n/ capacity of each vehicle /\n%let capacity = 3000;\n/ node, x coordinate, y coordinate, demand /\ndata vrpdata;\ninput node x y demand;\ndatalines;\n1 145 215 0\n2 151 264 1100\n3 159 261 700\n4 130 254 800\n5 128 252 1400\n6 163 247 2100\n7 146 246 400\n8 161 242 800\n9 142 239 100\n10 163 236 500\n11 148 232 600\n12 128 231 1200\n13 156 217 1300\n14 129 214 1300\n15 146 208 300\n16 164 208 900\n17 141 206 2100\n18 147 193 1000\n19 164 193 900\n20 129 189 2500\n21 155 185 1800\n22 139 182 700\n;\n\nproc optmodel;\n/ read the node location and demand data /\nset NODES;\nnum x {NODES};\nnum y {NODES};\nnum demand {NODES};\nnum capacity = &capacity;\nnum num_vehicles = &num_vehicles;\nread data vrpdata into NODES=[node] x y demand;\nset ARCS = {i in NODES, j in NODES: i ne j};\nset VEHICLES = 1..num_vehicles;\n\n/ define the depot as node 1 /\nnum depot = 1;\n\n/ define the arc cost as the rounded Euclidean distance /\nnum cost {<i,j> in ARCS} = round(sqrt((x[i]-x[j])^2 + (y[i]-y[j])^2));\n\n/ Flow[i,j,k] is the amount of demand carried on arc (i,j) by vehicle k /\nvar Flow {ARCS, VEHICLES} >= 0 <= capacity;\n/ UseNode[i,k] = 1, if and only if node i is serviced by vehicle k /\nvar UseNode {NODES, VEHICLES} binary;\n/ UseArc[i,j,k] = 1, if and only if arc (i,j) is traversed by vehicle k /\nvar UseArc {ARCS, VEHICLES} binary;\n\n/ minimize the total distance traversed /\nmin TotalCost = sum {<i,j> in ARCS, k in VEHICLES} cost[i,j] UseArc[i,j,k];\n\n/* each non-depot node must be serviced by at least one vehicle /\ncon Assignment {i in NODES diff {depot}}:\nsum {k in VEHICLES} UseNode[i,k] >= 1;\n\n/ each vehicle must start at the depot node /\nfor{k in VEHICLES} fix UseNode[depot,k] = 1;\n\n/ some vehicle k traverses an arc that leaves node i\nif and only if UseNode[i,k] = 1 /\ncon LeaveNode {i in NODES, k in VEHICLES}:\nsum {<(i),j> in ARCS} UseArc[i,j,k] = UseNode[i,k];\n\n/ some vehicle k traverses an arc that enters node i\nif and only if UseNode[i,k] = 1 /\ncon EnterNode {i in NODES, k in VEHICLES}:\nsum {<j,(i)> in ARCS} UseArc[j,i,k] = UseNode[i,k];\n\n/ the amount of demand supplied by vehicle k to node i must equal demand\nif UseNode[i,k]=1; otherwise, it must equal 0 /\ncon FlowBalance {i in NODES diff {depot}, k in VEHICLES}:\nsum {<j,(i)> in ARCS} Flow[j,i,k] - sum {<(i),j> in ARCS} Flow[i,j,k]\n= demand[i] UseNode[i,k];\n\n/* if UseArc[i,j,k] = 1, then the flow on arc (i,j) must be at most capacity\nif UseArc[i,j,k] = 0, then no flow is allowed on arc (i,j) /\ncon VehicleCapacity {<i,j> in ARCS, k in VEHICLES}:\nFlow[i,j,k] <= Flow[i,j,k].ub UseArc[i,j,k];\n\n/* decomp by vehicle /\nfor {i in NODES, k in VEHICLES} do;\nLeaveNode[i,k].block = k;\nEnterNode[i,k].block = k;\nend;\nfor {i in NODES diff {depot}, k in VEHICLES} FlowBalance[i,k].block = k;\nfor {<i,j> in ARCS, k in VEHICLES} VehicleCapacity[i,j,k].block = k;\n\n/ solve using decomp (aggregate formulation) /\nsolve with MILP / varsel=ryanfoster decomp=(logfreq=20);\n\n/ create solution data set */\nstr color {k in VEHICLES} =\n['red' 'green' 'blue' 'black' 'orange' 'gray' 'maroon' 'purple'];\ncreate data node_data from [i] x y;\ncreate data edge_data from [i j k]=\n{<i,j> in ARCS, k in VEHICLES: UseArc[i,j,k].sol > 0.5}\nx1=x[i] y1=y[i] x2=x[j] y2=y[j] linecolor=color[k];\nquit;\n\ndata sganno(drop=i j);\nretain drawspace \"datavalue\" linethickness 1;\nset edge_data;\nfunction = 'line';\nrun;\n\nproc sgplot data=node_data sganno=sganno;\nscatter x=x y=y / datalabel=i;\nxaxis display=none;\nyaxis display=none;\nrun;\n\n```" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.637938,"math_prob":0.9132806,"size":4032,"snap":"2020-24-2020-29","text_gpt3_token_len":1368,"char_repetition_ratio":0.17204568,"word_repetition_ratio":0.07594936,"special_character_ratio":0.42237103,"punctuation_ratio":0.1740113,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9978336,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-05-28T15:29:01Z\",\"WARC-Record-ID\":\"<urn:uuid:a1d04b37-7ddc-4c5e-9d29-6b188e468e27>\",\"Content-Length\":\"15040\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b3c4ee90-cc85-408a-b9d2-2cce73ea8222>\",\"WARC-Concurrent-To\":\"<urn:uuid:df9bba08-4a84-492f-9e17-2d61c9f38e2c>\",\"WARC-IP-Address\":\"149.173.160.38\",\"WARC-Target-URI\":\"https://support.sas.com/documentation/onlinedoc/or/ex_code/132/dcmpe08.html\",\"WARC-Payload-Digest\":\"sha1:WNR2Z2VRCFHRQ64YAWNAAPXZAX7D3BO2\",\"WARC-Block-Digest\":\"sha1:FR75FSKCWC2FHODVI7NEXO3BBLHJ5B7V\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-24/CC-MAIN-2020-24_segments_1590347399820.9_warc_CC-MAIN-20200528135528-20200528165528-00010.warc.gz\"}"}
http://pandas.pydata.org/pandas-docs/dev/reference/api/pandas.DataFrame.index.html	[ "# pandas.DataFrame.index#\n\nDataFrame.index#\n\nThe index (row labels) of the DataFrame.\n\nThe index of a DataFrame is a series of labels that identify each row. The labels can be integers, strings, or any other hashable type. The index is used for label-based access and alignment, and can be accessed or modified using this attribute.\n\nReturns:\npandas.Index\n\nThe index labels of the DataFrame.\n\n`DataFrame.columns`\n\nThe column labels of the DataFrame.\n\n`DataFrame.to_numpy`\n\nConvert the DataFrame to a NumPy array.\n\nExamples\n\n```>>> df = pd.DataFrame({'Name': ['Alice', 'Bob', 'Aritra'],\n... 'Age': [25, 30, 35],\n... 'Location': ['Seattle', 'New York', 'Kona']},\n... index=([10, 20, 30]))\n>>> df.index\nIndex([10, 20, 30], dtype='int64')\n```\n\nIn this example, we create a DataFrame with 3 rows and 3 columns, including Name, Age, and Location information. We set the index labels to be the integers 10, 20, and 30. We then access the index attribute of the DataFrame, which returns an Index object containing the index labels.\n\n```>>> df.index = [100, 200, 300]\n>>> df\nName Age Location\n100 Alice 25 Seattle\n200 Bob 30 New York\n300 Aritra 35 Kona\n```\n\nIn this example, we modify the index labels of the DataFrame by assigning a new list of labels to the index attribute. The DataFrame is then updated with the new labels, and the output shows the modified DataFrame." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.57682157,"math_prob":0.7991934,"size":1344,"snap":"2023-40-2023-50","text_gpt3_token_len":338,"char_repetition_ratio":0.20223881,"word_repetition_ratio":0.0,"special_character_ratio":0.296875,"punctuation_ratio":0.21754386,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9855325,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-23T10:44:49Z\",\"WARC-Record-ID\":\"<urn:uuid:e20867ce-0506-46dd-abec-60b6d233f169>\",\"Content-Length\":\"50335\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f92630bd-4b98-4f76-8a5c-ad0bb531d5ce>\",\"WARC-Concurrent-To\":\"<urn:uuid:39c5ecef-5782-4b2e-b42a-180bef9adbbe>\",\"WARC-IP-Address\":\"104.26.1.204\",\"WARC-Target-URI\":\"http://pandas.pydata.org/pandas-docs/dev/reference/api/pandas.DataFrame.index.html\",\"WARC-Payload-Digest\":\"sha1:QW6HXXDGRU7X65NJCNMGKLMA3VZKHFJR\",\"WARC-Block-Digest\":\"sha1:5OZW47BHTH7AIDXMW7IYH34WUJNAGKFJ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233506480.7_warc_CC-MAIN-20230923094750-20230923124750-00562.warc.gz\"}"}
http://www.aei.com.mm/shop/thickness-distance-measurement/extech-dt200-laser-distance-meter/	[ "", null, "# Extech DT200: Laser Distance Meter\n\nPrice Each\n\nQuote Only\n\n(exc. GST)\n\n## Laser measurements up to 115ft (35m)\n\nThe DT200 features a laser measurement accurate to 0.08 in. at 32 ft. Measures from 2 in. to 115 ft. (0.05 to 35 m). Historical Storage recalls the previous 10 records (measurements or calculations). Automatically calculates Area and Volume. Indirect measurement using Pythagorean Theorem. Continuous measurement function with Min/Max distance tracking updates every 5 seconds. Addition/Subtraction, Front or rear edge reference. Low battery indicator, Auto power off. Complete with carrying case and 2 AAA batteries.\n\n??\n\n• Measures from 2″ to 115′ (0.05 to 35m)\n• Laser measurement accurate to 0.08 inches at 32 feet\n• Historical Storage recalls the previous 10 records (measurements or calculated results)\n• Automatically calculates Area and Volume\n• Indirect measurement using Pythagorean theorem\n• Continuous measurement function with Min/Max distance tracking updates every 5 seconds\n• Addition/Subtraction, Front or rear edge reference\n• Low battery indicator, Auto power off\n• Complete with carrying case and 2 AAA batteries\nSpecifications Range\nMeasurement Range 2″ to 115′ (0.05 to 35m)\nAccuracy (up to 32’/10m) ±0.08″ (±2mm)\nResolution 0.001″ (0.001m)" ]	[ null, "http://www.aei.com.mm/wp-content/uploads/2017/09/DT200-350x350.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.756834,"math_prob":0.87671244,"size":1624,"snap":"2021-04-2021-17","text_gpt3_token_len":450,"char_repetition_ratio":0.120987654,"word_repetition_ratio":0.14225942,"special_character_ratio":0.29987684,"punctuation_ratio":0.13220339,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95752084,"pos_list":[0,1,2],"im_url_duplicate_count":[null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-01-28T03:04:24Z\",\"WARC-Record-ID\":\"<urn:uuid:0f24a6cb-7a0d-4df2-851e-a751c62ce4ce>\",\"Content-Length\":\"161903\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:496436f9-5c92-4083-8745-f83428798d0c>\",\"WARC-Concurrent-To\":\"<urn:uuid:577f6738-0ec7-46db-9d35-603b06276708>\",\"WARC-IP-Address\":\"101.100.226.163\",\"WARC-Target-URI\":\"http://www.aei.com.mm/shop/thickness-distance-measurement/extech-dt200-laser-distance-meter/\",\"WARC-Payload-Digest\":\"sha1:6JCH66PU7C3IQFNQRMBSYICYAIPMERMO\",\"WARC-Block-Digest\":\"sha1:7WFEIRGJXE54LNTQBGDDTB6WIVSOWLH4\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-04/CC-MAIN-2021-04_segments_1610704835583.91_warc_CC-MAIN-20210128005448-20210128035448-00119.warc.gz\"}"}
https://ee-paper.com/design-of-signal-frequency-counting-function-based-on-at89s51-single-chip-microcomputer/	[ "The frequency counting of the input signal is completed by using the t0 and T1 timing counter function of AT89S51 single chip microcomputer. The counted frequency results are displayed by 8-bit dynamic nixie tube. It is required to be able to accurately count the signal frequency of 0-250KHZ, and the counting error shall not exceed ± 1Hz.\n\n1. Circuit schematic diagram", null, "Figure 4.31.1\n\n2. Hardware connection on the system board\n\n（1）。 Connect p0.0-p0.7 in the “single chip microcomputer system” area with abcdefgh port in the “dynamic digital display” area with 8-core flat cable.\n\n（2）。 Connect p2.0-p2.7 in the “single chip microcomputer system” area with s1s2s3s4s5s6s7s8 port in the “dynamic digital display” area with 8-core flat cable.\n\n（3）。 Connect p3.4 (T0) terminal in “single chip microcomputer system” area to wave terminal in “frequency generator” area with wire.\n\n3. Program design content\n\n（1）。 The working mode settings of timer / counter t0 and T1 can be seen from the figure. T0 counts the input frequency signal when working in the counting state, but for t0 working in the counting state, the maximum count value is FOSC / 24. Since FOSC = 12Mhz, the maximum count frequency of t0 is 250kHz. The concept of frequency is to count the number of pulses in one second, that is, the frequency value. Therefore, T1 works in the timing state, stops the counting of t0 every 1 second of timing, reads the counted value from the counting unit of T0, and then performs data processing. Send it to the nixie tube for display.\n\n（2）。 T1 works in the timing state, and the maximum timing time is 65ms, which can not reach the timing of 1s. Therefore, the timing of 50ms is adopted, with a total of 20 times, and the timing function of 1s can be completed.\n\n4. C language source program\n\n#include\n\nunsigned char code dispbit［］={0xfe，0xfd，0xfb，0xf7，0xef，0xdf，0xbf，0x7f};\n\nunsigned char code dispcode［］={0x3f，0x06，0x5b，0x4f，0x66，\n\n0x6d，0x7d，0x07，0x7f，0x6f，0x00，0x40};\n\nunsigned char dispbuf［8］={0，0，0，0，0，0，10，10};\n\nunsigned char temp［8］;\n\nunsigned char dispcount;\n\nunsigned char T0count;\n\nunsigned char timecount;\n\nbit flag;\n\nunsigned long x;\n\nvoid main（void）\n\n{\n\nunsigned char i;\n\nTMOD=0x15;\n\nTH0=0;\n\nTL0=0;\n\nTH1=（65536-4000）/256;\n\nTL1=（65536-4000）%256;\n\nTR1=1;\n\nTR0=1;\n\nET0=1;\n\nET1=1;\n\nEA=1;\n\nwhile（1）\n\n{\n\nif（flag==1）\n\n{\n\nflag=0;\n\nx=T0count65536+TH0256+TL0;\n\nfor（i=0;i《8;i++）\n\n{\n\ntemp=0;\n\n}\n\ni=0;\n\nwhile（x/10）\n\n{\n\ntemp=x%10;\n\nx=x/10;\n\ni++;\n\n}\n\ntemp=x;\n\nfor（i=0;i《6;i++）\n\n{\n\ndispbuf=temp;\n\n}\n\nTImecount=0;\n\nT0count=0;\n\nTH0=0;\n\nTL0=0;\n\nTR0=1;\n\n}\n\n}\n\n}\n\nvoid t0（void） interrupt 1 using 0\n\n{\n\nT0count++;\n\n}\n\nvoid t1（void） interrupt 3 using 0\n\n{\n\nTH1=（65536-4000）/256;\n\nTL1=（65536-4000）%256;\n\nTImecount++;\n\nif（TImecount==250）\n\n{\n\nTR0=0;\n\nTImecount=0;\n\nflag=1;\n\n}\n\nP0=dispcode［dispbuf［dispcount］］;\n\nP2=dispbit［dispcount］;\n\ndispcount++;\n\nif（dispcount==8）\n\n{\n\ndispcount=0;\n\n}\n\n}" ]	[ null, "https://imgs.ee-paper.com/imgs/o4YBAF1KgNqAbovGAAGGrrnNboA066.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.648015,"math_prob":0.974148,"size":2790,"snap":"2021-43-2021-49","text_gpt3_token_len":963,"char_repetition_ratio":0.13137114,"word_repetition_ratio":0.07055961,"special_character_ratio":0.33584228,"punctuation_ratio":0.15033785,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95938575,"pos_list":[0,1,2],"im_url_duplicate_count":[null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-19T11:58:53Z\",\"WARC-Record-ID\":\"<urn:uuid:fc2d6fbc-b544-4396-93ea-fb8410671b0b>\",\"Content-Length\":\"40703\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9a7c810c-082f-4f81-864f-f962e11e1374>\",\"WARC-Concurrent-To\":\"<urn:uuid:e48df640-15cd-4878-8377-8d82cda8d8a1>\",\"WARC-IP-Address\":\"172.67.208.193\",\"WARC-Target-URI\":\"https://ee-paper.com/design-of-signal-frequency-counting-function-based-on-at89s51-single-chip-microcomputer/\",\"WARC-Payload-Digest\":\"sha1:CCT2735HTGAFD24FIOGWPKWSCTKSC6HX\",\"WARC-Block-Digest\":\"sha1:QSRXF5EVNLC5BC7ZORM44BXWIY4V7IUQ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585265.67_warc_CC-MAIN-20211019105138-20211019135138-00677.warc.gz\"}"}
http://gitlinux.net/2020-03-15-(985)-sum-of-even-numbers-after-queries/	[ "## Description\n\nWe have an array A of integers, and an array queries of queries.\n\nFor the i-th query val = queries[i], index = queries[i], we add val to A[index]. Then, the answer to the i-th query is the sum of the even values of A.\n\n(Here, the given index = queries[i] is a 0-based index, and each query permanently modifies the array A.)\n\nReturn the answer to all queries. Your answer array should have answer[i] as the answer to the i-th query.\n\nExample 1:\n\nInput: A = [1,2,3,4], queries = [[1,0],[-3,1],[-4,0],[2,3]]\nOutput: [8,6,2,4]\nExplanation:\nAt the beginning, the array is [1,2,3,4].\nAfter adding 1 to A, the array is [2,2,3,4], and the sum of even values is 2 + 2 + 4 = 8.\nAfter adding -3 to A, the array is [2,-1,3,4], and the sum of even values is 2 + 4 = 6.\nAfter adding -4 to A, the array is [-2,-1,3,4], and the sum of even values is -2 + 4 = 2.\nAfter adding 2 to A, the array is [-2,-1,3,6], and the sum of even values is -2 + 6 = 4.\n\n\nNote:\n\n1. 1 <= A.length <= 10000\n2. -10000 <= A[i] <= 10000\n3. 1 <= queries.length <= 10000\n4. -10000 <= queries[i] <= 10000\n5. 0 <= queries[i] < A.length\n\n## Solutions\n\n题意比较简单，每次 query，都把对应的数加到对应位置上，然后计算数列中所有偶数的和。\n\n### 1. Array\n\n# Time: O(n)\n# Space: O(1)\nclass Solution:\ndef sumEvenAfterQueries(self, A: List[int], queries: List[List[int]]) -> List[int]:\nres = []\npre_sum = self.get_even_sum(A)\nfor val, idx in queries:\nif A[idx] % 2 == 0 and val % 2 == 0:\npre_sum += val\nelif A[idx] % 2 != 0 and val % 2 != 0:\npre_sum += val + A[idx]\nelif A[idx] % 2 == 0 and val % 2 != 0:\npre_sum -= A[idx]\nelse: # A[idx] % 2 != 0 and val % 2 == 0\npass\nA[idx] += val\nres.append(pre_sum)\nreturn res\n\ndef get_even_sum(self, A):\nif not A:\nreturn 0\nsum_v = 0\nfor a in A:\nif a % 2 == 0:\nsum_v += a\nreturn sum_v\n\n# 58/58 cases passed (548 ms)\n# Your runtime beats 82.58 % of python3 submissions\n# Your memory usage beats 5.88 % of python3 submissions (18.8 MB)" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.62413365,"math_prob":0.99892235,"size":1882,"snap":"2023-14-2023-23","text_gpt3_token_len":706,"char_repetition_ratio":0.1315229,"word_repetition_ratio":0.14404432,"special_character_ratio":0.42986184,"punctuation_ratio":0.19409283,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.998961,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-03-31T02:45:58Z\",\"WARC-Record-ID\":\"<urn:uuid:21a83b01-633e-4b1b-b938-0ef374bce4d3>\",\"Content-Length\":\"29550\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0386e7ab-3582-4975-8179-b12b06e0c2f8>\",\"WARC-Concurrent-To\":\"<urn:uuid:290aad1f-8e0a-4aa3-8186-304eb8c1f9ca>\",\"WARC-IP-Address\":\"185.199.111.153\",\"WARC-Target-URI\":\"http://gitlinux.net/2020-03-15-(985)-sum-of-even-numbers-after-queries/\",\"WARC-Payload-Digest\":\"sha1:XKZI5JREDSTJREQFBUNGQW7GAM3SWV5G\",\"WARC-Block-Digest\":\"sha1:EJ5WXMIYGKKWMC5UKGAYXED7LRT3TXAZ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-14/CC-MAIN-2023-14_segments_1679296949533.16_warc_CC-MAIN-20230331020535-20230331050535-00369.warc.gz\"}"}
https://courses.lumenlearning.com/atd-austincc-physics1/chapter/10-6-collisions-of-extended-bodies-in-two-dimensions/	[ "## Collisions of Extended Bodies in Two Dimensions\n\n### Learning Objectives\n\nBy the end of this section, you will be able to:\n\n• Observe collisions of extended bodies in two dimensions.\n• Examine collision at the point of percussion.\n\nBowling pins are sent flying and spinning when hit by a bowling ball—angular momentum as well as linear momentum and energy have been imparted to the pins. (See Figure 1). Many collisions involve angular momentum. Cars, for example, may spin and collide on ice or a wet surface. Baseball pitchers throw curves by putting spin on the baseball. A tennis player can put a lot of top spin on the tennis ball which causes it to dive down onto the court once it crosses the net. We now take a brief look at what happens when objects that can rotate collide.\n\nConsider the relatively simple collision shown in Figure 2, in which a disk strikes and adheres to an initially motionless stick nailed at one end to a frictionless surface. After the collision, the two rotate about the nail. There is an unbalanced external force on the system at the nail. This force exerts no torque because its lever arm r is zero. Angular momentum is therefore conserved in the collision. Kinetic energy is not conserved, because the collision is inelastic. It is possible that momentum is not conserved either because the force at the nail may have a component in the direction of the disk’s initial velocity. Let us examine a case of rotation in a collision in Example 1.", null, "Figure 1. The bowling ball causes the pins to fly, some of them spinning violently. (credit: Tinou Bao, Flickr)", null, "Figure 2. (a) A disk slides toward a motionless stick on a frictionless surface. (b) The disk hits the stick at one end and adheres to it, and they rotate together, pivoting around the nail. Angular momentum is conserved for this inelastic collision because the surface is frictionless and the unbalanced external force at the nail exerts no torque.\n\n### Example 1. Rotation in a Collision\n\nSuppose the disk in Figure 2 has a mass of 50.0 g and an initial velocity of 30.0 m/s when it strikes the stick that is 1.20 m long and 2.00 kg. (a) What is the angular velocity of the two after the collision? (b) What is the kinetic energy before and after the collision? (c) What is the total linear momentum before and after the collision?\n\n#### Strategy for (a)\n\nWe can answer the first question using conservation of angular momentum as noted. Because angular momentum is , we can solve for angular velocity.\n\n#### Solution for (a)\n\nConservation of angular momentum states\n\nL′,\n\nwhere primed quantities stand for conditions after the collision and both momenta are calculated relative to the pivot point. The initial angular momentum of the system of stick-disk is that of the disk just before it strikes the stick. That is,\n\n,\n\nwhere I is the moment of inertia of the disk and ω is its angular velocity around the pivot point. Now, = mr2 (taking the disk to be approximately a point mass) and ω v/r, so that\n\n$L={{mr}^{2}}\\frac{v}{r}={mvr}\\\\$.\n\nAfter the collision,\n\nL′ = Iω.\n\nIt is ω′ that we wish to find. Conservation of angular momentum gives\n\n$I'\\omega'={mvr}\\\\$.\n\nRearranging the equation yields\n\n$\\omega' =\\frac{mvr}{I'}\\\\$,\n\nwhere I′ is the moment of inertia of the stick and disk stuck together, which is the sum of their individual moments of inertia about the nail. Figure 3 gives the formula for a rod rotating around one end to be I=Mr2/3. Thus,\n\n$I'={mr}^{2}+\\frac{{Mr}^{2}}{3}=\\left(m+\\frac{M}{3}\\right){r}^{2}\\\\$.", null, "Figure 3. Some rotational inertias.\n\nEntering known values in this equation yields,\n\n$I'=\\left(0.0500\\text{ kg}+0.667\\text{ kg}\\right){\\left(1.20\\text{ m}\\right)}^{2}=1.032\\text{ kg}\\cdot{\\text{m}}^{2}\\\\$.\n\nThe value of I′ is now entered into the expression for ω′, which yields\n\n$\\begin{array}{lll}\\omega'&=& \\frac{mvr}{I'}=\\frac{\\left(0.0500\\text{ kg}\\right)\\left(30.0\\text{ m/s}\\right)\\left(1.20\\text{ m}\\right)}{1.032\\text{ kg}\\cdot\\text{m}^{2}}\\\\ & =& 1.744\\text{ rad/s}\\approx 1.74\\text{ rad/s}\\end{array}\\\\$.\n\n#### Strategy for (b)\n\nThe kinetic energy before the collision is the incoming disk’s translational kinetic energy, and after the collision, it is the rotational kinetic energy of the two stuck together.\n\n#### Solution for (b)\n\nFirst, we calculate the translational kinetic energy by entering given values for the mass and speed of the incoming disk.\n\n$\\text{KE}=\\frac{1}{2}{mv}^{2}=\\left(0.500\\right)\\left(0.0500\\text{ kg}\\right){\\left(30.0\\text{ m/s}\\right)}^{2}=22.5 \\text{ J}\\\\$\n\nAfter the collision, the rotational kinetic energy can be found because we now know the final angular velocity and the final moment of inertia. Thus, entering the values into the rotational kinetic energy equation gives\n\n$\\begin{array}{lll}\\text{KE'}& =& \\frac{1}{2}I'{\\omega'^{2}}=\\left(0.5\\right)\\left(1.032\\text{kg}\\cdot\\text{m}^{2}\\right)\\left(1.744\\frac{\\text{rad}}{\\text{s}}\\right)^{2}\\\\ & =& 1.57\\text{ J}\\end{array}\\\\$.\n\n#### Strategy for (c)\n\nThe linear momentum before the collision is that of the disk. After the collision, it is the sum of the disk’s momentum and that of the center of mass of the stick.\n\n#### Solution of (c)\n\nBefore the collision, then, linear momentum is\n\n$p=\\text{mv}=\\left(0\\text{.}\\text{0500}\\text{kg}\\right)\\left(\\text{30}\\text{.}0\\text{m/s}\\right)=1\\text{.}\\text{50}\\text{kg}\\cdot \\text{m/s}\\\\$.\n\nAfter the collision, the disk and the stick’s center of mass move in the same direction. The total linear momentum is that of the disk moving at a new velocity v′ = ′ plus that of the stick’s center of mass, which moves at half this speed because ${v}_{\\text{CM}}=\\left(\\frac{r}{2}\\right)\\omega' =\\frac{v'}{2}\\\\$\n\nThus,\n\n$p' ={mv}' +{{Mv}}_{\\text{CM}}=\\text{mv}' +\\frac{{Mv}'}{2}\\\\$.\n\nGathering similar terms in the equation yields,\n\n$p'=\\left(m+\\frac{M}{2}\\right)v'\\\\$\n\nso that\n\n$p' =\\left(m+\\frac{M}{2}\\right)r\\omega'\\\\$.\n\nSubstituting known values into the equation,\n\n$p' =\\left(1\\text{.}\\text{050 kg}\\right)\\left(1\\text{.}\\text{20}\\text{m}\\right)\\left(1\\text{.}\\text{744 rad/s}\\right)=\\text{2.20 kg}\\cdot \\text{m/s}\\\\$.\n\n#### Discussion\n\nFirst note that the kinetic energy is less after the collision, as predicted, because the collision is inelastic. More surprising is that the momentum after the collision is actually greater than before the collision. This result can be understood if you consider how the nail affects the stick and vice versa. Apparently, the stick pushes backward on the nail when first struck by the disk. The nail’s reaction (consistent with Newton’s third law) is to push forward on the stick, imparting momentum to it in the same direction in which the disk was initially moving, thereby increasing the momentum of the system.\n\nThe above example has other implications. For example, what would happen if the disk hit very close to the nail? Obviously, a force would be exerted on the nail in the forward direction. So, when the stick is struck at the end farthest from the nail, a backward force is exerted on the nail, and when it is hit at the end nearest the nail, a forward force is exerted on the nail. Thus, striking it at a certain point in between produces no force on the nail. This intermediate point is known as the percussion point. An analogous situation occurs in tennis as seen in Figure 4. If you hit a ball with the end of your racquet, the handle is pulled away from your hand. If you hit a ball much farther down, for example, on the shaft of the racquet, the handle is pushed into your palm. And if you hit the ball at the racquet’s percussion point (what some people call the “sweet spot”), then little or no force is exerted on your hand, and there is less vibration, reducing chances of a tennis elbow. The same effect occurs for a baseball bat.", null, "Figure 4. A disk hitting a stick is compared to a tennis ball being hit by a racquet. (a) When the ball strikes the racquet near the end, a backward force is exerted on the hand. (b) When the racquet is struck much farther down, a forward force is exerted on the hand. (c) When the racquet is struck at the percussion point, no force is delivered to the hand.\n\n### Check Your Understanding\n\nIs rotational kinetic energy a vector? Justify your answer.\n\n#### Solution\n\nNo, energy is always scalar whether motion is involved or not. No form of energy has a direction in space and you can see that rotational kinetic energy does not depend on the direction of motion just as linear kinetic energy is independent of the direction of motion.\n\nSection Summary\n\n• Angular momentum L is analogous to linear momentum and is given by $L=I\\omega\\\\$ .\n• Angular momentum is changed by torque, following the relationship $\\text{net }\\tau =\\frac{\\Delta L}{\\Delta t}\\\\$.\n• Angular momentum is conserved if the net torque is zero $L=\\text{constant}\\left(\\text{net }\\tau =\\text{0}\\right)\\\\$ or $L=L′\\left(\\text{net }\\tau =0\\right)\\\\$. This equation is known as the law of conservation of angular momentum, which may be conserved in collisions.\n\n### Conceptual Questions\n\n1. Describe two different collisions—one in which angular momentum is conserved, and the other in which it is not. Which condition determines whether or not angular momentum is conserved in a collision?\n\n2. Suppose an ice hockey puck strikes a hockey stick that lies flat on the ice and is free to move in any direction. Which quantities are likely to be conserved: angular momentum, linear momentum, or kinetic energy (assuming the puck and stick are very resilient)?\n\n3. While driving his motorcycle at highway speed, a physics student notices that pulling back lightly on the right handlebar tips the cycle to the left and produces a left turn. Explain why this happens.\n\n### Problems & Exercises\n\n1. Repeat Example 1. Rotation in a Collision in which the disk strikes and adheres to the stick 0.100 m from the nail.\n\n2. Repeat Example 1. Rotation in a Collision in which the disk originally spins clockwise at 1000 rpm and has a radius of 1.50 cm.\n\n3. Twin skaters approach one another as shown in Figure 5 and lock hands. (a) Calculate their final angular velocity, given each had an initial speed of 2.50 m/s relative to the ice. Each has a mass of 70.0 kg, and each has a center of mass located 0.800 m from their locked hands. You may approximate their moments of inertia to be that of point masses at this radius. (b) Compare the initial kinetic energy and final kinetic energy.", null, "Figure 5. Twin skaters approach each other with identical speeds. Then, the skaters lock hands and spin.\n\n4. Suppose a 0.250-kg ball is thrown at 15.0 m/s to a motionless person standing on ice who catches it with an outstretched arm as shown in Figure 6.\n\n(a) Calculate the final linear velocity of the person, given his mass is 70.0 kg.\n\n(b) What is his angular velocity if each arm is 5.00 kg? You may treat the ball as a point mass and treat the person’s arms as uniform rods (each has a length of 0.900 m) and the rest of his body as a uniform cylinder of radius 0.180 m. Neglect the effect of the ball on his center of mass so that his center of mass remains in his geometrical center.\n\n(c) Compare the initial and final total kinetic energies.", null, "Figure 6. The figure shows the overhead view of a person standing motionless on ice about to catch a ball. Both arms are outstretched. After catching the ball, the skater recoils and rotates.\n\n5. Repeat Example 1. Rotation in a Collision in which the stick is free to have translational motion as well as rotational motion.\n\n### Selected Solutions to Problems & Answers\n\n1. (a) 0.156 rad/s (b) 1.17 × 10-2 J (c) 0.188 kg ⋅ m/s\n\n3. (a) 3.13 rad/s (b) Initial KE = 438 J, final KE = 438 J\n\n5. (a) 1.70 rad/s (b) Initial KE = 22.5 J, final KE = 2.04 J (c) 1.50 kg ⋅ m/s" ]	[ null, "https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/222/2014/12/20103601/Figure_11_06_01a.jpg", null, "https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/222/2014/12/20103602/Figure_11_06_02a.jpg", null, "https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/648/2016/11/04025642/Figure_11_03_06-1.jpg", null, "https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/222/2014/12/20103612/Figure_11_06_03.jpg", null, "https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/1322/2015/12/03210407/Figure_11_06_04a.jpg", null, "https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/1322/2015/12/03210408/Figure_11_06_05a.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.877886,"math_prob":0.98839325,"size":10505,"snap":"2022-40-2023-06","text_gpt3_token_len":2752,"char_repetition_ratio":0.1597943,"word_repetition_ratio":0.026829269,"special_character_ratio":0.26539743,"punctuation_ratio":0.098794065,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99800855,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12],"im_url_duplicate_count":[null,7,null,7,null,null,null,7,null,7,null,7,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-10-07T15:13:45Z\",\"WARC-Record-ID\":\"<urn:uuid:ea9d8e74-f1cc-463a-a4e8-e56dfd8047a8>\",\"Content-Length\":\"45569\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:76d5a15c-15ad-4379-b26e-b5970c518484>\",\"WARC-Concurrent-To\":\"<urn:uuid:8d93ca5e-06c0-4aea-92ab-35a3684af278>\",\"WARC-IP-Address\":\"23.185.0.1\",\"WARC-Target-URI\":\"https://courses.lumenlearning.com/atd-austincc-physics1/chapter/10-6-collisions-of-extended-bodies-in-two-dimensions/\",\"WARC-Payload-Digest\":\"sha1:X2BTXXE6FPR65O4ZHNJVRIPDHSWVTRW3\",\"WARC-Block-Digest\":\"sha1:HELDDF35KS4H4QNB6CCCBVTQ2ED443JC\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030338213.55_warc_CC-MAIN-20221007143842-20221007173842-00602.warc.gz\"}"}
https://www.hackmath.net/en/calculator/linear-regression	[ "# Linear regression calculator\n\nThis linear regression calculator uses the least squares method to find the line of best fit for a set of paired data. The line of best fit is described by the equation f(x) = Ax + B, where A is the slope of the line and B is the y-axis intercept.\nAll you need is enter paired data into the text box, each pair of x and y each line (row).\n\nAlso calculate coefficient of correlation Pearson product-moment correlation coefficient (PPMCC or PCC or R). The Pearson correlation coefficient is used to measure the strength of a linear association between two variables, where the value R = 1 means a perfect positive correlation and the value R = -1 means a perfect negataive correlation." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8897447,"math_prob":0.9951277,"size":2461,"snap":"2023-14-2023-23","text_gpt3_token_len":687,"char_repetition_ratio":0.1029711,"word_repetition_ratio":0.0,"special_character_ratio":0.3043478,"punctuation_ratio":0.14901257,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99985147,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-03-27T22:29:10Z\",\"WARC-Record-ID\":\"<urn:uuid:141b1be6-e400-43fd-8b46-51d5a1eaf02f>\",\"Content-Length\":\"11712\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2bf2026f-9d81-48bb-9bb1-db4beb527271>\",\"WARC-Concurrent-To\":\"<urn:uuid:ebdbc59d-1fdd-4bd1-bddc-616ca42acabf>\",\"WARC-IP-Address\":\"172.67.134.123\",\"WARC-Target-URI\":\"https://www.hackmath.net/en/calculator/linear-regression\",\"WARC-Payload-Digest\":\"sha1:RUEJDQFHTJEAULA2H73GEGQU4RAGA6AO\",\"WARC-Block-Digest\":\"sha1:GZ7JWZK6QYOXIA53UERX4JHVKKNYIOQY\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-14/CC-MAIN-2023-14_segments_1679296948708.2_warc_CC-MAIN-20230327220742-20230328010742-00146.warc.gz\"}"}
https://math.stackexchange.com/questions/300803/what-theorems-examples-will-make-me-really-understand-representation-theory	[ "# What theorems/examples will make me really understand representation theory?\n\nOkay, so I've been through some basic results on representation theory. I've gone over the proof of Burnside's $pq$ theorem using characters. I've also read though the basics of Lie groups and algebras. However, I still haven't come across a theorem or any examples which set off an \"Aha!\" moment in which I understand what representations really are and when their use would be appropriate.\n\nFor example, when studying group actions, in my opinion the orbit-stabilizer theorem gives me a good idea of what's going on when we study actions on finite groups - I haven't found any such analogue in representation theory.\n\nI suppose part of the problem is that representations are useful in so many distinct ways (finite groups, Lie groups, harmonic analysis, combinatorics, etc.) that I have a hard time synthesizing a coherent picture. Anyone have any good recommendations for books/topic/theorem/examples?\n\nLet $G$ be a finite solvable group and let $K/L$ be a chief factor of $G$ (this means that $K$ is a normal subgroup of $G$ and $L$ is a subgroup of $K$ which is maximal among proper subgroups of $K$ which are normal in $G$). Since this corresponds to $K$ being a minimal normal subgroup of the solvable group $G/L$ we see that $K/L$ is elementary abelian, so it is isomorphic to $(\\mathbb{F}_p)^n$ for some prime $p$ and some natural number $n$, so it is a vector space over the finite field $\\mathbb{F}_p$.\nNow $G$ acts on $K/L$ via conjugation, and due to the maximality of $L$, this gives us an irreducible representation of $G$ on the vector space $(\\mathbb{F}_p)^n$.\nOne question is then how this relates to those representations one is usually introduced to, which are over the complex numbers. But it turns out that since $G$ is solvable, we can from the existence of the above representation deduce that there is an irreducible complex character of $G$ of degree at most $n$ and such that a $p$-regular element of $G$ (ie, an element of $G$ whose order is not divisible by $p$) is in the kernel of this irreducible character iff it acts trivially on $K/L$. Further, if the representation from above is faithful (which means that the only $g\\in G$ that acts trivially on $K/L$ is the neutral element), then there exists a faithful irreducible complex character of $G$ of degree at most $n$." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.95756906,"math_prob":0.99755937,"size":903,"snap":"2021-31-2021-39","text_gpt3_token_len":186,"char_repetition_ratio":0.12458287,"word_repetition_ratio":0.0,"special_character_ratio":0.19712071,"punctuation_ratio":0.103658535,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99977344,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-07-24T15:01:03Z\",\"WARC-Record-ID\":\"<urn:uuid:b0c5c9e6-e829-4bbd-a7a4-9da82c846179>\",\"Content-Length\":\"163309\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:27d6892f-74a7-4afb-82a9-4ac677f1ff8a>\",\"WARC-Concurrent-To\":\"<urn:uuid:3315376d-e9d8-49a3-9fcd-eca4b0528a4f>\",\"WARC-IP-Address\":\"151.101.193.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/300803/what-theorems-examples-will-make-me-really-understand-representation-theory\",\"WARC-Payload-Digest\":\"sha1:BYWQUUCXAPDGPXCU445MKPZBSRORUCN4\",\"WARC-Block-Digest\":\"sha1:N5MDAEHZ2T4OCHXE7R2L5CBI7LIWYVEJ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-31/CC-MAIN-2021-31_segments_1627046150266.65_warc_CC-MAIN-20210724125655-20210724155655-00208.warc.gz\"}"}
https://www.sportsbookreview.com/picks/tools/sports-betting-functions-for-excel/	[ "# Sports Betting Functions for Excel\n\n• SBRVer(): Displays current template version number.\n• US2Dec(USOdds) (usage 1): Converts US-style to decimal. Example: US2Dec(-110) ≈ 1.909090909\n• US2Dec(range of USOdds) (usage 2): Converts an array or Excel range of US-style odds to decimal parlay odds. Example: US2Dec(-110,-110) ≈ 3.644628099\n• US2Par(range of USOdds): Converts an array or Excel range of US-style odds to US-style true parlay odds. Example: US2Par(-110,-110,-110) ≈ +595.7926.\n• Dec2US(DecimalOdds): Converts decimal odds to US. Example: US2Dec(1.909090909) = -110.\n• US2Win(USOdds, WagerQuantity {{default = 1}) or Dec2Win(DecOdds, WagerQuantity {{default = 1}): Converts US or Decimal odds and wager size to potential win quantity. Note that by using the default value of 1 for wager size, these two functions effectively convert from US/decimal odds to fractional odds. Example, US2Win(-120,120) = 100, or US2Win(-110) ≈ 0.90909.\n• US2Res(USOdds, WagerQuantity {{default = 1}, Result) or Dec2Win(DecOdds, WagerQuantity {{default = 1}, Result): Converts US or Decimal odds, wager size, and result (where \"WIN\", \"W\", or \"1\", corresponds to a win; \"LOSS\", \"L\", or -1 corresponds to a loss; and \"PUSH\", \"P\", or 0 corresponds to a push). Example, US2Res(-120,120,\"P\") = 0, or =US2Res(-110, 200,\"Win\") ≈ 181.82.\n• US2Prob(USOdds) or Dec2Prob(DecimalOdds): Converts from US or decimal odds to probability. Example: US2Prob(+100) = Dec2Prob(2.0000) = 50%.\n• US2Hold(range of US Odds) or Dec2Hold(range of Decimal Odds): Calculates theoretical hold based on an Excel range of US or decimal odds. Example: if cells A1 and A2 are both -110, US2Hold(A1:A2) = 4.54545%.\n• {US2Real(range of US Odds)} or {Dec2Real(range of Decimal Odds)}: (array function) Returns an array of zero-vig probabilities based on an Excel range of US or decimal odds. Example: if cells A1 and A2 are both -110, if B1, B2, and B3 were set to the array formula {=US2Real(A1:A2)}, B1 and B2 would both have the value of 50%, and B3 would have the value of the theoretical hold (4.54545%).\n• {US2Fair(range of US Odds)} or {Dec2Fair(range of Decimal Odds)}: (array function) Returns an array of fair value zero-vig odds based on an Excel range of US or decimal odds. Example: if cells A1 and A2 are -200 and +176, respectively, and if B1 and B2 were set to the array formula {=US2Fair(A1:A2)}, B1 and B2 would display the values of -184 and +184, respectively.\n• ProbUS2Edge(Probability, USOdds) or ProbDec2Edge(Probability, DecimalOdds): Calculates edge based on win probability and US or decimal odds. Example: ProbUS2Edge(55%,-110) = 5%\n• EdgeUS2Prob(Edge, USOdds) or EdgeDec2Prob(Edge, DecimalOdds): Calculates win probability based on edge and US or decimal odds. Example: ProbUS2Edge(5%,-110) = 55%.\n• ProbEdge2US(Probability, Edge) or ProbEdge2Dec(Probability, Edge): Calculates US or decimal odds based on probability and edge. Example: ProbEdge2US(55%,5%) = -110.\n• USRisk2Win(USOdds, RiskQuantity {default=1}) or DecRisk2Win(DecimalOdds, RiskQuantity {default=1}): Calculates resultant win quantities given US/Decimal odds and risk quantity. (These functions are also aliased as USR2W(·) and DecR2W(·), respectively). Example: USRisk2Win(-110,22) = USR2W(-110,22) = \\$20.\n• USWin2Risk(USOdds, WinQuantity{default=1}) or DecWin2Risk(DecimalOdds, WinQuantity {default=1}): Calculates required risk given US/Decimal odds and desired risk amount. (These functions are also aliased as USW2R(·) and DecW2R(·), respectively). Example: USWin2Risk(-110,20) = USW2R(-110,20) = \\$22.\n• Exch2US(US Exchange Odds, Commission {default = 2%}) or Exch2Dec(Decimal Exchange Odds, Commission {default = 2%}): Calculate sportsbook equivalent US or decimal odds given US or decimal exchange odds and commission. Example: Exch2US(-110, 1%) would refer to the sportsbook equivalent odds of betting at -110 given 1% sportsbook commission (~-111.11).\n• E2S(US exchange odds, exchange commission {default = 2%}): Shortcut to Exch2US(US Exchange Odds, Commission).\n• ExchUS2Hold(range of US Odds, Commission) or ExchDec2Hold(range of Decimal Odds, Commission): Calculates theoretical hold including sports betting exchange commissions based on an Excel range of US or decimal odds. Example: if the values of cells A1 and A2 both equal -102 ExchUS2Holds(A1:A2,2%) would equal the theoretical hold theoretical on the -102/-012 market inclusive of 2% exchange commission (a value of 1.961%).\n• KUtil(bankroll, Kelly multiplier {default = 1}): Calculates Kelly criterion utility for a given bankroll (expressed in percent terms) and Kelly multiplier. Example: KUtil(1.05, 0.5) would yield half-Kelly utility for a bankroll of 105% of initial.\n• InvKUtil(utilily, Kelly multiplier {default = 1}): The inverse Kelly Utility function. Calculates the bankroll (expressed in percent terms) implied by a given Kelly criterion utility and Kelly multiplier. Example: InvKutil(KUtil(X, KellyMult),KellyMult) would just equal X (provided X > 0).\n• SBKelly(Probability, Odds, Kelly Multiplier {default = 1}, Decimal Odds Flag {default = FALSE}): Calculates single bet Kelly stake given an expected win probability, paypout odds, and optional Kelly Multiplier. If the \"Decimal Odds Flag\" isn't set or is set to FALSE, then the function will use a \"best guess\" as to whether the odds specified are US or decimal-style (if absolute value<100, it will assume decimal). Setting the flag to TRUE will cause the function to always assume decimal-style odds (this could be helpful when using decimal-style odds at very high payout levels).\n• ProbUS2Edge(Probability, USOdds) or ProbDec2Edge(Probability, DecimalOdds): Calculates edge based on win probability and US or decimal odds. Example: ProbUS2Edge(55%,-110) = 5%\n• EdgeUS2Prob(Edge, USOdds) or EdgeDec2Prob(Edge, DecimalOdds): Calculates win probability based on edge and US or decimal odds. Example: ProbUS2Edge(5%,-110) = 55%.\n• ProbEdge2US(Probability, Edge) or ProbEdge2Dec(Probability, Edge): Calculates US or decimal odds based on probability and edge. Example: ProbEdge2US(55%,5%) = -110.\n• USRisk2Win(USOdds, RiskQuantity {default=1}) or DecRisk2Win(DecimalOdds, RiskQuantity {default=1}): Calculates resultant win quantities given US/Decimal odds and risk quantity. (These functions are also aliased as USR2W(·) and DecR2W(·), respectively). Example: USRisk2Win(-110,22) = USR2W(-110,22) = \\$20.\n• USWin2Risk(USOdds, WinQuantity{default=1}) or DecWin2Risk(DecimalOdds, WinQuantity {default=1}): Calculates required risk given US/Decimal odds and desired risk amount. (These functions are also aliased as USW2R(·) and DecW2R(·), respectively). Example: USWin2Risk(-110,20) = USW2R(-110,20) = \\$22.\n• Exch2US(US Exchange Odds, Commission {default = 2%}) or Exch2Dec(Decimal Exchange Odds, Commission {default = 2%}): Calculate sportsbook equivalent US or decimal odds given US or decimal exchange odds and commission. Example: Exch2US(-110, 1%) would refer to the sportsbook equivalent odds of betting at -110 given 1% sportsbook commission (~-111.11).\n• E2S(US exchange odds, exchange commission {default = 2%}): Shortcut to Exch2US(US Exchange Odds, Commission).\n• ExchUS2Hold(range of US Odds, Commission) or ExchDec2Hold(range of Decimal Odds, Commission): Calculates theoretical hold including sports betting exchange commissions based on an Excel range of US or decimal odds. Example: if the values of cells A1 and A2 both equal -102 ExchUS2Holds(A1:A2,2%) would equal the theoretical hold theoretical on the -102/-012 market inclusive of 2% exchange commission (a value of 1.961%).\n• KUtil(bankroll, Kelly multiplier {default = 1}): Calculates Kelly criterion utility for a given bankroll (expressed in percent terms) and Kelly multiplier. Example: KUtil(1.05, 0.5) would yield half-Kelly utility for a bankroll of 105% of initial.\n• InvKUtil(utilily, Kelly multiplier {default = 1}): The inverse Kelly Utility function. Calculates the bankroll (expressed in percent terms) implied by a given Kelly criterion utility and Kelly multiplier. Example: InvKutil(KUtil(X, KellyMult),KellyMult) would just equal X (provided X > 0).\n• SBKelly(Probability, Odds, Kelly Multiplier {default = 1}, Decimal Odds Flag {default = FALSE}): Calculates single bet Kelly stake given an expected win probability, paypout odds, and optional Kelly Multiplier. If the \"Decimal Odds Flag\" isn't set or is set to FALSE, then the function will use a \"best guess\" as to whether the odds specified are US or decimal-style (if absolute value<100, it will assume decimal). Setting the flag to TRUE will cause the function to always assume decimal-style odds (this could be helpful when using decimal-style odds at very high payout levels).\n• {P2L(range of win probabilities)}: (array function) Returns an array of likelihoods such that the ith element of the output array (i ∈ [0, 1, 2, ..., n], where n = the number of probabilities in the input range) corresponds to to the likelihood of exactly n-i wins and i losses given the n event win probabilities in the input range. Example: if cells (A1, A2, A3) = (75%, 70%, 65%), and cells B1, B2, B3, and B4 were set to the array formula {=P2L(A1:A3)}, cell B1 would correspond to the probability of 3 wins and 0 losses (~ 34.13%), cell B the probability of 2 wins and 1 loss (~44.38%), B3 the probability of 1 win and 2 losses (~18.88%), and B4 the probability of 0 wins and 3 losses (2.63%). Note that this function may be rather slow to calculate for large input sets.\n• {EnumCombin(range of items, size)}: (array function) Returns a 2-D array of every possible combination of of the specified size of the input range. Example: if cells (A1, A2, A3, A4, A5) = (\"A\", \"B\", \"C\", \"D\", \"E\"), then {=ENUMCOMBIN(A1:A5, 2)} would return the 6-row, 2-column array of {(\"A\",\"B\"),(\"A\",\"C\"),(\"A\",\"D\"),(\"B\",\"C\"),(\"B\",\"D\"),(\"C\",\"D\")}. Note that this function may be rather slow to calculate for large input sets. EnumComin is short for \"Enumerate Combinations\".\n• lg(p): Calculates the logit function of probability p, where lg(p) is defined as ln(p) - ln(1-p) ∀ 0<p<1.\n• invlg(x): Calculates the inverse logit function of x where invlg(x) is defined as invlg(x) = Exp(x) / (1 + Exp(x)).\n• MB2US(US Matchbook Exchange Odds, Commission {default=1%}) or MB2DEC(Decimal Matchbook Exchange Odds, Commission {default=1%}): Calculate sportsbook equivalent US or Decimal odds given US or Decimal Matckbook exchange odds and commission. This references a commission structure where the player pays a set percentage of the lesser of risk or win irrespective of bet outcome. Example: MB2US(-110, 1%) would refer to the sportsbook equivalent odds of betting at -110 given 1% Matchbook commission (~-112.12).\n• {Bets2Stats(range of Odds, range of Wager Quantities {default=1}, range of Outcomes, range of Edges {default=0%), Decimal Odds Flag = {default = FALSE})}: Array function. Takes a range of betting odds (US odds are the default, but will take decimal odds if the Decimal Odds Flag argument) is set to TRUE, an optional range of wager quantities (if not provided then 1 unit per wager is assumed), a range of outcomes (1 or a string starting with 'W' for a win, -1 or a string starting with 'L' for a loss, anything else for a push/no action), and a range of expected edges (defaults to 0). Returns an array with the following values:\n1. Number of Non-Pushed Bets\n2. Number of Wins\n3. Win %\n4. Unit Return\n5. % Return\n6. Unit Standard Deviation\n7. % Standard Deviation\n8. Standard Score\n9. p-value (from t-distribution)\n\nIf you want these functions to be available every time you start Excel you'll need to save the Book.xlt template file in your Excel XLStart directory (”C:\\Program Files\\Microsoft Office\\OFFICE11\\XLSTART\\” by default for Excel 2003 -- \\Office12\\XLSTART\\ for Excel2007, \\Office10\\XLSTART\\ for Excel 2002, and \\Office\\XLSTART\\ for Excel 2000 and 97). If the file already exists you shouldn't overwrite it unless you know the preexisting file to be empty but instead choose a different name as in the next paragraph. Alternatively, if you know what you're doing, you could manually add the Excel VBA functions or module to your preexisting Book.xlt file.\n\nIf you want these functions to only be available only by request then save the file under a different name in the XLStart directory. For example, if you saved the file as Ganchrow.xlt, then by clicking ”New” on the ”File” menu, you'd be able to select the template ”Ganchrow” and have all the above functions available." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7497906,"math_prob":0.98265505,"size":12482,"snap":"2019-35-2019-39","text_gpt3_token_len":3432,"char_repetition_ratio":0.15883955,"word_repetition_ratio":0.5201939,"special_character_ratio":0.28673288,"punctuation_ratio":0.15760408,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9981993,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-08-18T12:55:18Z\",\"WARC-Record-ID\":\"<urn:uuid:7d281e03-54e8-4025-87df-9b2aa51f0833>\",\"Content-Length\":\"166448\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:326680fa-6dfd-4c45-b7ca-31cbb3671a48>\",\"WARC-Concurrent-To\":\"<urn:uuid:1cfd11e9-1818-465a-a6c8-4af9bd9c87be>\",\"WARC-IP-Address\":\"69.172.200.156\",\"WARC-Target-URI\":\"https://www.sportsbookreview.com/picks/tools/sports-betting-functions-for-excel/\",\"WARC-Payload-Digest\":\"sha1:QFKLFJHVOPXPDHDONGXEAA53H4YGQ5QJ\",\"WARC-Block-Digest\":\"sha1:PBS2VQQ3YBWJ6YVUZVP2TLXGSBI3EDPW\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-35/CC-MAIN-2019-35_segments_1566027313889.29_warc_CC-MAIN-20190818124516-20190818150516-00452.warc.gz\"}"}
https://www.sparrho.com/item/antenna-apparatus-utilizing-aperture-of-transmission-line/d2c032/	[ "", null, "# Antenna apparatus utilizing aperture of transmission line\n\nImported: 13 Feb '17 \| Published: 18 Jan '11\n\nKanji Otsuka, Tamotsu Usami, Yutaka Akiyama, Chihiro Ueda\n\nUSPTO - Utility Patents\n\n## Abstract\n\nAn antenna apparatus utilizing an aperture of transmission line, which is connected to a first transmission line having a predetermined characteristic impedance, includes a tapered line portion, and an aperture portion. The tapered line portion is connected to one end of the transmission line, and the tapered line portion includes a second transmission line including a pair of line conductors. The tapered line portion keeps a predetermined characteristic impedance constant and expands at least one of a width of the transmission line and an interval in a tapered shape at a predetermined taper angle. The aperture portion has a radiation aperture connected to one end of the tapered line portion. A size of one side of the aperture end plane of the aperture portion is set to be equal to or higher than a quarter wavelength of the minimum operating frequency of the antenna apparatus.\n\n## Description\n\n### BACKGROUND OF THE INVENTION\n\n1. Field of the Invention\n\nThe present invention relates to an antenna apparatus utilizing an aperture of transmission line, and in particular, to an antenna apparatus which can be used in frequency bands such as bands of microwaves, quasi millimeter waves, millimeter waves, or the like.\n\n2. Description of the Related Art\n\nAn antenna has been always used in wireless communication systems such as portable telephones. The concept of the conventional antenna has such a structure for resonating at a specified frequency, and a typical dipole antenna resonates at a half of an operating wavelength thereof.\n\nIn the dipole antenna, electromagnetic waves having the TM (Transverse Magnetic) mode are generated concentrically around the pole. However, electromagnetic waves that have reached a distance several times the wavelength interfere with one another at the boundary portions thereof, and the electromagnetic wave mode is transformed into TEM (Transverse Electro-Magnetic) mode (the radio waves thereof is called as transverse waves), and is radiated almost in a form of spherical waves. When the radius of curvature is increased, the electromagnetic waves become plane waves. The electromagnetic waves travel as group waves where a lots of electromagnetic waves travel so as to be distributed in a transverse straight line (namely, distributed averagely to a plane perpendicular to the traveling direction) concurrently travel. The documents which are related to the present invention are as follows:\n\n• Patent Document 1: Japanese patent laid-open publication No. JP 2005-244733 A;\n• Non-Patent Document 1: Kanji Otsuka, et al, “Measurement Potential Swing by Electric Field on Package Transmission Lines”, Proceedings of ICEP, pp. 490-495, April 2001;\n• Non-Patent Document 2: Kanji Otsuka, et al, “Measurement Evidence of Mirror Potential Traveling on Transmission Lines”, Technical Digest of 5th VLSI Packaging Workshop of Japan, pp. 27-28, December 2000; and\n• Non-Patent Document 3: Kanji Otsuka et al, “Stacked pair line”, Journal of Japan Institute of Electronics Packaging (JIEP), Vol. 4, No. 7, pp. 556-561, November, 2001.\n\nSince the group waves fill up the space, the group waves need not only frequency allocation by the Radio Law but also a sufficient protection circuit against resonant mode noises leaking from the band, and the high-frequency circuit substantially becomes a circuit having a large overhead. Furthermore, the group waves are heavily attenuated even in the air at high frequencies in the bands equal to or higher than GHz band and become a level, that is larger than such an attenuation theorem at lower frequencies that the energy becomes weak in inverse proportion to the square of the distance (because of expansion in a spherical shape), and that is in inverse proportion to the cube of the distance by approximation, and this leads to that it is difficult to perform long distance communications.\n\n### SUMMARY OF THE INVENTION\n\nAn object of the present invention is to solve the above-mentioned problems and provide an antenna apparatus, that is connected to a transmission line and has a simple configuration and a directivity of almost no change in frequency characteristics, and that is capable of performing communications even at a comparatively long distance.\n\nIn order to achieve the aforementioned objective, according to one aspect of the present invention, there is provided an antenna apparatus utilizing an aperture of transmission line, and the antenna apparatus is connected to a first transmission line having a predetermined characteristic impedance. The antenna apparatus includes a tapered line portion, and an aperture portion. The tapered line portion is connected to one end of the transmission line, and the tapered line portion includes a second transmission line including a pair of line conductors. The tapered line portion keeps a predetermined characteristic impedance constant and expands at least one of a width of the transmission line and an interval in a tapered shape at a predetermined taper angle. The aperture portion has a radiation aperture connected to one end of the tapered line portion. A size of one side of the aperture end plane of the aperture portion is set to be equal to or higher than a quarter wavelength of the minimum operating frequency of the antenna apparatus.\n\nThe antenna apparatus preferably further includes a first support member that short-circuits and supports the second transmission line including the pair of line conductors substantially in a center portion in a width direction of the transmission line of the aperture portion.\n\nIn addition, the antenna apparatus preferably further includes a pair of second support members that short-circuit and support the second transmission line including the pair of line conductors substantially at both ends in a width direction of the transmission line of the aperture portion.\n\nIn the above-mentioned antenna apparatus, the aperture portion is preferably constituted by expanding a width of the transmission line in a tapered shape.\n\nIn the above-mentioned antenna apparatus, a space located between the pair of line conductors of the first transmission line in the tapered line portion is preferably filled with a predetermined dielectric.\n\nIn the above-mentioned antenna apparatus, a space located between the pair of line conductors of the second transmission line in the aperture portion is preferably filled with a predetermined dielectric.\n\nThe above-mentioned antenna apparatus preferably further includes a first support member for supporting both end portions in a width direction of the transmission line of the first transmission line in the tapered line portion with interposition of a predetermined interval.\n\nThe above-mentioned antenna apparatus preferably further includes a second support member for supporting both end portions in the width direction of the transmission line of the second transmission line in the aperture portion with interposition of a predetermined interval.\n\nIn the above-mentioned antenna apparatus, the taper angle is preferably set to a predetermined value which is larger than zero degree and equal to or smaller than 30 degrees.\n\nIn the above-mentioned antenna apparatus, the characteristic impedance is preferably set to a predetermined value that is set within a range from 50Ω to 100 Ω.\n\nAccordingly, the antenna apparatus utilizing the aperture of transmission line according to the present invention is connected to the transmission line, and has a configuration much simpler than that of the prior art and a narrow directivity with almost no change in frequency characteristics, thereby making it possible to achieve a large antenna gain and to perform communications even at a comparatively long distance.\n\n### DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS\n\nPreferred embodiments of the present invention will be described below with reference to the drawings. In each of the following preferred embodiments, like components are denoted by like reference numerals.\n\nThe present invention is derived from a fundamental principle quite different from that of the antenna that resonates at a specified frequency such as a dipole antenna, and has a theoretically novel configuration as described in detail below.\n\nFIG. 1 is a perspective view showing an appearance of the antenna apparatus utilizing the aperture of transmission line according to the first preferred embodiment of the present invention. The antenna apparatus utilizing the aperture of transmission line of the first preferred embodiment is connected to a stacked pair line 1 including a pair of line conductors 1a and 1b each having a predetermined characteristic impedance such as 50Ω, where a pair of line conductors 1a and 1b oppose to each other.\n\nThe antenna apparatus of the present preferred embodiment is characterized by including the following:\n\n(A) a tapered line portion 2, which is connected to one end of the stacked pair line 1 and includes a transmission line including a pair of planar line conductors 2a and 2b opposing to each other, where both (this may be at least one) of a width of the transmission line and an interval between the line conductors 2a and 2b (referred to as an antenna interval hereinafter) are expanded in a tapered shape at predetermined taper angles θ and Φ with a predetermined characteristic impedance maintained or kept to be constant; and\n\n(B) an aperture portion 3 having a radiation aperture connected to one end of the tapered line portion 2, and including a pair of parallel planar line conductors 3a and 3b opposing to each other, and\n\n(C) where the size of one side of the aperture end plane of the aperture portion 3 is set to equal to or higher than the quarter wavelength of the minimum operating frequency.\n\nThe configuration and operation of the antenna apparatus utilizing the aperture of transmission line of the present preferred embodiment will be described below.\n\nThe electromagnetic waves that travel in the structure of the transmission line are confined in the structure in the traveling direction with respect to group waves, and therefore, enter the state of one line of electromagnetic waves limited by the structure. When only an electromagnetic wave having a certain frequency travels, the electromagnetic wave is also aligned in phase. If the electromagnetic wave is radiated to the space, a line of electromagnetic waves (whose frequency is lower than that of light) similar to a beam of laser light is formed. The present invention was derived from this conceptual origin. The frequency of the electromagnetic wave is, of course, lower than that of light, and it is impossible to keep a non-dispersed state unlike a beam of laser light, and the electromagnetic wave finally becomes TEM wave. However, the dispersion is suppressed at a short distance, and the square term of the distance can be ignored. Within this interval, the effect obtained by only the influence of attenuation, i.e., the square of the distance is effected, and the electromagnetic wave reaches a far destination point with weaker energy. If the distance is relatively short, the transmitting direction can be specified if the receiving location and the azimuth are determined in a manner similar to that of the case of a pencil of light, and the radio waves do not leak in azimuths other than those of transmitting and receiving. Therefore, not only applications different from those according to the conventional Radio Law can be considered, but also clean waves can be obtained since the spatial noise level (ground level) is not raised. So to speak, this results in the concept equivalent to providing conductor wiring of an electronic circuit in the air. Further, only TEM waves are in far locations, and able to be handled similar to the conventional antenna, whereas this results in a configuration in which the waves can effectively reach far locations by the distance of the convergent part in near locations.\n\nThe above description has been made on the assumption that the electromagnetic waves in the electromagnetic state of the transmission line are emitted to the space, and it becomes possible by making the transmission line abruptly or suddenly has an aperture portion in a manner similar to that of the stub wiring. This is the phenomenon conventionally known as stub noise radiation. The present invention provides an antenna that positively utilizes the phenomenon and takes its efficiency into consideration.\n\nThe structures shown in FIGS. 11 to 14 can be considered as the transmission line. Although the transmission lines, each including two pairs of lines, are shown in the present cases, the fundamental structure may be a pair of lines. These are, of course, well known in the art. A planar pair line (FIG. 13), a coplanar line (FIG. 14), a stacked pair line (FIG. 11), a split strip line (FIG. 12) and so on can be considered. It has been well known as a common sense that these transmission lines have no frequency characteristic.\n\nThe principle of the present invention is simply shown below. First of all, assuming that an inductance per unit length of the line is L0 [H/m], a capacitance per unit length is C0 [F/m], a resistance per unit length is R0 [Ω/m], and a leakage conductance per unit length is G0 [S/m], then the characteristic impedance Z0 [Ω] of the transmission line having a predetermined length is expressed by the following Equation (1):\n\n$Z 0 = R 0 + jω ⁢ ⁢ L 0 G 0 + jω ⁢ ⁢ C 0 . ( 1 )$\n\nAssuming that the line is enough short and the resistance R0 and the leakage conductance G0 can be ignored, then R0=G0=0, and the Equation (1) is expressed by the following Equation (2):\n\n$Z 0 = jω ⁢ ⁢ L 0 jω ⁢ ⁢ C 0 = L 0 C 0 = L C . ( 2 )$\n\nThe frequency dependence and length dependence are removed, and this is equivalent to the parameter of the total line length. That is, the defined characteristic impedance is identical even if the transmission line is short or extremely long. This is expressed metaphorically as the reciprocal of a conductance corresponding to the cross section area of a water pipe. It can be expressed by the physical concept of the reciprocal of the conductance of every section of the transmission line. Therefore, it can be expressed by a dimensional structure of the section which the electromagnetic waves can pass through. That is, it is expressed by the following Equation (3):\n\n$Z 0 = L 0 C 0 = 1 K ⁢ μ r ⁢ μ 0 ɛ r ⁢ ɛ 0 ⁢ ( d w ) = 377 ⁢ 1 K ⁢ μ r ɛ r ⁢ ( d w ) . ( 3 )$\n\nFor example, the structure and the parameters of the transmission line are shown in FIG. 8 such as the stacked pair line. A fringe coefficient K shown in the following Table 1 is the ratio of an electromagnetic energy (numerator) in a material (this may be, for example, air or a dielectric 1c having a specific inductive capacity ∈r and a relative magnetic permeability μr) having a width “w” of the transmission line and an interval (or aperture size) “d” interposed between a pair of line conductors 1a and 1b of FIG. 8 to the total electromagnetic energy (denominator) including energies distributed outside the same line.\n\nTABLE 1 Fringe Coefficient K w/d εr = 1, μr = 1 εr = 4.5, μr = 1 0.100 14.33 9.30 0.125 12.08 7.90 0.200 8.51 5.68 0.250 7.25 4.86 0.500 4.25 3.14 1.000 2.98 2.17 2.500 1.92 1.50 5.000 1.52 1.27 10.00 1.29 1.14\n\nIf the distribution of the electromagnetic field of the pair line is described a little more, the distribution becomes as shown in FIGS. 9 and 10. When there are positive charges in the top line conductor 1a as shown in FIG. 9, negative charges correspond to them in the bottom line conductor 1b. The electric lines of force are generated so as to make connections from the positive charges to the negative charges and come to have expansion so as to spatially minimize the mutual interference. Although the electric lines of force expand into the infinite space because of innumerable numbers of electric charges existing in the conductor, it is preferable to handle only the space that cannot be ignored by approximation. When the electric charges move in the depth direction of the sheet plane, the magnetic lines of force are generated surrounding the line conductors 1a and 1b and perpendicularly intersect the electric lines of force. Since the positive charges travel in the depth direction of the sheet plane, the magnetic fields are clockwise in the upper half and counterclockwise in the lower half. They help each other like mutually meshing gears at the center. The elements operate as negative mutual inductances M12 and M21 that cancel the self-inductances L1 and L2 of the conductors. In this case, the effective inductance L0 per unit length is expressed by the following Equation (4):\n\n$L 0 = L 1 + L 2 - M 12 - M 21 = ( μ 0 ⁢ μ r K ) ⁢ d w . ( 4 )$\n\nAs the interval between the top and bottom line conductors 1a and 1b is narrowed, the mutual inductances M12 and M21 increase, and the effective inductance L0 per unit length decreases. On the other hand, as the top and bottom line conductors 1a and 1b become closer to each other, the electric lines of force are reduced in length, and the coupling is intensified, consequently increasing the capacitance per unit length. That is, it is expressed by the following Equation (5):\n\n$C 0 = ɛ 0 ⁢ ɛ r ⁢ K ⁢ w d . ( 5 )$\n\nAs a result, as the line conductors 1a and 1b come closer to each other, the characteristic impedance is reduced as indicated by the Equation (3).\n\nFIG. 10 shows a distribution of the electromagnetic lines of force on the stacked pair line when seen in the traveling direction. The signal has the maximum amplitude so that the state of FIG. 9 is established when the right-end plane is regarded as the aperture portion, and the electromagnetic vector is perpendicular to the sheet plane as illustrated. The interval d of the aperture and the quarter wavelength (accurately ¼ of the guide wavelength λg of the transmission line) of the signal frequency are illustrated by same dimensions. The present inventor and others discovered that time corresponding to the time of vector change was the time of passing through the interval “d”, and the electromagnetic radiation could efficiently be achieved with the dimensions. The present inventor further discovered that only required was (¼)λg≦d, and the frequencies having (¼)λg shorter than the interval “d” were all efficiently radiated. In other words, it is the discovery of a directional antenna having no frequency characteristic.\n\nAs well known, in the transmission line, the electromagnetic waves cause energy reflection depending on the degree of change at the change in the characteristic impedance. When electromagnetic waves travel from a port 1 (suffix 1) to a port 2 (suffix 2), the reflectance Γ is expressed by the following Equation (6):\n\n$Γ = Z 02 - Z 01 Z 02 + Z 01 . ( 6 )$\n\nIf the transmission line has an open end, then impedance seen from charges is infinite, and therefore, the reflectance Γ=+1 in the Equation (6), which results in total reflection with no radiation of electromagnetic waves into the air. The reflection occurs when Γ=−1 at a short-circuited end, or the energy is totally consumed by the matching resistance at an impedance-matched end and emitted as thermal energy, which results in a complete failure in producing the effect of the antenna. However, it is presumed that the time space relaxation condition that satisfies the spatial radiation condition is established when (¼)λg≦d as shown in FIG. 10. An open-end transmission line structure that maintains the condition of (¼)λg≦d is the fundamental structure of the antenna apparatus utilizing the aperture of transmission line of the present invention.\n\nThe transmission line propagates a line of electromagnetic waves, which are aligned in phase in the case of a single frequency. Therefore, the radiated electromagnetic waves additionally have such an advantageous effect that they travel in a line of electromagnetic waves that hardly disperse in a manner similar to that of the case of a beam of laser light. Since the transmission line has no frequency characteristic, it becomes possible to radiate even a composite wave like a pulse without changing the composition ratio. Accordingly, an antenna structure is proposed which needs none of generally so-called high-frequency circuits such as the oscillator circuit, the frequency converter circuit in the receiver circuit.\n\nAlthough innumerable numbers of derivative structures can be naturally considered from the fundamental structure, another fundamental structure of the present invention is to concurrently serve as means having a structure for adjusting the dimensions so that the characteristic impedance Z0 of the transmission line in the circuit is uniform up to the aperture portion 3 in order to secure the line conductor interval “d” that satisfies (¼)λg≦d. In order to maintain the characteristic impedance Z0, the width “w” of the transmission line, which is a function of the interval “d”, i.e., the Equation (3), is automatically determined. When using a method for making constant the sectional parameters of the width “w” of the transmission line, the interval “d” and the time “t” by extension and contraction in similitude, a shape as shown in FIG. 1 can be obtained. In order to minimize the turbulence of the electromagnetic fields, taper angles θ and φ (note that θ is the taper angle in the width direction of the transmission line, and φ is the taper angle in the longitudinal direction of the transmission line) should be preferably larger than zero degree and equal to or smaller than 30 degrees. Since the extension and contraction of similarity can freely be achieved, a gigantic antenna and a minute micro antenna are both possible and applied to all the sorts of applications. FIG. 7 shows one example of a structure of leading from a stacked pair line 1 of a width “w1” of the transmission line including a pair of line conductors 1a and 1b surrounded by a dielectric 10 to a stacked pair line (including a pair of line conductors 1c and 1d) of a width “w2 (>w1)” of the transmission line in the air.\n\nAnother important thing is that the electromagnetic waves traveling in the transmission line are in the TEM mode as understood from FIGS. 7 and 9 to 10, and then, it is necessary to provide means for accurately keeping the same state. One example is a method for embedding the whole transmission line configuration in the dielectric, and FIG. 7 shows a conceptual illustration. An electromagnetic wave velocity c is expressed by the following Equation (7):\n\n$c = c 0 μ r ⁢ ɛ r . ( 7 )$\n\nIn the Equation (7), μr is the relative magnetic permeability of the dielectric, and ∈r is the specific inductive capacity of the dielectric. If a portion of a changed or different relative magnetic permeability or/and specific inductive capacity is formed in the transmission line, i.e., within the range in the cross section of traveling of the distribution of the effective electromagnetic lines of force in FIGS. 9 and 10, the electromagnetic lines of force in the portion are advanced or delayed, and this leads to collapse of the TEM mode. This is called the pseudo TEM mode, in which the spatial radiation efficiency is degraded by this coefficient as a result of time dispersion. It is desirable to completely enclose the transmission line with an insulator as shown in FIG. 7. For practical dimensional specifications, it is preferable to additionally expand the dielectric 10 by the width “w” of the transmission line on both sides in the plane and to expand the vertical dimension by the line length “d” in the profile.\n\nFIG. 2 is a perspective view showing an appearance of an antenna apparatus utilizing an aperture of transmission line according to the second preferred embodiment of the present invention. Referring to FIG. 2, the antenna apparatus utilizing the aperture of transmission line of the second preferred embodiment is characterized in that support members 4a and 4b made of a metal or a dielectric for short-circuiting and supporting a pair of line conductors of the transmission line at both ends in the width direction of the transmission line of the aperture portion 3 are further provided as compared with that of the first preferred embodiment.\n\nFIG. 3 is a perspective view showing an appearance of an antenna apparatus utilizing an aperture of transmission line according to the third preferred embodiment of the present invention. Referring to FIG. 3, the antenna apparatus utilizing the aperture of transmission line of the third preferred embodiment is characterized in that a support member 4c made of a metal or a dielectric for short-circuiting and supporting a pair of line conductors 3a and 3b of the transmission line substantially in a center portion in the width direction of the transmission line of the aperture portion 3 is further provided as compared with that of the first preferred embodiment.\n\nFIG. 4 is a perspective view showing an appearance of an antenna apparatus utilizing an aperture of transmission line according to the fourth preferred embodiment of the present invention. The antenna apparatus utilizing the aperture of transmission line of the fourth preferred embodiment is characterized in that a pair of parallel planar line conductors 5a and 5b of the aperture portion 3 have its width of the transmission line expanded in a tapered shape as compared with that of the first preferred embodiment.\n\nFIG. 5 is a perspective view showing an appearance of an antenna apparatus utilizing an aperture of transmission line according to the fifth preferred embodiment of the present invention. Referring to FIG. 5, the antenna apparatus utilizing the aperture of transmission line of the fifth preferred embodiment is characterized in that support members 4a and 4b made of a metal or a dielectric for short-circuiting and supporting a pair of line conductors 5a and 5b of the transmission line at both ends in the width direction of the transmission line of the aperture portion 3 are further provided as compared with that of the fourth preferred embodiment.\n\nFIG. 6 is a perspective view showing an appearance of an antenna apparatus utilizing an aperture of transmission line according to the sixth preferred embodiment of the present invention. Referring to FIG. 6, the antenna apparatus utilizing the aperture of transmission line of the sixth preferred embodiment is characterized in that a support member 4c made of a metal or a dielectric for short-circuiting and supporting a pair of line conductors 5a and 5b of the transmission line substantially in a center portion in the width direction of the transmission line of the aperture portion 3 is further provided as compared with that of the fourth preferred embodiment.\n\nAlthough the stacked pair line 1 is employed as an input line in each of the above-mentioned preferred embodiments, the present invention is not limited to this, and it is acceptable to connect another unbalanced type cable or transmission line, such as a coaxial cable, via an unbalanced connector.\n\nFurthermore, in each of the above-mentioned preferred embodiments, it is acceptable to fill a space located between a pair of line conductors 3a and 3b or 5a and 5b of the aperture portion 3 with a predetermined dielectric that supports the line conductors. Moreover, it is acceptable to further provide a support member 6 made of a dielectric for supporting both ends in the width direction of the transmission line of the tapered line portion 2 with interposition of a predetermined interval, as illustrated in FIG. 30. Furthermore, it is acceptable to further provide a support member made of a dielectric for supporting both ends in the width direction of the transmission line of the aperture portion 3 with interposition of a predetermined interval.\n\nThe taper angles θ and φ is preferably set to a predetermined value that is larger than zero degree and equal to or smaller than 30 degrees. Moreover, the characteristic impedance Z0 of the stacked pair line 1, the tapered line portion 2 and the aperture portion 3 are preferably set to a predetermined value that is equal to or larger than 50Ω and equal to or smaller than 100 Ω.\n\nAlthough it is preferable to provide the setting of (¼)λg≦d in each of the above-mentioned preferred embodiments, the similar action and advantageous effects can be obtained even with the setting of (¼)λg≦w.\n\n### IMPLEMENTAL EXAMPLES\n\nNext, the simulations and results conducted by the present inventors and others will be described below.\n\nFIG. 15 is a perspective view showing an antenna aperture plane of the antenna apparatus utilizing the aperture of transmission line used in the simulations of the present preferred embodiment. FIG. 16 is a perspective view showing an electric field [V/m] of a port on the antenna aperture plane of the antenna apparatus utilizing the aperture of transmission line used in the simulations of the present preferred embodiment. In addition, FIG. 17 is a graph showing a frequency characteristic (the frequency ranging from 0 to 10 GHz) of the reflection coefficient S11 [dB] of the antenna apparatus utilizing the aperture of transmission line used in the simulations of the present preferred embodiment. Further, FIG. 18 is a Smith chart showing an impedance characteristic at the input terminal of the antenna apparatus utilizing the aperture of transmission line used in the simulations of the present preferred embodiment.\n\nFIGS. 15 to 18 show the reflection and the impedance characteristic of the antenna apparatus utilizing the aperture of transmission line of an aperture plane of 1 m×1 m, in which the characteristics toward the space when the aperture portion 3 is used as a port are shown. As is apparent from FIG. 16, it can be understood that there are TEM waves with uniform field intensities (at the waveguide port) throughout the entire aperture plane. It is preferably better for the reflection energy (reflection coefficient S11 when indicated by the S parameter) to be smaller when directed toward the space. As is apparent from FIG. 17, (¼)λ=1000 mm and λ=75 MHz result because w=d=1 m. At the frequency, the reflection coefficient S11=−23 dB, which is very small, and the level equal to or smaller than −30 dB is maintained at higher frequency bands. We have never seen such any antenna having almost no frequency characteristic and emitting electromagnetic radiation so efficiently as described above. Moreover, as is apparent from the Smith chart of FIG. 18, although the aperture portion 3 has a characteristic impedance of 194Ω (o in FIG. 18), it can be understood that the characteristic impedance is 376Ω (● in FIG. 18) so as to be impedance-matched with that of the spatial electromagnetic impedance at 10 GHz due to the electromagnetic resonance (imaginary part) of reflection.\n\nFIG. 19A is a graph showing a directional pattern at 1 GHz of the antenna apparatus utilizing the aperture of transmission line used in the simulations of the present preferred embodiment. FIG. 19A shows the directional pattern having directivity gains in a range from 21.6 to −18.4 dBi, and the antenna apparatus has the maximum directivity gain of about 22 dBi.\n\nFIG. 19B is a graph showing a directional pattern at 2.5 GHz of the antenna apparatus utilizing the aperture of transmission line used in the simulations of the present preferred embodiment. FIG. 19B shows the directional pattern having directivity gains in a range from 29.5 to −10.5 dBi, and the antenna apparatus has the maximum directivity gain of about 30 dBi.\n\nFIG. 19C is a graph showing a directional pattern at 5 GHz of the antenna apparatus utilizing the aperture of transmission line used in the simulations of the present preferred embodiment. FIG. 19C shows the directional pattern having directivity gains in a range from 35.5 to −4.45 dBi, and the antenna apparatus has the maximum directivity gain of about 36 dBi.\n\nFIG. 19D is a graph showing a directional pattern at 7.5 GHz of the antenna apparatus utilizing the aperture of transmission line used in the simulations of the present preferred embodiment. FIG. 19D shows the directional pattern having directivity gains in a range from 39.0 to −0.977 dBi, and the antenna apparatus has the maximum directivity gain of about 39 dBi.\n\nFIG. 19E is a graph showing a directional pattern at 10 GHz of the antenna apparatus utilizing the aperture of transmission line used in the simulations of the present preferred embodiment. FIG. 19E shows the directional pattern having directivity gains in a range from 41.5 to 1.48 dBi, and the antenna apparatus has the maximum directivity gain of about 41 dBi.\n\nAs is apparent from FIGS. 19A to 19E, the gain is about 22 dBi even with the directivity at a comparatively low frequency of 1 GHz, and the antenna apparatus having such an excellent directivity has not been conventionally found out.\n\nFIGS. 20A, 20B and 20C are graphs showing spatial distributions of electromagnetic radiations at operating frequencies of 2, 5 and 10 GHz from the antenna apparatus utilizing the aperture of transmission line having a characteristic impedance of 50Ω used in the simulations of the present preferred embodiment, where the antenna apparatus has the aperture of d=70 mm, w=460 mm, and Zo=50Ω, at (¼)λ=1.11 GHz, and the electric field strength is in a range to about 158 V/m.\n\nFIG. 21A is a graph showing an energy distribution of an electromagnetic radiation field at 2 GHz at a location apart by 200 mm from the aperture plane of the antenna apparatus utilizing the aperture of transmission line used in the simulations of the present preferred embodiment. FIG. 21A shows radiation distributions of electric field strengths in a range to 49.8 V/m.\n\nFIG. 21B is a graph showing an energy distribution of an electromagnetic radiation field at 5 GHz at a location apart by 200 mm from the aperture plane of the antenna apparatus utilizing the aperture of transmission line used in the simulations of the present preferred embodiment. FIG. 21B shows radiation distributions of electric field strengths in a range to 77.7 V/m.\n\nFIG. 21C is a graph showing an energy distribution of an electromagnetic radiation field at 10 GHz at a location apart by 200 mm from the aperture plane of the antenna apparatus utilizing the aperture of transmission line used in the simulations of the present preferred embodiment. FIG. 21C shows radiation distributions of electric field strengths in a range to 87.6 V/m.\n\nFIG. 21D is a graph showing an energy distribution of an electromagnetic radiation field at 10 GHz at a location apart by 400 mm from the aperture plane of the antenna apparatus utilizing the aperture of transmission line used in the simulations of the present preferred embodiment. FIG. 21D shows radiation distributions of electric field strengths in a range to 69.3 V/m.\n\nFIGS. 21A to 21D show the degree of energy concentrations in the space of the antenna apparatus utilizing the aperture of transmission line of d=70 mm and w=460 mm. As is apparent from FIGS. 21A to 21D, although (¼)λ is 1.11 GHz and the reflection is equal to or smaller than −20 dB at 2 GHz, the state of dispersion doubled or more has already occurred at a location 200 mm apart from the aperture plane. However, almost no dispersion occurs in the transverse direction. Although the dispersion is reduced when the frequency is raised, side lobes are observed on the upper and lower sides. Almost no dispersion occurs in the transverse direction also in the case, and it can be anticipated that an azimuth on a map can be sufficiently taken with respect to the ground surface of communications. The aperture areas and the directivity gains are brought together and shown in Table 2.\n\nTABLE 2 Aperture Plane Dimensions and Directivity Gains [dBi] Aperture Sizes [mm] 1 GHz 5 GHz 10 GHz 100 × 100 3.967 18.42 23.58 300 × 300 11.15 26.32 32.13 500 × 500 16.11 30.40 36.23 1000 × 1000 21.72 35.91 41.81\n\nIt can be understood from the results in Table 2 that more excellent antenna characteristics can be obtained with respect to the directivity, as the aperture plane is larger.\n\nFIG. 22 is a graph showing an aperture area and a frequency characteristic of the reflection coefficient S11 on the aperture plane of the antenna apparatus utilizing the aperture of transmission line having a characteristic impedance Z0=100Ω used in the simulations of the present preferred embodiment. FIG. 22 shows the frequency characteristic of reflection when the aperture area is changed with the characteristic impedance Z0 of the antenna apparatus utilizing the aperture of transmission line maintained to be 100 Ω.\n\nFIG. 23A is a waveform chart showing an incident received signal waveform of Gaussian pulses of the antenna apparatus utilizing the aperture of transmission line used in the simulations of the present preferred embodiment, FIG. 23B is a waveform chart showing a received signal waveform of Gaussian pulses of the antenna apparatus with an antenna interval of 10 mm, FIG. 23C is a waveform chart showing a received signal waveform of Gaussian pulses of the antenna apparatus with an antenna interval of 30 mm, and FIG. 23D is a waveform chart showing a received signal waveform of Gaussian pulses of the antenna apparatus with an antenna interval of 60 mm. Namely, FIGS. 23A to 23D shows the receiving characteristic at the time of Gaussian pulse by using an antenna apparatus utilizing an aperture of transmission line of d=32 mm is used for transmitting and receiving. The Gaussian pulse receiving characteristics when a pair of transmission line aperture type antenna apparatuses of d=32 mm and w=80 mm are opposed to each other and used for transmitting and receiving are shown with the antenna interval changed to 10 mm, 30 mm, and 60 mm. Moreover, the frequency components of the Gaussian pulse were set to composite waves that flatly contain energies from 0.01 GHz to 20 GHz. In this case, waveforms receivable at d=60 mm are shown. However, as is apparent from FIG. 22, the reflection coefficient S11 becomes −20 dB at 6.5 GHz, and therefore, the frequency characteristic does not become flat, meaning that the transmitting characteristic is not so good.\n\nFIG. 24A is a chart showing transmitting waveforms indicated by the time-domain electric field strength (for 830 pico seconds) of the antenna apparatus with an antenna interval of 10 mm utilizing the aperture of transmission line used in the simulations of the preferred embodiment with Gaussian pulses, FIG. 24B is a chart showing transmitting waveforms indicated by the time-domain electric field strength (for 830 pico seconds) of the antenna apparatus with an antenna interval of 30 mm, and FIG. 24C is a chart showing transmitting waveforms indicated by the time-domain electric field strength (for 830 pico seconds) of the antenna apparatus with an antenna interval of 60 mm. Namely, FIGS. 24A to 24C shows transmitting waveforms indicated by the time-domain electric field strength in a range to 200 V/m (for 830 pico seconds) of the antenna apparatus utilizing the aperture of transmission line used in the simulations of the preferred embodiment with Gaussian pulses.\n\nFIG. 25A is a chart showing transmitting waveforms indicated by the time-domain electric field strength (for 1050 pico seconds) of the antenna apparatus with an antenna interval of 10 mm utilizing the aperture of transmission line used in the simulations of the preferred embodiment with Gaussian pulses, FIG. 25B is a chart showing transmitting waveforms indicated by the time-domain electric field strength (for 1050 pico seconds) of the antenna apparatus with an antenna interval of 30 mm, and FIG. 25C is a chart showing transmitting waveforms indicated by the time-domain electric field strength (for 1050 pico seconds) of the antenna apparatus with an antenna interval of 60 mm.\n\nIn FIGS. 24A to 24C and 25A to 25C, the transmitting characteristics of the antenna apparatus utilizing the aperture of transmission line of d=32 mm and w=80 mm are expressed by electric field energies.\n\nFIG. 26A is a signal waveform chart of the antenna apparatus with an antenna interval of 60 mm for showing differences when the antenna interval is changed as shown in FIGS. 23A to 23D, and FIG. 26B is a signal waveform chart of the antenna apparatus with an antenna interval of 100 mm for showing the same differences. As is apparent from FIGS. 26A and 26B, if the receiving characteristics with respect to the aperture plane of d=65 mm are compared with each other, the characteristic that is better than when d=32 mm is obtained in spite of separation by 100 mm. This means that the relation of (¼)λg≦d can be confirmed by the signal transmission simulations. It was discovered from FIGS. 23A to 23D that the antenna apparatus utilizing the aperture of transmission line was able to achieve almost same efficiencies in transmitting and receiving.\n\nFIG. 27A is a chart showing a top view of a tapered expanded field distribution when the characteristic impedance is made constant in the antenna apparatus utilizing the aperture of transmission line used in the simulations of the present preferred embodiment, and FIG. 27B is a chart showing a side view thereof, where the electric field strength is in a range to 1628 V/m. Namely, FIGS. 27A and 27B show transmitting states of electromagnetic waves of 10-GHz sinusoidal waves when a taper angle θ of 120 degrees is added with the characteristic impedance Z0 maintained or kept to be constant. The dispersion in the state of a circular arc originating at the expansion starting point is found out. The aperture portion 3 cannot perform TEM wave transmission as an antenna as a consequence of time dispersion due to the circular arc shape. The dispersion angle was about 60 degrees, and it was considered that the taper angle θ (or φ), expanding in such a state that the electromagnetic coupling on the transmission line was completely achieved, was 30 degrees, and this was adopted as the feature of the present invention.\n\nFIG. 28 is a graph showing a frequency characteristic (reference value (horizontal axis located one scale smaller than the upper limit value) in a range from 0.05 GHz to 20 GHz is 0 dB, and one scale represents 5 dB) of the reflection coefficient S11 of the antenna apparatus utilizing the aperture of transmission line used in the simulations of the present preferred embodiment. In FIG. 28, f1 denote 3 GHz, f2 denotes 5 GHz, f3 denotes 10 GHz, f4 denotes 15 GHz, and f5 denotes 20 GHz. FIG. 28 is an experimental example, which has such a structure that the expanded tapered line portion 2 is formed of an acrylic plate and floated partway. The specifications of the aperture portion 3 are as follows: d=20 mm, w=30 mm, (¼)λ=3.75 GHz. In the present experiment, the stacked pair line 1 is not provided, and the tapered line portion 2 is connected in series with a BNC connector. The characteristic impedance Z0 of the BNC connector is 50Ω, the characteristic impedance Z0 of the tapered line portion 2 formed of the acrylic part is 83.5Ω, and the characteristic impedance Z0 of the aperture portion 3 is 139.4Ω, which constitute such a structure that a large relation attenuation occurs under the conditions far from a constant characteristic impedance. Reviewing the frequency characteristic of the reflection coefficient S11 of FIG. 28, such results that are not so bad can be obtained as the radiation characteristics having substantially flat frequency characteristics and that S11=approx. −10 dB at frequencies equal to or higher than 3.75 GHz.\n\nFIG. 29A is a signal waveform chart showing signal waveforms of 10 GHz when the distance between aperture planes is changed in a pair of transmission line aperture type antenna apparatuses used in the simulations of the present preferred embodiment. In FIG. 29A, 291 denotes a case of a transmission distance of 10 cm and an amplitude of 42.76 mV, 292 denotes a case of a transmission distance of 50 cm and an amplitude of 10.95 mV, and 293 denotes a case of a transmission distance of 100 cm and an amplitude of 10.54 mV.\n\nFIG. 29B is a signal waveform chart showing signal waveforms at 10 GHz when a displacement distance from the center line is changed in a pair of transmission line aperture type antenna apparatuses used in the simulations of the present preferred embodiment. In FIG. 29B, 294 denotes a case of no displacement and an amplitude of 10.95 mV, 295 denotes a case of 5 cm displacement and an amplitude of 11.72 mV, 296 denotes a case of 10 cm displacement and an amplitude of 9.70 mV, 297 denotes a case of 20 cm displacement and an amplitude of 5.5 mV. Namely, FIGS. 29A and 29B show the transmitting and receiving characteristics when a pair of transmission line aperture type antenna apparatuses having the dimensions of d=20 mm and w=30 mm at the time of input of 10-GHz sine waves (having an amplitude of 1 V) are put in a mutually opposed state. FIG. 29A shows 10-GHz sine wave transmitting characteristics (receiving waveforms) when the antenna apparatus utilizing the aperture of transmission lines of FIG. 28 are opposed to each other and used for transmitting and receiving. Since the input voltage to the transmitting antenna apparatus is 1 V, an antenna gain of −40 dB is obtained by transmitting with the antenna apparatus with an antenna interval of 1 m. An advantageous effect similar to that of the simulations can presumably be expected so long as the characteristic impedance Z0 of the antenna is constant. Moreover, FIG. 29B shows receiving characteristics when the central axes of the transmitting and receiving antennas at an antenna interval of 50 cm are shifted in parallel in the width direction of the transmission line, and this leads to that the antenna apparatus has a substantial directivity.\n\nThe present inventor and others further conducted simulations of the antenna apparatus utilizing the aperture of transmission lines of the preferred embodiments of FIGS. 2 to 6. As a result, it was confirmed that the passing coefficient S21 and the reflection coefficient S11 scarcely suffered influences so long as the width was sufficiently small (e.g., a width of 1 μm with respect to d=100 μm) even when the support members 4a, 4b or 4c was provided. Moreover, in FIGS. 4 to 6 where the aperture portion 3 was expanded in the tapered shape, the results of a reduction in the reflection coefficient S11, a slight increase in the antenna gain and a consequent increase in the total antenna radiation efficiency were obtained.\n\n### INDUSTRIAL APPLICABILITY\n\nAs described in detail above, the antenna apparatus utilizing the aperture of transmission line of the present invention is the antenna apparatus, which is connected to the transmission line and has an extremely simple configuration as compared with that of the prior art and a narrow directivity with almost no change in the frequency characteristics, thereby allowing a remarkably large antenna gain to be achieved. Therefore, the communications can be achieved even at a comparatively long distance. The antenna apparatus utilizing the aperture of transmission line of the present invention is applicable to various applications as follows.\n\n(1) It is applicable to transmitting and receiving between IP terminals of global interconnections or wirings on an IC chip.\n\n(2) It is applicable to means for communications between IC chips.\n\n(3) It is applicable to means for communications between LSI packages.\n\n(4) It is applicable to communications between boards.\n\n(5) It is applicable to long-distance communications.\n\n(6) It is applicable to a system in which UWB or digital signals are subject to direct communications without modulation because of almost no frequency characteristic.\n\n(7) It is applicable to distance measurement and shape measurement of reflective objects.\n\n(8) It is applicable to transmitting and receiving for RFID or the like on the base station side.\n\n(9) It is applicable to transmitting with scanning the frequency, transmitting and receiving intended for scanning receiving, and applications intended for reflection receiving by utilizing the narrow directivity.\n\n(10) It is applicable to MEMS communications, communications inside a living body for medical use and satellite communications with gigantic antennas and power transmission because the principle of the characteristic impedance can be expanded and contracted in similitude.\n\n(11) It is applicable to applications having no relation to the allocation of radio frequencies because of the narrow directivity.\n\nAlthough the present invention has been fully described in connection with the preferred embodiments thereof with reference to the accompanying drawings, it is to be noted that various changes and modifications are apparent to those skilled in the art. Such changes and modifications are to be understood as included within the scope of the present invention as defined by the appended claims unless they depart therefrom.\n\n## Claims\n\n1. An antenna apparatus which is connected to a first transmission line having a predetermined characteristic impedance, the first transmission line including a pair of first planar line conductors electrically separated from each other such that magnetic lines of force are generated surrounding the first planar line conductors and perpendicularly intersect electric lines of force when electromagnetic waves travel on the first transmission line, the antenna apparatus comprising:\na tapered line portion connected to one end of the first transmission line, the tapered line portion including a second transmission line including a pair of second planar line conductors electrically separated from each other such that magnetic lines of force are generated surrounding the second planar line conductors and perpendicularly intersect electric lines of force when electromagnetic waves travel on the second transmission line, the tapered line portion keeping a predetermined characteristic impedance constant and expanding at least one of a width of the second transmission line and an interval in a tapered shape at a predetermined taper angle; and\nan aperture portion having a radiation aperture connected to one end of the tapered line portion, the aperture portion including a pair of parallel planar line conductors separated from each other,\nwherein a size of one side of the aperture end plane of the aperture portion is set to be equal to or higher than a quarter wavelength of the minimum operating frequency of the antenna apparatus.\na tapered line portion connected to one end of the first transmission line, the tapered line portion including a second transmission line including a pair of second planar line conductors electrically separated from each other such that magnetic lines of force are generated surrounding the second planar line conductors and perpendicularly intersect electric lines of force when electromagnetic waves travel on the second transmission line, the tapered line portion keeping a predetermined characteristic impedance constant and expanding at least one of a width of the second transmission line and an interval in a tapered shape at a predetermined taper angle; and\nan aperture portion having a radiation aperture connected to one end of the tapered line portion, the aperture portion including a pair of parallel planar line conductors separated from each other,\nwherein a size of one side of the aperture end plane of the aperture portion is set to be equal to or higher than a quarter wavelength of the minimum operating frequency of the antenna apparatus.\n2. The antenna apparatus as claimed in claim 1, further comprising a support member that short-circuits and supports the second transmission line including the pair of second planar line conductors substantially in a center portion in a width direction of the second transmission line of the aperture portion.\n3. The antenna apparatus as claimed in claim 1, further comprising a pair of support members that short-circuit and support the second transmission line including the pair of second planar line conductors substantially at both ends in a width direction of the second transmission line of the aperture portion.\n4. The antenna apparatus as claimed in claim 1,\nwherein the aperture portion is constituted by expanding a width of the second transmission line in a tapered shape.\nwherein the aperture portion is constituted by expanding a width of the second transmission line in a tapered shape.\n5. The antenna apparatus as claimed in claim 1,\nwherein a space located between the pair of first planar line conductors of the first transmission line in the tapered line portion is filled with a predetermined dielectric.\nwherein a space located between the pair of first planar line conductors of the first transmission line in the tapered line portion is filled with a predetermined dielectric.\n6. The antenna apparatus as claimed in claim 1,\nwherein a space located between the pair of second planar line conductors of the second transmission line in the aperture portion is filled with a predetermined dielectric.\nwherein a space located between the pair of second planar line conductors of the second transmission line in the aperture portion is filled with a predetermined dielectric.\n7. The antenna apparatus as claimed in claim 1, further comprising a support member for supporting both end portions in a width direction of the first transmission line in the tapered line portion with interposition of a predetermined interval.\n8. The antenna apparatus as claimed in claim 1, further comprising a support member for supporting both end portions in the width direction of the second transmission line in the aperture portion with interposition of a predetermined interval.\n9. The antenna apparatus as claimed in claim 1,\nwherein the taper angle is set to a predetermined value which is larger than zero degree and equal to or smaller than 30 degrees.\nwherein the taper angle is set to a predetermined value which is larger than zero degree and equal to or smaller than 30 degrees.\n10. The antenna apparatus as claimed in claim 1,\nwherein the characteristic impedance is set to a predetermined value that is set within a range from 50Ω to 100Ω.\nwherein the characteristic impedance is set to a predetermined value that is set within a range from 50Ω to 100Ω." ]	[ null, "https://pixel.quantserve.com/pixel/p-S5j449sRLqmpu.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9245282,"math_prob":0.93323684,"size":47509,"snap":"2020-34-2020-40","text_gpt3_token_len":9602,"char_repetition_ratio":0.22073466,"word_repetition_ratio":0.29833612,"special_character_ratio":0.19981477,"punctuation_ratio":0.074352056,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9505994,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-09-29T16:28:20Z\",\"WARC-Record-ID\":\"<urn:uuid:d0c09410-5085-4169-a744-abafe51a6a88>\",\"Content-Length\":\"231212\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:fd467e36-bc98-4b0d-ae64-01be7c9415d6>\",\"WARC-Concurrent-To\":\"<urn:uuid:0d54d0cf-602f-4206-b728-b2fe3d94a568>\",\"WARC-IP-Address\":\"54.86.205.222\",\"WARC-Target-URI\":\"https://www.sparrho.com/item/antenna-apparatus-utilizing-aperture-of-transmission-line/d2c032/\",\"WARC-Payload-Digest\":\"sha1:2W2GNOJGO4QZ7AT5PI3INE4IPVKMR65I\",\"WARC-Block-Digest\":\"sha1:WARUEHK44BHAMRX5NJQC46XC3HDHXJZK\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600400202418.22_warc_CC-MAIN-20200929154729-20200929184729-00244.warc.gz\"}"}
https://rajeshshuklacatalyst.in/class-12-practical-file-programs/	[ "# Class 12 : Practical File Programs\n\n## Write a python script to take input for a number and print its table?\n\n```n=int(input(\"Enter any no \"))\ni=1\nwhile(i<=10):\nt=ni\nprint(n,\" \",i,\" = \",t)\ni=i+1 ```\n\nOutput:\n\nEnter any no 5\n5 * 1 = 5\n5 * 2 = 10\n5 * 3 = 15\n5 * 4 = 20\n5 * 5 = 25\n5 * 6 = 30\n5 * 7 = 35\n5 * 8 = 40\n5 * 9 = 45\n5 * 10 = 50\n>>>\n\n## Write a python script to take input for a number and print its factorial?\n\n```n=int(input(\"Enter any no \"))\ni=1\nf=1\nwhile(i<=n):\nf=fi\ni=i+1\nprint(\"Factorial = \",f) ```\n\nOutput:\n\nEnter any no 5\nFactorial = 120\n>>>\n\n## Write a python script to take input for a number check if the entered number is Armstrong or not.\n\n```n=int(input(\"Enter the number to check : \"))\nn1=n\ns=0\nwhile(n>0):\nd=n%10;\ns=s + (d d * d)\nn=int(n/10)\nif s==n1:\nprint(\"Armstrong Number\")\nelse:\nprint(\"Not an Armstrong Number\") ```\n\nOutput:\n\nEnter the number to check : 153\nArmstrong Number\n>>>\n\nOutput:\n\nEnter the number to check : 152\nNot an Armstrong Number\n>>>\n\n## Write a python script to take input for a number and print its factorial using recursion?\n\nSol:\n\n```#Factorial of a number using recursion\ndef recur_factorial(n):\nif n == 1:\nreturn n\nelse:\nreturn n*recur_factorial(n-1)\n#for fixed number\nnum = 7\n#using user input\nnum=int(input(\"Enter any no \"))\ncheck if the number is negative\nif num < 0:\nprint(\"Sorry, factorial does not exist for negative numbers\")\nelif num == 0:\nprint(\"The factorial of 0 is 1\")\nelse:\nprint(\"The factorial of\", num, \"is\", recur_factorial(num))```\n\nOutput:\n\nEnter any no 5\nThe factorial of 5 is 120\n>>>\n\n## Write a python script to Display Fibonacci Sequence Using Recursion?\n\nSol:\n\n```#Python program to display the Fibonacci sequence\ndef recur_fibo(n):\nif n <= 1:\nreturn n\nelse:\nreturn(recur_fibo(n-1) + recur_fibo(n-2))\nnterms = 10\n#check if the number of terms is valid\nif (nterms <= 0):\nprint(\"Plese enter a positive integer\")\nelse:\nprint(\"Fibonacci sequence:\")\nfor i in range(nterms):\nprint(recur_fibo(i)) ```\n\nOutput:\n\nFibonacci sequence:\n0\n1\n1\n2\n3\n5\n8\n13\n21\n34\n>>>\n\n### CBSE Class 12 @ Python\n\n• Class 12 @ Python Theory Syllabus\n• Class 12 @ Python Practical Syllabus\n• Revision Tour\n• Functions (Funcations, Inbuilt Functions, Python Modules, Python Packages)\n• Inbuilt Functions\n• Python Modules\n• Python Packages\n• Using Python Libraries\n• Python Data File Handling\n• Program Efficiency\n• Data Structures In Python\n• Data Visualization Using Pyplot\n• Computer Networks\n• MySQL\n• Interface Python with SQL\n• Society, Law and Ethics\n• Web Development with Django\n• Class 12 @ Python Sample Practical File\n• Class 12 @ Python Sample Papers\n• Class 12 @ Python Projects\n\n### Interview Questions\n\nC Programming\nC++ Programming\nClass 11 (Python)\nClass 12 (Python)\nC Language\nC++ Programming\nPython\n\nC Interview Questions\nC++ Interview Questions\nC Programs\nC++ Programs" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5197713,"math_prob":0.9539637,"size":2225,"snap":"2022-40-2023-06","text_gpt3_token_len":700,"char_repetition_ratio":0.14768122,"word_repetition_ratio":0.11675127,"special_character_ratio":0.34651685,"punctuation_ratio":0.10822511,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99854803,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-01-28T23:20:18Z\",\"WARC-Record-ID\":\"<urn:uuid:290153d3-4428-49c2-99c6-42067cd6b281>\",\"Content-Length\":\"73886\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:132910fe-fc32-4795-ada3-a9e007f2b8b6>\",\"WARC-Concurrent-To\":\"<urn:uuid:7126ab51-c05a-4480-940c-39f3f24920a1>\",\"WARC-IP-Address\":\"103.92.235.92\",\"WARC-Target-URI\":\"https://rajeshshuklacatalyst.in/class-12-practical-file-programs/\",\"WARC-Payload-Digest\":\"sha1:L5XERTUCR3TVWYQSGECH3XNUVCRVYVHK\",\"WARC-Block-Digest\":\"sha1:RDD5ZWTP6MTMZQ7TE3BNI3SRRXFJ5TEC\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764499695.59_warc_CC-MAIN-20230128220716-20230129010716-00586.warc.gz\"}"}
https://www.colorhexa.com/074132	[ "# #074132 Color Information\n\nIn a RGB color space, hex #074132 is composed of 2.7% red, 25.5% green and 19.6% blue. Whereas in a CMYK color space, it is composed of 89.2% cyan, 0% magenta, 23.1% yellow and 74.5% black. It has a hue angle of 164.5 degrees, a saturation of 80.6% and a lightness of 14.1%. #074132 color hex could be obtained by blending #0e8264 with #000000. Closest websafe color is: #003333.\n\n• R 3\n• G 25\n• B 20\nRGB color chart\n• C 89\n• M 0\n• Y 23\n• K 75\nCMYK color chart\n\n#074132 color description : Very dark cyan - lime green.\n\n# #074132 Color Conversion\n\nThe hexadecimal color #074132 has RGB values of R:7, G:65, B:50 and CMYK values of C:0.89, M:0, Y:0.23, K:0.75. Its decimal value is 475442.\n\nHex triplet RGB Decimal 074132 `#074132` 7, 65, 50 `rgb(7,65,50)` 2.7, 25.5, 19.6 `rgb(2.7%,25.5%,19.6%)` 89, 0, 23, 75 164.5°, 80.6, 14.1 `hsl(164.5,80.6%,14.1%)` 164.5°, 89.2, 25.5 003333 `#003333`\nCIE-LAB 23.855, -22.04, 4.136 2.553, 4.056, 3.666 0.249, 0.395, 4.056 23.855, 22.424, 169.372 23.855, -18.773, 6.936 20.139, -12.611, 3.305 00000111, 01000001, 00110010\n\n# Color Schemes with #074132\n\n• #074132\n``#074132` `rgb(7,65,50)``\n• #410716\n``#410716` `rgb(65,7,22)``\nComplementary Color\n• #074115\n``#074115` `rgb(7,65,21)``\n• #074132\n``#074132` `rgb(7,65,50)``\n• #073341\n``#073341` `rgb(7,51,65)``\nAnalogous Color\n• #411507\n``#411507` `rgb(65,21,7)``\n• #074132\n``#074132` `rgb(7,65,50)``\n• #410733\n``#410733` `rgb(65,7,51)``\nSplit Complementary Color\n• #413207\n``#413207` `rgb(65,50,7)``\n• #074132\n``#074132` `rgb(7,65,50)``\n• #320741\n``#320741` `rgb(50,7,65)``\n• #164107\n``#164107` `rgb(22,65,7)``\n• #074132\n``#074132` `rgb(7,65,50)``\n• #320741\n``#320741` `rgb(50,7,65)``\n• #410716\n``#410716` `rgb(65,7,22)``\n• #000000\n``#000000` `rgb(0,0,0)``\n• #02130f\n``#02130f` `rgb(2,19,15)``\n• #052a20\n``#052a20` `rgb(5,42,32)``\n• #074132\n``#074132` `rgb(7,65,50)``\n• #095844\n``#095844` `rgb(9,88,68)``\n• #0c6f55\n``#0c6f55` `rgb(12,111,85)``\n• #0e8667\n``#0e8667` `rgb(14,134,103)``\nMonochromatic Color\n\n# Alternatives to #074132\n\nBelow, you can see some colors close to #074132. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #074124\n``#074124` `rgb(7,65,36)``\n• #074128\n``#074128` `rgb(7,65,40)``\n• #07412d\n``#07412d` `rgb(7,65,45)``\n• #074132\n``#074132` `rgb(7,65,50)``\n• #074137\n``#074137` `rgb(7,65,55)``\n• #07413c\n``#07413c` `rgb(7,65,60)``\n• #074141\n``#074141` `rgb(7,65,65)``\nSimilar Colors\n\n# #074132 Preview\n\nThis text has a font color of #074132.\n\n``<span style=\"color:#074132;\">Text here</span>``\n#074132 background color\n\nThis paragraph has a background color of #074132.\n\n``<p style=\"background-color:#074132;\">Content here</p>``\n#074132 border color\n\nThis element has a border color of #074132.\n\n``<div style=\"border:1px solid #074132;\">Content here</div>``\nCSS codes\n``.text {color:#074132;}``\n``.background {background-color:#074132;}``\n``.border {border:1px solid #074132;}``\n\n# Shades and Tints of #074132\n\nA shade is achieved by adding black to any pure hue, while a tint is created by mixing white to any pure color. In this example, #010c09 is the darkest color, while #f9fefd is the lightest one.\n\n• #010c09\n``#010c09` `rgb(1,12,9)``\n• #031e17\n``#031e17` `rgb(3,30,23)``\n• #052f24\n``#052f24` `rgb(5,47,36)``\n• #074132\n``#074132` `rgb(7,65,50)``\n• #095340\n``#095340` `rgb(9,83,64)``\n• #0b644d\n``#0b644d` `rgb(11,100,77)``\n• #0d765b\n``#0d765b` `rgb(13,118,91)``\n• #0f8868\n``#0f8868` `rgb(15,136,104)``\n• #119a76\n``#119a76` `rgb(17,154,118)``\n• #12ab84\n``#12ab84` `rgb(18,171,132)``\n• #14bd91\n``#14bd91` `rgb(20,189,145)``\n• #16cf9f\n``#16cf9f` `rgb(22,207,159)``\n``#18e0ad` `rgb(24,224,173)``\n• #25e7b5\n``#25e7b5` `rgb(37,231,181)``\n• #36e9bb\n``#36e9bb` `rgb(54,233,187)``\n• #48ebc1\n``#48ebc1` `rgb(72,235,193)``\n• #5aedc7\n``#5aedc7` `rgb(90,237,199)``\n• #6cefcd\n``#6cefcd` `rgb(108,239,205)``\n• #7df1d3\n``#7df1d3` `rgb(125,241,211)``\n• #8ff3d9\n``#8ff3d9` `rgb(143,243,217)``\n• #a1f5df\n``#a1f5df` `rgb(161,245,223)``\n• #b2f7e5\n``#b2f7e5` `rgb(178,247,229)``\n• #c4f9eb\n``#c4f9eb` `rgb(196,249,235)``\n• #d6fbf1\n``#d6fbf1` `rgb(214,251,241)``\n• #e7fcf7\n``#e7fcf7` `rgb(231,252,247)``\n• #f9fefd\n``#f9fefd` `rgb(249,254,253)``\nTint Color Variation\n\n# Tones of #074132\n\nA tone is produced by adding gray to any pure hue. In this case, #232525 is the less saturated color, while #014735 is the most saturated one.\n\n• #232525\n``#232525` `rgb(35,37,37)``\n• #202826\n``#202826` `rgb(32,40,38)``\n• #1d2b27\n``#1d2b27` `rgb(29,43,39)``\n• #1a2e29\n``#1a2e29` `rgb(26,46,41)``\n• #18302a\n``#18302a` `rgb(24,48,42)``\n• #15332b\n``#15332b` `rgb(21,51,43)``\n• #12362d\n``#12362d` `rgb(18,54,45)``\n• #0f392e\n``#0f392e` `rgb(15,57,46)``\n• #0d3b2f\n``#0d3b2f` `rgb(13,59,47)``\n• #0a3e31\n``#0a3e31` `rgb(10,62,49)``\n• #074132\n``#074132` `rgb(7,65,50)``\n• #044433\n``#044433` `rgb(4,68,51)``\n• #014735\n``#014735` `rgb(1,71,53)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #074132 is perceived by people affected by a color vision deficiency. This can be useful if you need to ensure your color combinations are accessible to color-blind users.\n\nMonochromacy\n• Achromatopsia 0.005% of the population\n• Atypical Achromatopsia 0.001% of the population\nDichromacy\n• Protanopia 1% of men\n• Deuteranopia 1% of men\n• Tritanopia 0.001% of the population\nTrichromacy\n• Protanomaly 1% of men, 0.01% of women\n• Deuteranomaly 6% of men, 0.4% of women\n• Tritanomaly 0.01% of the population" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.52338296,"math_prob":0.722846,"size":3679,"snap":"2021-31-2021-39","text_gpt3_token_len":1625,"char_repetition_ratio":0.12761904,"word_repetition_ratio":0.011029412,"special_character_ratio":0.5710791,"punctuation_ratio":0.23730685,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9940069,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-08-06T04:34:07Z\",\"WARC-Record-ID\":\"<urn:uuid:7874377c-9914-4aa8-a591-0987cb707b25>\",\"Content-Length\":\"36085\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a93b56e5-f616-4ca7-b437-9b659fdaaafb>\",\"WARC-Concurrent-To\":\"<urn:uuid:6ff77d5c-992e-4569-9ca7-f24a7cff8d94>\",\"WARC-IP-Address\":\"178.32.117.56\",\"WARC-Target-URI\":\"https://www.colorhexa.com/074132\",\"WARC-Payload-Digest\":\"sha1:EHONTMS6N2OH7KDHOHJ6N7KLYPRO4CQF\",\"WARC-Block-Digest\":\"sha1:BTSBPZF3I5FIOSVOQ6BP5B3FLZBQITKT\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-31/CC-MAIN-2021-31_segments_1627046152112.54_warc_CC-MAIN-20210806020121-20210806050121-00269.warc.gz\"}"}
https://math.stackexchange.com/questions/3417896/find-the-cdf-and-pdf-of-u2-is-the-distribution-of-u2-uniform-on-0-1	[ "# Find the CDF and PDF of $U^2$. Is the distribution of $U^2$ Uniform on $(0, 1)$?\n\nI have the following problem:\n\nLet $$U$$ be a $$\\text{Unif}(−1,1)$$ random variable on the interval $$(−1,1)$$.\n\nFind the CDF and PDF of $$U^2$$. Is the distribution of $$U^2$$ Uniform on $$(0, 1)$$?\n\nThe solution is as follows:\n\nLet $$X = U^2$$, for $$0 < x < 1$$, $$P(X \\le x) = P(−\\sqrt{x} \\le U \\le \\sqrt{x}) = P(U \\le \\sqrt{x}) - P(U \\le -\\sqrt{x}) = \\dfrac{\\sqrt{x} + 1}{2} - \\dfrac{-\\sqrt{x} + 1}{2} = \\sqrt{x}$$ (Note that $$P(U \\le u) = \\dfrac{u + 1}{2}$$ for $$-1 \\le u \\le 1$$. The density is then given by $$f_X(x) = \\dfrac{d}{dx} P(X \\le x) = \\dfrac{d}{dx}x^{1/2} = \\dfrac{1}{2} x^{-1/2}$$. The distribution of $$X = U^2$$ is not Unif$$(0, 1)$$ on the interval $$(0, 1)$$ as the PDF is not a constant on this interval.\n\nThe first fact that I am confused about is how the author got that the interval is now $$0 < x < 1$$ instead of $$-1 < x < 1$$.\n\nThe second fact that I am confused about, which it seems is related to the first, is how the author calculated that $$P(U \\le \\sqrt{x}) - P(U \\le -\\sqrt{x}) = \\dfrac{\\sqrt{x} + 1}{2} - \\dfrac{-\\sqrt{x} + 1}{2}$$. Did they just insert the values into the formula for the CDF (since the formula for the CDF is $$\\dfrac{x - a}{b - a}$$), or did they otherwise somehow calculate the CDF? I ask because I'm unsure if it's just a matter of memorization, or whether there is some mathematical understanding here that I am missing.\n\nI would greatly appreciate it if people could please take the time to clarify this.\n\n• Isn't it $X = \|U\|$? In this case for $0<x<1$ we have $P(X \\leq x) = P(\|U\| \\leq x) = P(-x \\leq U \\leq x)$ – G. Gare Nov 1 at 15:23\n• @G.Gare My apologies!!! I made a transcription error; it should be $U^2$. I'm really sorry!!! – The Pointer Nov 1 at 15:25\n• Then the square roots make more sense. There's another edit you missed: $P(-x\\leq U^2 \\leq x)$. – G. Gare Nov 1 at 15:26\n• @G.Gare How's that? – The Pointer Nov 1 at 15:28\n• It's ok now! Don't worry – G. Gare Nov 1 at 15:28\n\nExplaining fact 1: while $$U$$ has support $$[-1,\\,1]$$, $$X=U^2$$ has support $$[0,\\,1]$$.\n• Thanks for the answer. Can you please explain the reasoning for how we go from support $[-1, 1]$ to $[0, 1]$? I ask because it is very simple to naively just reason that, since $X = U^2$, we have $(-1)^2$ and $1^2$, and so the new support is $[1, 1]$ (which is obviously nonsense). I recognize that this would be due to a misunderstanding of what \"support\" is, but I'm curious what explanation you would use to describe the reasoning behind how we get $[0, 1]$, so that it is clear in my mind as a person who is new to this concept. – The Pointer Nov 1 at 15:32\n• @ThePointer The support is the set of allowed values. Note that $\\{u^2\|u\\in[-1,\\,1]\\}=[0,\\,1]$. (For example, the graph $x=u^2$ has $x$-coordinates $\\in[0,\\,1]$ for $u\\in[-1,\\,1]$.) – J.G. Nov 1 at 15:46" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8415259,"math_prob":0.9999901,"size":1425,"snap":"2019-51-2020-05","text_gpt3_token_len":496,"char_repetition_ratio":0.14637579,"word_repetition_ratio":0.08301887,"special_character_ratio":0.3817544,"punctuation_ratio":0.07570978,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000013,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-12-14T13:40:02Z\",\"WARC-Record-ID\":\"<urn:uuid:b704fcd3-55e0-448a-b428-98db2d3874ae>\",\"Content-Length\":\"142988\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ccd0be52-9344-464c-91ca-7b05651f0912>\",\"WARC-Concurrent-To\":\"<urn:uuid:94f1cd48-2875-44d9-b5ae-c99f1d221751>\",\"WARC-IP-Address\":\"151.101.65.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/3417896/find-the-cdf-and-pdf-of-u2-is-the-distribution-of-u2-uniform-on-0-1\",\"WARC-Payload-Digest\":\"sha1:HQHEFOSD3UGWLU3EUYNEGNUKFC7UCFEN\",\"WARC-Block-Digest\":\"sha1:M7UKUMPLH5Z5P7AHQB5MVO3YVPKGABUK\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-51/CC-MAIN-2019-51_segments_1575541157498.50_warc_CC-MAIN-20191214122253-20191214150253-00380.warc.gz\"}"}
https://moneyexchangerate.org/currencyexchange/chf/dzd	[ "Swiss Franc to Algerian Dinar Exchange Rate and Currency Converter\n\nView current exchange rates for the pair Swiss Franc (CHF) and Algerian Dinar (DZD). This page shows a rate of Algerian Dinar for 1 Swiss Franc and compare local money of Switzerland and Algeria. Currency exchange rates updates every day and use average rates based on Trusted International exchange rate. Use Currency converter to calculate any amount of CHF to DZD exchange rate. On this page available money conversion tables of popular amounts, compare tables, history chart, popular money converter and list of live conversion of Swiss Franc in Algerian Dinar:\n\nToday exchange rate:\n\n1 CHF =\n121.87 DZD\n\nBy today rate (2019-06-26) CHF to DZD equal 121.870285\n\nInvert: DZD to CHF Currency rate\n\nCurrency converter\n\nSwiss Franc Algerian Dinar 1 CHF 1 DZD 100 Swiss Francs 100 Algerian Dinars\n\nSwiss Franc Currency Exchange Table\n\nCHF Value: Currency\n1 CHF\n=\n1.0258 USD\n1 CHF\n=\n0.9023 EUR\n1 CHF\n=\n0.8083 GBP\n1 CHF\n=\n1 CHF\n=\n1.4737 AUD\n1 CHF\n=\n1 CHF\n1 CHF\n=\n6.7362 DKK\n1 CHF\n=\n8.7629 NOK\n1 CHF\n=\n9.5243 SEK\n1 CHF\n=\n3.7678 AED\n1 CHF\n=\n7.0571 CNY\n1 CHF\n=\n8.0119 HKD\n1 CHF\n=\n109.9394 JPY\n1 CHF\n=\n71.1196 INR\n1 CHF\n=\n14486.094 IDR\n1 CHF\n=\n1.3894 SGD\n1 CHF\n=\n1186.2479 KRW\n1 CHF\n=\n14.7087 ZAR\n1 CHF\n=\n0.0001 BTC\n\nSwiss Franc currency rate vs major currencies Conversion table\n\nSwiss Franc vs other currencies\n\nAlgerian Dinar Currency Exchange Table\n\nDZD Value: Currency\n1 DZD\n=\n0.0084 USD\n1 DZD\n=\n0.0074 EUR\n1 DZD\n=\n0.0066 GBP\n1 DZD\n=\n1 DZD\n=\n0.0121 AUD\n1 DZD\n=\n0.0082 CHF\n1 DZD\n=\n0.0553 DKK\n1 DZD\n=\n0.0719 NOK\n1 DZD\n=\n0.0782 SEK\n1 DZD\n=\n0.0309 AED\n1 DZD\n=\n0.0579 CNY\n1 DZD\n=\n0.0657 HKD\n1 DZD\n=\n0.9021 JPY\n1 DZD\n=\n0.5836 INR\n1 DZD\n=\n118.8649 IDR\n1 DZD\n=\n0.0114 SGD\n1 DZD\n=\n9.7337 KRW\n1 DZD\n=\n0.1207 ZAR\n1 DZD\n=\n0 BTC\n\nAlgerian Dinar currency rate vs major currencies Conversion table\n\nAlgerian Dinar vs other currencies\n\nSwiss Franc compared to Algerian Dinar\n\nx1 x100 x1000\n1 Swiss Franc = 121.87 Algerian Dinar 100 Swiss Franc = 12187.03 Algerian Dinar 1000 Swiss Franc = 121870.28 Algerian Dinar\n2 Swiss Franc = 243.74 Algerian Dinar 200 Swiss Franc = 24374.06 Algerian Dinar 2000 Swiss Franc = 243740.57 Algerian Dinar\n3 Swiss Franc = 365.61 Algerian Dinar 300 Swiss Franc = 36561.09 Algerian Dinar 3000 Swiss Franc = 365610.85 Algerian Dinar\n4 Swiss Franc = 487.48 Algerian Dinar 400 Swiss Franc = 48748.11 Algerian Dinar 4000 Swiss Franc = 487481.14 Algerian Dinar\n5 Swiss Franc = 609.35 Algerian Dinar 500 Swiss Franc = 60935.14 Algerian Dinar 5000 Swiss Franc = 609351.42 Algerian Dinar\n6 Swiss Franc = 731.22 Algerian Dinar 600 Swiss Franc = 73122.17 Algerian Dinar 6000 Swiss Franc = 731221.71 Algerian Dinar\n7 Swiss Franc = 853.09 Algerian Dinar 700 Swiss Franc = 85309.2 Algerian Dinar 7000 Swiss Franc = 853091.99 Algerian Dinar\n8 Swiss Franc = 974.96 Algerian Dinar 800 Swiss Franc = 97496.23 Algerian Dinar 8000 Swiss Franc = 974962.28 Algerian Dinar\n9 Swiss Franc = 1096.83 Algerian Dinar 900 Swiss Franc = 109683.26 Algerian Dinar 9000 Swiss Franc = 1096832.56 Algerian Dinar\n\nSwiss Franc in Algerian Dinars History Chart\n\nDuring last 30 days average exchange rate of Swiss Franc in Algerian Dinars was 120.17954 DZD for 1 CHF. The highest price of Swiss Franc in Algerian Dinar was Tue, 25 Jun 2019 when 1 Swiss Franc = 122.1104 Algerian Dinar. The lowest change rate in last month between Swiss Francs and Algerian Dinar currencies was on Tue, 25 Jun 2019. On that day 1 CHF = 118.7362 DZD.\n\nen" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.549259,"math_prob":0.9015219,"size":2327,"snap":"2019-26-2019-30","text_gpt3_token_len":986,"char_repetition_ratio":0.339647,"word_repetition_ratio":0.013953488,"special_character_ratio":0.5345939,"punctuation_ratio":0.13690476,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95474076,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-06-26T12:13:23Z\",\"WARC-Record-ID\":\"<urn:uuid:b8a8b64f-ae12-4d55-a5f3-48b39cf8e412>\",\"Content-Length\":\"71171\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2736cd4e-0cb5-4a7b-a418-4922f0455fa2>\",\"WARC-Concurrent-To\":\"<urn:uuid:4aa7d158-55d2-47c0-9f56-f8d563116bbd>\",\"WARC-IP-Address\":\"45.55.84.94\",\"WARC-Target-URI\":\"https://moneyexchangerate.org/currencyexchange/chf/dzd\",\"WARC-Payload-Digest\":\"sha1:VQPXXZNCFVPM4UW6SHY6TVS4NIC6WUUS\",\"WARC-Block-Digest\":\"sha1:GCLBCKSJW5YUIKB2JLEYAIXW5HU2C5TT\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-26/CC-MAIN-2019-26_segments_1560628000306.84_warc_CC-MAIN-20190626114215-20190626140215-00523.warc.gz\"}"}
https://onlinemonks.com/web-stories/hedgepay-brief-price-prediction-analysis-forecast-29-Dec-2022/	[ "LIVE CURRENT PRICE (29 DEC 2022)\n\nALL TIME HIGH\n\nTOTAL MARKET CAP\n\n## HEDGEPAY (HPAY)\n\n### Current Rank #6606\n\nPRICE PREDICTION 2022\n\nthe HPAY price could reach a maximum possible level of \\$0.00093199 with the average forecast price of \\$0.00090611.\n\n## HEDGEPAY\n\nPRICE FORECAST 2023\n\nIn 2023 the price of HedgePay is expected to reach at a minimum price value of \\$0.001. The HPAY price can reach a maximum price value of \\$0.002 with the average value of \\$0.001.\n\n## HEDGEPAY\n\nPRICE ANALYSIS 2024\n\nthe HPAY price could reach a maximum possible level of \\$0.002 with the average forecast price of \\$0.002.\n\n## HEDGEPAY\n\nPRICE TARGET 2025\n\nIn 2025 the price of HedgePay is forecasted to be at around a minimum value of \\$0.003. The HedgePay price value can reach a maximum of \\$0.003 with the average trading value of \\$0.003 in USD.\n\n## HEDGEPAY\n\nPRICE PREDICTION 2026\n\nThe price of HedgePay is predicted to reach at a minimum level of \\$0.004 in 2026. The HedgePay price can reach a maximum level of \\$0.005 with the average price of \\$0.004 throughout 2026.\n\n## HEDGEPAY\n\nPRICE FORECAST 2027\n\nThe price of 1 HedgePay is expected to reach at a minimum level of \\$0.006 in 2027. The HPAY price can reach a maximum level of \\$0.007 with the average price of \\$0.006 throughout 2027.\n\n## HEDGEPAY\n\nPRICE ANALYSIS 2028\n\nIn 2028 the price of HedgePay is predicted to reach at a minimum level of \\$0.009. The HPAY price can reach a maximum level of \\$0.010 with the average trading price of \\$0.009.\n\n## HEDGEPAY\n\nPRICE TARGET 2029\n\nThe price of HedgePay is predicted to reach at a minimum value of \\$0.011 in 2029. The HedgePay price could reach a maximum value of \\$0.015 with the average trading price of \\$0.012 throughout 2029.\n\n## HEDGEPAY\n\nPRICE PREDICTION 2030\n\nThe price of HedgePay is predicted to reach at a minimum value of \\$0.017 in 2030. The HedgePay price could reach a maximum value of \\$0.020 with the average trading price of \\$0.018 throughout 2030.\n\n## HEDGEPAY\n\nPRICE FORECAST 2031\n\nIn 2031 the price of HedgePay is predicted to reach at a minimum level of \\$0.025. The HPAY price can reach a maximum level of \\$0.030 with the average trading price of \\$0.026.\n\n## HEDGEPAY\n\nHedgepay Brief, Price Prediction, Analysis & Forecast 2022-2031" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.88275373,"math_prob":0.6658028,"size":2033,"snap":"2023-14-2023-23","text_gpt3_token_len":572,"char_repetition_ratio":0.23016264,"word_repetition_ratio":0.32492998,"special_character_ratio":0.3271028,"punctuation_ratio":0.10983982,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96370196,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-03-23T04:06:23Z\",\"WARC-Record-ID\":\"<urn:uuid:7e9c4bd5-33ca-4204-9997-b0c763002dd0>\",\"Content-Length\":\"113300\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:63f25ff1-289b-45bb-8165-055e040318f0>\",\"WARC-Concurrent-To\":\"<urn:uuid:0145c08a-13d1-4914-a817-2d081c1ac79b>\",\"WARC-IP-Address\":\"137.184.178.87\",\"WARC-Target-URI\":\"https://onlinemonks.com/web-stories/hedgepay-brief-price-prediction-analysis-forecast-29-Dec-2022/\",\"WARC-Payload-Digest\":\"sha1:TCMI2GRCPIPJAJI5YCXGJWFGNHL7NEAF\",\"WARC-Block-Digest\":\"sha1:LGVEWMG2JNYGWCVBNHUDZQH7YLWLTMP3\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-14/CC-MAIN-2023-14_segments_1679296944996.49_warc_CC-MAIN-20230323034459-20230323064459-00126.warc.gz\"}"}
https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Book%3A_Electromagnetics_I_(Ellingson)/05%3A_Electrostatics/5.13%3A_Electric_Potential_Field_due_to_a_Continuous_Distribution_of_Charge	[ "# 5.13: Electric Potential Field due to a Continuous Distribution of Charge\n\nThe electrostatic potential field at $${\\bf r}$$ associated with $$N$$ charged particles is\n\n$V({\\bf r}) = \\frac{1}{4\\pi\\epsilon} \\sum_{n=1}^N { \\frac{q_n}{\\left\|{\\bf r}-{\\bf r}_n\\right\|} } \\label{m0065_eCountable}$\n\nwhere $$q_n$$ and $${\\bf r_n}$$ are the charge and position of the $$n^{\\mbox{th}}$$ particle. However, it is more common to have a continuous distribution of charge as opposed to a countable number of charged particles. We now consider how to compute $$V({\\bf r})$$ three types of these commonly-encountered distributions. Before beginning, it’s worth noting that the methods will be essentially the same, from a mathematical viewpoint, as those developed in Section [m0104_E_due_to_a_Continuous_Distribution_of_Charge]; therefore, a review of that section may be helpful before attempting this section.\n\n## Continuous Distribution of Charge Along a Curve\n\nConsider a continuous distribution of charge along a curve $$\\mathcal{C}$$. The curve can be divided into short segments of length $$\\Delta l$$. Then, the charge associated with the $$n^{\\mbox{th}}$$ segment, located at $${\\bf r}_n$$, is\n\n$q_n = \\rho_l({\\bf r}_n)~\\Delta l$\n\nwhere $$\\rho_l$$ is the line charge density (units of C/m) at $${\\bf r}_n$$. Substituting this expression into Equation \\ref{m0065_eCountable}, we obtain\n\n${\\bf V}({\\bf r}) = \\frac{1}{4\\pi\\epsilon} \\sum_{n=1}^{N} { \\frac{\\rho_l({\\bf r}_n)}{\\left\|{\\bf r}-{\\bf r}_n\\right\|} \\Delta l}$\n\nTaking the limit as $$\\Delta l\\to 0$$ yields:\n\n$V({\\bf r}) = \\frac{1}{4\\pi\\epsilon} \\int_{\\mathcal C} { \\frac{\\rho_l(l)}{\\left\|{\\bf r}-{\\bf r}'\\right\|} dl} \\label{m0065_eLineCharge}$\n\nwhere $${\\bf r}'$$ represents the varying position along $${\\mathcal C}$$ with integration along the length $$l$$.\n\n## Continuous Distribution of Charge Over a Surface\n\nConsider a continuous distribution of charge over a surface $$\\mathcal{S}$$. The surface can be divided into small patches having area $$\\Delta s$$. Then, the charge associated with the $$n^{\\mbox{th}}$$ patch, located at $${\\bf r}_n$$, is\n\n$q_n = \\rho_s({\\bf r}_n)~\\Delta s$\n\nwhere $$\\rho_s$$ is surface charge density (units of C/m$$^2$$) at $${\\bf r}_n$$. Substituting this expression into Equation \\ref{m0065_eCountable}, we obtain\n\n$V({\\bf r}) = \\frac{1}{4\\pi\\epsilon} \\sum_{n=1}^{N} { \\frac{\\rho_s({\\bf r}_n)}{\\left\|{\\bf r}-{\\bf r}_n\\right\|}~\\Delta s}$\n\nTaking the limit as $$\\Delta s\\to 0$$ yields:\n\n$V({\\bf r}) = \\frac{1}{4\\pi\\epsilon} \\int_{\\mathcal S} { \\frac{\\rho_s({\\bf r}')}{\\left\|{\\bf r}-{\\bf r}'\\right\|}~ds} \\label{m0065_eSurfCharge}$\n\nwhere $${\\bf r}'$$ represents the varying position over $${\\mathcal S}$$ with integration.\n\n## Continuous Distribution of Charge in a Volume\n\nConsider a continuous distribution of charge within a volume $$\\mathcal{V}$$. The volume can be divided into small cells (volume elements) having area $$\\Delta v$$. Then, the charge associated with the $$n^{\\mbox{th}}$$ cell, located at $${\\bf r}_n$$, is\n\n$q_n = \\rho_v({\\bf r}_n)~\\Delta v$\n\nwhere $$\\rho_v$$ is the volume charge density (units of C/m$$^3$$) at $${\\bf r}_n$$. Substituting this expression into Equation \\ref{m0065_eCountable}, we obtain\n\n$V({\\bf r}) = \\frac{1}{4\\pi\\epsilon} \\sum_{n=1}^{N} { \\frac{\\rho_v({\\bf r}_n)}{\\left\|{\\bf r}-{\\bf r}_n\\right\|}~\\Delta v}$\n\nTaking the limit as $$\\Delta v\\to 0$$ yields:\n\n$V({\\bf r}) = \\frac{1}{4\\pi\\epsilon} \\int_{\\mathcal V} { \\frac{\\rho_v({\\bf r}')}{\\left\|{\\bf r}-{\\bf r}'\\right\|}~dv}$\n\nwhere $${\\bf r}'$$ represents the varying position over $${\\mathcal V}$$ with integration." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.78866565,"math_prob":0.9999951,"size":3996,"snap":"2019-51-2020-05","text_gpt3_token_len":1293,"char_repetition_ratio":0.1503006,"word_repetition_ratio":0.16829745,"special_character_ratio":0.34134135,"punctuation_ratio":0.081491716,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000094,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-12-12T23:58:04Z\",\"WARC-Record-ID\":\"<urn:uuid:17c1e2c4-41ca-4a1b-8f57-10f56bbae798>\",\"Content-Length\":\"81762\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2ae7025a-3893-4730-98c7-3ddfe9d1acf1>\",\"WARC-Concurrent-To\":\"<urn:uuid:e4733517-7cf7-4d0d-a89c-550c7c6d0d44>\",\"WARC-IP-Address\":\"34.232.212.106\",\"WARC-Target-URI\":\"https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Book%3A_Electromagnetics_I_(Ellingson)/05%3A_Electrostatics/5.13%3A_Electric_Potential_Field_due_to_a_Continuous_Distribution_of_Charge\",\"WARC-Payload-Digest\":\"sha1:6YZ2BH4T2FRKGNB5OHUIEEGGPCB7FMNG\",\"WARC-Block-Digest\":\"sha1:OBTYSFAJFYMAXCGU3XR7JL2YJT2AZT7M\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-51/CC-MAIN-2019-51_segments_1575540547536.49_warc_CC-MAIN-20191212232450-20191213020450-00030.warc.gz\"}"}
https://fr.mathworks.com/help/mpc/ref/mpcstate.html	[ "Documentation\n\n### This is machine translation\n\nMouseover text to see original. Click the button below to return to the English version of the page.\n\nNote: This page has been translated by MathWorks. Click here to see\nTo view all translated materials including this page, select Country from the country navigator on the bottom of this page.\n\n# mpcstate\n\nDefine MPC controller state\n\n## Syntax\n\n```xmpc = mpcstate(MPCobj) xmpc = mpcstate(MPCobj,xp,xd,xn,u,p) xmpc = mpcstate ```\n\n## Description\n\n`xmpc = mpcstate(MPCobj)` creates a controller state object compatible with the controller object, `MPCobj`, in which all fields are set to their default values that are associated with the controller’s nominal operating point.\n\n`xmpc = mpcstate(MPCobj,xp,xd,xn,u,p)` sets the state fields of the controller state object to specified values. The controller may be an implicit or explicit controller object. Use this controller state object to initialize an MPC controller at a specific state other than the default state.\n\n`xmpc = mpcstate` returns an `mpcstate` object in which all fields are empty.\n\n`mpcstate` objects are updated by `mpcmove` through the internal state observer based on the extended prediction model. The overall state is updated from the measured output ym(k) by a linear state observer (see State Observer).\n\n## Input Arguments\n\n `MPCobj` MPC controller, specified as either a traditional MPC controller (`mpc`) or explicit MPC controller (`generateExplicitMPC`). `xp` Plant model state estimates, specified as a vector with Nxp elements, where Nxp is the number of states in the plant model. `xd` Disturbance model state estimates, specified as a vector with Nxd elements, where Nxd is the total number of states in the input and output disturbance models. The disturbance model states are ordered such that input disturbance model states are followed by output disturbance model state estimates. `xn` Measurement noise model state estimates, specified as a vector with Nxn elements, where Nxn is the number of states in the measurement noise model. `u` Values of the manipulated variables during the previous control interval, specified as a vector with Nu elements, where Nu is the number of manipulated variables. `p` Covariance matrix for the state estimates, specified as an N-by-N matrix, where N is the sum of Nxp, Nxd and Nxn).\n\n## Output Arguments\n\n`xmpc`\n\nMPC state object, containing the following properties.\n\nProperty\n\nDescription\n\n`Plant`\n\nVector of state estimates for the controller’s plant model. Values are in engineering units and are absolute, i.e., they include state offsets.\n\nIf the controller’s plant model includes delays, the `Plant` field of the MPC state object includes states that model the delays. Therefore `length(Plant)` > order of undelayed controller plant model.\n\nDefault: controller’s `Model.Nominal.X` property.\n\n`Disturbance`\n\nVector of unmeasured disturbance model state estimates. This comprises the states of the input disturbance model followed by the states of the output disturbances model.\n\nDisturbance models may be created by default. Use the `getindist`and `getoutdist`commands to view the two disturbance model structures.\n\nDefault: zero, or empty if there are no disturbance model states.\n\n`Noise`\n\nVector of output measurement noise model state estimates.\n\nDefault: zero, or empty if there are no noise model states.\n\n`LastMove`\n\nVector of manipulated variables used in the previous control interval, u(k–1). Values are absolute, i.e., they include manipulated variable offsets.\n\nDefault: nominal values of the manipulated variables.\n\n`Covariance`\n\nn-by-n symmetrical covariance matrix for the controller state estimates, where n is the dimension of the extended controller state, i.e., the sum of the number states contained in the `Plant`, `Disturbance`, and `Noise` fields.\n\nDefault: If the controller is employing default state estimation the default is the steady-state covariance computed according to the assumptions in Controller State Estimation. See also the description of the `P` matrix in the Control System Toolbox `kalmd` command. If the controller is employing custom state estimation, this field is empty (not used).\n\n## Examples\n\ncollapse all\n\nCreate a Model Predictive Controller for a single-input-single-output (SISO) plant. For this example, the plant includes an input delay of 0.4 time units, and the control interval to 0.2 time units.\n\n```H = tf(1,[10 1],'InputDelay',0.4); MPCobj = mpc(H,0.2);```\n```-->The \"PredictionHorizon\" property of \"mpc\" object is empty. Trying PredictionHorizon = 10. -->The \"ControlHorizon\" property of the \"mpc\" object is empty. Assuming 2. -->The \"Weights.ManipulatedVariables\" property of \"mpc\" object is empty. Assuming default 0.00000. -->The \"Weights.ManipulatedVariablesRate\" property of \"mpc\" object is empty. Assuming default 0.10000. -->The \"Weights.OutputVariables\" property of \"mpc\" object is empty. Assuming default 1.00000. ```\n\nCreate the corresponding controller state object in which all states are at their default values.\n\n`xMPC = mpcstate(MPCobj)`\n```-->Converting the \"Model.Plant\" property of \"mpc\" object to state-space. -->Converting model to discrete time. -->Converting delays to states. -->Assuming output disturbance added to measured output channel #1 is integrated white noise. -->The \"Model.Noise\" property of the \"mpc\" object is empty. Assuming white noise on each measured output channel. MPCSTATE object with fields Plant: [0 0 0] Disturbance: 0 Noise: [1x0 double] LastMove: 0 Covariance: [4x4 double] ```\n\nThe plant model, `H`, is a first-order, continuous-time transfer function. The `Plant` property of the `mpcstate` object contains two additional states to model the two intervals of delay. Also, by default the controller contains a first-order output disturbance model (an integrator) and an empty measured output noise model.\n\nView the default covariance matrix.\n\n`xMPC.Covariance`\n```ans = 4×4 0.0624 0.0000 0.0000 -0.0224 0.0000 1.0000 -0.0000 -0.0000 0.0000 -0.0000 1.0000 0.0000 -0.0224 -0.0000 0.0000 0.2301 ```\n\n## See Also\n\n#### Implementing an Adaptive Cruise Controller with Simulink\n\nDownload technical paper" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8341904,"math_prob":0.9115717,"size":1984,"snap":"2019-13-2019-22","text_gpt3_token_len":460,"char_repetition_ratio":0.16414142,"word_repetition_ratio":0.071428575,"special_character_ratio":0.19455644,"punctuation_ratio":0.111428574,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9796032,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-05-23T11:04:48Z\",\"WARC-Record-ID\":\"<urn:uuid:30aefe91-79ad-4432-983e-ded291ed6105>\",\"Content-Length\":\"88564\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:8508444d-539b-4503-b50a-d0fe660b3974>\",\"WARC-Concurrent-To\":\"<urn:uuid:9d165aaf-14da-469e-8bf8-6d0633896735>\",\"WARC-IP-Address\":\"23.218.145.211\",\"WARC-Target-URI\":\"https://fr.mathworks.com/help/mpc/ref/mpcstate.html\",\"WARC-Payload-Digest\":\"sha1:CX4CWBSZDXUUYNBFI75BO4ZGXNSCJEYP\",\"WARC-Block-Digest\":\"sha1:YMKTDNWHBBBE6DE5TA5PVRBLB7QEGZ24\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-22/CC-MAIN-2019-22_segments_1558232257243.19_warc_CC-MAIN-20190523103802-20190523125802-00227.warc.gz\"}"}
http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=1382.msg5185	[ "NTNUJAVA Virtual Physics LaboratoryEnjoy the fun of physics with simulations! Backup site http://enjoy.phy.ntnu.edu.tw/ntnujava/", null, "", null, "January 29, 2020, 10:55:30 am", null, "", null, "Welcome, Guest. Please login or register.Did you miss your activation email? 1 Hour 1 Day 1 Week 1 Month Forever Login with username, password and session length", null, "Discovery consists of seeing what everybody has seen and thinking what nobody has thought. ...\"Albert von Szent-Gyorgyi(1893-1986, 1937 Nobel Prize for Medicine, Lived to 93)\"\n Pages: Go Down", null, "Author Topic: RC circuit + magnetic induction (B field) (Read 9129 times) 0 Members and 1 Guest are viewing this topic. Click", null, "to toggle author information(expand message area).\nFu-Kwun Hwang\nHero Member", null, "", null, "", null, "", null, "", null, "", null, "Offline\n\nPosts: 3085", null, "", null, "", null, "« Embed this message on: December 12, 2009, 01:26:11 pm » posted from:Taipei,T\\'ai-pei,Taiwan", null, "", null, "A capacitor has been charged to", null, "$V_o$ , a metal bar with mass m ,resistor R is located between two parallel wire (distance L) in magnetic field B, as shown in the above figure.\nAt t=0; the switch was turn from a to b.\nCurrent I will flow throught metal bar, which F=I L B will accelerate meta bar.", null, "$m\\frac{dv}{dt}=I L B$.\nWhen the metal bar is moving, the changing in magnetic flux will induce voltage", null, "$V_i= B L v$\nSo the equation for the loop is", null, "$Vc= \\frac{Q_c}{C} = I R + B L v$, where", null, "$I=-\\frac{dQ_c}{dt}$", null, "$\\frac{1}{C} \\frac{dQ_c}{dt}=-I =R \\frac{dI}{dt}+B L \\frac{dv}{dt}= R \\frac{dI}{dt}+B L \\frac{BLI}{m}$\n,", null, "$\\frac{dI}{dt}=-(\\frac{1}{RC}+\\frac{B^2L^2}{mR}) I$\nthe solution is", null, "$I(t)=\\frac{V_o}{R}(1-e^{-\\alpha t})$ ,where", null, "$\\alpha=\\frac{1}{RC}+\\frac{B^2L^2}{mR}$\n\nThe following is a simulation for the above case:\nThe charge", null, "$Q_c(t)$, the velocity", null, "$v(t)$ and the current", null, "$I(t)$ are shown (C=1 in the calculation).\n\nEmbed a running copy of this simulation\n\nEmbed a running copy link(show simulation in a popuped window)\nFull screen applet or Problem viewing java?Add http://www.phy.ntnu.edu.tw/ to exception site list", null, "", null, "", null, "Logged\n Related Topics Subject Started by Replies Views Last post", null, "", null, "magnetic induction communication system Electromagnetics Taona 5 11531", null, "May 04, 2009, 12:38:50 am by Taona", null, "", null, "Individual and Resultant Force Magnetic Field of a wire and external mag. field Request for physics Simulations lookang 8 18516", null, "December 10, 2009, 10:42:37 pm by Fu-Kwun Hwang", null, "", null, "RC circuit + magnetic induction (B field) electromagnetism ahmedelshfie 0 4699", null, "May 25, 2010, 06:48:45 pm by ahmedelshfie", null, "", null, "Falling Magnetic Field Line Collaborative Community of EJS engrg1 3 5483", null, "September 10, 2012, 08:42:57 pm by lookang", null, "", null, "Magnetic field H Electromagnetics itctrans 0 1646", null, "March 07, 2015, 03:26:51 am by itctrans\nPage created in 0.941 seconds with 22 queries.", null, "since 2011/06/15" ]	[ null, "http://www.phy.ntnu.edu.tw/ntnujava/Themes/default/images/rss.png", null, "http://www.phy.ntnu.edu.tw/demolab/smf/Themes/default/images/smflogo.gif", null, "http://www.phy.ntnu.edu.tw/demolab/smf/Themes/default/images/upshrink.gif", null, "http://www.phy.ntnu.edu.tw/ntnujava/logo.gif", null, "http://www.phy.ntnu.edu.tw/demolab/smf/Themes/default/images/filter.gif", null, "http://www.phy.ntnu.edu.tw/demolab/smf/Themes/default/images/topic/normal_post.gif", null, 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