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https://math.stackexchange.com/questions/3118542/combining-simultaneous-linear-recurrence-relations	[ "# Combining simultaneous linear recurrence relations\n\nI was studying the sequence A249665, which I will call $$a_n$$, and came up with a second sequence $$b_n$$ for which I could prove that the following recurrence holds. \\begin{align} a_n&=a_{n-1}+a_{n-4}+a_{n-5}+b_{n-2}+b_{n-3} \\\\b_n&=a_{n-1}+a_{n-2}+a_{n-3}+a_{n-4}+b_{n-2}+b_{n-3}+b_{n-4} \\end{align} Here are some initial values. \\begin{align} (a_0,...)&=(0,1,1,1,2,6,14,28,56,118,254,541,1140,...) \\\\(b_0,...)&=(0,0,1,2,4,8,17,37,79,166,349,738,1563,...) \\end{align}\n\nOn the OEIS they give a linear recurrence in three sequence, so when I saw that I felt good about myself for finding one in only two sequences. However a bit further on, they give a recurrence in $$a_n$$ only, which caught me by surprise. $$a_n=2a_{n-1}-a_{n-2}+2a_{n-3}+a_{n-4}+a_{n-5}-a_{n-7}-a_{n-8}$$ The companion matrix of this recurrence has the same dimension as the one for my recurrence, which leads me to believe there should be some method to find this recurrence out of mine.\n\nSo is there a general method to combine two simultaneous linear recurrences into one? Or can you at least find a way to deduce the one from OEIS from the simultaneous one I found?\n\nNote that from the first recurrence formula we get $$b_n+b_{n-1}=a_{n+2}-a_{n+1}-a_{n-2}-a_{n-3}$$. We find \\begin{align} a_n &=a_{n-1}+a_{n-4}+a_{n-5}+b_{n-2}+b_{n-3} \\\\&=a_{n-1}+a_{n-4}+a_{n-5}+b_{n-3} \\\\&+(a_{n-3}+a_{n-4}+a_{n-5}+a_{n-6}+b_{n-4}+b_{n-5}+b_{n-6}) \\\\&=a_{n-1}+a_{n-3}+2a_{n-4}+2a_{n-5}+a_{n-6} \\\\&+(b_{n-3}+b_{n-4})+(b_{n-5}+b_{n-6}) \\\\&=a_{n-1}+a_{n-3}+2a_{n-4}+2a_{n-5}+a_{n-6} \\\\&+(a_{n-1}-a_{n-2}-a_{n-5}-a_{n-6})+(a_{n-3}-a_{n-4}-a_{n-7}-a_{n-8}) \\\\&=2a_{n-1}-a_{n-2}+2a_{n-3}+a_{n-4}+a_{n-5}-a_{n-7}-a_{n-8}. \\end{align}\nTo get an idea of the general approach, consider the problem of finding a recurrence for $$b_n$$. By applying the first recurrence formula on every $$a$$ term in the second, we obtain \\begin{align} b_n &=b_{n-2}+b_{n-3}+b_{n-4} \\\\&+(a_{n-2}+a_{n-5}+a_{n-6}+b_{n-3}+b_{n-4}) \\\\&+(a_{n-3}+a_{n-6}+a_{n-7}+b_{n-4}+b_{n-5}) \\\\&+(a_{n-4}+a_{n-7}+a_{n-8}+b_{n-5}+b_{n-6}) \\\\&+(a_{n-5}+a_{n-8}+a_{n-9}+b_{n-6}+b_{n-7}). \\end{align} By the second recurrence formula we get $$a_n+a_{n-1}+a_{n-2}+a_{n-3}=b_{n+1}-b_{n-1}-b_{n-2}-b_{n-3}$$, giving \\begin{align} b_n &=b_{n-2}+2b_{n-3}+3b_{n-4}+2b_{n-5}+2b_{n-6}+b_{n-7} \\\\&+(b_{n-1}-b_{n-3}-b_{n-4}-b_{n-5}) \\\\&+(b_{n-4}-b_{n-6}-b_{n-7}-b_{n-8}) \\\\&+(b_{n-5}-b_{n-7}-b_{n-8}-b_{n-9}) \\\\&=b_{n-1}+b_{n-2}+b_{n-3}+3b_{n-4}+2b_{n-5}+b_{n-6}-b_{n-7}-2b_{n-8}-b_{n-9}. \\end{align}\nIf you want to practice this youself, I advise to try and come up with a recurrence formula for the partial sums of fibonacci numbers, so give a recurrence formula for $$s_n$$ when $$f_n=f_{n-1}+f_{n-2}$$ and $$s_n=s_{n-1}+f_n$$." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.90082514,"math_prob":0.9998667,"size":1132,"snap":"2022-27-2022-33","text_gpt3_token_len":407,"char_repetition_ratio":0.14007092,"word_repetition_ratio":0.0,"special_character_ratio":0.37897527,"punctuation_ratio":0.17966102,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000052,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-08-12T05:10:44Z\",\"WARC-Record-ID\":\"<urn:uuid:37ccb325-1e0e-439d-a7a6-dc978cc17387>\",\"Content-Length\":\"220162\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2fbd30f8-a65c-41dd-8ae0-a5347cf216ba>\",\"WARC-Concurrent-To\":\"<urn:uuid:c7b35eb3-e7ed-4fe5-a8e6-d87fb01a3b44>\",\"WARC-IP-Address\":\"151.101.65.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/3118542/combining-simultaneous-linear-recurrence-relations\",\"WARC-Payload-Digest\":\"sha1:BWPHCVW5VQGM6Z2BY3EWTKWQHW3TEA7I\",\"WARC-Block-Digest\":\"sha1:3XN7Y7JC4BUQTM7JTE5ZJZOZPETOSY7F\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-33/CC-MAIN-2022-33_segments_1659882571584.72_warc_CC-MAIN-20220812045352-20220812075352-00299.warc.gz\"}"}
http://montmirail.info/geometry-worksheets-for-elementary-students/geometry-worksheets-elementary-students-basic-geometrical-ideas-for-school/	[ "# Geometry Worksheets Elementary Students Basic Geometrical Ideas For School", null, "geometry worksheets elementary students basic geometrical ideas for school.\n\ngeometry worksheets elementary students for school ,geometry worksheets for second grade with kids elementary school on students ,library worksheets for elementary geometry school students ,geometry worksheets for elementary school students ,geometry worksheets elementary students for school grade 7 math, geometry worksheets elementary students for school multiplication,geometry worksheets elementary students grade 4 maths and shapes first properties of for school,measurement worksheets third grade geometry elementary students for school,elementary geometry worksheets for all download and printable school students, geometry worksheets for elementary school lines angles worksheet by all in one students." ]	[ null, "http://montmirail.info/wp-content/uploads/2019/11/geometry-worksheets-elementary-students-basic-geometrical-ideas-for-school.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.91359454,"math_prob":0.5734088,"size":787,"snap":"2019-51-2020-05","text_gpt3_token_len":116,"char_repetition_ratio":0.25159642,"word_repetition_ratio":0.05319149,"special_character_ratio":0.13977128,"punctuation_ratio":0.09821428,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9602264,"pos_list":[0,1,2],"im_url_duplicate_count":[null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-12-14T13:18:59Z\",\"WARC-Record-ID\":\"<urn:uuid:2e67d3d4-df9e-4e49-95c9-074fa4d08b20>\",\"Content-Length\":\"45139\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0a73eb1e-68fe-452f-98ee-ff251e9ce81c>\",\"WARC-Concurrent-To\":\"<urn:uuid:f753802c-beeb-41ac-a4f8-55acd432d142>\",\"WARC-IP-Address\":\"104.31.90.180\",\"WARC-Target-URI\":\"http://montmirail.info/geometry-worksheets-for-elementary-students/geometry-worksheets-elementary-students-basic-geometrical-ideas-for-school/\",\"WARC-Payload-Digest\":\"sha1:WIHTCNCCYFSX6JH2N4NTVPJXGSJUNI4J\",\"WARC-Block-Digest\":\"sha1:7XIDYVZCEREOBDJ34ADGNPWQU74GAB4I\",\"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-20191214122253-20191214150253-00276.warc.gz\"}"}
https://virtualnerd.com/algebra-2/linear-equations-functions/graphing-equations/graph-lines-find-points/slope-formula-missing-x-coordinate	[ "# How Do You Find the X-Coordinate of a Point on a Line If You Have Another Point and the Slope?\n\n### Note:\n\nHow do you find the x-coordinate of a point on a line if you have another point and the slope? You'll need to use the slope formula. Watch this tutorial and see how to find this missing coordinate!\n\n### Keywords:\n\n• problem\n• slope\n• find coordinate\n• coordinate\n• x coordinate\n• slope formula\n• point\n• find point" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9269219,"math_prob":0.6817075,"size":2035,"snap":"2020-34-2020-40","text_gpt3_token_len":442,"char_repetition_ratio":0.16838996,"word_repetition_ratio":0.029569892,"special_character_ratio":0.2113022,"punctuation_ratio":0.092857145,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99706966,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-09-22T21:34:45Z\",\"WARC-Record-ID\":\"<urn:uuid:bf3fd864-fddc-470b-a933-fa1796770f2d>\",\"Content-Length\":\"36584\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:adfc3f8d-5421-4534-b61f-61fb26cb6be1>\",\"WARC-Concurrent-To\":\"<urn:uuid:d56ef843-d9cc-4967-98b5-ce1932d967c7>\",\"WARC-IP-Address\":\"52.85.144.13\",\"WARC-Target-URI\":\"https://virtualnerd.com/algebra-2/linear-equations-functions/graphing-equations/graph-lines-find-points/slope-formula-missing-x-coordinate\",\"WARC-Payload-Digest\":\"sha1:ANQPR2676O5OW5JZYKHGRSP67NBTM7EJ\",\"WARC-Block-Digest\":\"sha1:FCN2VYJUCLUA352Z6NPASDDPW5IAYX77\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600400206763.24_warc_CC-MAIN-20200922192512-20200922222512-00571.warc.gz\"}"}
http://mathcs.chapman.edu/~jipsen/structures/doku.php/cyclic_involutive_fl-algebras	[ "## Cyclic involutive FL-algebras\n\nAbbreviation: CyInFL\n\n### Definition\n\nA cyclic involutive FL-algebra or cyclic involutive residuated lattice is a structure $\\mathbf{A}=\\langle A, \\vee, \\wedge, \\cdot, 1, \\sim\\rangle$ of type $\\langle 2, 2, 2, 0, 1\\rangle$ such that\n\n$\\langle A, \\vee, \\wedge\\rangle$ is a lattice\n\n$\\langle A, \\cdot, 1\\rangle$ is a monoid\n\n$\\sim$ is an involution: ${\\sim}{\\sim}x=x$ and\n\n$xy\\le z\\iff x\\le {\\sim}(y({\\sim}z))\\iff y\\le {\\sim}(({\\sim}z)x)$\n\n##### Morphisms\n\nLet $\\mathbf{A}$ and $\\mathbf{B}$ be involutive residuated lattices. A morphism from $\\mathbf{A}$ to $\\mathbf{B}$ is a function $h:A\\rightarrow B$ that is a homomorphism: $h(x \\vee y)=h(x) \\vee h(y)$, $h(x \\cdot y)=h(x) \\cdot h(y)$, $h({\\sim}x)={\\sim}h(x)$ and $h(1)=1$.\n\n### Definition\n\nAn involutive FL-algebra is an FL-algebra $\\mathbf{A}=\\langle A,\\vee,\\wedge,\\cdot,1,\\backslash,/,0\\rangle$ such that\n\ncyclic involution holds: $(0/x)\\backslash 0=x=0/(x\\backslash 0)$ and $0/x=x\\backslash 0$\n\nExample 1:\n\n### Properties\n\nClasstype Value Decidable 1) No $\\infty$ Yes Yes No\n\n### Finite members\n\n$\\begin{array}{lr} f(1)= &1\\\\ f(2)= &1\\\\ f(3)= &2\\\\ f(4)= &9\\\\ f(5)= &21\\\\ \\end{array}$ $\\begin{array}{lr} f(6)= &101\\\\ f(7)= &279\\\\ f(8)= &1433\\\\ f(9)= &\\\\ f(10)= &\\\\ \\end{array}$\n\n### Superclasses\n\ninvolutive FL-algebras supervariety\n\n##### Toolbox", null, "" ]	[ null, "http://mathcs.chapman.edu/~jipsen/structures/lib/exe/indexer.php", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5499483,"math_prob":0.9999982,"size":1355,"snap":"2019-35-2019-39","text_gpt3_token_len":530,"char_repetition_ratio":0.15544042,"word_repetition_ratio":0.0,"special_character_ratio":0.3601476,"punctuation_ratio":0.11313868,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99999714,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-09-22T05:43:53Z\",\"WARC-Record-ID\":\"<urn:uuid:f90ddf3d-65dd-453c-9ef6-7756da7e4068>\",\"Content-Length\":\"19354\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c5e86656-cb40-4460-924f-7a9b6368f4d9>\",\"WARC-Concurrent-To\":\"<urn:uuid:a94b99ae-7107-460e-a7c2-11dce16a4562>\",\"WARC-IP-Address\":\"192.77.116.236\",\"WARC-Target-URI\":\"http://mathcs.chapman.edu/~jipsen/structures/doku.php/cyclic_involutive_fl-algebras\",\"WARC-Payload-Digest\":\"sha1:ACMEKCNO7MQ3BLQD2EWE6IGT6Q7YFLYV\",\"WARC-Block-Digest\":\"sha1:XCX7NILB2VXP7TNFHFF2DWANJQ3PTXMW\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-39/CC-MAIN-2019-39_segments_1568514575168.82_warc_CC-MAIN-20190922053242-20190922075242-00055.warc.gz\"}"}
https://www.semanticscholar.org/paper/Levy-processes-and-stochastic-integrals-in-Banach-Applebaum/5ac84bb00f1626513112d707e2617badb73f6047	[ "• Corpus ID: 56302065\n\n# Levy processes and stochastic integrals in Banach spaces\n\n@inproceedings{Applebaum2007LevyPA,\ntitle={Levy processes and stochastic integrals in Banach spaces},\nauthor={David Applebaum},\nyear={2007}\n}\nWe review in¯nite divisibility and Levy processes in Banach spaces and discuss the relationship with notions of type and cotype. The Levy-It^o decomposition is described. Strong, weak and Pettis-style notions of stochastic integral are introduced and applied to construct generalised Ornstein-Uhlenbeck processes.\n\n### Maximal Inequalities of the Itô Integral with Respect to Poisson Random Measures or Lévy Processes on Banach Spaces\n\nWe are interested in maximal inequalities satisfied by a stochastic integral driven by a Poisson random measure in a general Banach space.\n\n### Martingales and stochastic calculus in Banach spaces\n\nIn this thesis we study martingales and stochastic integration of processes with values in UMD Banach spaces.\n\n### Stochastic Reaction-diffusion Equations Driven by Jump Processes\n\n• Mathematics\n• 2018\nWe establish the existence of weak martingale solutions to a class of second order parabolic stochastic partial differential equations. The equations are driven by multiplicative jump type noise,\n\n### The Itô integral for a certain class of Lévy processes and its application to stochastic partial differential equations\n\nStochastic integration with respect to a Wiener process in Banach spaces have been considered by several authors (Brzeźniak , Dettweiler , Neidhart and Van Neerven, Veraar and Weiss\n\n### Local characteristics and tangency of vector-valued martingales\n\nThis paper is devoted to tangent martingales in Banach spaces. We provide the definition of tangency through local characteristics, basic $L^p$- and $\\phi$-estimates, a precise construction of a\n\n### Itô isomorphisms for $L^{p}$-valued Poisson stochastic integrals\n\nMotivated by the study of existence, uniqueness and regularity of solutions to stochastic partial differential equations driven by jump noise, we prove It\\^{o} isomorphisms for $L^p$-valued" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.769464,"math_prob":0.9072584,"size":5146,"snap":"2022-40-2023-06","text_gpt3_token_len":1173,"char_repetition_ratio":0.1831972,"word_repetition_ratio":0.031988874,"special_character_ratio":0.19335406,"punctuation_ratio":0.06931964,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97383404,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-09-29T03:00:08Z\",\"WARC-Record-ID\":\"<urn:uuid:8fc4fe8d-0086-41d0-9494-408d3791d7be>\",\"Content-Length\":\"321557\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:aabe3296-0cc4-4852-9024-9389c76414be>\",\"WARC-Concurrent-To\":\"<urn:uuid:2aee57f9-8631-4cc7-8e25-bafca51f19ed>\",\"WARC-IP-Address\":\"18.154.227.127\",\"WARC-Target-URI\":\"https://www.semanticscholar.org/paper/Levy-processes-and-stochastic-integrals-in-Banach-Applebaum/5ac84bb00f1626513112d707e2617badb73f6047\",\"WARC-Payload-Digest\":\"sha1:2OSYOTPNLFGTOPDPW5WTZKYCPHIMMOTJ\",\"WARC-Block-Digest\":\"sha1:7DHE4DOK6YJRRMHN6EYSHFVLFNPOG6KM\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030335303.67_warc_CC-MAIN-20220929003121-20220929033121-00750.warc.gz\"}"}
https://docs.splunk.com/Documentation/Splunk/7.1.2/Search/Calculatesizesofdynamicfields	[ "Splunk Enterprise version 7.1 is no longer supported as of October 31, 2020. See the Splunk Software Support Policy for details. For information about upgrading to a supported version, see How to upgrade Splunk Enterprise.", null, "Download topic as PDF\n\n# Calculate sizes of dynamic fields\n\nThis search determines which fields in your events consume the most disk space, without any prior knowledge of field names and number of events.\n\n## Scenario\n\n```index=_internal earliest=-15m latest=now\n\| fieldsummary\n\| rex field=values max_match=0 \"value\\\":\\\"(?<values>[^\\\"])\\\",\"\n\| mvexpand values\n\| eval bytes=len(values)\n\| rex field=field \"^(?!date\|punct\|host\|hostip\|index\|linecount\|source\|sourcetype\|timeendpos\|timestartpos\|splunk_server)(?<FieldName>.)\"\n\| stats count sum(bytes) as SumOfBytesInField values(values) as Values max(bytes) as MaxFieldLengthInBytes by FieldName\n\| rename count as NumOfValuesPerField\n\| eventstats sum(NumOfValuesPerField) as TotalEvents sum(SumOfBytesInField) as TotalBytes\n\| eval PercentOfTotalEvents=round(NumOfValuesPerField/TotalEvents100,2)\n\| eval PercentOfTotalBytes=round(SumOfBytesInField/TotalBytes100,2)\n\| eval ConsumedMB=SumOfBytesInField/1024/1024\n\| eval TotalMB=TotalBytes/1024/1024\n\| table FieldName NumOfValuesPerField SumOfBytesInField ConsumedMB PercentOfTotalBytes PercentOfTotalEvents\n\| sort - PercentOfTotalEvents\n```\n\nThe results appear on the Statistics tab and look something like this:\n\nFieldName NumValuesPerField SumOfBytesInField ConsumedMB PercentOfTotalBytes PercentOfTotalEvents\nTotals 1802 45700 0.0436 99.87 100.29\ncumulative_hits 100 587 0.0006 1.28 5.55\neps 100 1862 0.0018 4.07 5.55\nkb 100 1159 0.0011 2.54 5.55\nkbps 100 1881 0.0018 4.12 5.55\nreq_time 100 3000 0.0029 6.56 5.55\nuri 100 10559 0.0101 23.11 5.55\nuri_query 100 3532 0.0034 7.73 5.55\nmessage 96 11633 0.0111 25.46 5.33\navg_age 76 280 0.0012 2.80 4.22\nev 62 140 0.0001 0.31 3.44\naverage_kbps 59 1071 0.0010 2.34 3.27\n\nThe totals do not add up to exactly 100% because this search is adding up the values after the values have been rounded. To avoid this, you can move the rounding to the end of the search string.\n\n## Walkthrough\n\nLet's walk through each part of the search.\n\n1. The example begins with a search to retrieve all events in `index=_internal` within the last 15 minutes.\n```index=_internal earliest=-15m latest=now\n```\n\nNote: You can replace this with any search string and time range.\n\n2. Next, add the the fieldsummary command to create a summary of all the fields in the previously retrieved events.\n```\| fieldsummary\n```\n\nThe results appear on the Statistics tab and look something like this:\n\nfield count distinct _count is _exact max mean min numeric _count stdev values\nabandoned _channels 29 1 1 0.0 0.00 0.0 29 0.00 [{\"value\":\"0\",\"count\":29}]\nactive 29 1 1 0.0 0.00 0.0 29 0.00 [{\"value\":\"0\",\"count\":29}]\nactive _hist _searches 31 2 1 1.0 0.13 0.0 31 0.34 [{\"value\":\"0\",\"count\":27},\n\n{\"value\":\"1\",\"count\":4}]\n\naverage _kbps 87 59 1 0.3 0.21 0.0 87 0.15 [{\"value\":\"0\",\"count\":29},\n\n{\"value\":\"0.31239045073685034\",\"count\":1}, {\"value\":\"0.31240549380412547\",\"count\":1}, {\"value\":\"0.3124557194522294\",\"count\":1}, {\"value\":\"0.3124707607545469\",\"count\":1}]\n\n3. The values of each field are extracted with a regex into a multivalue field called values, and then expanded. The length of each value is calculated in bytes.\n```\| rex field=values max_match=0 \"value\\\":\\\"(?<values>[^\\\"])\\\",\"\n\| mvexpand values\n\| eval bytes=len(values)\n```\n4. The values of the field are extracted with another regex, with some exceptions.\n```\| rex field=field \"^(?!date\|punct\|host\|hostip\|index\|linecount\|source\|sourcetype\|timeendpos\|timestartpos\|splunk_server)(?<FieldName>.)\"\n```\n5. The `stats` command is used to perform multiple calculations using stats functions, including the count and the sum of the bytes (SumOfBytesInField). The values function is used to returns the list of all distinct values of the `values` field as a multivalue entry (Values). The max function calculates the maximum field length in bytes (MaxFieldLengthInBytes). The results are organized by field name.\n```\| stats count sum(bytes) as SumOfBytesInField values(values) as Values max(bytes) as MaxFieldLengthInBytes by FieldName\n\| rename count as NumOfValuesPerField\n```\n6. The `eventstats` command is used to calculate several sums, the number of values in each field (TotalEvents) and the sum of the bytes in each field (Total Bytes).\n```\| eventstats sum(NumOfValuesPerField) as TotalEvents sum(SumOfBytesInField) as TotalBytes\n```\n7. Several `eval` commands are run to calculate the percentage of total events, the percentage of total bytes, the megabytes consumed, and the total megabytes.\n```\| eval PercentOfTotalEvents=round(NumOfValuesPerField/TotalEvents100,2)\n\| eval PercentOfTotalBytes=round(SumOfBytesInField/TotalBytes100,2)\n\| eval ConsumedMB=SumOfBytesInField/1024/1024\n\| eval TotalMB=TotalBytes/1024/1024\n```\n8. The `table` command is used to display on a specific set of fields. The `addfoltotals` command is used to calculate the total for each column. The `sort` command is used sort the list in descending order by the PercentageOfTotalEvents field.\n```\| table FieldName NumberOfValuesPerField SumOfBytesInField ConsumedMB PercentageOfTotalBytes PercentageOfTotalEvents\n\| sort - PercentageOfTotalEvents\n```\n\nThe results appear on the Statistics tab and look something like this:\n\nFieldName NumValuesPerField SumOfBytesInField ConsumedMB PercentOfTotalBytes PercentOfTotalEvents\nTotals 1802 45700 0.0436 99.87 100.29\ncumulative_hits 100 587 0.0006 1.28 5.55\neps 100 1862 0.0018 4.07 5.55\nkb 100 1159 0.0011 2.54 5.55\nkbps 100 1881 0.0018 4.12 5.55\nreq_time 100 3000 0.0029 6.56 5.55\nuri 100 10559 0.0101 23.11 5.55\nuri_query 100 3532 0.0034 7.73 5.55\nmessage 96 11633 0.0111 25.46 5.33\navg_age 76 280 0.0012 2.80 4.22\nev 62 140 0.0001 0.31 3.44\naverage_kbps 59 1071 0.0010 2.34 3.27" ]	[ null, "https://docs.splunk.com/skins/OxfordComma/images/acrobat-logo.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5977451,"math_prob":0.94171774,"size":6463,"snap":"2022-27-2022-33","text_gpt3_token_len":2505,"char_repetition_ratio":0.13794705,"word_repetition_ratio":0.33047736,"special_character_ratio":0.39935014,"punctuation_ratio":0.2635659,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95160556,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-07-01T23:12:16Z\",\"WARC-Record-ID\":\"<urn:uuid:be8de164-ebb6-4a45-b4a9-911cdd0a19d7>\",\"Content-Length\":\"193468\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:dd52572f-6feb-435b-a912-319e981a7914>\",\"WARC-Concurrent-To\":\"<urn:uuid:d142f5f9-f731-4045-b206-0e7a2f5a6e33>\",\"WARC-IP-Address\":\"44.242.93.73\",\"WARC-Target-URI\":\"https://docs.splunk.com/Documentation/Splunk/7.1.2/Search/Calculatesizesofdynamicfields\",\"WARC-Payload-Digest\":\"sha1:J32WH6LJZ2XT3XCYPKV3YAWSTDBYLV4H\",\"WARC-Block-Digest\":\"sha1:6LXJ7UU7RM4EJ462B35E72OCRTT7YYGR\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656103947269.55_warc_CC-MAIN-20220701220150-20220702010150-00672.warc.gz\"}"}
https://thedesigningfairy.com/a-probability-distribution-for-all-possible-values-of-a-sample-statistic-is-known-as/	[ "## What Is a Sampling Distribution?\n\nThe sampling circulation of a statistic is the circulation of the statistic because that all feasible samples indigenous the same population of a provided size.\n\nYou are watching: A probability distribution for all possible values of a sample statistic is known as\n\n### Key Takeaways\n\nKey PointsA an important part that inferential statistics requires determining how much sample statistics are most likely to vary from each other and from the population parameter.The sampling circulation of a statistic is the circulation of the statistic, considered as a random variable, when acquired from a arbitrarily sample of size extn.Sampling distributions allow analytical considerations come be based on the sampling circulation of a statistic quite than on the joint probability distribution of every the separation, personal, instance sample values.The sampling circulation depends on: the underlying circulation of the population, the statistic gift considered, the sampling procedure employed, and the sample size used.Key Termsinferential statistics: A branch of math that involves illustration conclusions about a populace based on sample data attracted from it.sampling distribution: The probability circulation of a provided statistic based upon a arbitrarily sample.\n\nSuppose friend randomly sampled 10 women between the ages of 21 and also 35 year from the populace of women in Houston, Texas, and then computed the mean height of her sample. You would certainly not mean your sample average to be same to the mean of all ladies in Houston. It can be somewhat reduced or higher, but it would certainly not same the populace mean exactly. Similarly, if you take it a 2nd sample that 10 women from the very same population, you would not suppose the median of this second sample to equal the typical of the first sample.\n\nHouston Skyline: expect you randomly sampled 10 civilization from the populace of females in Houston, Texas between the eras of 21 and also 35 years and computed the mean elevation of your sample. You would not expect your sample typical to be same to the typical of all females in Houston.\n\nInferential statistics entails generalizing from a sample come a population. A an important part that inferential statistics requires determining how much sample statistics are most likely to differ from each other and from the populace parameter. These determinations are based on sampling distributions. The sampling circulation of a statistic is the circulation of that statistic, thought about as a random variable, when obtained from a random sample of size extn. It might be taken into consideration as the circulation of the statistic because that all possible samples from the same population of a offered size. Sampling distributions enable analytical considerations to be based on the sampling circulation of a statistic fairly than top top the joint probability circulation of every the separation, personal, instance sample values.\n\nThe sampling circulation depends on: the underlying distribution of the population, the statistic being considered, the sampling procedure employed, and the sample size used. For example, take into consideration a normal populace with mean mu and also variance sigma. Assume we repetitively take samples the a provided size native this population and calculate the arithmetic mean for every sample. This statistic is then dubbed the sample mean. Each sample has actually its own mean value, and the distribution of this averages is dubbed the “sampling distribution of the sample mean. ” This circulation is normal since the underlying populace is normal, although sampling distributions may also often it is in close come normal also when the population distribution is not.\n\nAn alternative to the sample mean is the sample median. As soon as calculated from the very same population, it has actually a different sampling distribution to that of the mean and is usually not normal (but it might be nearby for huge sample sizes).\n\n## Properties of Sampling Distributions\n\nKnowledge that the sampling circulation can be really useful in making inferences about the in its entirety population.\n\n### Learning Objectives\n\nDescribe the general properties of sampling distributions and the use of traditional error in evaluating them\n\n### Key Takeaways\n\nKey PointsIn practice, one will collect sample data and, from this data, calculation parameters that the population distribution.Knowing the level to which method from various samples would certainly differ from every other and also from the population mean would give you a feeling of just how close your specific sample typical is likely to be to the populace mean.The conventional deviation of the sampling distribution of a statistic is described as the conventional error of the quantity.If all the sample method were really close to the populace mean, climate the typical error of the mean would it is in small.On the various other hand, if the sample means varied considerably, climate the conventional error the the average would it is in large.Key Termsinferential statistics: A branch of math that involves illustration conclusions about a population based top top sample data drawn from it.sampling distribution: The probability distribution of a provided statistic based upon a random sample.\n\n### Sampling Distributions and Inferential Statistics\n\nSampling distributions are important for inferential statistics. In practice, one will collect sample data and, from these data, calculation parameters of the populace distribution. Thus, knowledge of the sampling circulation can be very useful in do inferences about the overall population.\n\nFor example, discovering the degree to which way from different samples differ from every other and from the populace mean would give you a feeling of how close your specific sample median is likely to it is in to the population mean. Fortunately, this info is directly available from a sampling distribution. The most usual measure of exactly how much sample means differ native each various other is the typical deviation the the sampling distribution of the mean. This typical deviation is called the typical error of the mean.\n\n### Standard Error\n\nThe traditional deviation the the sampling circulation of a statistic is referred to as the typical error of the quantity. For the instance where the statistic is the sample mean, and samples are uncorrelated, the standard error is:\n\ndisplaystyle extSE_\bar extx= frac extssqrt extn\n\nWhere exts is the sample standard deviation and also extn is the dimension (number of items) in the sample. Vital implication the this formula is that the sample size must be quadrupled (multiplied by 4) come achieve fifty percent the measurement error. When creating statistical studies where cost is a factor, this may have actually a role in expertise cost-benefit tradeoffs.\n\nIf all the sample method were really close come the population mean, climate the standard error the the median would it is in small. ~ above the various other hand, if the sample means varied considerably, climate the typical error that the median would it is in large. To it is in specific, assume her sample mean is 125 and also you approximated that the conventional error that the mean is 5. If you had a regular distribution, then it would certainly be likely that your sample mean would be in ~ 10 systems of the population mean since most that a normal distribution is within two standard deviations of the mean.\n\n### More nature of Sampling Distributions\n\nThe in its entirety shape that the distribution is symmetric and approximately normal.There room no outliers or other crucial deviations indigenous the as whole pattern.The facility of the circulation is really close come the true population mean.\n\nA statistics study can be claimed to be biased when one outcome is systematically favored end another. However, the study can be claimed to it is in unbiased if the average of the sampling circulation is equal to the true value of the parameter gift estimated.\n\nFinally, the variability of a statistic is defined by the spread out of that sampling distribution. This spread out is determined by the sampling design and the size of the sample. Bigger samples offer smaller spread. As lengthy as the populace is much bigger than the sample (at least 10 times together large), the spread out of the sampling circulation is approximately the very same for any populace size\n\n## Creating a Sampling Distribution\n\nLearn to produce a sampling distribution from a discrete set of data.\n\n### Key Takeaways\n\nKey PointsConsider three pool balls, each v a number on it.Two of the balls are selected randomly (with replacement), and also the median of their numbers is computed.The family member frequencies are equal to the frequencies separated by nine since there space nine feasible outcomes.The distribution produced from these loved one frequencies is referred to as the sampling distribution of the mean.As the number of samples approaches infinity, the frequency distribution will approach the sampling distribution.Key Termssampling distribution: The probability circulation of a given statistic based upon a arbitrarily sample.frequency distribution: a representation, one of two people in a graphical or tabular format, which screens the number of observations in ~ a given interval\n\nWe will illustrate the principle of sampling distributions through a simple example. Take into consideration three pool balls, each through a number top top it. Two of the balls are selected randomly (with replacement), and the median of your numbers is computed. All feasible outcomes are shown below.\n\nPool Ball instance 1: This table reflects all the feasible outcome of choosing two swimming pool balls randomly from a populace of three.\n\nNotice the all the way are one of two people 1.0, 1.5, 2.0, 2.5, or 3.0. The frequencies the these method are presented below. The relative frequencies room equal to the frequencies divided by nine because there are nine feasible outcomes.\n\nPool Ball example 2: This table shows the frequency of method for extN=2.\n\nThe figure listed below shows a loved one frequency circulation of the means. This distribution is likewise a probability distribution because the exty-axis is the probability the obtaining a given mean native a sample of 2 balls in addition to being the relative frequency.\n\nRelative Frequency Distribution: family member frequency distribution of our pool sphere example.\n\nThe distribution shown in the above figure is called the sampling circulation of the mean. Specifically, the is the sampling circulation of the median for a sample dimension of 2 ( extN=2). Because that this basic example, the circulation of pool balls and also the sampling circulation are both discrete distributions. The pool balls have only the number 1, 2, and 3, and also a sample mean deserve to have one of only five possible values.\n\nThere is one alternative means of conceptualizing a sampling circulation that will certainly be valuable for more complicated distributions. Imagine that two balls room sampled (with replacement), and also the mean of the two balls is computed and also recorded. This procedure is recurring for a second sample, a third sample, and also eventually countless samples. After thousands of samples space taken and also the average is computed because that each, a relative frequency distribution is drawn. The an ext samples, the closer the relative frequency distribution will concerned the sampling distribution displayed in the over figure. Together the variety of samples philosophies infinity, the frequency circulation will method the sampling distribution. This means that girlfriend can develop of a sampling circulation as being a frequency distribution based upon a very huge number that samples. To it is in strictly correct, the sampling distribution only amounts to the frequency distribution exactly when there is an infinite variety of samples.\n\n## Continuous Sampling Distributions\n\nWhen we have actually a truly constant distribution, that is not only impractical but actually difficult to enumerate all possible outcomes.\n\n### Key Takeaways\n\nKey PointsIn continuous distributions, the probability that obtaining any single value is zero.Therefore, these worths are referred to as probability densities rather than probabilities.A probability thickness function, or thickness of a constant random variable, is a function that describes the family member likelihood for this random variable to take it on a offered value.Key Termsprobability density function: any role whose integral end a collection gives the probability the a random variable has a value in that set\n\nIn the ahead section, we created a sampling distribution out of a populace consisting of three pool balls. This distribution was discrete, since there to be a finite variety of possible observations. Currently we will consider sampling distributions as soon as the population distribution is continuous.\n\nWhat if we had a thousand swimming pool balls through numbers varying from 0.001 to 1.000 in equal steps? keep in mind that although this circulation is not really continuous, that is close enough to it is in considered constant for useful purposes. As before, we space interested in the circulation of the means we would acquire if we sampled two balls and also computed the median of this two. In the previous example, we began by computer the mean for every of the nine feasible outcomes. This would acquire a bit tedious because that our present example because there room 1,000,000 possible outcomes (1,000 for the very first ball multiplied by 1,000 because that the second.) Therefore, the is more convenient to usage our 2nd conceptualization the sampling distributions, i beg your pardon conceives of sampling distributions in terms of loved one frequency distributions — special, the loved one frequency distribution that would take place if samples of 2 balls were continuously taken and also the typical of each sample computed.\n\n### Probability density Function\n\nWhen we have a truly continuous distribution, the is not only impractical but actually difficult to enumerate all possible outcomes. Moreover, in consistent distributions, the probability that obtaining any single value is zero. Therefore, these values are dubbed probability densities quite than probabilities.\n\nA probability thickness function, or density of a consistent random variable, is a duty that explains the relative likelihood for this random variable to take it on a given value. The probability for the random variable to loss within a particular region is offered by the integral the this variable’s thickness over the region.", null, "Probability density Function: Boxplot and also probability density duty of a normal distribution extN(0, 2).\n\n### Key Takeaways\n\nKey PointsStatistical evaluation are an extremely often pertained to with the difference in between means.The mean of the sampling distribution of the average is μM1−M2 = μ1−2.The variance sum legislation states the the variance of the sampling circulation of the distinction between method is equal to the variance of the sampling distribution of the mean for population 1 to add the variance the the sampling circulation of the typical for population 2.Key Termssampling distribution: The probability distribution of a given statistic based on a random sample.\n\nStatistical analyses are, an extremely often, involved with the difference in between means. A common example is an experiment design to to compare the mean of a regulate group v the mean of an speculative group. Inferential statistics offered in the analysis of this kind of experiment depend on the sampling distribution of the difference in between means.\n\nThe sampling distribution of the distinction between way can be believed of as the circulation that would result if we recurring the complying with three actions over and over again:\n\nSample n1 scores from populace 1 and also n2 scores from populace 2;Compute the method of the 2 samples ( M1 and also M2);Compute the distinction between way M1M2. The circulation of the distinctions between means is the sampling distribution of the difference between means.\n\nThe average of the sampling circulation of the typical is:\n\nμM1−M2 = μ1−2,\n\nwhich claims that the median of the circulation of differences in between sample way is same to the difference between population means. Because that example, speak that median test score of all 12-year olds in a population is 34 and the typical of 10-year olds is 25. If many samples were taken from each period group and the mean distinction computed every time, the median of these many differences in between sample method would it is in 34 – 25 = 9.\n\nThe variance sum regulation states that the variance of the sampling distribution of the distinction between way is equal to the variance of the sampling distribution of the typical for population 1 add to the variance of the sampling distribution of the average for population 2. The formula for the variance that the sampling distribution of the difference between method is as follows:\n\nsigma _ extM _ 1 - extM ^ 2 _ 2 =frac sigma _ extM _ 1 ^ 2 extn _ 1 +frac sigma _ extM _ 2 ^ 2 extn _ 2 .\n\nRecall that the typical error the a sampling distribution is the traditional deviation of the sampling distribution, which is the square source of the over variance.\n\nLet’s look in ~ an applications of this formula to develop a sampling circulation of the difference between means. Assume there room two types of environment-friendly beings ~ above Mars. The mean elevation of species 1 is 32, if the mean elevation of varieties 2 is 22. The variances of the two species are 60 and also 70, respectively, and also the heights the both types are generally distributed. You randomly sample 10 members of types 1 and 14 members of types 2.\n\nThe distinction between means comes the end to be 10, and also the conventional error comes out to it is in 3.317.\n\nμM1−M2 = 32 – 22 = 10\n\nStandard error equates to the square root of (60 / 10) + (70 / 14) = 3.317.\n\nThe resulting sampling circulation as diagramed in, is normally dispersed with a median of 10 and also a conventional deviation of 3.317.\n\nSampling distribution of the Difference between Means: The circulation is normally dispersed with a typical of 10 and also a traditional deviation the 3.317.\n\n### Key Takeaways\n\nKey PointsThe principle of the form of a distribution refers come the form of a probability distribution.It most regularly arises in inquiries of finding an ideal distribution to use to version the statistical properties that a population, given a sample from the population.A sampling circulation is assumed to have actually no outliers or other crucial deviations from the overall pattern.When calculated native the very same population, the sample median has a different sampling distribution to that of the mean and is normally not normal; although, it might be nearby for huge sample sizes.Key Termsnormal distribution: A household of consistent probability distributions such the the probability density role is the normal (or Gaussian) function.skewed: Biased or distorted (pertaining to statistics or information).Pareto Distribution: The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution that is offered in summary of social, scientific, geophysical, actuarial, and also many other types of observable phenomena.probability distribution: A duty of a discrete arbitrarily variable yielding the probability that the variable will have actually a given value.\n\nThe “shape that a distribution” describes the shape of a probability distribution. The most frequently arises in questions of recognize an proper distribution to use in stimulate to version the statistical properties the a population, offered a sample from the population. The form of a distribution will fall somewhere in a continuum where a flat distribution might be considered central; and also where varieties of departure from this include:\n\nmounded (or unimodal)u-shapedj-shapedreverse-j-shapedmulti-modal\n\nThe shape of a distribution is sometimes characterized by the actions of the tails (as in a long or quick tail). Because that example, a flat distribution can be claimed either to have actually no tails or to have brief tails. A normal distribution is usually related to as having quick tails, when a Pareto distribution has lengthy tails. Even in the reasonably simple case of a mounded distribution, the distribution may be skewed to the left or skewed to the best (with symmetric corresponding to no skew).\n\nAs formerly mentioned, the all at once shape the a sampling distribution is meant to it is in symmetric and approximately normal. This is as result of the fact, or assumption, the there room no outliers or other important deviations indigenous the in its entirety pattern. This fact holds true as soon as we repetitively take samples the a provided size from a population and calculate the arithmetic average for every sample.\n\nAn alternate to the sample mean is the sample median. Once calculated native the very same population, it has actually a various sampling circulation to the of the mean and is typically not normal; although, it might be close for large sample sizes.", null, "The normal Distribution: Sample distributions, as soon as the sampling statistic is the mean, are typically expected to screen a regular distribution.\n\n### Key Takeaways\n\nKey PointsThe normal distribution has the same mean as the original distribution and also a variance that equates to the original variance separated by extn, the sample size. extn is the number of values that space averaged together not the number of times the experiment is done.The usefulness that the to organize is that the sampling distribution approaches normality regardless of the shape of the population distribution.Key Termssampling distribution: The probability circulation of a provided statistic based upon a random sample.central border theorem: The theorem the states: If the sum of independent identically distributed random variables has a finite variance, then it will certainly be (approximately) typically distributed.\n\nThe central limit theorem claims that, given particular conditions, the mean of a sufficiently big number the independent random variables, each with a well-defined mean and well-defined variance, will be (approximately) usually distributed. The central limit theorem has a variety of variants. In its typical form, the arbitrarily variables have to be identically distributed. In variants, convergence of the average to the normal distribution likewise occurs because that non-identical distributions, given that lock comply with particular conditions.\n\nThe main limit theorem for sample way specifically states that if girlfriend keep illustration larger and also larger samples (like rojo 1, 2, 5, and, finally, 10 dice) and calculating their means the sample means form their very own normal distribution (the sampling distribution). The normal circulation has the same typical as the original distribution and also a variance that equates to the original variance separated by extn, the sample size. extn is the number of values that room averaged with each other not the number of times the experiment is done.\n\n### Classical central Limit Theorem\n\nConsider a succession of independent and also identically spread random variables drawn from distributions of expected values given by mu and finite variances offered by sigma^2. Mean we space interested in the sample median of these arbitrarily variables. Through the law of large numbers, the sample averages converge in probability and nearly surely to the meant value mu together extn ightarrow infty. The classical central limit theorem defines the size and the distributional kind of the stochastic fluctuations about the deterministic number mu throughout this convergence. An ext precisely, it states that together extn gets larger, the circulation of the difference in between the sample median extS_ extn and also its limit mu almost right the normal distribution with average 0 and also variance sigma^2. For large enough extn, the distribution of extS_ extn is close come the normal circulation with mean mu and variance\n\ndisplaystyle frac sigma ^ 2 extn\n\nThe upshot is that the sampling circulation of the typical approaches a normal distribution as extn, the sample size, increases. The usefulness of the to organize is the the sampling circulation approaches normality regardless of the form of the populace distribution.\n\nSee more: Nfl Mock Draft 2017 Eagles, Some New Names Enter Mix For Eagles\n\nEmpirical main Limit Theorem: This figure demonstrates the central limit theorem. The sample way are generated using a random number generator, i beg your pardon draws numbers in between 1 and also 100 from a uniform probability distribution. That illustrates that increasing sample sizes result in the 500 measured sample way being an ext closely distributed around the population mean (50 in this case)." ]	[ null, "https://thedesigningfairy.com/a-probability-distribution-for-all-possible-values-of-a-sample-statistic-is-known-as/imager_1_8565_700.jpg", null, "https://thedesigningfairy.com/a-probability-distribution-for-all-possible-values-of-a-sample-statistic-is-known-as/imager_2_8565_700.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9412136,"math_prob":0.9898094,"size":20840,"snap":"2022-27-2022-33","text_gpt3_token_len":3890,"char_repetition_ratio":0.22297946,"word_repetition_ratio":0.0977169,"special_character_ratio":0.18397313,"punctuation_ratio":0.083493546,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99520475,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-06-29T04:10:27Z\",\"WARC-Record-ID\":\"<urn:uuid:d691ef2e-04be-4cbd-8898-42306d712afd>\",\"Content-Length\":\"36966\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:61287dda-728a-4f66-8784-5d2a164302a1>\",\"WARC-Concurrent-To\":\"<urn:uuid:f1b53fcd-1bf3-41cb-a589-d4333d53943e>\",\"WARC-IP-Address\":\"172.67.207.251\",\"WARC-Target-URI\":\"https://thedesigningfairy.com/a-probability-distribution-for-all-possible-values-of-a-sample-statistic-is-known-as/\",\"WARC-Payload-Digest\":\"sha1:4QGH2AQRWJOZ46IQOJXMIDGXOENILD3W\",\"WARC-Block-Digest\":\"sha1:5RKEL5JYR2V7MW2YLPNHEOC4PBH4TMYW\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656103620968.33_warc_CC-MAIN-20220629024217-20220629054217-00744.warc.gz\"}"}
https://math.stackexchange.com/questions/574308/proving-inequality-with-the-greatest-integer-function	[ "Proving Inequality with the Greatest Integer Function\n\nShow that\n\n$$[(m+n)x]+[(m+n)y] \\ge [mx+(n-1)y]+[my+(n-1)x]$$\n\nwhere $m,~n \\in \\Bbb{N}$ and $0\\le x,~y < 1$.\n\nI've tried everything for about half a day and still couldn't figure it out. Actually I don't know if it is true so should I say, 'show that if it is right or not'. This question was aroused from the question, show that\n\n$$\\frac{(5m)!(5n)!}{m!n!(3m+n)!(3n+m)!}$$\n\nis an integer where $m,~n$ are natural numbers. This question was so hard but I was able to prove it by seperating some cases. Then I began to wonder if I could generalize this, which is actually the same question I proposed above. At first, I thought I could replace $n-1$ into plain $n$, but I did find a counter-example for that case. So now I'm wondering if the very first proposition holds since for now I couldn't find any counter-example. So I'm guessing the inequality might be true.\n\nPlease help me! The question is killing me! It would be also glad if someone could suggest any idea to solve the second question in a simple(or brilliant) manner.\n\nEDIT The factorial question can be reduced to proving the given inequality for the special case when $m=3,~n=2$ as Ewan Delanoy had explained well below. This implies that proving the inequality is just the same as proving that\n\n$$\\frac{(mx)!(ny)!}{x!y!(mx+(n-1)y)!(my+(n-1)x)!}$$\n\nis an integer. We can also think about generalizing this question into proving for the case where the $x,~y$ in the denominator is something like $ax,~by$. But first I'd just be happy to know how to prove (or show counter-example) the first inequality I suggested.\n\n• Actually I don't need 'official' source as shown below. – Taxxi Nov 22 '13 at 7:46\n• What is the function $[\\cdot]$ in the desired inequality? From context I'm guessing that you mean the floor function $\\lfloor t \\rfloor$ = greatest integer $k \\leq t$; is that right? The title of the question says \"Gaussian Function\" but that's usually a multiple of $\\exp (-C(t-t_0)^2)$ for some real $C>0$ and $t_0$, and that can't be what you mean here. – Noam D. Elkies Nov 25 '13 at 17:31\n• @NoamD.Elkies I didn't know that for my short knowledge. Thanks for letting me know. – Taxxi Nov 26 '13 at 4:06\n\nLet $m=6,n=3,x=0.10,y=0.21$. $$\\lfloor(m+n)x\\rfloor+\\lfloor(m+n)y\\rfloor=\\lfloor0.90\\rfloor+\\lfloor1.89\\rfloor=0+1=1.$$ $$\\lfloor mx+(n-1)y\\rfloor+\\lfloor my+(n-1)x\\rfloor=\\lfloor1.02\\rfloor+\\lfloor1.46\\rfloor=1+1=2.$$\n\nI do not know if your inequality holds for every $m,n$, but I know two things :\n\n• It is true for $m=3,n=2$.\n• The case $m=3,n=2$ suffices to solve your second question.\n\nTo show that $r=\\frac{(5n)!(5m)!}{n!m!(n+3m)!(m+3n)!}$ is an integer, it suffices to show that the valuation $v_p(r)$ is nonnegative for every prime $p$. But by Legendre's theorem,\n\n$$v_p(n!)=\\sum_{k=1}^{\\infty} \\lfloor \\frac{x}{p^k} \\rfloor \\tag{1}$$\n\nSo it will suffice to show the following :\n\n$$\\lfloor \\frac{5n}{p^k} \\rfloor+ \\lfloor \\frac{5m}{p^k} \\rfloor \\geq \\lfloor \\frac{n}{p^k} \\rfloor+ \\lfloor \\frac{m}{p^k} \\rfloor+ \\lfloor \\frac{n+3m}{p^k} \\rfloor+ \\lfloor \\frac{m+3n}{p^k} \\rfloor \\tag{2}$$\n\nIf we put $x=\\frac{n}{p^k}$ and $y=\\frac{n}{p^k}$, (2) is equivalent to $d(x,y) \\geq 0$ where\n\n$$d(x,y)=\\big(\\lfloor 5x \\rfloor + \\lfloor 5y \\rfloor\\big)- \\big(\\lfloor x \\rfloor + \\lfloor y \\rfloor+ \\lfloor x+3y \\rfloor + \\lfloor y+3x \\rfloor \\big) \\tag{3}$$\n\nNow $d$ is $1$-periodic in both variables $x$ and $y$, so it will suffice to show that $d(x,y)\\geq 0$ when $x$ and $y$ are both in $[0,1)$. In which case (3) reduces to\n\n$$\\lfloor 5x \\rfloor + \\lfloor 5y \\rfloor \\geq \\lfloor x+3y \\rfloor + \\lfloor y+3x \\rfloor \\ (\\ \\text{for} \\ x,y\\in [0,1)) \\tag{4}$$\n\nUPDATE 11/22/2013\n\nLet us put\n\n$$i=\\lfloor 5x \\rfloor, j=\\lfloor 5y \\rfloor, k=\\lfloor x+3y \\rfloor, l=\\lfloor 3x+y \\rfloor \\tag{5}$$\n\nand also\n\n$$\\begin{array}{lcl} \\mu&=&i+j-(k+l), \\\\ \\varepsilon&=&{\\sf min}(i+1-5x,j+1-5y,k+1-(x+3y),l+1-(3x+y)) > 0. \\\\ \\end{array} \\tag{6}$$\n\nWe then have\n\n$$\\begin{array}{lcl} 4(i+j)-5(k+l) &\\geq& 4(5x+5y-2+2\\varepsilon)-5(4x+4y)=-8+8\\varepsilon \\\\ 3i+j-5l &\\geq& 3(5x-1+\\varepsilon)+(5y-1+\\varepsilon)-5(3x+y)=-4+4\\varepsilon \\\\ i+3j-5k &\\geq& (5y-1+\\varepsilon)+3(5y-1+\\varepsilon)-5(x+3y)=-4+4\\varepsilon \\end{array}$$\n\nSince the left-hand sides are all integers, we deduce\n\n$$\\begin{array}{lclc} 4(i+j)-5(k+l) &\\geq& -7 & (7) \\\\ 3i+j-5l &\\geq& -3 & (8) \\\\ i+3j-5k &\\geq& -3 & (9) \\end{array}$$\n\nWe deduce from (7) that $4(i+j)-4(k+l) \\geq -7$, so $\\mu \\geq -\\frac{7}{4}$. Since $\\mu$ is an integer, we must have $\\mu \\geq -1$. So the only thing we need to show is that $\\mu\\neq -1$. Suppose, by contradiction, that $\\mu=(-1)$. Then $l=i+j-k+1$, and (8) and (9) can be rewritten\n\n$$\\begin{array}{lclc} -2i-4j+5k &\\geq& 2 & (8') \\\\ -4i-2j+5l &\\geq& 2 & (9') \\end{array}$$\n\nSo $k$ and $l$ are both $\\geq\\frac{2}{5}$ ; we deduce\n\n$$k\\geq 1, l \\geq 1 \\tag{10}$$\n\nsince they are integers. Consider then the numbers\n\n$$t_1=3i+j-5l+3, \\ t_2=i+3j-5k+3, \\ t_3=k-1, \\ t_4=l-1$$\n\nThey are all nonnegative by (8), (9) and (10), but at the same time their sum is $4(i+j-(k+l))=-4$ and this is a contradiction, as wished.\n\n• Yes, I also used that method and without loss of generality, assumed $5x \\ge x+3y$. Then I had to check only the case when $5y \\le y+3x$. It was tedious but I was able to prove it by seperating cases like this. But for the problem I suggested, it was just too tedious and too confusing so I just got stuck... That was the problem. Any suggestions about proving it? – Taxxi Nov 20 '13 at 18:15\n• @TaxxiDriver Please see my update. – Ewan Delanoy Nov 22 '13 at 16:11\n• Sorry for late replying I was sick. I checked your proof but it was actually not much better than just seperating some cases. I also thought about using your proof to prove the question I raised, but I still couldn't find a way to use it. I'm sad :( – Taxxi Nov 26 '13 at 4:10" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.95614094,"math_prob":0.9995512,"size":1567,"snap":"2019-43-2019-47","text_gpt3_token_len":436,"char_repetition_ratio":0.11452335,"word_repetition_ratio":0.0,"special_character_ratio":0.27504787,"punctuation_ratio":0.11304348,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9998965,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-16T05:31:30Z\",\"WARC-Record-ID\":\"<urn:uuid:47211b3a-0595-41f8-a2d2-e498083fa57e>\",\"Content-Length\":\"154430\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:85daf0bb-b109-4a8d-876a-577b9893ef20>\",\"WARC-Concurrent-To\":\"<urn:uuid:d1233908-0ea1-4177-a7fa-9624369a0d9a>\",\"WARC-IP-Address\":\"151.101.129.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/574308/proving-inequality-with-the-greatest-integer-function\",\"WARC-Payload-Digest\":\"sha1:CP3XJFR43COYBEZ4SGCXIV4B47HJESRS\",\"WARC-Block-Digest\":\"sha1:X5ZVXFKAJSJEBVB3D7ZIETWVLRLGHBQJ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570986664662.15_warc_CC-MAIN-20191016041344-20191016064844-00005.warc.gz\"}"}
https://wiki.openmrs.org/exportword?pageId=34146770	[ "Date: Sun, 20 Oct 2019 09:56:26 +0000 (UTC) Message-ID: <[email protected]> Subject: Exported From Confluence MIME-Version: 1.0 Content-Type: multipart/related; boundary=\"----=_Part_14627_535347903.1571565386443\" ------=_Part_14627_535347903.1571565386443 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable Content-Location: file:///C:/exported.html 2011 Sprint Schedule History\n\n2011 Sprint Schedule History\n\nThis page lists off the sprints completed in 2011. There is a di= fferent page for the current Sprin= t Schedule\n\n=20\n=20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 = =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 = =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 = =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 = =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20\n\nFrom\n\nTo\n\nTopic\n\nDec 26\n\nJan 1\n\nIndividual work\n\nYou\n\nPlanning notes: 26th-1st will only have half= devs\n\nDec 12\n\nDec 25\n\nBen\n\n=20\n\nDec 5\n\nDec 11\n\nWyclif / Darius\n\nAddress 1.9 alpha bugs, and get the beta rel= eased\n\nNov 21\n\nDec 4\n\nTBD\n\n=20\n\nAddress 1.9 alpha bugs, and get the beta rel= eased\n\nNov 7\n\nNov 20\n\nTBD\n\nDocumentation, tooling, and automation to he= lp us release and test releases faster.\n\nOct 17\n\nNov 6\n\n1.9 alpha release\n\nOct 10\n\nOct 16\n\n--\n\n--\n\nOpenMRS Implementer's Conference\n\nSept 19\n\nOct 9\n\n(Three weeks) 1.9 features, fixes, code revi= ew, cleanup. Get alpha out the door in mid-October.\n\nSept 5\n\nSept 18\n\n=EF=BB=BFUnknown Us= er (bwolfe) and =EF=BB=BFUnknown User = (mcunderlik)\n\nFixes and features for the =EF=BB=BFSync Module\n\nAug 29\n\nSept 4\n\n=EF=BB=BFUnknown Us= er (darius)\n\nA 1 week push to polish off work on Visits a= nd Providers\n\nAug 15\n\nAug 28\n\nFinal features and fixes for the SMART Conta= iner module\nTickets/features/bugs related= to the upcoming 1.9 release\n\nAug 1\n\n=\n\nAug 14\n\nA push on features and bug fixes to get a be= ta release of the module and server published\n\nJul 18\n\nJul 31\n\n=EF=BB=BFUnknown Us= er (darius)\n\nImprovements to 2.x UI Framework\n\nJul 5\n\n=\n\nJul 17\n\n=EF=BB=BF1.9 Feature: Web Services\n\n=EF=BB=BFUnknown Us= er (bwolfe)\n\nSprint on web services to finalize user stor= ies and prepare the initial alpha module release\n\nJun 20\n\nJul 3\n\n=\n\n=EF=BB=BFUnknown Us= er (darius)\n\nWork on =EF=BB=BFAPI Support for Orde= r Entry (Design Page)\n\n(catchup week, no sprint. finish reviews, et= c from previous)\n\nApply volunteer patches, document, etc\n\nMay 30\n\nJun 12\n\n=EF=BB=BFUnknown Us= er (bwolfe)\n\nMay 16\n\nMay 29\n\n=EF=BB=BF1.9 Feature: Web Services\n\n=EF=BB=BFUnknown Us= er (bwolfe)\n\nMay 2\n\nMay 15\n\n=EF=BB=BFBundled Module: XForms\n\n=EF=BB=BFDaniel = Kayiwa\n\nAll about the =EF=BB=BFXForms Module\n\nApril 18\n\nMay 1\n\n=EF=BB=BFUnknown Us= er (darius)\n\nWork on =EF=BB=BFAdditional Attributes on Concept Map, = =EF=BB=BFSupport for V= isits (Design Page) and =EF=BB=BFMultiple providers per encounter (Design Page)\n\nApril 11\n\nApril 17\n\n=EF=BB=BFUnknown Us= er (bwolfe)\n\nTop-voted bug fixes, primarily from OpenMRS = core/trunk\n\nApril 4\n\nApril 10\n\n=EF=BB=BF2.x Dashboard Fragments\n\nLearn the 2.x UI Framework by writing patien= t dashboard fragments\n\nMarch 21\n\nApril 3\n\n------=_Part_14627_535347903.1571565386443--" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5901625,"math_prob":0.93367624,"size":3360,"snap":"2019-43-2019-47","text_gpt3_token_len":1369,"char_repetition_ratio":0.23122765,"word_repetition_ratio":0.12339743,"special_character_ratio":0.45416668,"punctuation_ratio":0.08219178,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99999964,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-20T09:56:26Z\",\"WARC-Record-ID\":\"<urn:uuid:6487c7f9-85c3-4052-9bdb-ced751576bde>\",\"Content-Length\":\"23660\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ffc2d36b-2eee-42bd-88be-3350d89da336>\",\"WARC-Concurrent-To\":\"<urn:uuid:14f0f91c-00b3-4dd5-8667-16e71cf1bce4>\",\"WARC-IP-Address\":\"149.165.169.4\",\"WARC-Target-URI\":\"https://wiki.openmrs.org/exportword?pageId=34146770\",\"WARC-Payload-Digest\":\"sha1:IR6F4MMM2EHPDQIJ5UML3HY2U6LILYLH\",\"WARC-Block-Digest\":\"sha1:YSJ6ZVJ3YVJMLGACYZXTQ53WTPBSBIIE\",\"WARC-Identified-Payload-Type\":\"message/rfc822\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570986705411.60_warc_CC-MAIN-20191020081806-20191020105306-00435.warc.gz\"}"}
http://mohamaddia.me/	[ "## Biography", null, "[email protected] I am currently a data scientist and machine learning course developer at the EPFL Extension School, where I am involved in the production of online learning materials, designing and leading workshops and hackathons, and promoting the EPFL Extension School data science and digital skills to a wide audience. I received the B.E. degree in Electrical and Computer Engineering and the B.A. degree in Economics with distinctions concurrently in 2012 from the American University of Beirut (AUB), Lebanon. I received the M.Sc. degree in Communication Systems in 2014 and the Ph.D. degree in Computer and Communication Sciences in 2018 both from the Swiss Federal Institute of Technology (EPFL), Switzerland. Between 2018 and 2020, I was a research scientist at the Institute for Data Science (i4DS) in the University of Applied Sciences Northwestern Switzerland (FHNW), where I worked in collaboration with the European Space Agency (ESA). I was a visiting researcher at Nokia Bell Labs, Germany in 2017 where I worked on the design of a novel coding scheme for the high-speed fiber optical communication systems. I was also a R&D engineer at the European Technology Center of SONY, Germany in 2014 working on the standardization of digital terrestrial TV broadcasting receivers. In 2011, I did my undergraduate internship at the University of California, Berkeley U.S.A. where I contributed to the “Mobile Millennium” traffic-monitoring project. My CV is available here.\n\n## Research Interests", null, "My research interests lie at the interface between statistical inference, machine learning, coding theory, and statistical physics of spin glasses. I was involved in developing new data science tools and applying deep learning techniques for the “Euclid” space mission project in order to investigate dark matter. My PhD research at EPFL's Information Processing Group (IPG) was principally focused on inference and learning over graphical models, which includes problems from error-correcting codes, compressed sensing, and community detection. My work spans both the practical and theoretical aspects of such problems. This covers the design and analysis of optimal low-complexity message-passing algorithms, the application of statistical physics methods, the derivation of information theoretic limits, and the development of rigorous proof techniques. (The wordcloud is based on my research statement 2018 - powered by wordclouds.com).\n\n#### Astronomical Data Processing - Euclid Space Mission\n\nThe stunning discovery of the accelerated expansion rate of the universe in the late 1990s, as opposed to the former prevailing belief on the decelerated expansion, has changed the modern perception of the cosmos and presented several challenges in astrophysics. Such acceleration can be attributed to the presence of a mysterious invisible “dark matter” inducing a repulsive gravitational force; so that Einstein's general relativity continues to hold on the cosmological scale. “Euclid” is the first satellite, scheduled for launch by the ESA (European Space Agency), to map the geometry of dark matter. It will provide images of 2 billion galaxies with unprecedented quality. The Euclid consortium includes 1400 scientists across Europe and the USA. My work in the astroinformatics group covers the crucial pre-launch period (2017-2020) with a focus on software and algorithmic development for the scientific ground-segment activities. My research within Euclid revolves around solving inverse problems in order to investigate dark matter using high-spatial resolution imagery. This includes the development of new data science tools and the application of deep learning techniques to find patterns in cosmic structure.\n\n#### Statistical Physics - Phase Transitions and Rigorous Predictions\n\nOver the last century, statistical physics techniques have developed with the aim to describe the behaviour of systems with a large number of degrees of freedom and to give predictions which would be very difficult to guess. One of these techniques is the Replica method, which was conjectured to predict the asymptotic mutual information of a random graphical model and to detect the algorithmic and optimal phase transitions (see figure below). We prove that the Replica formula is exact in many problems that have been studied in the context of error correcting codes, compressed sensing and machine learning (mainly the random linear estimation and low-rank matrix factorization problems). Hence, we are able to come up with rigorous information-theoretical limits for many open problems. Moreover, we prove that, for a large set of parameters, an efficient iterative algorithm called Approximate Message-Passing (AMP) is optimal in the Bayesian setting. Our proof technique has an interest of its own as it is transposable to various inference problems and it exploits three essential ingredients: the Guerra-interpolation method introduced in statistical physics, the analysis of the AMP algorithm through State Evolution (SE) and the theory of spatial coupling and threshold saturation in coding.", null, "#### Spatial Coupling - Algorithmic Tools and Proof Techniques\n\nSpatial coupling is a powerful graphical representation used to improve the algorithmic message-passing performance. It is the underlying principle behind the threshold saturation phenomenon (where the algoritmic threshold achieves the optimal one). Such representation was successfully applied to multiple graphical models ranging from LDPC codes to compressed sensing. Spatial coupling can be represented via a graphical model starting from the original factor graph. Assume that we have a factor graph of size $$N$$. We take several instances of this factor graph and we place them next to each other on a chain of length $$Γ$$. We then locally couple the underlying factor graphs with a coupling window $$w$$ to obtain a bigger factor graph of size $$Γ × N$$ (see figure below). In the resulting factor graph, each variable node is connected to the corresponding check nodes of the same underlying factor graph and to the check nodes of the neighboring factor graphs. This construction creates a spatial dimension, along the positions of the chain, that will help the algorithm. The second step in constructing efficient spatially coupled graphs is to introduce a seed at a certain position of the chain. This seed can be introduced as a side information which helps the algorithm at the boundaries and initiates a “wave” that propagates inwards and boosts the performance. Interestingly, spatial coupling can be used both as a “construction technique” to boost the algorithmic performance and as a “proof technique” to compute some information theoretic quantities. Therefore, even if the problem at hand does not provide the freedom of constructing a spatially coupled model in practice, one can still use spatial coupling for an auxiliary model. Intuitively speaking, since the low-complexity algorithm on the auxiliary model is optimal by the threshold saturation phenomenon, it is easier to compute the information theoretic quantities on that model and then apply them to the underlying model.", null, "## Selected Publications\n\n#### Journals\n\nNote: Authors are listed in alphabetical and/or affiliation order." ]	[ null, "http://mohamaddia.me/profile.png", null, "http://mohamaddia.me/Dia_wordcloud2.jpg", null, "http://mohamaddia.me/phaseTran1.png", null, "http://mohamaddia.me/spaCoup1.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9183721,"math_prob":0.7178688,"size":5810,"snap":"2020-45-2020-50","text_gpt3_token_len":1078,"char_repetition_ratio":0.116259046,"word_repetition_ratio":0.0046457606,"special_character_ratio":0.17607573,"punctuation_ratio":0.072186835,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.958176,"pos_list":[0,1,2,3,4,5,6,7,8],"im_url_duplicate_count":[null,3,null,3,null,3,null,3,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-11-27T00:41:23Z\",\"WARC-Record-ID\":\"<urn:uuid:49ac8b8a-1733-4ddb-a55c-0018b6181a93>\",\"Content-Length\":\"16960\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:73b5e6b4-aad6-47f9-adf5-c689e3370871>\",\"WARC-Concurrent-To\":\"<urn:uuid:e1297e5c-a09d-4d36-bea5-0f850bc3a647>\",\"WARC-IP-Address\":\"192.30.252.153\",\"WARC-Target-URI\":\"http://mohamaddia.me/\",\"WARC-Payload-Digest\":\"sha1:JMFABTYMO75DSOCALZKSZC6NY6FYUOCS\",\"WARC-Block-Digest\":\"sha1:WI4E2MLMXT5CBFBEXFWXO7YBZNWHV7WK\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-50/CC-MAIN-2020-50_segments_1606141189030.27_warc_CC-MAIN-20201126230216-20201127020216-00049.warc.gz\"}"}
https://www.helpteaching.com/tests/586465/measuring-objects-with-rulers	[ "", null, "##### Print Instructions\n\nNOTE: Only your test content will print.\nTo preview this test, click on the File menu and select Print Preview.\n\nSee our guide on How To Change Browser Print Settings to customize headers and footers before printing.\n\n# Measuring Objects with Rulers (Grade 3)\n\nPrint Test (Only the test content will print)\n\n## Measuring Objects with Rulers\n\n1.\nHow many inches is the stamp?", null, "1. $1$\n2. $1 1/4$\n3. $1 1/2$\n4. $1 3/4$\n2.\nHow many inches is the pencil?", null, "1. $3$\n2. $3 1/4$\n3. $3 1/2$\n4. $3 3/4$\n3.\nAbout how long is the rock in inches?", null, "", null, "1. $1$\n2. $1 1/2$\n3. $2$\n4. $2 1/2$\n4.\nHow many inches is the pencil?", null, "1. $5$\n2. $5 1/4$\n3. $5 1/2$\n4. $5 3/4$\n5.\nHow many inches is the paperclip?", null, "1. $1$\n2. $1 1/4$\n3. $1 1/2$\n4. $1 3/4$\n6.", null, "" ]	[ null, "https://assets.pinterest.com/images/pidgets/pin_it_button.png", null, "https://dru69sbqnarp.cloudfront.net/imgs/Rulers/inches-0-6-with-object-1.jpg", null, "https://dru69sbqnarp.cloudfront.net/imgs/Rulers/inches-0-6-with-object-4.jpg", null, "https://dru69sbqnarp.cloudfront.net/imgs/Earth_Science/200x200/Rock_Igneous_Scoria.jpg", null, "https://dru69sbqnarp.cloudfront.net/imgs/Rulers/600x100/inches-0-6-halves.jpg", null, "https://dru69sbqnarp.cloudfront.net/imgs/Rulers/inches-0-6-with-object-3.jpg", null, "https://dru69sbqnarp.cloudfront.net/imgs/Rulers/inches-0-6-with-object-2.jpg", null, "https://dru69sbqnarp.cloudfront.net/imgs/Rulers/600x100/inches-0-6-quarters.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6917887,"math_prob":0.9982232,"size":1242,"snap":"2020-34-2020-40","text_gpt3_token_len":417,"char_repetition_ratio":0.2609047,"word_repetition_ratio":0.27751195,"special_character_ratio":0.35185185,"punctuation_ratio":0.09022556,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99424005,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"im_url_duplicate_count":[null,null,null,2,null,2,null,2,null,2,null,6,null,2,null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-09-27T04:58:35Z\",\"WARC-Record-ID\":\"<urn:uuid:945d4726-1208-4dc6-b43e-67ad6e8d525d>\",\"Content-Length\":\"34003\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4c8c7250-33f8-467f-9190-f4d5b6bae97b>\",\"WARC-Concurrent-To\":\"<urn:uuid:56c7382b-7c3d-447f-a97e-4603c55c0cf4>\",\"WARC-IP-Address\":\"18.213.75.172\",\"WARC-Target-URI\":\"https://www.helpteaching.com/tests/586465/measuring-objects-with-rulers\",\"WARC-Payload-Digest\":\"sha1:6RBYVXQXEILLOZ4J4DY33UJDEIZRNNVE\",\"WARC-Block-Digest\":\"sha1:CLN2CO5HXT47B7PE6D75R3OS7W4BH2K6\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600400250241.72_warc_CC-MAIN-20200927023329-20200927053329-00543.warc.gz\"}"}
https://cs.stackexchange.com/questions/152509/nth-permutation-generator	[ "# nth Permutation generator\n\nI'm just trying to write a little algorithm. I've got nine objects, so there's 9! permutations. My question is, is there a way of turning a number from 1 to 9! into a permutation?\n\nfor example, f(1)=[1,2,3,4,5,6,7,8,9], f(2)=[1,2,3,4,5,6,7,9,8] or something similar.\n\nEach number should have a unique ordering, and vice versa.\n\nI know it's possible to write out everything into an array, but that's very memory taxing. Is there a simpler way of doing it?\n\n• This is called permutation unranking. See this Wikipedia section for an approach (see in particular the example for the 2982nd permutation of {0..6}). Essentially, divide your number by 8! to get the first digit. Then remove that digit from the set and proceed in similar fashion for the remaining 8 digits. Jun 20, 2022 at 8:36" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9427038,"math_prob":0.88921833,"size":451,"snap":"2023-40-2023-50","text_gpt3_token_len":141,"char_repetition_ratio":0.08948546,"word_repetition_ratio":0.0,"special_character_ratio":0.31707317,"punctuation_ratio":0.24603175,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96556437,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-11-29T02:35:31Z\",\"WARC-Record-ID\":\"<urn:uuid:ceb4896e-c092-4365-8917-b307a21d1630>\",\"Content-Length\":\"159008\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:01f92b12-7463-46cb-85ee-4bdc28a4b3cc>\",\"WARC-Concurrent-To\":\"<urn:uuid:c41a37f8-a503-47b0-bc67-0fcd275ecfa2>\",\"WARC-IP-Address\":\"104.18.43.226\",\"WARC-Target-URI\":\"https://cs.stackexchange.com/questions/152509/nth-permutation-generator\",\"WARC-Payload-Digest\":\"sha1:2AOFALDYRDZLBGKVV5D4CV5J2GIMJW5R\",\"WARC-Block-Digest\":\"sha1:BU572NRPSTB4AKPQEGBORRQQ6LU6TUH4\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100047.66_warc_CC-MAIN-20231129010302-20231129040302-00203.warc.gz\"}"}
https://rd.springer.com/article/10.1134%2FS0965542518060040	[ "# On Asymptotics for the Solution of a Singularly Perturbed Parabolic Problem with a Multizone Internal Transition Layer\n\nArticle\n\n### Abstract\n\nFor a singularly perturbed parabolic equation with Neumann boundary conditions, we construct and substantiate asymptotics of a time-periodic solution possessing a multizone internal transition layer. Multizonality of the transition layer is caused by the fact that the degenerate equation has three nonintersecting roots, two of which are simple and the third one has multiplicity two. The asymptotic decomposition of the solution is qualitatively different from the well-known decomposition in the case when all the three roots of the degenerate equation are simple.\n\n## Keywords:\n\nparabolic problem internal transition layer asymptotic solution steplike contrast structure.\n\n## REFERENCES\n\n1. 1.\nA. B. Vasil’eva, V. F. Butuzov, and N. N. Nefedov, “Contrast structures in singularly perturbed problems,” Fundament. Prikl. Mat. 4 (3), 799–851 (1998).\n2. 2.\nA. B. Vasil’eva, V. F. Butuzov, and N. N. Nefedov, “Singularly perturbed problems with boundary and internal layers,” Proc. Steklov Inst. Math. 268, 258–273 (2010).\n3. 3.\nA. B. Vasil’eva and V. F. Butuzov, Asymptotic Methods in the Theory of Singular Perturbations (Vysshaya Shkola, Moscow, 1990) [in Russian].\n4. 4.\nV. F. Butuzov, “On the special properties of the boundary layer in singularly perturbed problems with multiple root of the degenerate equation,” Math. Notes 94 (1), 60–70 (2013).\n5. 5.\nV. F. Butuzov and A. I. Bychkov, “Asymptotics of the solution of an initial–boundary value problem for a singularly perturbed parabolic equation in the case of double root of the degenerate equation,” Differ. Equations 49 (10), 1261–1273 (2013).\n6. 6.\nV. F. Butuzov, “On the stability and the attraction domain of the stationary solution of a singularly perturbed parabolic equation with a multiple root of the degenerate equation,” Differ. Equations 51 (12), 1569–1582 (2015).\n7. 7.\nV. F. Butuzov, “Singularly perturbed boundary value problem with a multizone internal transition layer,” Model. Anal. Inf. Sist. 22 (1), 5–22 (2015).\n8. 8.\nN. N. Nefedov, “Method of differential inequalities for certain classes of nonlinear singularly perturbed problems with internal layers,” Differ. Uravn. 31 (7), 1132–1139 (1995).Google Scholar\n9. 9.\nC. V. Pao, Nonlinear Parabolic and Elliptic Equations (Plenum, New York, 1992).\n10. 10.\nV. F. Butuzov, “On periodic solutions to singularly perturbed parabolic problems in the case of multiple roots of the degenerate equation,” Comput. Math. Math. Phys. 51 (1), 40–50 (2011).\n11. 11.\nV. F. Butuzov and A. I. Bychkov, “Asymptotics of the solution to an initial boundary value problem for a singularly perturbed parabolic equation in the case of a triple root of the degenerate equation,” Comput. Math. Math. Phys. 56 (4), 593–611 (2016)." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7547971,"math_prob":0.98720145,"size":3370,"snap":"2019-13-2019-22","text_gpt3_token_len":923,"char_repetition_ratio":0.14616756,"word_repetition_ratio":0.13129103,"special_character_ratio":0.24688427,"punctuation_ratio":0.21163167,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9843452,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-03-20T20:32:47Z\",\"WARC-Record-ID\":\"<urn:uuid:5ff307b5-bbc7-4b85-8a29-f276b251f681>\",\"Content-Length\":\"75807\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0c277162-44a8-4c66-9e53-4e684df5e549>\",\"WARC-Concurrent-To\":\"<urn:uuid:3cc5b897-59e8-4d1e-9959-d4dd53d400fc>\",\"WARC-IP-Address\":\"151.101.248.95\",\"WARC-Target-URI\":\"https://rd.springer.com/article/10.1134%2FS0965542518060040\",\"WARC-Payload-Digest\":\"sha1:GURV46QGO6HXPOGSRZCFUDSUV2CRPATT\",\"WARC-Block-Digest\":\"sha1:26BTJGO6EXQIIMF67SXXSRNUUQPGWSQI\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-13/CC-MAIN-2019-13_segments_1552912202450.86_warc_CC-MAIN-20190320190324-20190320212324-00494.warc.gz\"}"}
https://www.gradesaver.com/textbooks/math/algebra/algebra-2-1st-edition/chapter-2-linear-equations-and-functions-2-4-write-equations-of-lines-2-4-exercises-mixed-review-page-104/73	[ "## Algebra 2 (1st Edition)\n\n$x \\lt 3$ or $x \\geq 5$", null, "Given: $2x \\lt 6$ or $5x-9 \\geq16$ $x\\lt 3$ or $5x \\geq 25$ $x \\lt 3$ or $x \\geq 5$" ]	[ null, "https://gradesaver.s3.amazonaws.com/uploads/solution/d96d0719-96d6-48c9-8727-0a780c0895f0/result_image/1597401435.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.76749,"math_prob":1.00001,"size":381,"snap":"2021-21-2021-25","text_gpt3_token_len":124,"char_repetition_ratio":0.14058356,"word_repetition_ratio":0.11111111,"special_character_ratio":0.32020998,"punctuation_ratio":0.06097561,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99335223,"pos_list":[0,1,2],"im_url_duplicate_count":[null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-05-16T03:21:58Z\",\"WARC-Record-ID\":\"<urn:uuid:03dd24b7-4d64-4ec9-8843-72984c386712>\",\"Content-Length\":\"109531\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:5c791b0b-dfd4-4569-8db2-a5575c398280>\",\"WARC-Concurrent-To\":\"<urn:uuid:92ebbf36-a087-41ff-a5d9-b79d7d92b933>\",\"WARC-IP-Address\":\"54.164.42.204\",\"WARC-Target-URI\":\"https://www.gradesaver.com/textbooks/math/algebra/algebra-2-1st-edition/chapter-2-linear-equations-and-functions-2-4-write-equations-of-lines-2-4-exercises-mixed-review-page-104/73\",\"WARC-Payload-Digest\":\"sha1:6L3TOVSSWLKSM2ZTYDQYAMIY7CQVQEPX\",\"WARC-Block-Digest\":\"sha1:HL2NTIOEMS6HE6NB5I23JLPVLVX4AYPQ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-21/CC-MAIN-2021-21_segments_1620243991659.54_warc_CC-MAIN-20210516013713-20210516043713-00105.warc.gz\"}"}
https://arpi.unipi.it/handle/11568/6841	[ "A dominating set of a graph G = (N,E) is a subset S of nodes such that every node is either in S or adjacent to a node which is in S. The domatic number of G is the size of a maximum cardinality partition of N into dominating sets. The problems of finding a minimum cardinality dominating set and the domatic number are both NP-complete even for special classes of graphs. In the present paper we give an O(n{divides}E{divides}) time algorithm that finds a minimum cardinality dominating set when G is a circular arc graph (intersection graph of arcs on a circle). The domatic number problem is solved in O(n2 log n) time when G is a proper circular arc graph, and it is shown NP-complete for general circular arc graphs\n\n### DOMINATING SETS AND DOMATIC NUMBER OF CIRCULAR ARC GRAPHS\n\n#### Abstract\n\nA dominating set of a graph G = (N,E) is a subset S of nodes such that every node is either in S or adjacent to a node which is in S. The domatic number of G is the size of a maximum cardinality partition of N into dominating sets. The problems of finding a minimum cardinality dominating set and the domatic number are both NP-complete even for special classes of graphs. In the present paper we give an O(n{divides}E{divides}) time algorithm that finds a minimum cardinality dominating set when G is a circular arc graph (intersection graph of arcs on a circle). The domatic number problem is solved in O(n2 log n) time when G is a proper circular arc graph, and it is shown NP-complete for general circular arc graphs\n##### Scheda breve Scheda completa Scheda completa (DC)\n1985\nBonuccelli, MAURIZIO ANGELO\nFile in questo prodotto:\nNon ci sono file associati a questo prodotto.\n\nI documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.\n\nUtilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11568/6841`\n##### Attenzione\n\nAttenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo\n\n##### Citazioni\n•", null, "ND\n•", null, "49\n•", null, "38" ]	[ null, "https://arpi.unipi.it/sr/cineca/images/thirdparty/pmc_small.png", null, "https://arpi.unipi.it/sr/cineca/images/thirdparty/scopus_small.png", null, "https://arpi.unipi.it/sr/cineca/images/thirdparty/isi_small.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.88762355,"math_prob":0.93317187,"size":1654,"snap":"2023-40-2023-50","text_gpt3_token_len":394,"char_repetition_ratio":0.12,"word_repetition_ratio":0.87586206,"special_character_ratio":0.21160822,"punctuation_ratio":0.046153847,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9790664,"pos_list":[0,1,2,3,4,5,6],"im_url_duplicate_count":[null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-22T16:06:59Z\",\"WARC-Record-ID\":\"<urn:uuid:55786633-58aa-4ab7-a363-350f458e6003>\",\"Content-Length\":\"42645\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ef1a3cdb-29e2-4a21-81f4-9fa205c9f4e6>\",\"WARC-Concurrent-To\":\"<urn:uuid:be0bca82-e336-487f-876c-3e4262edbcd5>\",\"WARC-IP-Address\":\"130.186.29.4\",\"WARC-Target-URI\":\"https://arpi.unipi.it/handle/11568/6841\",\"WARC-Payload-Digest\":\"sha1:NPGRYXOVSGM33QP4FES657RTJITDUNCU\",\"WARC-Block-Digest\":\"sha1:H53HR3JKCLMLP3PPN7S7F27XBMFMLWA2\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233506420.84_warc_CC-MAIN-20230922134342-20230922164342-00343.warc.gz\"}"}
https://www.flightpedia.org/convert-gigalux-to-lumens-per-square-millimeter.html	[ "# Convert Gigalux to Lumen per Square Millimeter - 1 Glx to lm/mm2\n\n 1 Gigalux (Glx) = 1,000 Lumen per Square Millimeter (lm/mm2) 1 Glx = 1,000 lm/mm2 1 lm/mm2 = 1.0e-03 Glx\n\n• Q: How do you convert Gigalux to Lumen per Square Millimeter (Glx to lm/mm2)?\n\n1 Gigalux is equal to 1,000 Lumen per Square Millimeter. Formula to convert 1 Glx to lm/mm2 is 1 * 1000\n\n• Q: How many Gigalux in a Lumen per Square Millimeter?" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.60125965,"math_prob":0.99917614,"size":1013,"snap":"2021-43-2021-49","text_gpt3_token_len":366,"char_repetition_ratio":0.32309216,"word_repetition_ratio":0.12568305,"special_character_ratio":0.27147087,"punctuation_ratio":0.05882353,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9558301,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-20T13:15:13Z\",\"WARC-Record-ID\":\"<urn:uuid:32f1bdb6-2adb-4013-b355-4b785ca8e636>\",\"Content-Length\":\"13992\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b0a7216b-0aa8-459e-b1cc-b15b37fe8afa>\",\"WARC-Concurrent-To\":\"<urn:uuid:fe567b51-175f-496a-98e5-5fa81a5dd33f>\",\"WARC-IP-Address\":\"35.202.210.90\",\"WARC-Target-URI\":\"https://www.flightpedia.org/convert-gigalux-to-lumens-per-square-millimeter.html\",\"WARC-Payload-Digest\":\"sha1:3CCRTOXWGHS7ZDXN2YZ6T77HDG4OZFPH\",\"WARC-Block-Digest\":\"sha1:ALAYRUWN3AN7SKJZDLQ3YZ2ULFSXF66E\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585321.65_warc_CC-MAIN-20211020121220-20211020151220-00351.warc.gz\"}"}
https://www.paste.org/41475	[ "Welcome, guest! Login / Register - Why register?\n[email protected] webmail now available. Want one? Go here.\n\n## Paste\n\nPasted as VHDL by scrts ( 10 years ago )\n``````case(read_state) is\nwhen start =>\nif(mem_write = '0' and read_empty = '1') then -- if read FIFO is empty and other master is not writing...\nmem_read_addr <= mem_read_addr + burstcount; -- unsigned + unsigned here\navm_memrd_burstcount <= std_logic_vector(burstcount); -- constant burstcount: 0x5E packets\nmem_read <= '1';\nread_state <= begin_read;\nelse\nread_state <= start;\nend if;\nwhen begin_read =>\nif(avm_memrd_waitrequest = '0') then -- Wait until slave will be ready...\nmem_read <= '0'; -- and deassert the read signal\nread_state <= end_read;\nelse\nread_state <= begin_read;\nend if;\nwhen end_read =>\nif(avm_memrd_readdatavalid = '1') then\nif(read_valid_counter = 93) then\nread_valid_counter <= (others => '0');\nread_state <= start;\nelse\nread_valid_counter <= read_valid_counter + 1;\nread_state <= end_read;\nend if;\nend if;\nend case;\n--------------------------------------------------------------\n-- The code that works, but not correctly:\ncase(read_state) is\nwhen start =>\nif(mem_write = '0' and read_empty = '1') then -- if read FIFO is empty and other master is not writing...\nmem_read_addr <= mem_read_addr + burstcount; -- unsigned + unsigned here\navm_memrd_burstcount <= std_logic_vector(burstcount); -- constant burstcount: 0x5E packets\nmem_read <= '1';\nread_state <= begin_read;\nelse\nread_state <= start;\nend if;\nwhen begin_read =>\nif(avm_memrd_waitrequest = '0') then -- Wait until slave will be ready...\nmem_read <= '0'; -- and deassert the read signal\nread_state <= end_read;\nelse\nread_state <= begin_read;\nend if;\nwhen end_read =>\n--if(avm_memrd_readdatavalid = '1') then\nif(read_valid_counter = 93) then\nread_valid_counter <= (others => '0');\nread_state <= start;\nelse\nread_valid_counter <= read_valid_counter + 1;\nread_state <= end_read;\nend if;\n--end if;\nend case;``````\n\n## Revise this Paste\n\nChildren: 43032\nYour Name: Code Language:" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.66382855,"math_prob":0.85296583,"size":1971,"snap":"2021-21-2021-25","text_gpt3_token_len":527,"char_repetition_ratio":0.18912049,"word_repetition_ratio":0.7714286,"special_character_ratio":0.34246576,"punctuation_ratio":0.19723183,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99433315,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-17T18:20:34Z\",\"WARC-Record-ID\":\"<urn:uuid:f47159aa-7d0e-43eb-b3a8-dfe51882321d>\",\"Content-Length\":\"19385\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:6d9ca9d4-edc7-4040-a451-829dd3c03f71>\",\"WARC-Concurrent-To\":\"<urn:uuid:a851dfd4-5fa3-445f-a061-470b0b95c948>\",\"WARC-IP-Address\":\"172.67.141.168\",\"WARC-Target-URI\":\"https://www.paste.org/41475\",\"WARC-Payload-Digest\":\"sha1:WW2RFTGO33EL3PDPY43Q3IGXL6IQ5CNH\",\"WARC-Block-Digest\":\"sha1:I6DUTWMVJEMWAKRTV3SHAOF4NBE4HPSH\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623487630518.38_warc_CC-MAIN-20210617162149-20210617192149-00200.warc.gz\"}"}
https://mathematica.stackexchange.com/questions/95610/is-this-a-bug-in-mathematica-10-2	[ "# Is this a bug in Mathematica 10.2 [closed]\n\nI am using Mathematica 10.2.0.0\n\nFirst, I define the following function\n\nClear[fermi]\nfermi[ee_, EF_, T_] :=\n1/(E^((ee - EF)/(Subscript[k, B] T)) + 1) /.\nSubscript[k, B] -> (1.38 10^-23/(1.6 10^-19))\n\n\nthen I plot it with several parameter T as\n\nPlot[Evaluate@Table[fermi[ee, 3, T], {T, 0.5, 2000, 300}], {ee, -3,\n5}]\n\n\nthe result is pretty good, show here", null, "But strange thing happens when I change the plot interval of ee from {-3,5} to {-3,4},\n\nPlot[Evaluate@Table[fermi[ee, 3, T], {T, 0.5, 2000, 300}], {ee, -3,\n4}]\n\n\nMathematica gives", null, "What is wrong with these curves in the interval {3,4}?\n\n• Don't use the bugs tag unless it's confirmed a bug. In this case, you need to supply PlotRange -> All to get the behaviour you want. – Patrick Stevens Sep 27 '15 at 12:08\n• You haven't specified the output range at all, so Mathematica picks one it thinks shows the most important features and is aesthetically pleasing. It doesn't get it right here, but that's a matter of taste. The specification {ee, -3, 4} specifies the domain of the function, not its range. – Patrick Stevens Sep 27 '15 at 12:21\n• Read the docs for PlotRange: \"With the Automatic setting, the distribution of coordinate values is found, and any points sufficiently far out in the distribution are dropped. Such points are often produced as a result of singularities in functions being plotted.\" This is why, in order to get all the points plotted, you need to use PlotRange>All – bill s Sep 27 '15 at 12:21\n• @matheorem You'll be glad of this behaviour when you try Plot[1/x, {x, 0, 5}]. – Patrick Stevens Sep 27 '15 at 12:56\n• @matheorem Remember you can use SetOptions[Plot, PlotRange -> All] for that. – Mr.Wizard Sep 27 '15 at 14:54" ]	[ null, "https://i.stack.imgur.com/vJoG5.png", null, "https://i.stack.imgur.com/Q4Si6.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5883627,"math_prob":0.85248655,"size":953,"snap":"2021-04-2021-17","text_gpt3_token_len":352,"char_repetition_ratio":0.11169652,"word_repetition_ratio":0.14285715,"special_character_ratio":0.4207765,"punctuation_ratio":0.2278481,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9914591,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,4,null,4,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-01-25T14:34:52Z\",\"WARC-Record-ID\":\"<urn:uuid:5544bc4c-61ec-43c0-b6e8-0d758d359393>\",\"Content-Length\":\"128617\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:058f0a6a-60df-4aa7-bff5-1e2a67a9f531>\",\"WARC-Concurrent-To\":\"<urn:uuid:8e2e2efd-ee83-404c-8aa9-ef553cc84c65>\",\"WARC-IP-Address\":\"151.101.129.69\",\"WARC-Target-URI\":\"https://mathematica.stackexchange.com/questions/95610/is-this-a-bug-in-mathematica-10-2\",\"WARC-Payload-Digest\":\"sha1:WGJL4XMZTD2OC7TYC7ZFMS3FQ6AK6ZCP\",\"WARC-Block-Digest\":\"sha1:2Z3TOKM5ILSYP5PTBYPC4LKHGHRQIIIO\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-04/CC-MAIN-2021-04_segments_1610703581888.64_warc_CC-MAIN-20210125123120-20210125153120-00257.warc.gz\"}"}
https://ncatlab.org/nlab/show/spin%5Ec%20structure	[ "# nLab spin^c structure\n\nContents\n\n## Spin geometry\n\nspin geometry\n\nDynkin labelsp. orth. groupspin grouppin groupsemi-spin group\nSO(2)Spin(2)Pin(2)\nB1SO(3)Spin(3)Pin(3)\nD2SO(4)Spin(4)Pin(4)\nB2SO(5)Spin(5)Pin(5)\nD3SO(6)Spin(6)\nB3SO(7)Spin(7)\nD4SO(8)Spin(8)SO(8)\nB4SO(9)Spin(9)\nD5SO(10)Spin(10)\nB5SO(11)Spin(11)\nD6SO(12)Spin(12)\n$\\vdots$$\\vdots$\nD8SO(16)Spin(16)SemiSpin(16)\n$\\vdots$$\\vdots$\nD16SO(32)Spin(32)SemiSpin(32)\n\nstring geometry\n\ncohomology\n\n# Contents\n\n## Definition\n\n### Topological\n\nFor $n \\in \\mathbb{N}$ the Lie group spinc is a central extension\n\n$U(1) \\to Spin^c(n) \\to SO(n)$\n\nof the special orthogonal group by the circle group. This comes with a long fiber sequence\n\n$\\cdots \\to B U(1) \\to B Spin^c(n) \\to B SO(n) \\stackrel{W_3}{\\to} B^2 U(1) \\,,$\n\nwhere $W_3$ is the third integral Stiefel-Whitney class .\n\nAn oriented manifold $X$ has $Spin^c$-structure if the characteristic class $[W_3(X)] \\in H^3(X, \\mathbb{Z})$\n\n$W_3(X) \\coloneqq W_3(T X) \\;\\colon\\; X \\stackrel{T X}{\\to} B SO(n) \\stackrel{W_3}{\\to} B^2 U(1) \\simeq K(\\mathbb{Z},3)$\n\nis trivial. This is the Dixmier-Douady class of the circle 2-bundle/bundle gerbe that obstructs the existence of a $Spin^c$-principal bundle lifting the given tangent bundle.\n\nA manifold $X$ is equipped with $Spin^c$-structure $\\eta$ if it is equipped with a choice of trivializaton\n\n$\\eta : 1 \\stackrel{\\simeq}{\\to} W_3(T X) \\,.$\n\nThe homotopy type/∞-groupoid of $Spin^c$-structures on $X$ is the homotopy fiber $W_3 Struc(T X)$ in the pasting diagram of homotopy pullbacks\n\n$\\array{ W_3 Struc(T X) &\\to& W_3 Struc(X) &\\to& * \\\\ \\downarrow && \\downarrow && \\downarrow \\\\ * &\\stackrel{T X}{\\to}& Top(X, B SO(n)) &\\stackrel{W_3}{\\to}& Top(X,B^2 U(1)) } \\,.$\n\nIf the class does not vanish and if hence there is no $Spin^c$-structure, it still makes sense to discuss the structure that remains as twisted spinc structure .\n\n### Smooth\n\nSince $U(1) \\to Spin^c \\to SO$ is a sequence of Lie groups, the above may be lifted from the (∞,1)-topos $L_{whe}$ Top $\\simeq$ ∞Grpd of discrete ∞-groupoids to that of smooth ∞-groupoids, Smooth∞Grpd.\n\nMore in detail, by the discussion at Lie group cohomology (and smooth ∞-groupoid – structures) the characteristic map $W_3 : B SO \\to B^2 U(1)$ in $\\infty Grpd$ has, up to equivalence, a unique lift\n\n$\\mathbf{W}_3 : \\mathbf{B} SO \\to \\mathbf{B}^2 U(1)$\n\nto Smooth∞Grpd, where on the right we have the delooping of the smooth circle 2-group.\n\nAccordingly, the 2-groupoid of smooth $spin^c$-structures $\\mathbf{W}_3 Struc(X)$ is the joint (∞,1)-pullback\n\n$\\array{ \\mathbf{W}_3 Struc(T X) &\\to& \\mathbf{W}_3 Struc(X) &\\to& * \\\\ \\downarrow && \\downarrow && \\downarrow \\\\ * &\\stackrel{T X}{\\to}& Smooth \\infty Grpd(X, \\mathbf{B} SO(n)) &\\stackrel{\\mathbf{W}_3}{\\to}& Smooth \\infty Grpd(X,\\mathbf{B}^2 U(1)) } \\,.$\n\n### Higher $spin^c$-structures\n\nIn parallel to the existence of higher spin structures there are higher analogs of $Spin^c$-structures, related to quantum anomaly cancellation of theories of higher dimensional branes.\n\n## Properties\n\n### Of $Spin^c$\n\n###### Definition\n\nThe group $Spin^c$ is the fiber product\n\n\\begin{aligned} Spin^c & := Spin \\times_{\\mathbb{Z}_2} U(1) \\\\ & = (Spin \\times U(1))/{\\mathbb{Z}_2} \\,, \\end{aligned}\n\nwhere in the second line the action is the diagonal action induced from the two canonical embeddings of subgroups $\\mathbb{Z}_2 \\hookrightarrow \\mathbb{Z}$ and $\\mathbb{Z}_2 \\hookrightarrow U(1)$.\n\n###### Proposition\n\nWe have a homotopy pullback diagram\n\n$\\array{ \\mathbf{B} Spin^c &\\to& \\mathbf{B}U(1) \\\\ \\downarrow && \\downarrow^{\\mathrlap{\\mathbf{c}_1 mod 2}} \\\\ \\mathbf{B} SO &\\stackrel{w_2}{\\to}& \\mathbf{B}^2 \\mathbb{Z}_2 } \\,.$\n###### Proof\n\nWe present this as usual by simplicial presheaves and ∞-anafunctors.\n\nThe first Chern class is given by the ∞-anafunctor\n\n$\\array{ \\mathbf{B}(\\mathbb{Z} \\to \\mathbb{R}) &\\stackrel{c_1}{\\to}& \\mathbf{B}(\\mathbb{Z} \\to 1) = \\mathbf{B}^2 \\mathbb{Z} \\\\ \\downarrow^{\\mathrlap{\\simeq}} \\\\ \\mathbf{B} U(1) } \\,.$\n\nThe second Stiefel-Whitney class is given by\n\n$\\array{ \\mathbf{B}(\\mathbb{Z}_2 \\to Spin) &\\stackrel{w_2}{\\to}& \\mathbf{B}(\\mathbb{Z}_2 \\to 1) = \\mathbf{B}^2 \\mathbb{Z}_2 \\\\ \\downarrow^{\\mathrlap{\\simeq}} \\\\ \\mathbf{B} SO } \\,.$\n\nNotice that the top horizontal morphism here is a fibration.\n\nTherefore the homotopy pullback in question is given by the ordinary pullback\n\n$\\array{ Q &\\to& \\mathbf{B}(\\mathbb{Z} \\to \\mathbb{R}) \\\\ \\downarrow && \\downarrow \\\\ \\mathbf{B}(\\mathbb{Z}_2 \\to Spin) &\\to& \\mathbf{B}^2 \\mathbb{Z}_2 } \\,.$\n\nThis pullback is $\\mathbf{B}(\\mathbb{Z} \\stackrel{\\partial}{\\to} Spin \\times \\mathbb{R})$, where\n\n$\\partial : n \\mapsto ( n mod 2 , n) \\,.$\n\nThis is equivalent to\n\n$\\mathbf{B}(\\mathbb{Z}_2 \\stackrel{\\partial'}{\\to} Spin \\times U(1))$\n\nwhere now\n\n$\\partial' : \\sigma \\mapsto (\\sigma, \\sigma) \\,.$\n\nThis in turn is equivalent to\n\n$\\mathbf{B} (Spin \\times_{\\mathbb{Z}_2} U(1)) \\,,$\n\nwhich is the original definition.\n\nThis factors the above characterization of $\\mathbf{B}Spin^c$ as the homotopy fiber of $\\mathbf{W}_3$:\n\n###### Proposition\n\nWe have a pasting diagram of homotopy pullbacks of smooth infinity-groupoids of the form\n\n$\\array{ \\mathbf{B} Spin^c &\\to& \\mathbf{B}U(1) &\\to& \\ast \\\\ \\downarrow && \\downarrow^{\\mathrlap{c_1 \\, mod\\, 2}} && \\downarrow \\\\ \\mathbf{B}SO &\\stackrel{\\mathbf{w}_2}{\\to}& \\mathbf{B}^2 \\mathbb{Z}_2 &\\stackrel{\\mathbf{\\beta}_2}{\\to}& \\mathbf{B}^2 U(1) } \\,.$\n\nThis is discussed at Spin^c – Properties – As the homotopy fiber of smooth w3.\n\n### As $KU$-orientation\n\nFor $X$ an oriented manifold, the map $X \\to \\ast$ is generalized oriented in periodic complex K-theory precisely if $X$ has a $Spin^c$-structure.\n\nSee at K-orientation for more.\n\n### Relation to metaplectic structures\n\nLet $(X,\\omega)$ be a compact symplectic manifold equipped with a Kähler polarization $\\mathcal{P}$ hence a Kähler manifold structure $J$. A metaplectic structure of this data is a choice of square root $\\sqrt{\\Omega^{0,n}}$ of the canonical line bundle. This is equivalently a spin structure on $X$ (see the discussion at Theta characteristic).\n\nNow given a prequantum line bundle $L_\\omega$, in this case the Dolbault quantization of $L_\\omega$ coincides with the spinc quantization of the spinc structure induced by $J$ and $L_\\omega \\otimes \\sqrt{\\Omega^{0,n}}$.\n\nThis appears as (Paradan 09, prop. 2.2).\n\n## Examples\n\n### From almost complex structures\n\nAn almost complex structure canonically induces a $Spin^c$-structure:\n\n###### Proposition\n\nFor all $n \\in \\mathbb{N}$ we have a homotopy-commuting diagram\n\n$\\array{ \\mathbf{B}U(n) &\\to& \\mathbf{B}U(1) \\\\ \\downarrow &\\swArrow& \\downarrow^{\\mathrlap{\\mathbf{c_1} mod 2}} \\\\ \\mathbf{B}SO(2n) &\\stackrel{\\mathbf{w}_2}{\\to}& \\mathbf{B}^2 \\mathbb{Z}_2 } \\,,$\n\nwhere the vertical morphism is the canonical morphism induced from the identification of real vector spaces $\\mathbb{C} \\to \\mathbb{R}^2$, and where the top morphism is the canonical projection $\\mathbf{B}U(n) \\to \\mathbf{B}U(1)$ (induced from $U(n)$ being the semidirect product group $U(n) \\simeq SU(n) \\rtimes U(1)$).\n\n###### Proof\n\nBy the general relation between $c_1$ of an almost complex structure and $w_2$ of the underlying orthogonal structure, discussed at Stiefel-Whitney class – Relation to Chern classes.\n\n###### Remark\n\nBy prop. and the universal property of the homotopy pullback this induces a canonical morphism\n\n$k \\colon \\mathbf{B}U(n) \\to \\mathbf{B}Spin^c \\,.$\n\nand this is the universal morphism from almost complex structures:\n\n###### Definition\n\nFor $c \\colon X \\to \\mathbf{B}U(n)$ modulating an almost complex structure/complex vector bundle over $X$, the composite\n\n$k c \\colon X \\stackrel{c}{\\to} \\mathbf{B}U(n) \\stackrel{k}{\\to} \\mathbf{B}Spin^c$\n\nis the corresponding $Spin^c$-structure.\n\n### General\n\nA canonical textbook reference is\n\nOther accounts include\n\n• Blake Mellor, $Spin^c$-manifolds (pdf)\n\n• Stable complex and $Spin^c$-structures (pdf)\n\n• Peter Teichner, Elmar Vogt, All 4-manifolds have $Spin^c$-structures (pdf)\n\n• Robert Gompf, $Spin^c$ structures and homotopy equivalences, Geom. Topol. 1 (1997) 41-50 (arXiv:math/9705218)\n\n### As $KU$-orientation/anomaly cancellation in type II string theory\n\nThat the $U(1)$-gauge field on a D-brane in type II string theory in the absense of a B-field is rather to be regarded as part of a $spin^c$-structure was maybe first observed in\n\nThe twisted spinc structure (see there for more details) on the worldvolume of D-branes in the presence of a nontrivial B-field was discussed in\n\nA more recent review is provided in\n\n• Kim Laine, Geometric and topological aspects of Type IIB D-branes (arXiv:0912.0460)\n\n• Hisham Sati, Geometry of $Spin$ and $Spin^c$ structures in the M-theory partition function (arXiv:1005.1700)\n• O. Hijazi, S. Montiel, F. Urbano, $Spin^c$-geometry of Kähler manifolds and the Hodge Laplacian on minimal Lagrangian submanifolds (pdf)" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8346719,"math_prob":0.9993366,"size":5458,"snap":"2022-05-2022-21","text_gpt3_token_len":1411,"char_repetition_ratio":0.13953061,"word_repetition_ratio":0.002375297,"special_character_ratio":0.2213265,"punctuation_ratio":0.07652534,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9998517,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-05-22T00:46:44Z\",\"WARC-Record-ID\":\"<urn:uuid:25d618bf-5044-4a43-9702-3dcc15942e6c>\",\"Content-Length\":\"93540\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9bbaa2b6-7aa3-4216-a001-279876b2f381>\",\"WARC-Concurrent-To\":\"<urn:uuid:538e6d15-4157-4120-91d9-75db545a63df>\",\"WARC-IP-Address\":\"172.67.177.13\",\"WARC-Target-URI\":\"https://ncatlab.org/nlab/show/spin%5Ec%20structure\",\"WARC-Payload-Digest\":\"sha1:RRJE745SA342ZX7SNEVEW4YUKZWG2DQR\",\"WARC-Block-Digest\":\"sha1:4OLPXWIVUN55BDOHQES2XD4JFM62WHEY\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-21/CC-MAIN-2022-21_segments_1652662543264.49_warc_CC-MAIN-20220522001016-20220522031016-00646.warc.gz\"}"}
https://www.nag.com/numeric/nl/nagdoc_27.2/flhtml/f03/f03bnf.html	[ "# NAG FL Interfacef03bnf (complex_gen)\n\n## ▸▿ Contents\n\nSettings help\n\nFL Name Style:\n\nFL Specification Language:\n\n## 1Purpose\n\nf03bnf computes the determinant of a complex $n×n$ matrix $A$. f07arf must be called first to supply the matrix $A$ in factorized form.\n\n## 2Specification\n\nFortran Interface\n Subroutine f03bnf ( n, a, lda, ipiv, d, id,\n Integer, Intent (In) :: n, lda, ipiv(n) Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: id(2) Complex (Kind=nag_wp), Intent (In) :: a(lda,) Complex (Kind=nag_wp), Intent (Out) :: d\n#include <nag.h>\n void f03bnf_ (const Integer n, const Complex a[], const Integer lda, const Integer ipiv[], Complex d, Integer id[], Integer ifail)\nThe routine may be called by the names f03bnf or nagf_det_complex_gen.\n\n## 3Description\n\nf03bnf computes the determinant of a complex $n×n$ matrix $A$ that has been factorized by a call to f07arf. The determinant of $A$ is the product of the diagonal elements of $U$ with the correct sign determined by the row interchanges.\nWilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag\n\n## 5Arguments\n\n1: $\\mathbf{n}$Integer Input\nOn entry: $n$, the order of the matrix $A$.\nConstraint: ${\\mathbf{n}}>0$.\n2: $\\mathbf{a}\\left({\\mathbf{lda}},\\right)$Complex (Kind=nag_wp) array Input\nNote: the second dimension of the array a must be at least ${\\mathbf{n}}$.\nOn entry: the $n×n$ matrix $A$ in factorized form as returned by f07arf.\n3: $\\mathbf{lda}$Integer Input\nOn entry: the first dimension of the array a as declared in the (sub)program from which f03bnf is called.\nConstraint: ${\\mathbf{lda}}\\ge {\\mathbf{n}}$.\n4: $\\mathbf{ipiv}\\left({\\mathbf{n}}\\right)$Integer array Input\nOn entry: the row interchanges used to factorize matrix $A$ as returned by f07arf.\n5: $\\mathbf{d}$Complex (Kind=nag_wp) Output\nOn exit: the mantissa of the real and imaginary parts of the determinant.\n6: $\\mathbf{id}\\left(2\\right)$Integer array Output\nOn exit: the exponents for the real and imaginary parts of the determinant. The determinant, $d=\\left({d}_{r},{d}_{i}\\right)$, is returned as ${d}_{r}={D}_{r}×{2}^{j}$ and ${d}_{i}={D}_{i}×{2}^{k}$, where ${\\mathbf{d}}=\\left({D}_{r},{D}_{i}\\right)$ and $j$ and $k$ are stored in the first and second elements respectively of the array id on successful exit.\n7: $\\mathbf{ifail}$Integer Input/Output\nOn entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.\nA value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.\nIf halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\\mathbf{1}$ or $\\mathbf{1}$ is used it is essential to test the value of ifail on exit.\nOn exit: ${\\mathbf{ifail}}={\\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).\n\n## 6Error Indicators and Warnings\n\nIf on entry ${\\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).\nErrors or warnings detected by the routine:\n${\\mathbf{ifail}}=1$\nOn entry, ${\\mathbf{n}}=⟨\\mathit{\\text{value}}⟩$.\nConstraint: ${\\mathbf{n}}\\ge 1$.\n${\\mathbf{ifail}}=3$\nOn entry, ${\\mathbf{lda}}=⟨\\mathit{\\text{value}}⟩$ and ${\\mathbf{n}}=⟨\\mathit{\\text{value}}⟩$.\nConstraint: ${\\mathbf{lda}}\\ge {\\mathbf{n}}$.\n${\\mathbf{ifail}}=4$\nThe matrix $A$ is approximately singular.\n${\\mathbf{ifail}}=-99$\nSee Section 7 in the Introduction to the NAG Library FL Interface for further information.\n${\\mathbf{ifail}}=-399$\nYour licence key may have expired or may not have been installed correctly.\nSee Section 8 in the Introduction to the NAG Library FL Interface for further information.\n${\\mathbf{ifail}}=-999$\nDynamic memory allocation failed.\nSee Section 9 in the Introduction to the NAG Library FL Interface for further information.\n\n## 7Accuracy\n\nThe accuracy of the determinant depends on the conditioning of the original matrix. For a detailed error analysis, see page 107 of Wilkinson and Reinsch (1971).\n\n## 8Parallelism and Performance\n\nf03bnf is not threaded in any implementation.\n\nThe time taken by f03bnf is approximately proportional to $n$.\n\n## 10Example\n\nThis example calculates the determinant of the complex matrix\n $( 1 1+2i 2+10i 1+i 3i −5+14i 1+i 5i −8+20i ) .$\n\n### 10.1Program Text\n\nProgram Text (f03bnfe.f90)\n\n### 10.2Program Data\n\nProgram Data (f03bnfe.d)\n\n### 10.3Program Results\n\nProgram Results (f03bnfe.r)" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8649184,"math_prob":0.9953969,"size":3104,"snap":"2021-43-2021-49","text_gpt3_token_len":694,"char_repetition_ratio":0.12387097,"word_repetition_ratio":0.066019416,"special_character_ratio":0.21617268,"punctuation_ratio":0.12802768,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9994272,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-21T02:12:13Z\",\"WARC-Record-ID\":\"<urn:uuid:f804ef24-17b2-4110-833c-0ab4c0ca002c>\",\"Content-Length\":\"29825\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:fa34180d-5a97-4c2b-b80d-92ed65db1786>\",\"WARC-Concurrent-To\":\"<urn:uuid:97ad03e3-4b3a-4f12-b0d9-e6c7f7d35f22>\",\"WARC-IP-Address\":\"78.129.168.4\",\"WARC-Target-URI\":\"https://www.nag.com/numeric/nl/nagdoc_27.2/flhtml/f03/f03bnf.html\",\"WARC-Payload-Digest\":\"sha1:7ZHB2JZEF57JVTD4ISJKHNKEPOJGFVSW\",\"WARC-Block-Digest\":\"sha1:HQEHLTAM3C2QQASOED5AGTOMV55UXF4B\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585380.70_warc_CC-MAIN-20211021005314-20211021035314-00415.warc.gz\"}"}
https://www.wcschool.net/education-curriculum/curriculum-instruction/pre-algebra/	[ "", null, "### Pre-Algebra\n\nThis course prepares a student for a first year algebra course. It serves as a transition from arithmetic to algebra. Coursework includes a review of the concepts of arithmetic and an introduction to variables, expressions and equations, proportion, percent, and equations and inequalities and their graphs. It is designed to automate the use of fractions, mixed numbers, and decimal numbers in any and all operations. It also presents procedures for solving simple word problems. The concepts of area, percent, ratio, and order of operations are introduced. The beginning concepts of algebra are practiced thoroughly.\n\nContent Objectives\n\n• Operations\n• Rational vs. irrational numbers\n• Rational numbers in scientific notation\n• Whole-number powers\n• Convert fractions to decimals and percents\n• Fractions\n• Whole number exponents\n• Common denominators\n• Absolute value\n• Algebra\n• Use variables to write an equation or inequality\n• Order of operations\n• Two-step linear equations and inequalities in one variable\n• Integer powers and simple roots\n• Monomials: multiply, divide, take powers, and extract roots\n• Graph and interpret linear and some nonlinear functions\n• Slope\n• Measurement and Geometry\n• Use ratios to convert within and between measurement systems\n• Compute perimeter, area, and volume of common geometric objects\n• Understand and use the Pythagorean theorem\n• Identify and construct basic elements of geometric figures\n• Congruence\n• Statistics, Data Analysis, and Probability\n• Collect, organize, and represent data sets with one or more variable\n• Word Problems\n• Discounts, markups, commissions, and profit\n• Simple and compound interest\n• Percentage of increase and decrease of a quantity\n• Rate, average speed, distance, and time\n\nMethods of Assessment\nAssessment tools include the following but are not limited to:\n\n1. Research projects\n2. Portfolios\n3. Oral communication\n4. Student demonstrations" ]	[ null, "https://www.wcschool.net/wp-content/themes/westwood/images/headers/educational.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.84747636,"math_prob":0.95109504,"size":2272,"snap":"2020-34-2020-40","text_gpt3_token_len":482,"char_repetition_ratio":0.10890652,"word_repetition_ratio":0.0,"special_character_ratio":0.19982395,"punctuation_ratio":0.12430939,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9994679,"pos_list":[0,1,2],"im_url_duplicate_count":[null,7,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-08-09T02:31:06Z\",\"WARC-Record-ID\":\"<urn:uuid:e60ea9f3-16fa-42f9-b9bf-7b27e6cd13e3>\",\"Content-Length\":\"20799\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:69627fc8-247f-467b-a268-7ea173078693>\",\"WARC-Concurrent-To\":\"<urn:uuid:125190ed-d08b-460e-9b70-b8eee86f9e25>\",\"WARC-IP-Address\":\"194.1.147.16\",\"WARC-Target-URI\":\"https://www.wcschool.net/education-curriculum/curriculum-instruction/pre-algebra/\",\"WARC-Payload-Digest\":\"sha1:DT4G67VD4KYY5GHCJDGO7AAAAACE5C4D\",\"WARC-Block-Digest\":\"sha1:OBZI4OLHS64PZSW7LN7NGLBW2QPPGD4P\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-34/CC-MAIN-2020-34_segments_1596439738380.22_warc_CC-MAIN-20200809013812-20200809043812-00550.warc.gz\"}"}
https://www.htmlkick.com/ncert-solutions/ncert-solutions-for-class-10-maths-chapter-8/	[ "# NCERT Solutions For Class 10 Maths Chapter 8\n\nNCERT Solutions For Class 10 Maths Chapter 8: The NCERT Solutions are created and evaluated by subject specialists to ensure that they address all of the questions in the textbook. These NCERT Solutions are based on the most recent revisions to the CBSE Syllabus for 2021-22 and its accompanying guidelines, as well as the first-term exam pattern.\n\nStudents will benefit from NCERT Solutions for Class 10 Maths Chapter 8 Introduction to Trigonometry since it will help them understand the concepts and perform well in the CBSE Class 10 first term examination.\n\n## NCERT Solutions For Class 10 Maths Chapter 8\n\n### NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers\n\nActual Figures Class 10 comprises four exercises with a total of 18 problems. In earlier board exams, Prove Irrational, Problems based on Euclid’s division lemma, HCF and LCM, and Divisibility were frequently asked questions. The Fundamental Theorem of Arithmetic, important properties of positive integers, fraction to decimals, and decimals to a fraction are among the other topics covered.\n\n### NCERT Solutions for Class 10 Maths Chapter 2 Polynomials\n\nClass 10 Polynomials features a total of four exercises with a total of 13 questions. Finding polynomials problems, properties zeros and coefficients, long division of polynomials, finding a quadratic polynomial, and finding polynomial zeros are all scoring themes.\n\n### NCERT Solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables\n\nClass 10: Pair of Linear Equations has a total of seven exercises with a total of 55 problems. The questions will use topics such as two-variable linear equations, algebraic methods for solving linear equations, elimination method, cross-multiplication method, Time and Work, Age, Boat Stream, and equations reducible to a pair of linear equations. These solutions will make it easier for you to solve problems involving linear equations.\n\n### NCERT Solutions for Class 10 Maths Chapter 4 Quadratic Equations\n\nClass 10 Quadratic Equations features a total of four exercises with a total of 24 problems. Finding the roots of quadratic equations and converting world problems into quadratic equations are two topics that score well in board exams.\n\n### NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions\n\nClass 10 Arithmetic Progressions features a total of four exercises with a total of 49 problems. The sum of n successive phrases and finding the nth term are significant subjects in this chapter 5.\n\n### NCERT Solutions for Class 10 Maths Chapter 6 Triangles\n\nClass 10 Triangles features a total of six exercises with a total of 64 problems. The questions are based on triangle characteristics and nine key theorems that are critical for success in CBSE Class 10 exams.\n\n### NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry\n\nClass 10 Coordinate Geometry features a total of four exercises with a total of 33 problems. Important models in class 10 boards include questions about determining the distance between two points using their coordinates, the Area of a Triangle, and the Line Divided in Ratio (Section Formula).\n\n### NCERT Solutions for Class 10 Maths Chapter 8 Introduction to Trigonometry\n\nClass 10 of Introduction to Trigonometry has a total of four exercises with a total of 27 problems. The key subjects you will learn in this chapter are trigonometric ratios of specific angles, trigonometric identities, and trigonometric ratios of complementary angles. Trigonometry Formulas are critical in achieving a perfect score on board exams.\n\n### NCERT Solutions for Class 10 Maths Chapter 9 Some Applications of Trigonometry\n\nOne exercise in Trigonometry Applications Class 10 consists of 16 problems. In this chapter, you’ll learn about real-world applications of trigonometry, and the problems will be based on those applications.\n\n### NCERT Solutions for Class 10 Maths Chapter 10 Circles\n\nThere are 17 problems in total in Circle Class 10’s two exercises. Understand terms like tangent, secant, and number tangents from a point to a circle, as well as others.\n\n### NCERT Solutions for Class 10 Maths Chapter 11 Constructions\n\nClass 10 comprises a total of four exercises with a total of 14 problems. Drawing tangents and drawing comparable triangles are major themes in the questions.\n\n### NCERT Solutions for Class 10 Maths Chapter 12 Areas Related to Circles\n\nCircle-Related Subjects In class 10, there are three exercises totaling 35 problems. Solve problems using the ‘Perimeter and Area of a Circle,’ ‘Areas of Plane Figure Combinations,’ and ‘Areas of Sector and Segment of a Circle.’\n\n### NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas and Volumes\n\nVolumes and Surface Areas There are 36 problems in total in Class 10’s five exercises. The ‘Surface Areas and Volumes’ chapter is part of the mensuration subject in CBSE class 10 Maths. The issues revolve around determining the areas and volumes of various solids such as cubes, cuboid and cylinders, frustums, and solid combinations.\n\n### NCERT Solutions for Class 10 Maths Chapter 14 Statistics\n\nThere are a total of four exercises in Statistics Class 10 with a total of 25 problems. This chapter will look at how to find the mean, mode, and median of grouped data. Understanding the notion of cumulative frequency distribution will help you answer queries.\n\n### NCERT Solutions for Class 10 Maths Chapter 15 Probability\n\nClass 10 comprises a total of two exercises with a total of 30 problems. This chapter will look at questions based on the concept of theoretical probability.\n\n### Trigonometric formulae explored in this chapter\n\nLet’s consider the 3-points of a triangle that’s right-angled as OAB, where OA and OB make an angle of 90 degrees to every other.\n\nLet,\n\nOB be the opposite side\n\nAB be the hypotenuse side\n\nThe hypotenuse side of any triangle is normally the longest side and the other 2 sides are shorter.\n\nTherefore, from the provided information, the formula is gotten,\n\n• sin = opposite/hypotenuse\n\nClass 10 Maths Chapter 8 chapter has exercises that are explained using a right-angle triangle. The practice problems can go with this containing the derivation of cosine, sine, tangent, & other trigonometric functions.\n\nAnother part covered in a chapter is measurements that are related to trigonometric-ratios & the other section is about subjects that are related to trigonometry identities\n\n### Topics & Sub Topics exercises in Class 10 Maths Chapter 8\n\nIn triangle ABC that’s right-angled at B, BC=7cm and AB=24cm. Find:\n\n(a) sin A, cos A\n\n(b) sin C, cos C\n\nExplanation:\n\nAccording to Pythagoras Theorem,\n\nWith a right-angled triangle, the hypotenuse side squared is equal to the sum of the squares of the other two sides.\n\nThat is\n\nAC2=AB2+BC2\n\nAC2 = (24)2+72\n\nAC2 = (576+49)\n\nAC2 = 625cm2\n\nAC = √625\n\n25\n\nThus, AC = 25cm\n\nGetting Sin (A), Cos (A)\n\nWe understand that sine(Sin) function is equal to the ratio of the length of the opposite side to the hypotenuse side. Thus it becomes\n\nSin(A) = Opposite side divide with the Hypotenuse\n\nBC/AC = 7/25\n\nThe cosine (Cos) function is the same as the ratio of the length of the adjacent side to the hypotenuse side.\n\nThus;\n\nAB/AC = 24/25\n\nGetting Sin(C), Cos(C)\n\nSin(C) = AB/AC\n\n24/25\n\nCos(C) = BC/AC\n\n7/25\n\n### Find tan P – cot R from the figure below\n\nFrom the figure,\n\nPQ=12cm\n\nPQ=13cm\n\nFrom right triangle\n\nWith Pythagoras theorem getting the measure of adjacent side/base.\n\n(hypotenuse)2=(base)2+(perpendicular)2\n\n(PR)2=(PQ)2+(QR)2\n\n(13)2=(12)2+(QR)2\n\n169=144+(QR)2\n\n(QR)2=169−144\n\n(QR)2=25cm2\n\nQR=5cm\n\nThus;\n\ntanP=QR/PQ\n\ntanP=5/12\n\ncotR=QR/PQ\n\ncotR=5/12\n\ntanP−cotR=5/12−5/12\n\ntanP−cotR=0\n\n### If ∠A & ∠B are acute angles in that cos B = cos A, prove that ∠A = ∠B.\n\nExplanation:\n\nConsidering triangle ABC is a right-angled triangle\n\n∠A and ∠B are acute angles, in that\n\nCos(A) = cos(B)\n\nCos ratio is as\n\nInterchanging the terms, we will have\n\nTaking a constant value k\n\nLet’s have the equation as\n\nAD = k BD …. (1)\n\nAC = k BC …. (2)\n\nUsing Pythagoras theorem in triangle CAD & triangle CBD we will have,\n\nCD2 = BC2 – BD2 …. (3)\n\nFrom above equations (3) & (4) we will have,\n\nSubstituting equations (1) & (2) in (3) & (4)\n\nK2(BC2−BD2)=(BC2−BD2) k2=1\n\nPlacing this value in our equation, we get\n\nAC = BC\n\n∠A=∠B\n\n### Triangle ABC is right-angled (B), tan A=1/√3. Get the value of:\n\n(a) sinA cosC + cosA sinC\n\n(b) cosA cosC – sinA sinC\n\nExplanations:\n\nLet triangle ABC where ∠B=90°\n\ntanA = BC/AB = 1/√3\n\nLet BC = 1k & AB = √3 k,\n\nk is positive real number\n\nWith Pythagoras theorem in triangle ABC we will have:\n\nAC2=AB2+BC2\n\nAC2=(√3 k)2+(k)2\n\nAC2=3k2+k2\n\nAC2=4k2\n\nAC = 2k\n\nFinding the values of cosA, SinA\n\nSinA = BC/AC = 1/2\n\nCosA = AB/AC\n\n√3/2\n\nFinding the values of cosC & sinC\n\nSinC = AB/AC\n\n√3/2\n\nCosC = BC/AC\n\n1/2\n\nSubstituting the values to the questions\n\n(a) sin A cos C + cos A sin C\n\n(1/2) ×(1/2 )+ √3/2 ×√3/2\n\n1/4 + ¾\n\n1\n\n(b) cosA cosC – sinA sinC\n\n(√3/2 )(1/2) – (1/2) (√3/2 )\n\n0\n\n1. State if the following are false or true. Explain your answer.\n\n(i) Value of tanA is all the time less than 1.\n\nThe statement is false. The value of tanA relies on the length of the right triangle sides which can be of any value.\n\n(ii) Some value of angle A, sec A=12/5.\n\nWe understand that right triangle sec A = hypotenuse/adjacent side of ∠A\n\nThe hypotenuse of the right triangle is the biggest side.\n\nThus, the value of sec A needs to be bigger than 1.\n\nFrom the given statement sec A = 12/5, which is bigger than 1.\n\nThus, the statement is true.\n\n(iii) cos A is the acronym used for cosecant of angle-A.\n\nThe statement is false the reason being cos A is the acronym for cosine of angle-A. The abbreviation for cosecant of angle-A is cosecA.\n\n(iv) cot A is a product of cot & A.\n\ncotA is an abbreviation for cotangent of angle-A. Thus the statement is false.\n\n(v) For angle θ, sinθ=43.\n\nWe understand that in a right triangle sinθ=opposite side/hypotenuse.\n\nThe hypotenuse is the longest side.\n\nThus, the value of sinθ needs to be less than 1.\n\nFrom the statement given sinθ=43, is bigger than 1.\n\nHence, the statement is false.\n\n1. Show that:\n\ntan48° tan23° tan42° tan67° = 1\n\nSolution:\n\ntan48° tan23° tan42° tan67°\n\nWe need to simplify the problem by converting some tan functions to cot functions\n\nWe understand that, tan 48° = tan (90°–42°) = cot 42°\n\ntan 23° = tan (90°–67°) = cot 67°\n\ntan (90°–42°) tan (90°–67°) tan 42° tan 67°\n\nSubstituting the values\n\ncot 42° cot 67° tan 42° tan 67°\n\n(cot 42° tan 42°) (cot 67° tan 67°)\n\n1×1\n\n## FAQ\n\nHow can you understand the key ideas presented in the NCERT Solutions for Class 10 Maths?\n\nBYJU’S recommends that students who want to achieve well in their Class 10 exams download the NCERT Solutions. A group of academics with extensive experience in the field curates the solutions with great care. Each and every minor element is discussed in an interactive manner to aid pupils in their learning. The step-by-step answers are created with the marks weighted according to CBSE criteria in mind.\n\nIs it true that the NCERT Solutions for Class 10 Maths are good for CBSE students?\n\nSome of the applications of the NCERT Maths Solutions for Class 10 are listed here.\n1. It establishes a solid foundation of essential concepts and increases confidence in the ability to face term-based exams.\n2. It is simple to learn how to solve complex problems.\n3. It is an excellent study resource for students who want to complete their assignments on time and earn higher grades in their term exams." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.85954785,"math_prob":0.9782607,"size":13546,"snap":"2023-40-2023-50","text_gpt3_token_len":3643,"char_repetition_ratio":0.20381036,"word_repetition_ratio":0.16283108,"special_character_ratio":0.25638565,"punctuation_ratio":0.07926125,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99907833,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-09T08:18:22Z\",\"WARC-Record-ID\":\"<urn:uuid:461f515e-8e44-4627-b88d-1bde25c69024>\",\"Content-Length\":\"61993\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ab366383-05eb-4398-8c27-8e05aa270cbf>\",\"WARC-Concurrent-To\":\"<urn:uuid:6ea0ef9c-167f-434b-ae52-1fae651f1772>\",\"WARC-IP-Address\":\"195.179.238.49\",\"WARC-Target-URI\":\"https://www.htmlkick.com/ncert-solutions/ncert-solutions-for-class-10-maths-chapter-8/\",\"WARC-Payload-Digest\":\"sha1:KW557SQ55KBA53ZNZWIHEJPKKXFDYXGA\",\"WARC-Block-Digest\":\"sha1:OXZM2FDLUXOLAQ22VNPGAPZ2724OAMKU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100873.6_warc_CC-MAIN-20231209071722-20231209101722-00584.warc.gz\"}"}
https://slideplayer.com/slide/6381886/	[ "", null, "# General, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.1 Chapter 1 Measurements 1.1 Units of Measurement.\n\n## Presentation on theme: \"General, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.1 Chapter 1 Measurements 1.1 Units of Measurement.\"— Presentation transcript:\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.1 Chapter 1 Measurements 1.1 Units of Measurement\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.2 Measurement in Chemistry In chemistry, we  measure quantities  do experiments  calculate results  use numbers to report measurements  compare results to standards\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.3 Stating a Measurement In every measurement, a number is followed by a unit. Observe the following examples of measurements: Number and Unit 35 m 0.25 L 225 lb 3.4 h\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.4 The Metric System (SI) The metric system and SI (international system) are  related decimal systems based on 10  used in most of the world  used everywhere by scientists\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.5 Units in the Metric System In the metric and SI systems, one unit is used for each type of measurement: MeasurementMetricSI Lengthmeter (m)meter (m) Volumeliter (L)cubic meter (m 3 ) Massgram (g)kilogram (kg) Timesecond (s)second (s) TemperatureCelsius (  C)Kelvin (K)\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.6 Length Measurement Length  is measured using a meter stick  uses the unit meter (m) in both the metric and SI systems\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.7 Inches and Centimeters The unit of an inch  is equal to exactly 2.54 centimeters in the metric (SI) system 1 in. = 2.54 cm\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.8 Volume Measurement Volume  is the space occupied by a substance  uses the unit liter (L) in the metric system 1 L = 1.06 qt  uses the unit cubic meter (m 3 ) in the SI system  is measured using a graduated cylinder\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.9 Mass Measurement The mass of an object  is a measure of the quantity of material it contains  is measured on a balance  uses the unit gram (g) in the metric system  uses the unit kilogram (kg) in the SI system\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc. 10 Temperature Measurement The temperature  indicates how hot or cold a substance is  is measured on the Celsius (  C) scale in the metric system  in the SI system uses the Kelvin (K) scale  on this thermometer is 18 °C or 64 °F.\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.11 Time Measurement Time measurement  uses the unit second (s) in both the metric and SI systems.  is based on an atomic clock that uses a frequency emitted by cesium atoms\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.12 1.2 Scientific Notation Scientific notation  is used to write very large or very small numbers  for the width of a human hair (0.000 008 m) is written 8 x 10 -6 m  for a large number such as 4 500 000 s is written 4.5 x 10 6 s\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.13 Writing Numbers in Scientific Notation  A number in scientific notation contains a coefficient and a power of 10. coefficient power of ten coefficient power of ten 1.5 x 10 2 7.35 x 10 -4  To write a number in scientific notation, the decimal point is placed after the first digit.  The spaces moved are shown as a power of ten. 52 000. = 5.2 x 10 4 0.00378 = 3.78 x 10 -3 4 spaces left 3 spaces right\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.14 Some Powers of Ten\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.15 Comparing Numbers in Standard and Scientific Notation Here are some numbers written in standard format and in scientific notation. Number in Standard Format Scientific Notation Diameter of the Earth 12 800 000 m1.28 x 10 7 m Mass of a human 68 kg 6.8 x 10 1 kg Length of a virus 0.000 03 cm3 x 10 -5 cm\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.16 1.3 Measured Numbers A measuring tool  is used to determine a quantity such as height or the mass of an object  provides numbers for a measurement called measured numbers\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.17. l 2.... l.... l 3.... l.... l 4.. cm Reading a Meter Stick  The markings on the meter stick at the end of the blue line are read as The first digit 2 plus the second digit 2.7  The last digit is obtained by estimating.  The end of the line might be estimated between 2.7 and 2.8 as 0.05 or 0.06, which gives a reported length of 2.75 cm or 2.76 cm.\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.18 Known + Estimated Digits In the length reported as 2.76 cm:  The digits 2 and 7 are certain (known).  The final digit 6 was estimated (uncertain).  All three digits (2.76) are significant, including the estimated digit.\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.19  For this measurement, the first and second known digits are 4.5.  Because the line ends on a mark, the estimated digit in the hundredths place is 0.  This measurement is reported as 4.50 cm. Zero as a Measured Number. l 3.... l.... l 4.... l.... l 5.. cm\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.20 Significant Figures in Measured Numbers Significant figures  Significant figures obtained from a measurement include all of the known digits plus the estimated digit.  The number of significant figures reported in a measurement depends on the measuring tool.\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.21 Significant Figures\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.22 Significant Figures in Scientific Notation In scientific notation:  All digits, including zeros in the coefficient, are significant. Scientific NotationNumber of Significant Figures___________ 8 x 10 4 m1 8.0 x 10 4 m2 8.00 x 10 4 m3\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.23 Examples of Exact Numbers\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.24 Rounding Off Calculated Answers  When the first digit dropped is 4 or less, the retained numbers remain the same. To round 45.832 to 3 significant figures drop the digits 32 = 45.8  When the first digit dropped is 5 or greater, the last retained digit is increased by 1. To round 2.4884 to 2 significant figures drop the digits 884 = 2.5 (increase by 0.1)\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.25 Adding Significant Zeros  Sometimes a calculated answer requires more significant digits. Then one or more zeros are added. Calculated answerZeros added to give 3 significant figures 44.00 1.51.50 0.20.200 12 12.0\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.26 When multiplying or dividing use  the same number of significant figures (SF) as the measurement with the fewest significant figures  rounding rules to obtain the correct number of significant figures Example: 110.5 x 0.048 = 5.304 = 5.3 (rounded) 4SFs 2SFs calculator 2SFs Multiplication and Division\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.27 When adding or subtracting, use  the same number of decimal places as the measurement with the fewest decimal places  rounding rules to adjust the number of digits in the answer 25.2 one decimal place + 1.34 two decimal places 26.54calculated answer 26.5 final answer (with one decimal place) Addition and Subtraction\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.28 1.5 Prefixes A prefix  in front of a unit increases or decreases the size of that unit  makes units larger or smaller than the initial unit by one or more factors of 10  indicates a numerical value PrefixValue 1 kilometer=1000 meters 1 kilogram=1000 grams\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.29 Metric and SI Prefixes (continued)\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.30 An equality  states the same measurement in two different units  can be written using the relationships between two metric units Example: 1 meter is the same as 100 cm and 1000 mm. 1 m = 100 cm 1 m = 1000 mm Metric Equalities\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.31 1.6 Exact and Measured Numbers in Equalities Equalities between units of  the same system are definitions with numbers that are exact  different systems (metric and U.S.) are measurements with numbers that are significant figures\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.32 A conversion factor  is a fraction obtained from an equality Equality: 1 in. = 2.54 cm  is written as a ratio with a numerator and denominator  can be inverted to give two conversion factors for every equality 1 in. and 2.54 cm 2.54 cm 1 in. Conversion Factors\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.33 A conversion factor  may be obtained from information in a word problem  is written for that problem only Example 1: The price of one pound (1 lb) of red peppers is \\$1.39. 1 lb red peppers and\\$1.39 \\$1.391 lb red peppers Example 2: The cost of one gallon (1 gal) of gas is \\$2.25. 1 gallon of gasand \\$2.25 \\$2.251 gallon of gas Conversion Factors in a Problem\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.34 A percent factor  gives the ratio of the parts to the whole % =Parts x 100% Whole  uses the same units for the parts and whole  uses the value 100  can be written as two factors Example: A food contains 18% (by mass) fat. 18 g fat and100 g food 100 g food18 g fat Percent as a Conversion Factor\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.35 Percent Factor in a Problem The thickness of the skin fold at the waist indicates 11% body fat. What percent factors can be written for body fat using kg? Percent factors using kg 11 kg fat and 100 kg mass 100 kg mass11 kg fat\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.36 Chapter 1 Measurements 1.7 Problem Solving\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.37 To solve a problem  Identify the given unit.  Identify the needed unit. Problem: A person has a height of 2.0 meters. What is that height in inches? The given unit is meters (the unit of height). given unit = meters (m) The needed unit is inches (for the answer). needed unit = inches (in.) Initial and Final Units\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.38  Write the given and needed units.  Write a unit plan to convert the given unit to the needed unit.  Write equalities and conversion factors.  Use conversion factors to cancel the given unit and provide the needed unit. Unit 1 x Unit 2 = Unit 2 Unit 1 Given x Conversion= Needed unit factor unit Problem Setup\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.39 Guide to Problem Solving The steps in the Guide to Problem Solving are useful in setting up a problem with conversion factors.\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.40 Setting up a Problem How many minutes are 2.5 h? Given unit = 2.5 h Needed unit =? min Plan =h min Set up problem to cancel hours (h). Given Conversion Needed unit factor unit 2.5 h x 60 min = 150 min (2 SF) 1 h\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.41  Often, two or more conversion factors are required to obtain the unit needed for the answer. Unit 1 Unit 2 Unit 3  Additional conversion factors are placed in the setup to cancel each preceding unit. Given unit x factor 1 x factor 2 = Needed unit Unit 1 x Unit 2 x Unit 3 = Unit 3 Unit 1 Unit 2 Using Two or More Factors\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.42 How many minutes are in 1.6 days? Given unit: days Needed unit: min Factor 1 Factor 2 Plan: days h min Set up problem: 1.6 days x 24 h x 60 min = 2300 = 2.3 x 10 3 min 1 day 1 h 2 SFs Exact Exact = 2 SFs Example: Problem Solving\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.43  Be sure to check your unit cancellation in the setup.  The units in the conversion factors must cancel to give the correct unit for the answer. Example: What is wrong with the following setup? 1.4 day x 1 day x 1 h 24 h 60 min Units = day 2 /min, which is not the unit needed Units don’t cancel properly. Therefore, setup is wrong. Check the Unit Cancellation\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.44 Guide to Problem Solving What is 165 lb in kg? STEP 1 Given: 165 lb Need: kg STEP 2 Plan: lb kg STEP 3 Equalities/conversion factors: 1 kg = 2.20 lb 2.20 lb and 1 kg 1 kg 2.20 lb STEP 4 Set up problem: 165 lb x 1 kg = 74.8 kg (3SF) 2.20 lb\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.45 Percent Factor in a Problem If the thickness of the skin fold at the waist indicates an 11% body fat, how much fat is in a person with a mass of 86 kg? 11% body fat, 11 kg fat 100 kg percent factor 86 kg x 11 kg fat = 9.5 kg of fat 100 kg\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.46 Density  compares the mass of an object to its volume  is the mass of a substance divided by its volume Density expression: D = mass = g or g = g/cm 3 volume mL cm 3 Note: 1 mL = 1 cm 3 1.8 Density\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.47 Volume by Displacement  A solid completely submerged in water displaces its own volume of water.  The volume of the solid is calculated from the volume difference. 45.0 mL – 35.5 mL = 9.5 mL = 9.5 cm 3\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.48 Density Using Volume Displacement The density of the zinc object is calculated from its mass and volume. mass = 68.60 g = 7.2 g/cm 3 volume 9.5 cm 3\n\nGeneral, Organic, and Biological ChemistryCopyright © 2010 Pearson Education, Inc.49 Sink or Float  Ice floats in water because the density of ice is less than the density of water.  Aluminum sinks in water because its density is greater than the density of water." ]	[ null, "https://slideplayer.com/static/blue_design/img/slide-loader4.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.84763396,"math_prob":0.8433108,"size":14961,"snap":"2020-24-2020-29","text_gpt3_token_len":3733,"char_repetition_ratio":0.2573377,"word_repetition_ratio":0.11243544,"special_character_ratio":0.2705033,"punctuation_ratio":0.14309484,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9808271,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-10T06:57:14Z\",\"WARC-Record-ID\":\"<urn:uuid:c04a27ea-c28d-4543-8853-a009e6046cd1>\",\"Content-Length\":\"242267\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:29ff9ada-7691-4a04-a1e7-fe33421bc058>\",\"WARC-Concurrent-To\":\"<urn:uuid:88c03a2b-6fcd-4ede-8bd0-b77f289a50a8>\",\"WARC-IP-Address\":\"138.201.54.25\",\"WARC-Target-URI\":\"https://slideplayer.com/slide/6381886/\",\"WARC-Payload-Digest\":\"sha1:GT4PHZDOUNGN3MNZ3VQZSHJ2JEC4HYWW\",\"WARC-Block-Digest\":\"sha1:4JORRR2AYMFCWJNM4HCC3JRUOEUHBE6K\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593655906214.53_warc_CC-MAIN-20200710050953-20200710080953-00207.warc.gz\"}"}
https://m.hanspub.org/journal/paper/42457	[ " 基于ZigBee的关键区域人员定位技术研究\n\n# 基于ZigBee的关键区域人员定位技术研究Research on Personnel Location Technology in Key Areas Based on ZigBee\n\nAbstract: With the rapid development of social economy, agencies, large enterprises and safety management units are paying more attention to the realization of personnel positioning and trajectory manage-ment. Managers need to accurately grasp the actual location, entry and exit, and residence time of personnel to improve business management and emergency management efficiency. In order to achieve the problem of refined personnel management, this paper chooses to use ZigBee technology and RSSI weighted positioning algorithm for precise personnel positioning, providing an effective solution for relevant units to help relevant units grasp the real-time distribution of on-site person-nel in the production area and achieve early warning of personnel risks in areas with major haz-ards.\n\n1. 引言\n\n2. 无线定位机制", null, "Figure 1. Triangulation\n\n$\\left\\{\\begin{array}{l}d1=\\frac{T1a-T}{2}V\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\left(2\\text{-}1\\right)\\\\ d2=\\frac{T1b-T}{2}V\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\left(2\\text{-2}\\right)\\\\ d3=\\frac{T1c-T}{2}*V\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\left( 2 -3 \\right)\\end{array}$\n\n$\\left\\{\\begin{array}{l}RSSI=-\\left(10n\\mathrm{lg}d+A\\right)\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\left(3\\text{-}1\\right)\\\\ P\\left(d\\right)=P\\left({d}_{0}\\right)+10n\\mathrm{lg}\\left(\\frac{1}{2}\\right)+{\\mathcal{l}}_{\\partial }\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}\\left( 3 -2 \\right)\\end{array}$\n\n$n=\\frac{\\underset{i=1}{\\overset{w}{\\sum }}{a}_{i}×{n}_{i}}{\\underset{i=1}{\\overset{w}{\\sum }}{a}_{i}}$ (4-1)", null, "Figure 2. Weighted RSSI algorithm flowchart\n\n5. 系统设计方案", null, "Figure 3. CC2530 Peripheral circuits", null, "Table 1. Coordinate comparison between actual position and measured position\n\n6. 结论\n\n 刘勇. 基于ZigBee的无线网络传感器定位技术[J]. 计算机产品与流通, 2020(10): 104.\n\n 胡建芳, 胡志军, 李利飞. 无线传感器网络定位技术研究[J]. 信息与电脑(理论版), 2018(24): 146-148.\n\n 石欣, 印爱民, 陈曦. 基于RSSI的多维标度室内定位算法[J]. 仪器仪表学报, 2014, 34(2): 36-40.\n\n 高鹏, 石为人, 周伟, 等. 基于图论模糊聚类的室内自适应RSSI定位算法[J]. 仪器仪表学报, 2013, 34(9): 1998- 2003.\n\n 刘飞飞, 徐隆姬, 马礼然. 基于ZigBee的分布式农业环境监测系统设计[J]. 传感器与微系统, 2021, 40(3): 90-92.\n\n 李战明, 李振兴. ZigBee技术在人员搜救系统中的应用[J]. 电子测量与仪器学报, 2012, 25(5): 186-190.\n\n 王战备. 基于ZigBee的农田信息监测网络设计[J]. 国外电子测量技术, 2013, 32(8): 42-45.\n\n 吕宏, 黄钉劲. 基于ZigBee技术低功耗无线温度数据采集及传输[J]. 国外电子测量技术, 2012, 31(2): 58-60.\n\n 王森. 基于物联网的多功能智能家居系统设计[J]. 电子世界, 2020, 66(5): 122-123+126.\n\n 周洪伟, 罗建, 王韫, 等. 外场环境信息无线监测系统设计[J]. 国外电子测量技术, 2011, 30(4): 66-68.\n\nTop" ]	[ null, "https://html.hanspub.org/file/1-2780176x9_hanspub.png", null, "https://html.hanspub.org/file/1-2780176x17_hanspub.png", null, "https://html.hanspub.org/file/1-2780176x18_hanspub.png", null, "https://m.hanspub.org/journal/paper/Images/Table_Tmp.jpg", null ]	{"ft_lang_label":"__label__zh","ft_lang_prob":0.9250977,"math_prob":0.9859239,"size":4534,"snap":"2021-43-2021-49","text_gpt3_token_len":3830,"char_repetition_ratio":0.06799117,"word_repetition_ratio":0.0,"special_character_ratio":0.21195413,"punctuation_ratio":0.1688103,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.991661,"pos_list":[0,1,2,3,4,5,6,7,8],"im_url_duplicate_count":[null,1,null,1,null,1,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-12-07T18:27:22Z\",\"WARC-Record-ID\":\"<urn:uuid:777dbe1b-b883-4965-bb7c-27292a5c7f1a>\",\"Content-Length\":\"47226\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2694b6b3-b68c-40bb-a50c-77db7c75ad11>\",\"WARC-Concurrent-To\":\"<urn:uuid:a8b7d827-c653-4eb6-8267-148e9bc851e9>\",\"WARC-IP-Address\":\"98.126.19.91\",\"WARC-Target-URI\":\"https://m.hanspub.org/journal/paper/42457\",\"WARC-Payload-Digest\":\"sha1:HWLNRWRJPMJSRJHZHI3GU7EKTSOY4H3E\",\"WARC-Block-Digest\":\"sha1:HJSTAVO4V7QOZM6RM7L4CDVEJ3AAF7L2\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964363405.77_warc_CC-MAIN-20211207170825-20211207200825-00082.warc.gz\"}"}
https://shop.strato.de/epages/311141.sf/de_DE/?ObjectPath=/Shops/311141/Categories/Buecher	[ "Drucken\n\n# Bücher\n\nAnzeige pro Seite\nSortieren nach\n\n### Mathematica: Eine Beispielorientierte Einführung\n\nArtikel-Nr.: bu0201\nBuch von Dr. Marie-Luise Herrmann. 360 S.Deutschsprachige Einführung in Mathematica 4 mit vielen Beispielen, sehr gut geeignet auch für den Schulbereich (weil die Autorin Lehrerin für Mathematik/Informatik und Physik ist). Bitte Liefermöglichkeit erfragen !\n27,90 \n\n### The Mathematica Guidebook for Numerics\n\nArtikel-Nr.: Bu0301\nThis book concentrates on Mathematica's numerical mathematics capabilities. The available types of arithmetic (machine, high-precision, and interval) are introduced, discussed, and put to use. Fundamental numerical operations, such as compiling programs, fast Fourier transforms, minimization, numerical solutions of equations, and ordinary/partial differential equations, are analyzed in detail and are applied to a large number of examples in the main text and in the solutions to the exercises.\n87,99 \n\n### The Mathematica Guidebook for Symbolics\n\nArtikel-Nr.: bu0302\nThis book deals with Mathematica's symbolic mathematical capabilities. Structural and mathematical operations on single and systems of polynomials are fundamental to many symbolic calculations, and they are covered here in considerable detail. The solution of equations and differential equations, as well as the classical calculus operations (differentiation, integration, summation, series expansion, limits) are exhaustively treated. Generalized functions and their uses are discussed. In addition, this volume discusses and employs the classical orthogonal polynomials and special functions of mathematical physics. To demonstrate the symbolic mathematics power, a large variety of problems from mathematics and physics are presented.\n87,99 \n\nAuf Lager\n\n### The Mathematica Guidebook Graphics\n\nArtikel-Nr.: bu0303\nThis book provides a comprehensive step-by-step development of how to use Mathematica to visualize functions and data, manipulate graphics, and optimize their appearance. Two-dimensional graphics, contour plots, plots of surfaces, free-form three-dimensional surfaces, and animations are the core topics. Hundreds of detailed examples and programs show a large variety of visualization techniques, algorithms, methods, and tricks.\n87,99 \n\nAuf Lager\n\n### The Mathematica Guidebook Programming\n\nArtikel-Nr.: bu0304\nThis book provides a comprehensive, step-by-step development of Mathematica programming capabilities and contains an enormous collection of examples and worked exercises. It guides the reader to become fluent in the structure of Mathematica expressions, expression evaluation, pure and named functions, and in procedural, rule-based, and functional programming constructs.\n87,99 \n\nAuf Lager\n\n### The Mathematica Book, Version 5\n\nArtikel-Nr.: bu0202\n\nBuch von Dr. Stephen Wolfram, 1464 Seiten, englisch, Wolfram Media, 2003. Solange Vorrat reicht !\n5th edition \"Fully updated for Mathematica Version 5\". Über den Autor: \"Stephen Wolfram is the creator of Mathematica, and the founder and president of Wolfram Research. ...\"\n\n79,95 \n\nAuf Lager\ninnerhalb 5 Tagen lieferbar\n\n* Preise inkl. MwSt., zzgl. Versand" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6805297,"math_prob":0.7796631,"size":2652,"snap":"2020-45-2020-50","text_gpt3_token_len":575,"char_repetition_ratio":0.12877643,"word_repetition_ratio":0.051630434,"special_character_ratio":0.21304676,"punctuation_ratio":0.17607224,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99238086,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-29T12:08:14Z\",\"WARC-Record-ID\":\"<urn:uuid:525d5d0f-c793-461b-a1d5-c1b5d499b8d7>\",\"Content-Length\":\"64406\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f63301ca-cb87-4293-a115-2da88586baa5>\",\"WARC-Concurrent-To\":\"<urn:uuid:ba980162-4688-4dc1-8ef4-c80db15e7387>\",\"WARC-IP-Address\":\"81.169.145.119\",\"WARC-Target-URI\":\"https://shop.strato.de/epages/311141.sf/de_DE/?ObjectPath=/Shops/311141/Categories/Buecher\",\"WARC-Payload-Digest\":\"sha1:3KXPRDJSPUYHHGFJ7SSNX7M7ZCCYNQLG\",\"WARC-Block-Digest\":\"sha1:V6AY25U6JJMVWN63JTOINNMVT4QRYDCJ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-45/CC-MAIN-2020-45_segments_1603107904039.84_warc_CC-MAIN-20201029095029-20201029125029-00660.warc.gz\"}"}
https://patchwork.kernel.org/project/linux-arm-kernel/patch/abd4dbaedf226ad29461775bcd969810c87c305d.1312887860.git.viresh.kumar@st.com/	[ "# [4/6] spi/spi-pl022: calculate_effective_freq() must set rate <= requested rate\n\nMessage ID abd4dbaedf226ad29461775bcd969810c87c305d.1312887860.git.viresh.kumar@st.com (mailing list archive) New, archived show\n\n## Commit Message\n\nViresh KUMAR Aug. 9, 2011, 11:05 a.m. UTC\n```There were few issues with calculate_effective_freq() routine:\n- It was returning first rate found >= requested rate. Now, if system have spi's\nrate as 83 MHz, with possible prescaled rates as 83, 41.5, 20.75, 13.83 (as we\ncan prescale with multiples of 2). If user has given rate to be programmed as\n22 MHz, then driver programmes it to 41.5 MHz. This looks to be incorrect, as\nuser might have given the upper limit of the device, and we are programming it\nabove it.\n- Driver finds the first satisfying rate and programmes it, but with other\nvalues of scr & cpsdvsr, it is possible to get more closer rate.\n\nThis patch fixes these two issues, with some reformatting inside the code.\nThis also creates a macro to calculate prescaled rate based on spi's rate,\ncpsdvsr and scr.\n\nSigned-off-by: Viresh Kumar <[email protected]>\n---\ndrivers/spi/spi-pl022.c \| 98 ++++++++++++++++++++++------------------------\n1 files changed, 47 insertions(+), 51 deletions(-)\n```\n\n## Comments\n\nViresh KUMAR Aug. 10, 2011, 3:36 a.m. UTC \| #1\n```[Probably you missed reply-all by mistake, so adding them again]\n\nOn 08/10/2011 01:34 AM, Linus Walleij wrote:\n> On Tue, Aug 9, 2011 at 1:05 PM, Viresh Kumar <[email protected]> wrote:\n>\n>> There were few issues with calculate_effective_freq() routine:\n>> - It was returning first rate found >= requested rate. Now, if system have spi's\n>> rate as 83 MHz, with possible prescaled rates as 83, 41.5, 20.75, 13.83 (as we\n>> can prescale with multiples of 2). If user has given rate to be programmed as\n>> 22 MHz, then driver programmes it to 41.5 MHz. This looks to be incorrect, as\n>> user might have given the upper limit of the device, and we are programming it\n>> above it.\n>> - Driver finds the first satisfying rate and programmes it, but with other\n>> values of scr & cpsdvsr, it is possible to get more closer rate.\n>\n> Good that you found this bug!\n>\n>> +#define SPI_RATE(rate, cpsdvsr, scr) (rate / (cpsdvsr * (1 + scr)))\n>\n> Can you use a static inline instead of a macro? It is often preferred.\n> If you do, this is Acked-by.\n>\n\nNot a problem. Will surely do that.\n```\n\n## Patch\n\n```diff --git a/drivers/spi/spi-pl022.c b/drivers/spi/spi-pl022.c\nindex 452952b..e0cfb8c 100644\n--- a/drivers/spi/spi-pl022.c\n+++ b/drivers/spi/spi-pl022.c\n@@ -1791,67 +1791,63 @@ static int pl022_transfer(struct spi_device spi, struct spi_message msg)\nreturn 0;\n}\n\n-static int calculate_effective_freq(struct pl022 pl022,\n-\t\t\t\t int freq,\n-\t\t\t\t struct ssp_clock_params clk_freq)\n+#define SPI_RATE(rate, cpsdvsr, scr)\t(rate / (cpsdvsr * (1 + scr)))\n+static int calculate_effective_freq(struct pl022 pl022, int freq, struct\n+\t\t\t\t ssp_clock_params clk_freq)\n{\n/* Lets calculate the frequency parameters /\n-\tu16 cpsdvsr = 2;\n-\tu16 scr = 0;\n-\tbool freq_found = false;\n-\tu32 rate;\n-\tu32 max_tclk;\n-\tu32 min_tclk;\n+\tu16 cpsdvsr = CPSDVR_MIN, scr = SCR_MIN;\n+\tu32 rate, max_tclk, min_tclk, best_freq = 0, best_cpsdvsr = 0,\n+\t\tbest_scr = 0, tmp, found = 0;\n\nrate = clk_get_rate(pl022->clk);\n/ cpsdvscr = 2 & scr 0 /\n-\tmax_tclk = (rate / (CPSDVR_MIN (1 + SCR_MIN)));\n+\tmax_tclk = SPI_RATE(rate, CPSDVR_MIN, SCR_MIN);\n/* cpsdvsr = 254 & scr = 255 /\n-\tmin_tclk = (rate / (CPSDVR_MAX (1 + SCR_MAX)));\n-\n-\tif ((freq <= max_tclk) && (freq >= min_tclk)) {\n-\t\twhile (cpsdvsr <= CPSDVR_MAX && !freq_found) {\n-\t\t\twhile (scr <= SCR_MAX && !freq_found) {\n-\t\t\t\tif ((rate /\n-\t\t\t\t (cpsdvsr * (1 + scr))) > freq)\n-\t\t\t\t\tscr += 1;\n-\t\t\t\telse {\n-\t\t\t\t\t/\n-\t\t\t\t\t This bool is made true when\n-\t\t\t\t\t * effective frequency >=\n-\t\t\t\t\t * target frequency is found\n-\t\t\t\t\t /\n-\t\t\t\t\tfreq_found = true;\n-\t\t\t\t\tif ((rate /\n-\t\t\t\t\t (cpsdvsr (1 + scr))) != freq) {\n-\t\t\t\t\t\tif (scr == SCR_MIN) {\n-\t\t\t\t\t\t\tcpsdvsr -= 2;\n-\t\t\t\t\t\t\tscr = SCR_MAX;\n-\t\t\t\t\t\t} else\n-\t\t\t\t\t\t\tscr -= 1;\n-\t\t\t\t\t}\n-\t\t\t\t}\n-\t\t\t}\n-\t\t\tif (!freq_found) {\n-\t\t\t\tcpsdvsr += 2;\n-\t\t\t\tscr = SCR_MIN;\n-\t\t\t}\n-\t\t}\n-\t\tif (cpsdvsr != 0) {\n-\t\t\tdev_dbg(&pl022->adev->dev,\n-\t\t\t\t\"SSP Effective Frequency is %u\\n\",\n-\t\t\t\t(rate / (cpsdvsr * (1 + scr))));\n-\t\t\tclk_freq->cpsdvsr = (u8) (cpsdvsr & 0xFF);\n-\t\t\tclk_freq->scr = (u8) (scr & 0xFF);\n-\t\t\tdev_dbg(&pl022->adev->dev,\n-\t\t\t\t\"SSP cpsdvsr = %d, scr = %d\\n\",\n-\t\t\t\tclk_freq->cpsdvsr, clk_freq->scr);\n-\t\t}\n-\t} else {\n+\tmin_tclk = SPI_RATE(rate, CPSDVR_MAX, SCR_MAX);\n+\n+\tif (!((freq <= max_tclk) && (freq >= min_tclk))) {\ndev_err(&pl022->adev->dev,\n\"controller data is incorrect: out of range frequency\");\nreturn -EINVAL;\n}\n+\n+\t/\n+\t best_freq will give closest possible available rate (<= requested\n+\t * freq) for all values of scr & cpsdvsr.\n+\t /\n+\twhile ((cpsdvsr <= CPSDVR_MAX) && !found) {\n+\t\twhile (scr <= SCR_MAX) {\n+\t\t\ttmp = SPI_RATE(rate, cpsdvsr, scr);\n+\n+\t\t\tif (tmp > freq)\n+\t\t\t\tscr++;\n+\t\t\t/\n+\t\t\t * If found exact value, update and break.\n+\t\t\t * If found more closer value, update and continue.\n+\t\t\t */\n+\t\t\telse if ((tmp == freq) \|\| (tmp > best_freq)) {\n+\t\t\t\tbest_freq = tmp;\n+\t\t\t\tbest_cpsdvsr = cpsdvsr;\n+\t\t\t\tbest_scr = scr;\n+\n+\t\t\t\tif (tmp == freq)\n+\t\t\t\t\tbreak;\n+\t\t\t}\n+\t\t\tscr++;\n+\t\t}\n+\t\tcpsdvsr += 2;\n+\t\tscr = SCR_MIN;\n+\t}\n+\n+\tclk_freq->cpsdvsr = (u8) (best_cpsdvsr & 0xFF);\n+\tclk_freq->scr = (u8) (best_scr & 0xFF);\n+\tdev_dbg(&pl022->adev->dev,\n+\t\t\"SSP Target Frequency is: %u, Effective Frequency is %u\\n\",\n+\t\tfreq, best_freq);\n+\tdev_dbg(&pl022->adev->dev, \"SSP cpsdvsr = %d, scr = %d\\n\",\n+\t\tclk_freq->cpsdvsr, clk_freq->scr);\n+\nreturn 0;\n}\n\n```" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7405719,"math_prob":0.98847497,"size":3832,"snap":"2021-04-2021-17","text_gpt3_token_len":1287,"char_repetition_ratio":0.1415883,"word_repetition_ratio":0.04360465,"special_character_ratio":0.40944678,"punctuation_ratio":0.19617225,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97928256,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-04-13T01:00:38Z\",\"WARC-Record-ID\":\"<urn:uuid:73b60893-6f83-4dcb-940e-3899dbd3b6b6>\",\"Content-Length\":\"19848\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:3aef1af3-015e-469a-adcb-d0caeab1cea1>\",\"WARC-Concurrent-To\":\"<urn:uuid:b1748d3f-07bd-4b8b-8220-c9c0d31c5d62>\",\"WARC-IP-Address\":\"198.145.29.83\",\"WARC-Target-URI\":\"https://patchwork.kernel.org/project/linux-arm-kernel/patch/abd4dbaedf226ad29461775bcd969810c87c305d.1312887860.git.viresh.kumar@st.com/\",\"WARC-Payload-Digest\":\"sha1:VEARXAKIK2MBXF4PAKMDUJTUXNFT6EFM\",\"WARC-Block-Digest\":\"sha1:EHT7IMPV6VNIBPABWBOPTZATBQR23X77\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-17/CC-MAIN-2021-17_segments_1618038071212.27_warc_CC-MAIN-20210413000853-20210413030853-00150.warc.gz\"}"}
https://www.mersenneforum.org/showthread.php?s=9452ee03b3bac27664113ac615410fc0&t=17407	[ "", null, "mersenneforum.org", null, "Number of distinct prime factors of a Double Mersenne number\n Register FAQ Search Today's Posts Mark Forums Read", null, "2012-11-09, 19:49 #1 aketilander \"Åke Tilander\" Apr 2011 Sandviken, Sweden 10001101102 Posts", null, "Number of distinct prime factors of a Double Mersenne number I am trying to figure out a way to estimate the number of distinct prime factors of a Double Mersenne number. If I understand it rightly, for a specific number n the number of distinct prime factors x are: ω (n) which is asymptotically equal to ln (ln n) for a Double Mersenne number n=MMp: ln(ln (2^(2^p-1)-1)) ignoring both \"-1\" since those parts will be infinitesimally small with growing p. ln(2^p * ln(2)) = ln(ln(2)) + pln(2) = -0.367 + pln(2) = -0.367 + 0.693*p i.e. for MM127 (i.e. p=127) x=87.64 Maybe it may be argued that since both p and Mp are prime x may be a little smaller? Have I understood this rightly or have I done something wrong? If this is right the nice thing is that the estimated number of distinct prime factors of a MMp are directely proportional to p.", null, "Last fiddled with by aketilander on 2012-11-09 at 20:09", null, "", null, "", null, "2012-11-09, 21:16 #2 ewmayer ∂2ω=0 Sep 2002 República de California 2D6B16 Posts", null, "Any such estimate needs to take into account the special \"restricted\" form of Mersenne factors, which in general will cause M(p) (and by extension M(M(p) for M(p) prime) to have a lower expected number of factors than a general odd number of similar size. Here is a paper I found via cursory online search - the paper itself is not so much of interest in the present context as are the references, several of which appear to have investigated the question you ask.", null, "", null, "Thread Tools", null, "Show Printable Version", null, "Email this Page", null, "Similar Threads Thread Thread Starter Forum Replies Last Post michael Math 31 2015-09-04 05:57 henryzz Math 7 2012-05-23 01:13 princeps Miscellaneous Math 18 2011-11-30 00:16 kurtulmehtap Math 12 2010-05-03 14:02 flouran Math 10 2009-04-29 03:57\n\nAll times are UTC. The time now is 16:10.\n\nFri May 7 16:10:42 UTC 2021 up 29 days, 10:51, 1 user, load averages: 3.18, 3.17, 3.52" ]	[ null, "https://www.mersenneforum.org/images/misc/navbits_start.gif", null, "https://www.mersenneforum.org/images/misc/navbits_finallink_ltr.gif", null, "https://www.mersenneforum.org/images/statusicon/post_old.gif", null, "https://www.mersenneforum.org/images/icons/icon1.gif", null, "https://www.mersenneforum.org/images/smilies/extra/smile.gif", null, "https://www.mersenneforum.org/images/statusicon/user_offline.gif", null, "https://www.mersenneforum.org/images/buttons/quote.gif", null, "https://www.mersenneforum.org/images/statusicon/post_old.gif", null, "https://www.mersenneforum.org/images/icons/icon1.gif", null, "https://www.mersenneforum.org/images/statusicon/user_offline.gif", null, "https://www.mersenneforum.org/images/buttons/quote.gif", null, "https://www.mersenneforum.org/images/buttons/printer.gif", null, "https://www.mersenneforum.org/images/buttons/sendtofriend.gif", null, "https://www.mersenneforum.org/images/buttons/collapse_tcat.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8410347,"math_prob":0.8428723,"size":2071,"snap":"2021-21-2021-25","text_gpt3_token_len":626,"char_repetition_ratio":0.13981616,"word_repetition_ratio":0.048991356,"special_character_ratio":0.32303235,"punctuation_ratio":0.07363421,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9606251,"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],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-05-07T16:10:42Z\",\"WARC-Record-ID\":\"<urn:uuid:ed2b5f06-0d93-4364-93de-6054734caf00>\",\"Content-Length\":\"32159\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:bdd12666-18d1-4cc2-bbec-c0952f93f076>\",\"WARC-Concurrent-To\":\"<urn:uuid:948c238c-ab18-4e80-a9ad-2b20a40c26dd>\",\"WARC-IP-Address\":\"216.92.123.112\",\"WARC-Target-URI\":\"https://www.mersenneforum.org/showthread.php?s=9452ee03b3bac27664113ac615410fc0&t=17407\",\"WARC-Payload-Digest\":\"sha1:OUGWXFEV4QOTJYVAJ7INWY6Z4M7FYH4H\",\"WARC-Block-Digest\":\"sha1:SBD7XQWYDRLZDN5HDAQUIFZFWXCTEEZ6\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-21/CC-MAIN-2021-21_segments_1620243988796.88_warc_CC-MAIN-20210507150814-20210507180814-00354.warc.gz\"}"}
https://convertoctopus.com/18-5-kilometers-per-hour-to-meters-per-second	[ "## Conversion formula\n\nThe conversion factor from kilometers per hour to meters per second is 0.277777777778, which means that 1 kilometer per hour is equal to 0.277777777778 meters per second:\n\n1 km/h = 0.277777777778 m/s\n\nTo convert 18.5 kilometers per hour into meters per second we have to multiply 18.5 by the conversion factor in order to get the velocity amount from kilometers per hour to meters per second. We can also form a simple proportion to calculate the result:\n\n1 km/h → 0.277777777778 m/s\n\n18.5 km/h → V(m/s)\n\nSolve the above proportion to obtain the velocity V in meters per second:\n\nV(m/s) = 18.5 km/h × 0.277777777778 m/s\n\nV(m/s) = 5.138888888893 m/s\n\nThe final result is:\n\n18.5 km/h → 5.138888888893 m/s\n\nWe conclude that 18.5 kilometers per hour is equivalent to 5.138888888893 meters per second:\n\n18.5 kilometers per hour = 5.138888888893 meters per second\n\n## Alternative conversion\n\nWe can also convert by utilizing the inverse value of the conversion factor. In this case 1 meter per second is equal to 0.19459459459444 × 18.5 kilometers per hour.\n\nAnother way is saying that 18.5 kilometers per hour is equal to 1 ÷ 0.19459459459444 meters per second.\n\n## Approximate result\n\nFor practical purposes we can round our final result to an approximate numerical value. We can say that eighteen point five kilometers per hour is approximately five point one three nine meters per second:\n\n18.5 km/h ≅ 5.139 m/s\n\nAn alternative is also that one meter per second is approximately zero point one nine five times eighteen point five kilometers per hour.\n\n## Conversion table\n\n### kilometers per hour to meters per second chart\n\nFor quick reference purposes, below is the conversion table you can use to convert from kilometers per hour to meters per second\n\nkilometers per hour (km/h) meters per second (m/s)\n19.5 kilometers per hour 5.417 meters per second\n20.5 kilometers per hour 5.694 meters per second\n21.5 kilometers per hour 5.972 meters per second\n22.5 kilometers per hour 6.25 meters per second\n23.5 kilometers per hour 6.528 meters per second\n24.5 kilometers per hour 6.806 meters per second\n25.5 kilometers per hour 7.083 meters per second\n26.5 kilometers per hour 7.361 meters per second\n27.5 kilometers per hour 7.639 meters per second\n28.5 kilometers per hour 7.917 meters per second" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.706479,"math_prob":0.9929166,"size":2313,"snap":"2021-43-2021-49","text_gpt3_token_len":604,"char_repetition_ratio":0.34820268,"word_repetition_ratio":0.058666665,"special_character_ratio":0.32338953,"punctuation_ratio":0.115226336,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9982594,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-12-03T09:41:18Z\",\"WARC-Record-ID\":\"<urn:uuid:47c4763e-6250-4bfa-a12b-03b29971e75f>\",\"Content-Length\":\"32361\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d3447a1f-0e45-4f2b-8fd9-6529487587bb>\",\"WARC-Concurrent-To\":\"<urn:uuid:5465ba4c-3d5e-4270-9a84-aae2a3c3f295>\",\"WARC-IP-Address\":\"104.21.56.212\",\"WARC-Target-URI\":\"https://convertoctopus.com/18-5-kilometers-per-hour-to-meters-per-second\",\"WARC-Payload-Digest\":\"sha1:DRWJ7UVQASUNFGIEC4J4IG6K6LHINVXM\",\"WARC-Block-Digest\":\"sha1:S5PP57HIA4QIATUI6NEJSBKF7NH7EP6O\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964362619.23_warc_CC-MAIN-20211203091120-20211203121120-00010.warc.gz\"}"}
https://cse.engin.umich.edu/event/on-the-quantitative-hardness-of-the-closest-vector-problem	[ "# On the Quantitative Hardness of the Closest Vector Problem\n\nHuck BennettPostdoctoral ResearcherNorthwestern University\nSHARE:\n\nComputational problems on lattices have a wealth of applications in\ncomputer science, and, in recent years, lattice-based cryptography has\nemerged as the leading candidate for post-quantum cryptography. In\norder to ensure the security of such cryptosystems, it is crucial to\nunderstand the precise, quantitative computational complexity of\nlattice problems.\n\nIn this talk, I will discuss work that initiates the study of the\nquantitative hardness of lattice problems. As our main result, we\nprove that for almost every p \\geq 1 and every constant eps > 0 there\nis no 2^{(1-\\eps)n}-time algorithm for the Closest Vector Problem with\nrespect to the \\ell_p norm (CVP_p) assuming the Strong Exponential\nTime Hypothesis (SETH). This comes tantalizingly close to settling the\nquantitative time complexity of the important special case of CVP_2\n(i.e., CVP in the Euclidean norm) for which a 2^{n + o(n)}-time\nalgorithm is known, and is necessary (but not sufficient) for ensuring\nthe security of about-to-be-deployed lattice-based cryptosystems. If,\nfor example, there existed a 2^{n/20}-time algorithm for CVP then\nthese schemes would be insecure in practice.\n\nBased on joint work with Alexander Golovnev and Noah\nStephens-Davidowitz (https://arxiv.org/abs/1704.03928)." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.85130423,"math_prob":0.7897501,"size":1358,"snap":"2021-04-2021-17","text_gpt3_token_len":334,"char_repetition_ratio":0.10044313,"word_repetition_ratio":0.0,"special_character_ratio":0.2113402,"punctuation_ratio":0.094262294,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98090214,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-01-17T13:36:02Z\",\"WARC-Record-ID\":\"<urn:uuid:7f93f19e-9521-4dbd-9376-eef75af5316d>\",\"Content-Length\":\"50497\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:5d360197-5773-490a-8394-aca2e2036fe2>\",\"WARC-Concurrent-To\":\"<urn:uuid:396c1fdc-9b36-4de1-ab1f-b99cccf3e602>\",\"WARC-IP-Address\":\"34.67.162.33\",\"WARC-Target-URI\":\"https://cse.engin.umich.edu/event/on-the-quantitative-hardness-of-the-closest-vector-problem\",\"WARC-Payload-Digest\":\"sha1:GQOOJPEHX6C4UQ6BUNO6RDT5XXWOWHOL\",\"WARC-Block-Digest\":\"sha1:XY5UZS53SLPOOQSAKF7AVWSXIJBOSKUW\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-04/CC-MAIN-2021-04_segments_1610703512342.19_warc_CC-MAIN-20210117112618-20210117142618-00726.warc.gz\"}"}
https://ch.mathworks.com/matlabcentral/cody/problems/424-snakes-on-a-plane/solutions/909506	[ "Cody\n\n# Problem 424. Snakes on a plane\n\nSolution 909506\n\nSubmitted on 15 Jun 2016 by hubert andre\nThis solution is locked. To view this solution, you need to provide a solution of the same size or smaller.\n\n### Test Suite\n\nTest Status Code Input and Output\n1 Pass\nm = 7; n = 10; y_correct = 26; assert(isequal(sum(sum(your_fcn_name(m,n))),y_correct))\n\ny = Columns 1 through 16 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Columns 17 through 26 1 1 1 1 1 1 1 1 1 1\n\n2 Pass\nm = 5; n = 10; y_correct = 17; assert(isequal(sum(sum(your_fcn_name(m,n))),y_correct))\n\ny = Columns 1 through 16 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Column 17 1\n\n3 Pass\nm = 9; n = 6; y_correct = 19; assert(isequal(sum(sum(your_fcn_name(m,n))),y_correct))\n\ny = Columns 1 through 16 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Columns 17 through 19 1 1 1" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.518634,"math_prob":0.9963831,"size":753,"snap":"2020-34-2020-40","text_gpt3_token_len":337,"char_repetition_ratio":0.24833111,"word_repetition_ratio":0.4409938,"special_character_ratio":0.48339975,"punctuation_ratio":0.0952381,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97670686,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-09-21T07:37:05Z\",\"WARC-Record-ID\":\"<urn:uuid:1c98c6d1-5d26-4e24-b2ba-6c25a3269181>\",\"Content-Length\":\"76861\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:dd2d54e9-489a-4848-ad90-3de9934eb2c2>\",\"WARC-Concurrent-To\":\"<urn:uuid:e67315da-187a-4e57-9fb3-9e19dc6ea1f5>\",\"WARC-IP-Address\":\"23.212.144.59\",\"WARC-Target-URI\":\"https://ch.mathworks.com/matlabcentral/cody/problems/424-snakes-on-a-plane/solutions/909506\",\"WARC-Payload-Digest\":\"sha1:3PGOV7SL4MN6ROO47XS43BZ53R6R3POO\",\"WARC-Block-Digest\":\"sha1:QLLDBNQAABTTP2QVAJVFVT2G6TMQFVER\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600400198942.13_warc_CC-MAIN-20200921050331-20200921080331-00551.warc.gz\"}"}
https://visual-eiffel.com/what-are-some-examples-of-uniform-motion/	[ "# What are some examples of uniform motion?\n\nSuch a motion is uniform in nature. Eg: A car moving on a straight road without any change in its velocity is one of the examples. A Ball rolling on a floor without changing its velocity is again an example of uniform Motion.\n\nHerein, is constant velocity and uniform motion same?\n\nMotion is typically described in terms of displacement, distance, velocity, acceleration, time and speed. The definition of uniform motion is that the object is supposed to cover equal distances in equal intervals of time. This means that the object in motion will have constant velocity.\n\nWhat is the difference between uniform and constant motion?\n\nThe body can be said to be moving with constant velocity if it travels on a straight path with constant speed. When a body performs motion with uniform velocity, magnitude of the velocity will remain same but direction may change.\n\nIs constant acceleration uniform motion?\n\nThis is called uniform circular motion. No, when a body is under uniform acceleration doesn’t mean that it is under uniform motion. Uniform acceleration means, that the velocity of body is changing with a constant rate. On the other hand, uniform motion means that the body is in motion with constant velocity.\n\n## What is the uniform motion?\n\nUNIFORM MOTION AND NON-UNIFORM MOTION:A body is said to be in uniform motion, if it travels equal distances in equal intervals of time.A body is said to have non-uniform motion, if it travels unequal distances in equal intervals of time. We can say that the motion is called uniform motion.\n\n## What is the definition of uniform motion?\n\nDefinition: Uniform motion is defined as the motion of an object in which the object travels in a straight line and its velocity remains constant along that line as it covers equal distances in equal intervals of time, irrespective of the length of the time.\n\n## How many types of motions are there?\n\nThere are four basic types of motion in mechanical systems: Rotary motion is turning round in a circle, such as a wheel turning. Linear motion is moving in a straight line, such as on a paper trimmer. Reciprocating motion is moving backwards and forwards in a straight line, as in cutting with a saw.\n\n## What is the uniform?\n\nA uniform is a type of clothing worn by members of an organization while participating in that organization’s activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, security guards, in some workplaces and schools and by inmates in prisons.\n\n## What is meant by the term retardation?\n\nDefinition of retardation. 1 : an act or instance of retarding. 2 : the extent to which something is retarded. 3 : a musical suspension; specifically : one that resolves upward. 4 a : an abnormal slowness of thought or action; especially : mental retardation.\n\n## What is the SI unit of speed?\n\nThe SI unit of time is the second. The SI unit of speed and velocity is the ratio of two — the meter per second. This unit is only rarely used outside scientific and academic circles. Most people on this planet measure speeds in kilometer per hour (km/h or sometimes kph).\n\n## What is the definition of uniform speed?\n\nDefine uniform speed. Answer: If a body covers equal distance in equal intervals of time then it is said to be moving at uniform speed. Mathematically speed is defined as v = S⁄t here v is speed of object, S is the distance covered and t is time taken.\n\n## What is the uniformly accelerated motion?\n\nUniform or constant acceleration is a type of motion in which the velocity of an object changes by an equal amount in every equal time period. A frequently cited example of uniform acceleration is that of an object in free fall in a uniform gravitational field.\n\n## What is uniform and non uniform acceleration?\n\nNon-uniform acceleration constitutes the most general description of motion. It refers to variation in the rate of change in velocity. Simply put, it means that acceleration changes during motion.\n\n## What is the definition of uniform acceleration?\n\nBSL Physics Glossary – uniform acceleration – definition. Translation: If an object’s speed (velocity) is increasing at a constant rate then we say it has uniform acceleration. The rate of acceleration is constant. If a car speeds up then slows down then speeds up it doesn’t have uniform acceleration.\n\n## What is the shape of the path of the body when it is in uniform motion?\n\nA body is said to be in uniform motion, if it travels equal distances in equal intervals of time, no matter how small these time intervals may be. The path of the body may be straight line or curved or zig-zag. Its direction may differ but its magnitude remains fixed.\n\n## What is uniform and non uniform velocity?\n\nThe body should travel with the constant speed which means it covers equal displacement with equal time intervals. Also, for the non – uniform velocity: 1. The body should travel with constant speed but constantly changing the direction of the motion.\n\n## What is accelerated motion?\n\nIf an object is changing its velocity -whether by a constant amount or a varying amount – then it is an accelerating object. And an object with a constant velocity is not accelerating. The data tables below depict motions of objects with a constant acceleration and a changing acceleration.\n\n## What is non uniform linear motion?\n\nThe linear motion can be of two types: uniform linear motion with constant velocity or zero acceleration; non uniform linear motion with variable velocity or non-zero acceleration. The motion of a particle (a point-like object) along a line can be described by its position , which varies with (time).\n\n## What is the difference between speed and velocity?\n\nIn the end, this is major difference between Speed and Velocity. Though both are calculated using the same units (km/h, m/s, mph, etc.), the two are different in that one is described using numerical values alone (i.e. a scalar quantity) whereas the other describes both magnitude and direction (a vector quantity).\n\n## What is the average speed of a?\n\nAverage speed is a rate. In kinematics, a rate is always a quantity divided by the time taken to get that quantity (the elapsed time). Since average speed is the rate position changes, average speed = distance traveled/time taken.\n\n## What do you mean by oscillatory motion?\n\nOscillatory motion. Oscillatory motion can be termed as the repeated motion in which an object repeats the same movement over and over.\n\n## What is the definition of constant motion?\n\nDefinition of Constant Motion. Constant motion refers to any type of motion when either the distance traveled by the object is the same for each second, or the speed of the object changes by the same amount each second.\n\n## What is a position vs time graph?\n\nNote that a motion described as a changing, positive velocity results in a line of changing and positive slope when plotted as a position-time graph. The position vs. time graphs for the two types of motion – constant velocity and changing velocity (acceleration) – are depicted as follows.\n\nOriginally posted 2021-12-27 18:30:40.\n\nCategories FAQ" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.94008046,"math_prob":0.98125887,"size":6983,"snap":"2022-40-2023-06","text_gpt3_token_len":1392,"char_repetition_ratio":0.19988537,"word_repetition_ratio":0.04556962,"special_character_ratio":0.19719319,"punctuation_ratio":0.1051051,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9967683,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-02-07T19:02:26Z\",\"WARC-Record-ID\":\"<urn:uuid:72b9b5b4-0106-4896-91a6-fad3fbe7394c>\",\"Content-Length\":\"50953\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:444c1fd0-abd2-496e-969b-191a2b1b2cef>\",\"WARC-Concurrent-To\":\"<urn:uuid:371000c5-ed1c-46a8-b2e0-adf6ce64dbf5>\",\"WARC-IP-Address\":\"45.77.216.158\",\"WARC-Target-URI\":\"https://visual-eiffel.com/what-are-some-examples-of-uniform-motion/\",\"WARC-Payload-Digest\":\"sha1:KXQDVBF67JJXYAZFTFKJIRYWPO2XLILR\",\"WARC-Block-Digest\":\"sha1:Z6PEGNVKMYYP25ECLVPSZZY5MVI6CLEV\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764500628.77_warc_CC-MAIN-20230207170138-20230207200138-00515.warc.gz\"}"}
https://edurev.in/studytube/How-to-prepare-for-Class-7-Maths-Tips-Tricks-for-M/4ed13abe-11a0-45c4-a1f5-e95df0ee9360_t	[ "Courses\n\n# How to prepare for Class 7 Maths: Tips & Tricks for Maths Class 7 Notes \| EduRev\n\n## Class 7 : How to prepare for Class 7 Maths: Tips & Tricks for Maths Class 7 Notes \| EduRev\n\nThe document How to prepare for Class 7 Maths: Tips & Tricks for Maths Class 7 Notes \| EduRev is a part of Class 7 category.\nAll you need of Class 7 at this link: Class 7\n\nOne thing to be kept in mind all along the way for Maths preparation is that practice is the key here. The more you practice for every topic, the better understanding is achieved.\n\nThe main topics in Class 7 Maths are basic arithmetic, algebra, geometry and mensuration, probability, etc. These topics should be practiced as much as possible. The key to solve problems is to by-heart the formulas, understand the theorems, and practice as many times as possible. Maths can become your favorite subject with practice and dedication.", null, "Mathematics for Class 7 (VII) - CBSE & NCERT Curriculum\n\nIncludes 69 docs, 220 videos & 30 tests\n\n## Tips & Tricks for Mathematics :\n\n1. Study from the Mathematics NCERT Textbook and practice all the solved examples and exercise questions thoroughly before moving forward to extra practice from additional books. Refer to NCERT Solutions in case of any doubt.\n\n## NCERT Textbooks and solutions\n\nChapter 1 - Integers\n\nChapter 2 - Fractions and Decimals\n\nChapter 3 - Data Handling\n\nChapter 4 - Simple Equations\n\nChapter 5 - Lines and Angles\n\nChapter 6 - The Triangles and its properties\n\nChapter 7 - Congruence of Triangles\n\nChapter 8 - Comparing Quantities\n\nChapter 9 - Rational Numbers\n\nChapter 10 - Practical Geometry\n\nChapter 11 - Perimeter and Area\n\nChapter 12 - Algebraic Expressions\n\nChapter 13 - Exponents and Powers\n\nChapter 14 - Symmetry\n\nChapter 15 - Visualising Shapes\n\n2. Practice, above all. The range of questions in class 7 is limited. With substantial practice, most question types will be internalized, and you won’t have to think for most questions.\n\n3. Develop a good understanding of topics. It will help you with those challenging questions or derive formulae in the rare chance you forget. This can be done by watching videos for various topics that will help you develop an interest in the topic.\n\n4. Understand all the concepts in as much depth as you can. And then solve different varieties of questions and problems. Attempt various topic-wise and chapter-wise tests. Also, practice mre with sample papers and previous year papers. Use your knowledge to solve some real life questions as well.\n\n## Practice tests\n\n5. Give more importance to the topics which have high weightage. Solve the previous year exam papers which will increase your confidence and give you a pre-exam feel. It also helps to brush up all your concepts before the exam.\n\n6. Don’t just read or mug up the topics but practice them. This is the most important thing which you need to keep in mind as reading maths does not help you. Instead, make a routine and give some time for practice.\n\n## Important Topics for Each Chapter\n\nChapter 1 – Integers\n\n• Multiplication of Integers\n• Multiplicative and Additive Identities for Integers\n• Properties of Division of Integers\n\nChapter 2 - Fractions and Decimals\n\n• Multiplication of Fractions with Fractions\n• Division of Fractions by Whole Numbers or Fractions\n• Multiplication of Decimals\n• Division of Decimals by Decimals\n\nChapter 3 - Data Handling\n\n• Median of Data Sets Having Odd Number of Terms\n• Construction of Double Bar Graphs\n• Interpretation of Double Bar Graphs\n• Probabilities in Simple Experiments without Using Formula\n\nChapter 4 - Simple Equations\n\n• Solution of Equations by Transposing Terms\n\nChapter 5 - Lines and Angles\n\n• Linear Pair of Angles\n• Transversal on Two Lines\n• Corresponding Angles Axiom and Its Converse\n• Alternate Angle Axiom and Its Converse\n• Using the Theorem that the Interior Angles on the Same Side of Transversal are Supplementary and Its Converse\n\nChapter 6 - The Triangles and its properties\n\n• Properties of Equilateral Triangles\n• Properties of Isosceles Triangles\n• Triangle Inequalities\n• Pythagoras Theorem and Its Converse\n\nChapter 7 - Congruence of Triangles\n\n• SSS Congruence Rule\n• SAS Congruence Rule\n• ASA Congruence Rule\n• RHS Congruence Rule\n\nChapter 8 - Comparing Quantities\n\n• Application of Ratios in Solving Problems\n• Word Problems on Percentages\n• Percentage Increase and Decrease\n• Concept of Simple Interest\n\nChapter 9 - Rational Numbers\n\n• Finding Rational Numbers between Given Rational Numbers\n• Comparing and Ordering Rational Numbers\n• Multiplication and Division of Rational Numbers\n\nChapter 10 - Practical Geometry\n\n• Construction of a Triangle when the Lengths of Its Sides Are Given\n• Construction of a Triangle when the Lengths of Two Sides and Angle Between Them Are Given\n• Construction of a Triangle when Two Angles and the Length of Side Between Them Are Given\n• Construction of a Right-angled Triangle when the Length of One Leg and Hypotenuse Are Given\n\nChapter 11 - Perimeter and Area\n\n• Area of Rectangle by Dividing It into Congruent Parts\n• Area of Parallelogram\n• Area of Triangle\n• Circumference of Circle\n• Area of Circle\n\nChapter 12 - Algebraic Expressions\n\n• Addition and Subtraction of Polynomials\n• Values of Algebraic Expressions at Different Points\n\nChapter 13 - Exponents and Powers\n\n• Exponential Form of Numbers\n• Law of Multiplying Powers with the Same Base\n• Law of Dividing Powers with the Same Base\n• Law of Taking Power of a Power\n• Law of Multiplying Powers with the Same Exponent\n• Law of Dividing Powers with the Same Exponent\n\nChapter 14 – Symmetry\n\n• Rotational Symmetry and Order of Symmetry\n• Line of Symmetry and Rotational Symmetry of Figures\n\nChapter 15 - Visualising Shapes\n\n• Nets of Three-dimensional Figures\n• Drawing Isometric Sketches of Solids\n• Arrangements of Cubes\n• Cross Sections of Solids\n• Solid Objects Viewed from Different Angles\n\n## Other Subjects of Class 7:", null, "Science for Class 7 (VII) - CBSE & NCERT Curriculum\n\n106 docs, 162 videos & 36 tests", null, "Social Science Studies (SST) Class 7 (VII) - CBSE, NCERT\n\n105 docs, 91 videos & 60 tests", null, "Hindi (Vasant II) Class 7 (VII) - CBSE & NCERT Curriculum\n\n53 docs & 21 videos", null, "English Honeycomb for Class 7 - CBSE and NCERT Curriculum\n\n48 docs", null, "English An Alien Hand Class 7 (VII) - CBSE, NCERT Curriculum\n\n35 docs & 1 video\n\nOffer running on EduRev: Apply code STAYHOME200 to get INR 200 off on our premium plan EduRev Infinity!\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n,\n\n;" ]	[ null, "https://cdn3.edurev.in/ApplicationImages/Temp/1421561_2761612b-1d57-4047-9d37-23df228b9369_lg.png", null, "https://cdn3.edurev.in/ApplicationImages/Temp/1421561_6e704fdf-4a62-4fa7-bfaf-46d39996c38f_lg.png", null, "https://cdn3.edurev.in/ApplicationImages/Temp/1421561_609d75c4-61b1-47ee-ad4d-00b3b97af666_lg.png", null, "https://cdn3.edurev.in/ApplicationImages/Temp/1421561_8bb7eadf-99c9-49e0-8e36-fe61aac55df2_lg.png", null, "https://cdn3.edurev.in/ApplicationImages/Temp/1421561_c54c8afa-c1cc-42a2-8f4d-65a42c33ed02_lg.png", null, "https://cdn3.edurev.in/ApplicationImages/Temp/1421561_763e07fc-2d08-4186-b0ba-82a5054a3529_lg.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8671002,"math_prob":0.63752025,"size":3658,"snap":"2021-21-2021-25","text_gpt3_token_len":884,"char_repetition_ratio":0.15736179,"word_repetition_ratio":0.2783019,"special_character_ratio":0.23400766,"punctuation_ratio":0.08231708,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99223566,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12],"im_url_duplicate_count":[null,4,null,4,null,4,null,4,null,4,null,4,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-21T23:23:29Z\",\"WARC-Record-ID\":\"<urn:uuid:82c164fb-27c8-41e9-8c01-3eca47609c89>\",\"Content-Length\":\"402638\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9c6dd0fc-9ebc-42f0-920b-d1addcac2973>\",\"WARC-Concurrent-To\":\"<urn:uuid:75600d00-20d5-4b1b-96e2-a0a598375056>\",\"WARC-IP-Address\":\"35.198.207.72\",\"WARC-Target-URI\":\"https://edurev.in/studytube/How-to-prepare-for-Class-7-Maths-Tips-Tricks-for-M/4ed13abe-11a0-45c4-a1f5-e95df0ee9360_t\",\"WARC-Payload-Digest\":\"sha1:JIVPCP4PEWZ3WONZT32AVQX4OSTLWETX\",\"WARC-Block-Digest\":\"sha1:GQ6OMHVD6UXHNSAPOCNZDD4MSOCU3LJ2\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623488504838.98_warc_CC-MAIN-20210621212241-20210622002241-00408.warc.gz\"}"}
http://e-booksdirectory.com/details.php?ebook=3037	[ "", null, "# Fundamentals of Model Theory", null, "Fundamentals of Model Theory\nby\n\nPublisher: University of Toronto\nNumber of pages: 64\n\nDescription:\nThis book provides an introduction to Model Theory which can be used as a text for a reading course or a summer project at the senior undergraduate or graduate level. It is also a primer which will give someone a self contained overview of the subject, before diving into one of the more encyclopedic standard graduate texts.\n\n(PDF/PS/DVI)\n\n## Similar books", null, "Hack, Hack, Who's There? A Gentle Introduction to Model Theory\nby - Smashwords\nThe skeleton of this book is a science fiction story without the usual mangling of physics. The flesh is composed of non-technical mainstream explanations and examples of the field of mathematics which deals with meaning, called Model Theory.\n(5082 views)", null, "Model Theory\nby - The Pennsylvania State University\nAn important branch of mathematical logic is model theory, the study of first-order theories and the classes of models defined by such theories. This course will include numerous applications of model theory to algebra, especially ordered fields ...\n(3876 views)", null, "Metalogic\nby - ESSLLI\nThis text provides the basic information about the metatheory of formal system. It will start with a brief information about propositional calculus and first-order logic. Then, fundamental theorems about elementary logic will be stated and proved.\n(8046 views)", null, "Model Theory\nby - University of South Carolina\nThe purpose of this text is to give a thorough introduction to the methods of model theory for first order logic. Model theory is the branch of logic that deals with mathematical structures and the formal languages they interpret.\n(9939 views)" ]	[ null, "http://e-booksdirectory.com/img/ebd-logo.png", null, "http://e-booksdirectory.com/images/3037.jpg", null, "http://e-booksdirectory.com/images/11266.jpg", null, "http://e-booksdirectory.com/images/blank.gif", null, "http://e-booksdirectory.com/images/6666.jpg", null, "http://e-booksdirectory.com/images/4732.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8679554,"math_prob":0.6913241,"size":3020,"snap":"2022-40-2023-06","text_gpt3_token_len":617,"char_repetition_ratio":0.114721484,"word_repetition_ratio":0.7169811,"special_character_ratio":0.19834436,"punctuation_ratio":0.085877866,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9724295,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-01-30T12:21:13Z\",\"WARC-Record-ID\":\"<urn:uuid:67f85cb3-59c0-455e-a3c4-fe834194b016>\",\"Content-Length\":\"11183\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:cb432e0f-623f-491b-b7f5-fa8b475063f0>\",\"WARC-Concurrent-To\":\"<urn:uuid:90fb698e-3031-4d27-a591-3cb02afa8f42>\",\"WARC-IP-Address\":\"209.59.191.64\",\"WARC-Target-URI\":\"http://e-booksdirectory.com/details.php?ebook=3037\",\"WARC-Payload-Digest\":\"sha1:D3QGMGA425573ASUG6IAICDBYMCCQJUH\",\"WARC-Block-Digest\":\"sha1:352TOPZ564PZMWM27K35LHL7OGIALD7C\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764499816.79_warc_CC-MAIN-20230130101912-20230130131912-00259.warc.gz\"}"}
https://awrcorp.com/download/faq/english/docs/simulation/apa.html	[ "# Appendix A. Advanced Analysis Topics\n\nThis appendix includes advanced analysis topics for users who want more detailed information on these methods.\n\nSimulated Load Pull functionality supports the AWR multi-dimensional A/B Wave Load Pull file format. Variable sweeps (for example; input power, bias, or frequency) are controlled via the Load Pull Template schematic. For more information see “Load Pull Script”.\n\n### A.1.2. Using Load Pull Files\n\nSeveral Load Pull file formats encompassing both swept and non-swept data are supported. You can import all of these file types using the Scripts > Load Pull > Import_Load_Pull_File script. This file supports the following file types:\n\n• `.cst`\n\n• `.spl`\n\n• `.lpc`\n\n• `.mdf`, `.mdif` (“A/B Wave Format”)\n\n• `.sp`\n\n• `.lp`\n\n• `.lpd`\n\n### A.1.3. Load Pull File De-embedding\n\nThe NI AWR Design Environment platform supports applying a de-embedding network to the output pin of a load pull file DUT, allowing the data reference plane to be moved to either side of the specified de-embedding network. For example, moving from an external device pin, through a package model, to an internal device pin. The de-embedding is done at the load pull data file level so all measurements applied to that document show the de-embedded result. See “Project Options Dialog Box: Interpolation/Passivity Tab ” for details on configuring the de-embedding network.\n\n### A.1.4. Using Load Pull Measurements\n\nLoad Pull measurements are often more easily visualized through a marker-defined reference, allowing the coupling of measurements together and quickly parsing through the individual sweeps. For information on setting up a marker-defined reference, see “Using a Marker to Define a Sweep Index”.\n\nLoad Pull measurements are differentiated by a naming convention that separates swept and non-swept based measurements. The measurement names are interpreted as follows:\n\n• Swept Load Pull measurements are prefixed with \"G_\", while non-swept measurements have no prefix.\n\n• Measurements that align data to a specified value end in “A”.\n\n• Measurements that plot a single value contain “_V”.\n\n• Measurements that plot a real value contain “_R”.\n\n• Measurements that plot a complex value contain “_C”.\n\nThe following is a list of commonly used measurements:\n\n• For plotting contours\n\n• For plotting contour interpolated min or max value\n\n• plots the contour min value\n\n• plots the contour max value\n\nTo plot a single contour at a specified value\n\n• plots a single contour at a specified value\n\n• For plotting all the gamma points in your Load Pull file\n\n• plots gamma points for a swept load impedance\n\n• For plotting swept data on a rectangular plot (like PAE vs. output power)\n\n• plots real data with a derived value chosen for the x and y axis in the measurement setup\n\nFor measurement details, click the button in the Add/Modify Measurement dialog box.\n\n## A.2. APLAC HB Simulator Convergence\n\nWhen working on convergence problems, you need to change APLAC simulator settings. You can set simulator-specific settings globally (choose Options > Default Circuit Options) or locally for each schematic (right-click a schematic in the Project Browser and choose Options, and for each option tab you can clear Use project defaults to override the global options.)\n\nAPLAC settings are controlled on the Circuit Options or Options dialog box APLAC Sim tab as shown in the following figure.", null, "The dialog box splits the simulator settings into Common Simulator Options, and APLAC Simulator Options. Select an option to view additional information about it in the lower window of the dialog box.\n\nAlways begin with default settings, as these are most commonly successful. If you copy an existing project as a starting point for a new one, the new project may \"inherit\" simulator settings that are inappropriate for your new design(s).\n\n### A.2.1. DC Analysis\n\nDC convergence problems are very rare as various approaches are taken to achieve DC convergence.\n\nA common problem arises when using S-parameters for models that do not have a DC point defined, and the simulator must extrapolate to DC. You can try to change the extrapolation settings. Choose Options > Project Options and click the Interpolation/Passivity tab. You can try Rational function as the interpolation method or switch between Polar and Cartesian coordinate systems.", null, "### A.2.2. APLAC Harmonic Balance\n\nHarmonic balance can have convergence problems in some cases. If this occurs you can use the following steps to try to help the circuit converge.\n\n1. Ensure that all nodes of every element are connected.\n\n• You can inspect your circuit visually or use the Design_Checker script to check for you.\n\n• To access the Design_Checker script choose Scripts > Project > Design_Checker and configure the settings as follows.", null, "2. Change the HB Matrix Solver to Sparse or GMRES (whichever is not currently selected).", null, "3. The default HB Algorithm setting is Auto select, which analyzes the circuit topology and then uses either Piecewise or Nodal. You can switch to each of these settings specifically to see if it helps.", null, "4. Use a power sweep. Instead of using power levels that force the circuit into strongly nonlinear modes of operation with a port set to a single power, use a port that sweeps power from a small signal region of operation into the strongly nonlinear region.\n\n5. Try changing the Maximum Voltage or Current Change parameter to \"`1e6`\" or decreasing it to \"`1e-3`\".", null, "6. Use transient assisted harmonic balance by selecting the Transient Assisted HB check box.", null, "7. Answer these simple questions to help determine why the simulator is not converging:\n\n• Are your models valid at the highest frequency at which harmonic balance needs to simulate, at the highest tone product specified by your number of tones and highest tone order? One way to check is to make a plot of Pharm of a simple circuit (maybe just a resistor) to see the highest frequency the simulator needs.\n\n• If you have an APLAC Transient license, use it with enough periods to reach steady state to see if the answers look reasonable in that simulator. If the answers don't look good, it could signal that there is a model problem.\n\n## A.3. APLAC Transient Simulator Convergence\n\nWhen working on convergence problems, you need to change APLAC simulator settings. You can set simulator-specific settings globally (choose Options > Default Circuit Options) or locally for each schematic (right-click a schematic in the Project Browser and choose Options, and for each option tab you can clear Use project defaults to override the global options.)\n\nAPLAC settings are controlled on the Circuit Options or Options dialog box APLAC Sim tab as shown in the following figure.", null, "The dialog box splits the simulator settings into Common Simulator Options, and APLAC Simulator Options. Select an option to view additional information about it in the lower window of the dialog box.\n\nAlways begin with default settings, as these are most commonly successful. If you copy an existing project as a starting point for a new one, the new project may \"inherit\" simulator settings that are inappropriate for your new design(s).\n\n### A.3.1. DC Analysis\n\nDC convergence problems are very rare as various approaches are taken to achieve DC convergence.\n\nA common problem arises when using S-parameters for models that do not have a DC point defined, and the simulator must extrapolate to DC. You can try to change the extrapolation settings. Choose Options > Project Options and click the Interpolation/Passivity tab. You can try Rational function as the interpolation method or switch between Polar and Cartesian coordinate systems.", null, "### A.3.2. Transient Analysis\n\nTransient analysis can have convergence problems in some cases. If this occurs you can use the following steps to try to help the circuit to converge.\n\n1. Change the truncation error method by changing the Truncation Error Mode to either Voltage or Charge.", null, "2. Change the integration method by changing Integration Method to either Euler or Gear.", null, "3. Try increasing the Maximum Voltage or Current Change parameter to \"`10 ... 100`\" or decreasing it to \"`0.5`\".", null, "4. Change the Step time under Transient Options to smaller or larger time steps.", null, "5. Use fixed time steps by changing Time Stepping to Fixed and specifying the fixed step size by typing \"`TMIN`\" in Free Text under Miscellaneous Options; otherwise the Step time under Transient Options is used.", null, "6. Answer these simple questions to help determine why the simulator isn't converging:\n\n• Are there potentially bad component values (for example, unrealistically large capacitors or inductors)? For example, for a DC block, you could use a very large capacitor (such as 1 mF) for harmonic balance and it simulates correctly. For transient analysis, this large capacitor causes problems.\n\n• Are there short transmission lines in the circuit that limit the time step? Can you replace these with RLC networks? The lines might be inside models so you may need to check the input netlist.\n\n• Are there huge voltages in your design? If so, you may need to increase Maximum Voltage or Current Change.\n\n• What does HB say about the circuit? If the answers do not look correct, this could signal a model problem.\n\n## A.4. HB Simulator Convergence\n\nConvergence failures, though rare, are usually a consequence of very strongly nonlinear circuit behavior. In such cases, one difficulty encountered by the simulator is a lack of a \"good guess\" at the solution. Some of the common causes for convergence failure are circuit instabilities, too few frequency components, discontinuities in nonlinear device modeling, and too few simulation iterations.\n\nSome of the most common messages in the Status Window are \"Simulation only partially completed\" and \"Step size for source stepping has decreased below a minimum allowed value\". The following figure shows an example warning message for a circuit that has failed convergence.", null, "The following section is a generic procedure for finding convergence problems. These are basic guidelines based on NI AWR experience.\n\n### A.4.1. Circuit Operation Checks (Not changing simulator options)\n\n1. Simulate a DC annotation to test if the circuit is converging at DC, and to provide a check for the proper operating condition of your circuit (for example if you accidentally specified 50 volts for the supply voltage instead of 5 volts). See “Creating a New Annotation” for information about adding an annotation.\n\n2. Simulate at low input levels (for example, -30 dBm for power, 1mV) to see if you get a solution at low powers.\n\n3. If the previous step works, try sweeping your input source. Many times a circuit converges better if you sweep from a low input level to a high power rather than just setting a high input level.\n\n### A.4.2. Harmonic Balance Settings Options\n\n1. Verify whether your Harmonic Balance settings use the project default settings or are set at the schematic level. Right-click the top level schematic in the Project Browser, choose Options and then click the appropriate simulator tab in the Options dialog box. If Use project defaults is selected in the Harmonic Balance Options section, the schematic uses the project settings. You can access project settings in the Circuit Options dialog box (choose Options > Default Circuit Options).", null, "2. In Iteration Settings, increase the Max iterations to 100 or 200 and see if it helps. For strongly nonlinear circuits you may need to set this value higher. Another way of increasing the number of iterations is by increasing the number of sweep steps. For example, a power sweep from -20 db to 20 db in steps of 1 db is more likely to converge than that with steps of 5 db.", null, "3. Increase the number of harmonics. If the circuit you are analyzing is strongly nonlinear, it is important for both accuracy and convergence to specify a sufficient number of frequencies in the simulation. This is the first step in troubleshooting convergence problems.\n\n4. Add a small conductance across the nonlinear elements. Choose Options > Default Circuit Options and on the Circuit Options dialog box AWR Sim tab under Convergence Aids verify your results are correct.", null, "5. Also under Convergence Aids, remove voltage limiting by clearing the Limit step size check box.\n\n6. Use a predefined set of convergence settings. In many cases selecting some of these predefined convergence settings along with tolerance settings helps the circuit to converge. If the circuit always needs to source step, you can speed up the simulation by selecting Start with source stepping under Configuration Wizard.", null, "7. Under Advanced HB Options, select any linearization method along with a degree of tolerance to speed up the convergence in Linearization Mode and Linearization Tolerance.", null, "8. Change the simulation accuracy. The simulation terminates when the absolute current error at each nonlinear element and at each harmonic is below Absolute Tolerance, or when the Relative Tolerance criterion is satisfied. The default value of Absolute Tolerance is \"1e-9\" and sometimes this number is very low for highly nonlinear circuits. You can increase this number to a value such as \"1e-7\" while leaving Relative Tolerance as is.", null, "## A.5. Stability Analysis Methods\n\nThe standard method for checking the stability of an active two-port is to use the stability factors K and B1 or some other measure of stability.\n\nAlthough these measures are useful for analyzing the stability of a two-port, there are many situations where the satisfaction of the standard criteria for stability (K and B1) do not guarantee that the circuit is stable. Microwave Office or Analog Office programs offer some alternative stability analysis techniques to reduce the likelihood of an unstable design.\n\nCAUTION: Stability analysis methods try to prove a negative-- the absence of instability. No one method, or even collection of methods, can guarantee stability. NI AWR does not claim that the following approaches find all stability issues. You should consider all stability analysis approaches.\n\n### A.5.1. Normalized Determinant Function (NDF) Analysis\n\nTwo examples of circuits where K and B1 do not detect instability are described in the article by Platzker and associates. The article demonstrates how NDF detects the instability of these circuits (NDF finds poles in the right half plane).\n\nThe basics of NDF analysis in the Microwave Office program are:\n\n• NDF analysis is only available with the APLAC simulator, so all models must function with APLAC.\n\n• NDF is a complex measurement vs. frequency at schematic load impedance. It must be measured over a very wide frequency range (\"DC to daylight\"). At low frequencies and as the frequency approaches infinity, NDF is real. The circuit is unstable if the measurement circles the origin of a polar plot more times clockwise than counter-clockwise; i.e. if the change in the unwrapped phase (AngleU) of the NDF measurement (from near DC to approaching infinity) decreases by more than 360 degrees.\n\n• For optimization, use Output Equations to calculate the difference between the initial and final values of AngU(NDF), and set a goal for this value to be less than 360 degrees.\n\n• IMPORTANT: Like other measurements, the frequency sweep for NDF is set on the schematic, or in the project options, so it is \"fixed\", not adaptive. You should set the frequencies to ensure that the measurement is well resolved, and encirclements of the origin can be detected for each simulation, even over parametric sweeps and optimization. The most efficient frequency sweep may involve varying step sizes over different frequency ranges. For this reason, it is usually best to have a separate frequency sweep for the NDF measurement. There are two ways to do this:\n\n• Create a new top level schematic with its own frequencies for the NDF measurement, and place the circuit of interest in it as a subcircuit.\n\n• Add a FRQSWP block, and set its Values parameter to an equation that concatenates multiple sweeps together. (See the table at “Built-in Functions ”). The frequencies are automatically collated for simulation.\n\n• To function properly, NDF needs to identify individual dependent (voltage- or current-controlled) sources in the circuit. Active S-parameter files, or compiled linear models with controlled sources inside them do not allow this and are therefore not included in the measurement calculation.\n\n• The schematic on which NDF is measured must include a PORT element. If the schematic is drawn with the (source and) load impedances represented by other elements, you need to place a PORT element anywhere on the schematic, without anything connected to it.\n\n### A.5.2. Stability Envelope Analysis\n\nThe basics of stability envelope analysis are:\n\n• Stability envelope analysis is only available with the APLAC simulator, so all models must function with APLAC.\n\n• APLAC computes the envelope of the NDFs corresponding to all possible passive source and load impedances at the ports.\n\n• StabEnv is the complex envelope of the NDFs corresponding to all possible passive source and load impedances at the ports. The circuit is stable if, at each frequency, StabEnv does not circle the origin of a polar plot (if the unwrapped phase (AngleU) of the measurement has a range of less than 360 degrees).\n\n• For optimization, use Output Equations to calculate the difference between the maximum and minimum values of AngU(StabEnv), and set a goal for this value to be less than 360 degrees.\n\n• The number of points used to approximate the impedances is set by choosing Options > Default Circuit Options to display the Circuit Options dialog box. Click the APLAC tab and under Stability in StabEnvelopePoints, increase this number to ensure the measurement is well resolved, and encirclements of the origin can be detected for each simulation, even over frequency/parametric sweeps and optimization.\n\nFor more information see the following source: T. Narhi and M. Valtonen, “Stability Envelope - New Tool For Generalized Stability Analysis,” IEEE MTT-S International Microwave Symposium Digest, pp. 623–626, June 1997.\n\n### A.5.3. Loop Gain\n\nThe complex loop gain of a feedback network, such as an amplifier, can be used to check for instability. When loop gain is all real (angle=0) and greater than 1, the circuit is unstable. The LoopGain and NLLoopGain measurements calculate small-signal (linear) loop gain under DC bias or large-signal drive conditions, respectively. These measurements work without modifying the design, by accessing the internal controlled source of nonlinear transistor models. Consider the following linearized model for a FET transistor:", null, "Loop gain measurements internally break the feedback loop that includes this transistor by replacing the controlling voltage of the transconductance gm with a test signal, Vin, and measuring the resulting output voltage, Vgs, where the controlling voltage should be:", null, "The loop gain is defined as the voltage gain Vout/Vin. All the other measurements see the unmodified circuit, allowing other measurements to work normally for the same schematic. You are required to point to the place in the circuit where the feedback loop is broken by selecting an internal branch of a nonlinear device as the Measurement Component in the Add/Modify Measurement dialog box. For example, the CURTICE nonlinear FET model has three internal branches: \"ds\", \"gd\" and \"gs\". The drain-source branch \"ds\" is a good place to break the loop for a FET amplifier. Note that the feedback loop can be formed by internal elements like Cgd in the previous example, by external elements like biasing networks, or both.\n\nTo add a loop gain measurement, begin by selecting the appropriate top level schematic as the Data Source Name. To identify the internal branch of a nonlinear transistor model as a Measurement Component, click the browse button (\"...\") as shown in the following figure:", null, "See “Measurement Location Selection” for information on using the window that opens to navigate hierarchy and open the subcircuit schematic (or netlist) containing the desired transistor. Next, select the appropriate branch of that transistor as the Testpoint. For example, select the collector-emitter branch of the BJT model in a subcircuit netlist, as shown in the following figure.", null, "The LoopGain measurement is only possible if the selected transistor exposes its internal branches, so that the transconductance branch can be selected. Some transistors do not expose these branches so therefore cannot be used for a LoopGain measurement.\n\n#### LoopGain and NLLoopGain Measurements\n\nThe linear LoopGain measurement works as described in “Loop Gain”. As with any linear measurement on a nonlinear circuit, an initial DC bias analysis is performed to determine the operating point of all nonlinear devices. This allows the nonlinear transistors to be linearized (represented by an equivalent linear network like the one shown previously), so that the LoopGain measurement can be made using a linear small-signal analysis. For more information see “Loop Gain: LoopGain”.\n\nNLLoopGain is a nonlinear extension to LoopGain that takes into account the large-signal drive of the circuit. First, the nonlinear steady-state solution of the circuit is determined with Harmonic Balance. The circuit is linearized at this operating point and the feedback loop is broken at a user-specified point. Next, a linear small-signal analysis is performed to compute the loop gain. The NLLoopGain measurement requires that an NLSTABILITY control element is placed in the schematic, and it can only be used with the APLAC HB simulator. For more information see “Nonlinear Loop Gain: NLLoopGain”.\n\n### A.5.4. Microwave Office Approach to Internal Stability Analysis\n\nAnother example of the K and B1 requirements not ensuring stability is that of a two-stage amplifier. K and B1 may indicate the amplifier is stable, but it is still possible for the amplifier to be unstable due to oscillation conditions that may exist between the two amplifier stages. This \"internal\" instability in the interstage of the amplifier may not be detectable by a simple measure of K and B1 of the entire amplifier.\n\nOne approach to solving this problem is to make each device unconditionally stable before adding the matching. This is usually accomplished by adding lossy matching or feedback to the potentially unstable device. Although this may work, there are a drawbacks to this approach.\n\nThe first drawback is that it does not guarantee that the amplifier remains stable if there is any amount of feedback between the two stages. Although there is usually not any intentional feedback between the stages, there is often unintentional feedback that arises from the bias circuitry that may be connected together with less than ideal isolation. A thorough stability analysis should attempt to include the bias network as part of the analysis. The standard K and B1 stability factors do not detect instabilities caused by this type of feedback.\n\nThe second drawback is that the addition of the lossy matching (or feedback) usually reduces the performance of the device significantly. If a more thorough technique is used to analyze the stability of the interstage circuitry, then it is usually possible to get much better performance from the same devices with an acceptable margin for stability. One such method for predicting internal stability is presented in an S-probe article.. This method requires the measurement of internal reflection coefficients within the circuit, and it also requires the ability to analyze the circuit with arbitrary termination impedances.\n\nMicrowave Office and Analog Office programs have several unique features that greatly simplify the analysis of the internal stability:\n\n• An element for sampling the internal reflection coefficients at internal nodes in a circuit.\n\n• The ability to terminate any port with an arbitrary reflection coefficient (not restricted to two ports).\n\n• Built-in measurements that compute the stability without needing to use output equations.\n\n#### Internal Stability Background\n\nSome background information on internal stability is useful for understanding the analysis method. The following figure shows two networks connected together. The network on the left has S-parameters [S] and the network on the right has S-parameters [S']. The lower part of the figure shows a portion of the signal flow graph that includes the interface between the two networks. The use of a signal flow graph allows the stability to be analyzed using the same techniques as those used for analyzing the stability in control systems.", null, "You can determine the stability of the previous system by viewing the loop in the signal flow graph as a feedback loop. This allows the Nyquist stability criteria to be applied to the open loop frequency domain response given by\n\n G=-Γ1Γ2 (A.1)\n\nThe Nyquist stability criteria states that if the open loop function G, when plotted on the complex plane, encircles the -1 point in the clockwise direction, then the closed loop system will be unstable. The following polar plot of G shows an unstable system (G encircles the -1 point in a clockwise sense).", null, "The Microwave Office STABN_GP2 measurement allows the function G to be plotted on a polar graph for inspection of the stability. When plotting STABN_GP2, the frequency should be swept over the entire range where instability could occur.\n\nAn approximate simplification of the Nyquist stability criteria allows the computation of a single stability index that can be plotted as a single real number over frequency. The use of a real stability index makes it easier to include internal stability as an optimization goal.\n\nThe meaning of the stability index is shown in the following figure.", null, "The stability index is taken to be the negative of the component of G along the real axis. A value of the stability index greater than 1 is then used to indicate possible instability. The term \"possible\" is used because it is possible for the stability index to be greater than 1 without an encirclement of the -1 point, as shown in the following figure.", null, "Although the stability index may indicate instability when the device is actually stable, it is still a very useful measure in practice since it does not predict that the circuit is stable when the Nyquist criteria indicates that the circuit is not stable (it is a conservative measure). Usually if the stability index predicts that a stable circuit is not stable, then margin of stability for the circuit is not very high (minor changes in the response could cause instability). Also, if the stability index predicts an instability, then the more rigorous Nyquist criteria can be used to verify the instability. The stability index indicated here is the same as the stability index presented in an S-probe article.. You can use the Microwave Office STAB_GP2 measurement to plot this stability index (STAB_GP2 <1 indicates stability). For more information see “(Obsolete) Stability Index Measured with Gamma-Probe: STAB_GP”.\n\n### A.5.5. Performing Internal Stability Analysis\n\nThe following schematic shows the GPROBE element inserted into the schematic at an internal node where the stability is to be analyzed.", null, "The GPROBE element is used to measure the internal reflection coefficients at the reference plane indicated. The stability measurements STAB_GP2 and STABN_GP2 and the internal gamma measurements made with GAM_GP2 require that you select a specific GPROBE2 to identify the reference plane.\n\n### A.5.6. Termination Impedances\n\nOne of the drawbacks of the proposed method used to analyze the stability is that it is only valid when the circuit is terminated with the same termination impedances that are used during the analysis. The termination impedances that make the circuit the \"most unstable\" are generally the impedances with a reflection coefficient of magnitude one, so you should test the stability when the circuit is terminated with impedances that lie on the edge of the Smith Chart.\n\nThe port element PORTG allows the termination impedance of the port to be specified as a magnitude and angle of the reflection coefficient. In practice, it is not possible to compute the response of the circuit with perfect magnitude one reflection coefficient on the ports, so a value close to one (0.99 for example) is used instead. You can change a normal port to a PORTG port by editing the port element and selecting the Specify Source Gamma check box on the Port tab of the Port properties dialog box. You can also add the PORTG element through the Element Browser.\n\nOne method for testing the stability for a wide range of port terminations takes advantage of the yield analysis feature. If all the terminal ports use PORTG elements, and the magnitudes of the reflection coefficients are set to something close to 1 (0.99 for example) then the angle parameter can be set to 180 degrees, and the statistical properties can be set so the angle has a uniform distribution with a tolerance of 180 degrees. When the yield analysis is run, the angle takes on random values from 0 to 360 degrees which covers all points on the outer radius of the Smith Chart. The stability index can be plotted during the yield analysis and any trace that exceeds a value of one indicates that the circuit can become unstable.\n\n### A.5.7. Nonlinear Internal Stability Analysis\n\nLinear internal stability analysis can be generalized to nonlinear circuits. First, a large-signal solution is computed using the APLAC HB simulator. Next, the circuit is linearized at the found HB solution and the internal reflection coefficients are solved at each small-signal frequency. If there are sweeps in the simulation setup, the large-signal solution and the following internal stability analysis are performed at each sweep point. This allows analyzing the stability of the circuit as a function of some swept quantity, for example input power.\n\nThe NLGAM_GP2, NLSTABN_GP2, and NLSTAB_GP2 measurements are straightforward nonlinear generalizations of their linear counterparts. They can only be used with the APLAC HB simulator. Analyzing nonlinear stability requires that control element NLSTABILITY is placed in the schematic. The Fstart, Fend, and Fsteps parameters define the range of small-signal frequencies to sweep over. Note that these are absolute frequency values, not offsets to the large-signal fundamental frequency. SwpType defines the type of small-signal frequency sweep (linear or log).\n\n## A.6. Antenna Analysis Methods\n\nBy default, the Microwave Office program places a perfect electric conductor (PEC) on the top and bottom of the enclosure. Since the side-walls are always PEC, the default configuration is completely enclosed by the perfect conductor and no radiation is possible. To allow radiation into an infinite region, you must model the top of the enclosure as an open boundary.\n\nIf the top of the enclosure is made to have a boundary condition that approximates the boundary condition of an open box, then you can use the tangential electric field at the top of the enclosure (the tangential field at the absorbing boundary) to compute the far-field radiation pattern. A detailed description of the method used for determining the radiation pattern from the tangential electric field is included in Chapter 12 of `Antenna Theory Analysis and Design`. This process involves two steps:\n\n1. The first step is the computation of the tangential electric field. This is performed using an approximation to the aperture boundary condition at the top (and/or bottom) of the enclosure (currently there are two different types of boundaries you can use). Using these boundary conditions, the currents on the internal conductors are determined for a given excitation using standard EMSight methods. After these currents are known, the tangential E-fields are determined on the top surface of the enclosure. These fields are used in step two of the process.\n\n2. The second step in the process involves re-radiating the equivalent currents of these tangential electric fields into the upper hemisphere, assuming that these equivalent currents are present upon a perfectly conducting half plane. Using the equivalence theorem and the image theorem, you can represent the radiation problem as a sheet of magnetic current that is infinitely close to a perfectly conducting half plane. From this sheet of magnetic currents, you can obtain a far-field radiation pattern.\n\nNote that the problem used to find the sheet of equivalent currents is not exactly the problem used to compute the radiation. As a result, errors in the problem solution can result from these differences. You can minimize these errors by properly selecting the enclosure size and the location of the free space boundary or boundaries.\n\n### A.6.1. Selection of the Boundary Condition\n\nThe first type of boundary is a resistive boundary. You can use a resistive material of 377 ohms per square to provide an approximation to the free-space boundary. The 377 ohms is an approximation because the boundary would only have an impedance of 377 ohms at normal incidence and without the presence of the side-walls. A problem that can arise when using this type of boundary condition involves the absorption of power from the near fields of the antenna. This occurs when the resistive boundary is located too close to the radiating structure, artificially absorbing the stored energy of the structure. Nevertheless, the use of the 377 ohm boundary condition seems to give reasonable results when it located an appropriate distance from the radiating conductors. To test that this boundary is not absorbing stored energy, you should conduct two simulations with the boundary located at slightly different (for example λ/10) distances from the radiating structure. If no significant impact is seen in the S-parameters of the structure, then you can assume that the boundary is not absorbing substantial amounts of stored energy.\n\nThe second type of boundary is the boundary condition that you would see if the side-walls extended to infinity. This boundary condition is equivalent to the boundary seen looking into an infinitely long waveguide.\n\nWhat is important for getting an accurate radiation pattern is that the computed tangential E-field at the top of the enclosure looks as much like the E field of an equivalent problem radiating into free-space. The infinite waveguide termination provides a reasonable approximation for the tangential E-field as long as the boundary is not too far from the radiation element. The tangential E-field far from the radiating element always tends to the field pattern of the dominant waveguide modes as the distance between the enclosure top and the radiator increases. When the radiator is far enough away, you always obtain the radiation pattern of an open-ended rectangular waveguide when using the infinite waveguide termination.\n\nSince the EM simulator solves for the fields inside a conductive box, several assumptions are made that allow for the computation of the radiation pattern. One of these assumptions is that the side-walls of the enclosure are far enough away that they do not have a significant effect on the electric field on the top boundary of the enclosure. You should always view the electric field on the top of the boundary when working with antennas. This allows a quick check on the validity of this assumption. If the electric field has a very low magnitude near the edges of the enclosure, then the assumption is valid. If the electric field is relatively high near the edges, then the fields are interacting with the side-walls and the assumption is not valid.\n\n### A.6.2. Re-Radiation of Equivalent Currents into Free Space\n\nThe second part of the solution involves the determination of the radiation pattern from the tangential E-field present on the top surface of the enclosure. Using the equivalence theorem and the image theorem, you can represent the radiation problem as a sheet of magnetic current that is infinitely close to a perfectly conducting half plane. From this sheet of magnetic currents, you can obtain a far field radiation pattern. Again, note that the problem used to find the sheet of magnetic currents is not exactly equivalent to the problem used to compute the radiation. As a result, difficulties arise in forcing conservation of power, so the far-field radiation patterns are normalized to an average radiated power which is determined via an integration of all of the power radiated in the upper hemisphere. The normalization does not account for mismatch or resistive losses and thus results in a polarization sensitive directivity for the antenna.\n\nUnfortunately, while the S-parameters of the structure reveal the mismatch losses of the antenna, the resistive losses associated with the antenna cannot be determined due to the previously mentioned lack of conservation of power due to the imposed boundary condition. For antennas constructed with perfect conductors, the radiated power equals the power into the structure (easily computed from the S-parameters). If you want to compute the ohmic or dielectric loss of an antenna, two problems should be solved. One of the problems should use a PEC radiator (and no dielectric loss) and the other should use the lossy conductor and/or lossy dielectric that the true antenna uses. You can use the difference between the radiated power in the two cases to estimate the ohmic or dielectric losses in the antenna.\n\n### A.6.3. Antenna Measurements and Drawing Coordinate Systems\n\nTwo coordinate systems are used in an EM antenna simulation, one for drawing the structure and another for the radiation patterns. The second coordinate system is conforms to the standard coordinate system used in antenna analysis.\n\nThe coordinate system for the antenna measurements is shown in Figure A.1, “3D View of Coordinate System Used For Antenna Measurements”. This system is a right-hand coordinate system with the origin located at the center of the top of the enclosure. When viewing a two-dimensional image, the y-axis of the antenna coordinate system extends upward, the x-axis extends to the right, and the z-axis extends out of the image toward the viewer.\n\nFigure A.1. 3D View of Coordinate System Used For Antenna Measurements", null, "The antenna coordinate system is in contrast to the coordinate system used to draw structures in the Microwave Office program. The drawing coordinate system is a Left Hand Coordinate system with its origin located in the upper left-hand corner of the Microwave Office enclosure as viewed in a two-dimensional view of an EM simulation. Figure A.2, “2D View of Structure Showing Antenna and Drawing Coordinates ” contrasts these two coordinate system in a two-dimensional view.\n\nFigure A.2. 2D View of Structure Showing Antenna and Drawing Coordinates", null, "Physically, an antenna radiates energy at all frequencies in all directions simultaneously. To visualize the radiation, measurements that fix all but one of the independent parameters (Freq, θ and φ) must be implemented to allow a two-dimensional plot. Further, the phase and magnitude of the radiation is affected by the polarization of antenna used to measure the antenna under test. For this reason, you can make three basic types of antenna measurements that fix all but one of the independent axes. Further, each of these measurements can be performed for five common polarizations.\n\n### A.6.4. Antenna Measurement Types\n\nThe following are the antenna measurements types:\n\n• Principal Plane Cut (PPC): Also known as a Theta or an Elevation Cut, this antenna measurement type fixes the values of Frequency and φ to user-specified values. Theta is then swept to cover an entire sweep of the upper hemisphere (-90 to 90 degs or -π/2 to π/2 rads) if there is an infinite ground plane below the antenna, or to cover an entire sweep of the lower hemisphere (90 to 270 degs or π/2 to 3π/2 rads) if there is an infinite ground plane above the antenna, or from -180 to 180 (-π to π radians) if there is not an infinite ground plane. An example of a Principal Plane Cut is shown in Figure A.3, “Example of a Principal Plane Cut”.\n\n• Conic Cut (CON): Also known as a Phi or Azimuth Cut, this antenna measurement type fixes the values of Frequency and θ to user-specified values. Phi is then swept to cover an entire sweep of the upper hemisphere (-180 to 180 degs or -π to π rads). An example of a Conic Cut is shown in Figure A.4, “Example of a Conic Cut”.\n\n• Swept Frequency (SF): This antenna measurement type fixes the values of φ and θ to user-specified values. Frequency is then swept over a user-defined range.\n\n### A.6.5. Antenna Measurement Polarizations\n\nThe following are the antenna measurement polarizations:\n\n• E-Phi (Eφ): This represents signals received or transmitted by the test antenna if it is linearly polarized with its E-field aligned with the unit vector dφ in the previously mentioned antenna coordinate system. Importantly, the positive direction of dφ is in the increasing direction of φ. You should notice the dependence of dφ on the current value of φ and θ.\n\n• E-Theta (Eθ): This represents signals received or transmitted by the test antenna if it is linearly polarized with its E-field aligned with the unit vector dθ in the previously mentioned antenna coordinate system. Importantly, the positive direction of dθ is in the increasing direction of θ. You should notice the dependence of dθ on the current value of φ and θ.\n\n• Right-Hand Circular Polarization (RHCP): This polarization is a linear combination of Eθ and Eφ; it is defined as: RHCP=(Eθ+jEφ)/√2\n\n• Left-Hand Circular Polarization (LHCP): This polarization is a linear combination of Eθ and Eφ; it is defined as: LHCP=(Eθ-jEφ)/√2\n\n• Total Power (TPwr): Although this is not strictly a polarization, it is a very useful measure. TPwr represents the total power available regardless of polarization, and is obtained by summing the powers available from Eθ and Eφ. This measurement is purely real and does not have a phase associated with it.\n\n• Actually, antenna measurements represent the square root of the partial directivity in the specified direction that retain the phase of the corresponding electric field component (barring total power measurements that represent the square root of total directivity in the specified direction).\n\nFigure A.3. Example of a Principal Plane Cut", null, "Figure A.4. Example of a Conic Cut", null, "### A.6.6. Guidelines for Antenna Analysis\n\nTwo options are available for radiation pattern analysis:\n\n• Setting the top boundary of the enclosure to the impedance boundary condition (Approximate Open)\n\n• Setting the top boundary of the enclosure to an Infinite Waveguide termination.\n\n#### Placing Side-walls Far Enough from Antenna\n\nSince the EM simulator solves for the fields inside a conducting box, several assumptions are made that allow for the computation of the radiation pattern. One of these assumptions is that the side-walls of the enclosure are far enough away that they do not have a significant affect on the electric field on the top boundary of the enclosure. It is best to view the electric field on the top boundary after solving when working with antennas. This allows a quick check of the validity of this assumption. If the electric field has a very low magnitude near the edges of the enclosure, the assumption is valid. If the electric field is relatively high near the edges, then the fields are interacting with the side-walls and the assumption is not valid.\n\n#### Approximate Open: Select Optimal Elevation of Enclosure Top Cover\n\nIf an impedance boundary condition (such as an approximate open) is used to terminate the enclosure above an antenna, it is possible for the boundary to be close enough to interact with the near field (read: stored energy) of the antenna, thus resistively loading the antenna and causing significant undesired changes. This undesired effect occurs when the top of the enclosure is too close to the antenna surface.\n\nMoving the top of the enclosure away from the antenna reduces the resistive loading, but exposes more of the metallic side-walls for reflections. Although the input impedance of the antenna has stabilized, you are now sampling the field for re-radiation much further from the antenna. In this region, the interactions with the side-walls is converting the radiated fields into the modes of the waveguide.\n\nElevation of the enclosure top above the upper layer of dielectric stack may be crucial for obtaining correct results. You should select this height equal to approximately one quarter of the wavelength at the central frequency of operation. Control of the electric field on the top boundary may help to validate your choice. If the enclosure top is too close to radiating elements, distribution of the electric field displays dips just above the radiator locations. If the enclosure top is too far, distribution of the electric field develops pronounced oscillations from the center to the edges; whereas a reasonable selection of enclosure height provides distribution with a smooth slope towards all edges.\n\n#### Infinite Waveguide: Horizon Radiation Limitations\n\nA better approach is to sample the fields very close to the antenna surface without resistively loading the structure with an impedance boundary condition. You can do so by replacing the surface impedance boundary condition with an infinite waveguide termination. This type of boundary condition does not resistively load the near field of the antenna. Since you are sampling the electric fields very close to the antenna, they have minimal corruption due to side-wall locations.\n\nUnfortunately, antennas that direct significant amounts of energy toward the horizon still have significant degradation of the sampled electric field due to side-wall reflections.\n\n#### Calculation of Antenna Characteristics\n\nThe following sections include information on calculating antenna far-field radiation patterns, directivity, and gain.\n\n##### Calculation of the Antenna Far-field Radiation Pattern\n\nConic Cut or Phi Sweep (2D plot with specified constant value of θ and φ swept from -180 to 180 degrees): Use Con_EPhi or Con_ETheta to plot the normalized radiation pattern of Eφ or Eθ components of the E-field in the far zone correspondingly.\nPrincipal Plane Cut or Theta Sweep (2D plot with specified constant value of φ and θ swept from -90 to 90 if there is an infinite ground plane below the antenna, or from 90 to 180 if there is an infinite ground plane above the antenna, or from -180 to 180 if there is not an infinite ground plane): Use PPC_EPhi or PPC_ETheta to plot the normalized radiation pattern of Eφ or Eθ components of the E-field in the far zone correspondingly.\n\n##### Calculation of the Antenna Directivity\n\nDirectivity: Use SF_TPwr to calculate the antenna directivity in a given direction (defined by specified values of θ and φ).\nPartial directivity of an antenna for a given polarization: Use SF_EPhi or SF_ETheta correspondingly for Eφ or Eθ to calculate the partial antenna directivity in a given direction (defined by specified values of θ and φ).\nNOTE: The Include Resistive Losses and Include Reflection/Coupling Losses options are not selected.\n\n##### Calculation of the Antenna Gain\n\nUse SF_TPwr to calculate the antenna gain in a given direction (defined by specified values of θ and φ). Ensure that you select the Include Resistive Losses option.\n\n## A.7. Using Temperature in Simulations\n\nThis section illustrates the methods you can use to perform noise and nonlinear simulations when the operating temperature needs to be accounted for, and discusses how to create projects that use temperature as a variable in Microwave Office simulations, including:\n\n1. A brief discussion of how the built-in variables _TEMP and _TEMPK are used\n\n2. A description of how component models use temperature for simulation\n\nTemperature controls the noise generation processes and the static operating point of nonlinear components and their dynamic behavior. With the necessary models for these components, it is possible to calculate the DC operating point, noise figure, and both AC small and large signal properties of components as a function of temperature.\n\nThe NI AWR Design Environment software uses two built-in variables _TEMP and _TEMPK, and functions such as ctok(x) and ktoc(x) to assist designers in projects to perform temperature sensitive simulations. _TEMP is a built-in variable in the NI AWR Design Environment suite intended for models that have a specific temperature parameter. _TEMP uses the units specified as global units (choose Options > Project Options and click the Global Units tab to select degree Kelvin, degree Celsius, or degree Fahrenheit as the global units. You can overwrite the default value of this variable using an equation; for example the equation \"_TEMP = 30\" sets this global variable to 30 degrees Celsius if these are the set units. _TEMPK is in degrees Kelvin and retains these units regardless of the global units setting. This variable is used to adjust the temperature for models without a temperature parameter. This variable only affects noise simulation of linear elements.\n\nYou can use both _TEMP and _TEMPK in equations to define the operating temperature of components. You can use equations to tie both linear and nonlinear temperature to the same value, and to assign one temperature value to all elements through hierarchy. Also, you can assign different temperatures to passive circuits, small signal amplifiers, Power Amplifier drivers and Power Amplifiers, with a global temperature used to define the base-plate or housing temperature, and equations added to define the unique temperatures of the high temperature components using dissipation and thermal resistance calculations.\n\n### A.7.1. Displaying Temperature Values Used in Simulation\n\nYou can view the current value of any variable with the `\"variable:\"` notation. To return or expose the value of _TEMP, create the following equation in the Global Definitions window or any schematic window: \"_TEMP:\" and then simulate. The variable _TEMP is set to 25 and interpreted as 25 degrees Celsius. Now change the global units for temperature to degree Kelvin and resimulate. _TEMP is set to 298.1 degrees Kelvin. As previously explained, you can define the value of _TEMP by using the equation \"_TEMP = 30\". To see the default value, overwrite this value with a user-specified value, and confirm the new value, create the following equations:\n\n`_TEMP:`\n\n`_TEMP=30`\n\n`_TEMP:`\n\nand simulate. The value of _TEMPK always returns the current value in degrees Kelvin. If no value is explicitly set, the value is 290 degrees Kelvin.\n\n### A.7.2. Different Temperature Situations in NI AWR\n\nHow these components use the built-in variables for temperature and derived temperatures depends upon the origin of the model parameters for the component. There are four different situations when setting temperature in simulation:\n\n• passive elements\n\n• nonlinear elements without _TEMP assigned\n\n• nonlinear elements with _TEMP assigned\n\n• netlists.\n\n### A.7.3. Temperature and Noise for Passive Structures\n\nThe NI AWR Design Environment suite supports both implicit and explicit methods when using temperature in simulations. The implicit method uses the built-in variable _TEMPK; the use of this approach is illustrated in the following figure. The schematic consists of an attenuator whose shunt and series elements are designed using the standard equations for a PI attenuator. This particular building block is chosen for this example because the NF equals the insertion loss when the physical temperature of a matched attenuator is set to the reference temperature. Here you can see that the resistors responsible for loss and noise generation do not have an explicit temperature defined. The built-in global variable _TEMPK controls the temperature of all such elements that possess loss, and therefore can generate noise.\n\nThe simulation uses a local copy of the built-in variable _TEMPK which is controlled by the Swept Variable (SWPVAR) element. The temperature is swept between 0 and 400 degrees K. The X axis of the graph is set to use the sweep variable, _TEMPK. See “Swept Variable Control: SWPVAR” for more information about SWPVAR.", null, "When a specific temperature of an element needs to be defined other than that defined by the built-in global variable _TEMPK, you can create local variables and use elements with an explicit temperature parameter. In the following schematic the REST element is used in place of the RES element. The REST element has a parameter for the temperature of the resistor. The `localTemp` variable is used to control the temperature of the element and uses the global units setting, which in this example is in degrees Celsius. Since the sweep is still in degrees Kelvin, a conversion is used for the temperature setting for each resistor, `T=ktoc(localTemp)`. The `ktoc` function converts from Kelvin to Celsius. Note that you can use the built-in _TEMP variable, or alternatively, you can not use the ktoc conversion and sweep the `localTemp` variable in degrees Celsius instead of degrees Kelvin. In every other respect, this design is identical to the previous design. The simulation results for these schematics are identical.", null, "For passive devices defined by port parameter data files, like S-parameters, use the NPORT_F or NPORT_F_MDIF element, which has a secondary parameter for setting its temperature. SUBCKT blocks that refer to passive port parameter data files use the global _TEMPK setting, which you can change in the Global Definitions document. (Note that the Default Linear simulator overrides the global setting if the schematic in which the SUBCKT block is placed has an equation that sets _TEMPK;. The APLAC Linear simulator only uses the global setting.)\n\nEM structures have schematics in which their individual _TEMPK can be specified using an equation, as previously described.\n\n### A.7.4. Temperature and Active Device Operation\n\nThe principle of defining the temperature of circuit elements can be extended to active devices. The following is a schematic for a bipolar amplifier used later in the system noise figure calculations. The device is modeled using the nonlinear model and several elements to define the package parasitics. The schematic that defines the device model is shown following the test schematic. The top level schematic has _TEMP defined as a variable, and then the swept variable block is used to sweep the temperature from -270 to 100 degrees C in steps of 10 degrees C. The variable is set to be passed down through hierarchy designated by the solid red line surrounding the variable. (You can toggle this setting by using the Tune Tool and holding down the Shift key).", null, "The following schematic defines the transistor parasitic elements.", null, "The following is the transistor model. The _TEMP variable replaces the default value of the device temperature (TEMP = _TEMP).", null, "The assignment of the temperature displays (it is hidden by default) by changing the display options of the parameter on the Display tab of the Element Options dialog box. You can change the default (hidden) by clearing the Default column check box as shown in the following figure.", null, "Many of the device models for Monolithic Microwave Integrated Circuit (MMIC), RFIC and hybrid circuit design use this form when defining temperature. The model variable TEMP is the universal variable used by models within SPICE and other EDA simulations.\n\nThe NI AWR Design Environment suite supports the simulation of linearized noise figure, whereby the temperature and bias point dependent operating point is calculated before the small signal noise properties are calculated. The following graph shows the noise and gain for the previous transistor circuit.", null, "In this example, the Gummel-Poon model is used from the NI AWR Design Environment Element Browser under Nonlinear > BJT category. The appropriate model parameters are entered to model this part and _TEMP is assigned to the TEMP parameter. In the PDKs available for NI AWR, the temperature parameter is set to _TEMP by default.\n\nIn this example you can set the temperature to any variable name. NI AWR recommends that you use the _TEMP variable to maintain consistency between designs. If you want to assign different temperatures to different elements in the design, you should set a different variable name, however.\n\n### A.7.5. Temperature and Active Device Operation with a Device Defined by a SPICE Netlist\n\nOften the device model is found from a component manufacturer's web site in the form of a SPICE compliant netlist. The following netlist was imported into the NI AWR Design Environment suite. Note that on import, the syntax of the file is modified to conform to the NI AWR Design Environment netlist standard. The netlist contains both a circuit description of the transistor and an inline device model section with the normal BJT parameters. In SPICE syntax, the units are always assumed to be in base units (Farads, Amps, Henrys, etc). The base unit for temperature in SPICE is Celsius. The units for each type of variable are shown as follows.\n\n```.subckt qNE52418_v161 c b e s\n---- Intrinsic NPN Model ---------------------------------\nq1 c b e s qmod\n\n.model qmod npn\n+ level= 3 afn= 3.2\n+ ajc= -0.5 aje= -0.5\n+ ajs= -0.5 avc1= 3\n+ avc2= 200 bfn= 1\n+ cbco= 5.46E-15 cbeo= 8.34E-15\n+ cjc= 4.62E-15 cjcp= 2.64E-14\n+ cje= 3.78E-14 cjep= 1.04E-14\n+ cth= 3.26E-10 ea= 1.133\n+ eaic= 1.133 eaie= 1.133\n+ eais= 1.133 eanc= 1.133\n+ eane= 1.133 eans= 1.133\n+ fc= 0.95 gamm= 1.50E-11\n+ hrcf= 1 ibci= 8.22E-20\n+ ibcip= 0 ibcn= 7.37E-15\n+ ibcnp= 0 ibei= 1.77E-19\n+ ibeip= 2.49E-18 iben= 1.10E-15\n+ ibenp= 0 ikf= 2.51E-02\n+ ikp= 9.96E-02 ikr= 5.90E-03\n+ is= 2.87E-17 isp= 1.99E-19\n+ itf= 6.59E-02 kfn= 1.96E-09\n+ mc= 0.18 me= 0.2\n+ ms= 0.47 nci= 1.002\n+ ncip= 1.005 ncn= 1.5\n+ ncnp= 1.9 nei= 1.02\n+ nen= 1.9 nf= 1.015\n+ nfp= 1 nr= 1.01\n+ pc= 0.86 pe= 0.92\n+ ps= 0.55 qco= 1.47E-16\n+ qtf= 0 rbi= 3.4088\n+ rbp= 1 rbx= 13.111\n+ rci= 316.11 rcx= 0\n+ re= 0.68275 rs= 50\n+ rth= 644.03 tavc= 8.50E-04\n+ td= 9.17E-13 tf= 1.17E-12\n+ tnf= 5.00E-05 tr= 2.50E-11\n+ tref= 25 vef= 90\n+ ver= 2.25 vo= 100\n+ vtf= 0.3 wbe= 1\n+ wsp= 1 xii= 3\n+ xin= 2.2 xis= 1.9\n+ xrb= 0 xrc= 0\n+ xre= 0 xrs= 0\n+ xtf= 10 xvo= 0\n\n.ends qNE52418_v161\n```\n\nTypically, the model does not explicitly define the temperature of the device. The SPICE standard temperature variable is `dtemp`. You must add this to the netlist to simulate at temperatures other than the default. Also, to enable this variable to pass into the netlist from the parent schematic that owns the netlist, you must add an additional equation to the line that specifies the element node numbers.\n\n```.subckt qNE52418_v161 c b e s DeviceTemp=1\n---- Intrinsic NPN Model ---------------------------------\nq1 c b e s qmod dtemp=DeviceTemp\n\n.model qmod npn\n+ level= 3 afn= 3.2\n+ ajc= -0.5 aje= -0.5\n+ ajs= -0.5 avc1= 3\n+ avc2= 200 bfn= 1\n+ cbco= 5.46E-15 cbeo= 8.34E-15\n+ cjc= 4.62E-15 cjcp= 2.64E-14\n+ cje= 3.78E-14 cjep= 1.04E-14\n+ cth= 3.26E-10 ea= 1.133\n+ eaic= 1.133 eaie= 1.133\n+ eais= 1.133 eanc= 1.133\n+ eane= 1.133 eans= 1.133\n+ fc= 0.95 gamm= 1.50E-11\n+ hrcf= 1 ibci= 8.22E-20\n+ ibcip= 0 ibcn= 7.37E-15\n+ ibcnp= 0 ibei= 1.77E-19\n+ ibeip= 2.49E-18 iben= 1.10E-15\n+ ibenp= 0 ikf= 2.51E-02\n+ ikp= 9.96E-02 ikr= 5.90E-03\n+ is= 2.87E-17 isp= 1.99E-19\n+ itf= 6.59E-02 kfn= 1.96E-09\n+ mc= 0.18 me= 0.2\n+ ms= 0.47 nci= 1.002\n+ ncip= 1.005 ncn= 1.5\n+ ncnp= 1.9 nei= 1.02\n+ nen= 1.9 nf= 1.015\n+ nfp= 1 nr= 1.01\n+ pc= 0.86 pe= 0.92\n+ ps= 0.55 qco= 1.47E-16\n+ qtf= 0 rbi= 3.4088\n+ rbp= 1 rbx= 13.111\n+ rci= 316.11 rcx= 0\n+ re= 0.68275 rs= 50\n+ rth= 644.03 tavc= 8.50E-04\n+ td= 9.17E-13 tf= 1.17E-12\n+ tnf= 5.00E-05 tr= 2.50E-11\n+ tref= 25 vef= 90\n+ ver= 2.25 vo= 100\n+ vtf= 0.3 wbe= 1\n+ wsp= 1 xii= 3\n+ xin= 2.2 xis= 1.9\n+ xrb= 0 xrc= 0\n+ xre= 0 xrs= 0\n+ xtf= 10 xvo= 0\n\n.ends qNE52418_v161\n```", null, "The transistor is defined by a subcircuit (a netlist in this example) and the deviceTemp variable has in turn been equated to a _TEMP variable.\n\nThe following graph shows the results of the gain and NF simulations.", null, "In this case, the project units for temperature are degrees C so assigning _TEMP to the deviceTemp parameter set the correct temperature in the netlist. If the project units for temperature are other than degrees C, the temperature passed to the netlist must be converted to Celsius. For example, the same results can be achieved when the global unit for temperature is set to degrees K with the schematic as follows. This schematic is shown in the following figure.", null, "The differences to note are that the swept values are now from 73 to 313 in steps of 20 since the units are degrees K. The deviceTemp value passed to the netlist is now converted to degrees C using the ktoc function. So ```deviceTemp = ktoc(_TEMP)``` converts _TEMP from Kelvin to Celsius before the value is passed to the netlist.\n\nAgain, you can use any variable value to pass temperature, however you should use _TEMP to maintain consistency between designs.\n\n### A.7.6. How to Sweep Passive and Active Temperature Through Hierarchy\n\nIn IC design, the temperature of each component is normally set to the same value. In this situation, each nonlinear model typically has its explicit temperature set to _TEMP in the PDK model set. The best approach is to assign the temperature at the top level schematic and pass the values down through the hierarchy. Assign _TEMP and _TEMPK to use the same swept values. Because _TEMP uses the temperature units set for the project and _TEMPK is always in Kelvin, built-in equations are used to make the two temperatures the same.\n\nIn this example, the unit for temperature is set to degrees Celsius. In the top level schematic, the value for _TEMP is set to sweep its value and pass down its value using the SWPVAR block _TEMP variable. Again, the global temperature unit is Celsius so the swept values are in Celsius. _TEMPK is assigned to follow the _TEMP value, but you must use the \"_TEMPK = ctok(_TEMP)\" equation to convert it from degrees Celsius to degrees Kelvin. Note that the temperature conversion depends on the global unit setting for temperature.", null, "The following schematic shows the lower level in the hierarchy where a resistor is added at the input to show the effects of _TEMPK.", null, "The following graph shows the swept Noise Figure versus temperature.", null, "Note that different results are achieved if the different temperature settings are not passed down through the hierarchy. When neither _TEMP nor _TEMPK are passed down, the result is shown on the following graph.", null, "The noise figure is flat because the swept temperature is not passed, and so the default values of 25 degrees C for _TEMP and 290 degrees for _TEMPK are used in simulation.\n\nWhen just _TEMP is passed through hierarchy, the result is shown on the following graph.", null, "When just _TEMPK is passed through hierarchy, the result is shown on the following graph.", null, "### A.7.7. Group Design Issues\n\nWhen sharing designs between different designers or different projects in the NI AWR Design Environment suite, units can be an issue. If you do not use any variables in a design, there are no issues. However, variables do not use unit scaling and so it is possible for designs to be passed between NI AWR projects that are not identical. The following are some techniques to address this issue:\n\n• Set up a common set of global units to use in all projects.\n\n• Agree to use base units for all projects.\n\n• Agree to select the Dependent parameters use base units check box on the Schematics/Diagrams tab of the Project Options dialog box prior to doing any design.\n\nThe following example demonstrates. Assume a designer uses capacitance units of nF and a variable assigned to a capacitance of 0.01. If this schematic is exported and then imported to a project with units of pF, the software does not know how to scale the value of a variable (variables can be combination of equations, etc. so this cannot be done in a general sense). When this schematic is imported the variable still has a value of 0.01. However, now that units are pF, the value is off by three orders of magnitude. Using base units for dependent parameters (Farads) in this example allows for easy design sharing.\n\nYou can set Dependent parameters use base units globally by choosing Options > Project Options and clicking the Schematics/Diagrams tab, or you can set it locally for each schematic. `This option is not set by default`. This setting sets any parameter using a variable value to base units. Base units are any unit type without a modifier, such as Farads, Henrys, and Amps, instead of the specified units in the global units setting. When designers share designs, the software does not know what the global units setting are in the original design. Always using base units for any parameter using a variable ensures that values are set appropriately any time a design is shared.\n\nIf you use the Dependent parameters use base units setting, the units for temperature display as degrees Kelvin, and you should sweep temperature values in Kelvin. If using a hierarchical design, you should make this setting for the entire project, if not you need to make the setting for any schematic in the hierarchy with a temperature setting. If using this setting in this example, _TEMP is always in degrees Kelvin and should use those values to sweep. This setting also keeps _TEMP and _TEMPK in the same units and no conversion is necessary.\n\nTo demonstrate the Dependent parameters use base units option, the linear simulation is repeated with the explicit temperature model. However, now this option is set locally (right-click on the schematic in the Project Browser, (choose Options and click the Schematic tab), overriding the default setting. Now the temperature units display in base units and the equation to convert from Celsius to Kelvin is not required. Again, the simulation results are identical.", null, "### A.7.8. Conclusions\n\nThe NI AWR Design Environment suite supports several methods to define and manipulate the temperature of electronic devices, and therefore to enable comprehensive simulation of noise processes, device operating point parameters such as gain and full nonlinear behavior.\n\n## A.8. Simulating in a Non 50 Ohm System\n\nBy default, the Microwave Office program assumes that you design in a 50 ohm system. Microwave Office software is not limited to 50 ohm systems, however, and changing the characteristic impedance of your design is easy. The following sections discuss how to operate in a non 50 ohm system.\n\n### A.8.1. Setting the Characteristic Impedance of a Schematic\n\nWhen working in a schematic in the NI AWR Design Environment suite, the impedance specified on the ports determines the characteristic impedance of that system. To change the impedance parameter you can either double-click the Z parameter or right-click the port and choose Properties, then change the Z parameter in the Element Options dialog box.\n\n### A.8.2. Non 50 Ohm Data Files\n\n#### Creating non 50 Ohm Output Files\n\nYou can generate output files in various formats. See “Working with Output Files ” for more information about generating output files. When creating these files, you must set the reference impedance to match the system impedance of the structure that it represents. If the reference impedance is not set correctly, there are unexpected results. See “Generate Touchstone, MDIF, or MATLAB File: NPORTF” for details on setting up this output file, particularly the Ref. Impedance setting.\n\nThe Ref. impedance sets the characteristic impedance of the circuit and overrides the impedance set by the ports in the structure that is simulated. For example, if the ports in the schematic are set to 75 ohms, but the reference impedance is set to 50 ohms, the results are based on a 50 ohm characteristic impedance, not 75 ohms. This is equivalent to changing the port impedance in your schematic to 50 ohms rather than 75.\n\n#### Plotting from Data Files\n\nIn the Microwave Office program, you can plot Port Parameters measurements directly from data files if the impedance specified in the file is normalized to the default characteristic impedance, 50 ohms. If the data file is normalized to a different impedance, you must place it in a schematic as a subcircuit, where the ports are set to the corresponding impedance of the data file. See “Adding Subcircuits to a Schematic or System Diagram ” for information on using a data file as a subcircuit in a schematic. When plotting directly from a data file, the characteristic impedance is always 50 ohms.\n\nFor example, if you have an S-parameter file normalized to 75 ohms for use in a 75 ohm system, plotting a measurement directly from this file is equivalent to placing it in a schematic with a 50 ohm characteristic impedance (50 ohm ports) and making the measurement. To achieve the desired measurements, you must place the data file in a schematic with 75 ohm ports and make the measurements from this schematic, not the data file directly.\n\n### A.8.3. Measurements on Non 50 Ohm EM Structures\n\nIf EM analysis is required in your design and the characteristic impedance of the EM structure is not 50 ohms, plotting Port Parameters measurements directly from the EM structure yields seemingly incorrect behavior. Plotting directly from an EM structure, like plotting directly from a data file, sets the characteristic impedance to 50 ohms. Although you can set the termination impedance for each port in the EM structure, these are not used when calculating the port parameters, but are instead used for computing the currents in the structure. See “Setting the Port Excitation or Termination” for more information.\n\nTo set the characteristic impedance for an EM structure to something other than 50 ohms for Port Parameters measurements, you can place the EM structure as a subcircuit in a schematic and change the impedance of the ports to the desired impedance.\n\n### A.8.4. Load Pull Analysis in non 50 Ohms Systems\n\nLoad pull analysis in Microwave Office software is simple when using the NI AWR Load Pull Script. See “Load Pull Script” for more information on using this script to perform a load pull analysis. Performing load pull on a non 50 ohm system only requires a few extra steps.\n\n1. Set the system impedance for the schematic by changing the impedance of all the ports used in the schematic to the new characteristic impedance. See “Setting the Characteristic Impedance of a Schematic” for information on how to make these changes.\n\n2. Set the System Impedance parameter (Z0) on the load pull tuner you are using (for example, LTUNER, LPTUNER, or HBTUNER). To change the system impedance parameter you can either double-click the Z0 parameter or right-click the tuner, choose Properties, and change the Z0 parameter in the Element Options Dialog Box.\n\n### A.8.5. Denormalizing Impedance on a Smith Chart\n\nGenerally, when you view port parameters on a Smith Chart they are normalized to a specific impedance. You can, however, view the un-normalized impedances directly on the Smith Chart. To change this option, right-click on the graph and choose Properties to display the Smith Chart Properties dialog box. Click the Markers tab and change the Z or Y display from Normalized to Denormalized to. By default, the normalization impedance is set to 50 ohms. For the impedances to be correct, it is important to change this to the characteristic impedance of your system.", null, "A. Platzker, W. Struble, and K. Hetzler, “Instabilities Diagnosis and the Role of K in the Microwave Circuits,” IEEE MTT-S International Microwave Symposium Digest, pp. 1185–1188, 1993.\n\n Wang, K., Jones, M., Nelson, S., \"The S-Probe. A New, Cost-Effective, 4-Gamma Method for Evaluating Multi-Stage Amplifier Stability\", IEEE MTT-Symposium Digest,1992, p. 829-832.\n\n Truxal, J. \"Introductory Systems Engineering\", McGraw Hill, 1972.\n\n C. A. Balanis, Antenna Theory Analysis and Design (2nd Edition), John Wiley & Sons, Inc.,1997." ]	[ null, "https://awrcorp.com/download/faq/english/docs/simulation/images/aplac_conv_1.png", null, "https://awrcorp.com/download/faq/english/docs/simulation/images/aplac_conv_3.png", null, "https://awrcorp.com/download/faq/english/docs/simulation/images/aplac_conv_4a.png", null, "https://awrcorp.com/download/faq/english/docs/simulation/images/aplac_conv_4.png", null, "https://awrcorp.com/download/faq/english/docs/simulation/images/aplac_conv_5.png", null, "https://awrcorp.com/download/faq/english/docs/simulation/images/aplac_conv_7.png", null, "https://awrcorp.com/download/faq/english/docs/simulation/images/aplac_conv_9.png", null, "https://awrcorp.com/download/faq/english/docs/simulation/images/aplac_conv_1.png", null, "https://awrcorp.com/download/faq/english/docs/simulation/images/aplac_conv_3.png", null, "https://awrcorp.com/download/faq/english/docs/simulation/images/aplac_conv_10a.png", 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http://export.arxiv.org/abs/1512.00057	[ "math.DS\n\nTitle: Continuous spectrum or measurable reducibility for quasiperiodic cocycles in $\\mathbb{T} ^{d} \\times SU(2)$\n\nAbstract: We continue our study of the local theory for quasiperiodic cocycles in $\\mathbb{T} ^{d} \\times G$, where $G=SU(2)$, over a rotation satisfying a Diophantine condition and satisfying a closeness-to-constants condition, by proving a dichotomy between measurable reducibility (and therefore pure point spectrum), and purely continuous spectrum in the space orthogonal to $L^{2}(\\mathbb{T} ^{d}) \\hookrightarrow L^{2}(\\mathbb{T} ^{d} \\times G)$. Subsequently, we describe the equivalence classes of cocycles under smooth conjugacy, as a function of the parameters defining their K.A.M. normal form. Finally, we derive a complete classification of the dynamics of one-frequency ($d=1$) cocycles over a Recurrent Diophantine rotation.\nAll theorems will be stated sharply in terms of the number of frequencies $d$, but in the proofs we will always assume $d=1$, for simplicity in expression and notation.\n Comments: 25 pages, 1 figure. arXiv admin note: text overlap with arXiv:1407.4763 Subjects: Dynamical Systems (math.DS) MSC classes: 37C55, 37A30 DOI: 10.1007/s00220-017-3034-3 Cite as: arXiv:1512.00057 [math.DS] (or arXiv:1512.00057v1 [math.DS] for this version)\n\nSubmission history\n\nFrom: Nikolaos Karaliolios [view email]\n[v1] Mon, 30 Nov 2015 21:47:49 GMT (101kb)\n\nLink back to: arXiv, form interface, contact." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.803896,"math_prob":0.96730405,"size":1303,"snap":"2019-43-2019-47","text_gpt3_token_len":339,"char_repetition_ratio":0.095458046,"word_repetition_ratio":0.033898305,"special_character_ratio":0.24251726,"punctuation_ratio":0.12554112,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97690713,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-18T14:17:16Z\",\"WARC-Record-ID\":\"<urn:uuid:08ad4e1a-82ef-496c-b7ef-e7c19f069e83>\",\"Content-Length\":\"14191\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a56daaf3-12db-4715-ba06-3e57eba2978e>\",\"WARC-Concurrent-To\":\"<urn:uuid:168d84bf-19fe-4165-9420-6d09d9a37194>\",\"WARC-IP-Address\":\"128.84.21.203\",\"WARC-Target-URI\":\"http://export.arxiv.org/abs/1512.00057\",\"WARC-Payload-Digest\":\"sha1:OUHTRMF2JYGC6WT5GJ5TZAJJNBXRSIGI\",\"WARC-Block-Digest\":\"sha1:ULQJMCL4A6MLQTGR4JI2CBIBWQIIIXRJ\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570986682998.59_warc_CC-MAIN-20191018131050-20191018154550-00180.warc.gz\"}"}
https://math.stackexchange.com/questions/1762147/why-do-we-study-real-numbers	[ "# Why do we study real numbers?\n\nI apologize if this is a somewhat naive question, but is there any particular reason mathematicians disproportionately study the field $\\mathbb{R}$ and its subsets (as opposed to any other algebraic structure)?\n\nIs this because $\\mathbb{R}$ is \"objectively\" more interesting in that studying it allows one to gain deep insights into mathematics, or is it sort of \"arbitrary\" in the sense that we are inclined to study $\\mathbb{R}$ due to historical reasons, real-world applications and because human beings have a strong natural intuition of real numbers?\n\nEdit: Note that I am not asking why $\\mathbb{Q}$ is insufficient as a number system; this has been asked and answered on this site and elsewhere. Rather, why, in a more deep sense, are $\\mathbb{N} \\subset \\mathbb{Z} \\subset \\mathbb{Q} \\subset \\mathbb{R}$ so crucial to mathematics? Would we be able to construct a meaningful study of mathematics with absolutely no reference to these sets, or are they fundamentally imperative?\n\n• The first question is do mathematicians study real numbers disproportionately? – Dr Xorile Apr 28 '16 at 3:16\n• @ASKASK: If one were in the mood, one could easily play devil's advocate and argue that, in fact, every feasible measurement in the real world is only a rational number, since measuring a real number would require \"infinite precision\". But there are already many discussions on the internet about this, so let's not get bogged down... – Will R Apr 28 '16 at 4:14\n• The OP may be interested in spending some time reading this dialogue (or should I say trialogue?) concerning the real number system, written by Timothy Gowers of Field's medal fame. I'm not sure it entirely answers your question, but it's certainly related and it's probably worth a read. – Will R Apr 28 '16 at 4:18\n• Concerning the recent edit: you appear to have changed the nature of the question. Before, you were (and in the title, you still are) asking specifically about real numbers. But in your edit, you are talking about the whole gamut: $\\mathbb{N}\\subset\\mathbb{Z}\\subset\\mathbb{Q}\\subset\\mathbb{R}$. This is a very different question: you are essentially now asking \"why is mathematics about numbers?\", to which my only answer is, \"what else are you expecting?\" Numbers are crucial to mathematics because mathematics, as a subject, has been built around them; they are part of the definition of the word. – Will R Apr 28 '16 at 4:38\n• You might be surprised by how few mathematicians study $\\mathbb R$. There are lots of areas of mathematics, and many of them don't have much to do with $\\mathbb R$. – Robert Israel Apr 28 '16 at 4:52\n\nThere's no way to do justice to \"Why is mathematics about real numbers?\" within the length constraints of a Math.SE post, but here are some relatively philosophical observations and opinions (meant to be a bit provocative, in the spirit of answering a soft question).\n\nFirst, as multiple people have commented, the real numbers are not universally regarded as the be-all/end-all number system. In The Road to Reality, for example, Penrose argues that complex numbers are more fundamental for physics.\n\nSetting that aside, why do we count, and where did natural numbers, integers, and rational numbers come from? I'm not a historian, so everything below should be regarded as a parable, biased by modern mathematical training.\n\nCounting (both the possibility and the ability) arises from the tension between variation and uniformity in the natural world:\n\n• Thanks to variation, there are \"different types of thing\": that piece of granite, that oak tree over there, the pine tree next to it.... If we look closely at the natural world, we find it to be made of unique objects, to occur in unique, irreproduceable events. In fact, the notion of \"event\" is our way of cutting the solid stream of existence into temporal and spatial chunks. As Heraclitus said, you cannot step in the same river twice.\n\n• Thanks to uniformity, there are recognizable \"classes of things\": rocks, trees, snowflakes, stars, sunrises.... No two rocks (or trees, or snowflakes...) are exactly alike. At the very least, they're \"in different places\" or \"at different times\" (otherwise they'd be identical).\n\nOnce the natural world is observed to contain classes of things, \"counting\" is a reasonable way to measure \"how many/how much\". In (a paraphrase of) Kronecker's famous quotation, The integers alone were created by God. All else [in mathematics] is the work of Man. To the contrary, the natural numbers (and therefore the integers) were created by us, as well, an abstraction for enumerating distinct objects similar enough to group together for some purpose.\n\nTo make a long story short:\n\n• An integer is a measure of additive comparison between two natural numbers. That is, it's a thing comprising a relationship between two other things. The standard construction of the integers in set theory merely formalizes this: An integer is an equivalence class of ordered pairs $(m_{1}, n_{1})$ of natural numbers, with $(m_{1}, n_{1}) \\sim (m_{2}, n_{2})$ if and only if $m_{1} + n_{2} = m_{2} + n_{1}$.\n\n• A rational number is a measure of multiplicative comparison between two non-zero integers. The standard construction of the rationals blah, blah, blah.\n\nIt's unsurprising that both abstractions were invented: If two people have flocks of sheep (say), it's natural to ask \"who has more sheep, and by how many?\". It's natural to represent debts as negative integers. If a flock of sheep (well...) must be divided among several people, it's natural to ask \"What is each person's portion?\", and to use rational numbers to represent the answer.\n\nThe real numbers obviously arose many centuries later, under pressures of Archimedes' method of exhaustion (for which one needs numbers representing \"limits of rational sequences\"), and were formalized two millennia after that in order to put calculus on a solid logical footing.\n\nI won't even touch the complex numbers, partly for last of time and space (heh), but mostly because Penrose (and many others) do so far more competently, with the depth the subject deserves.\n\n• There is some evidence that natural numbers are not a convention \"created by us\", by actually an a priori function of the brains of humans and some other animals. I've seen a video of Noam Chomsky hypothesizing that an evolutionary mutation was required to permit the brain to entertain limitless counting, and that language would be an extrapolated development after that. – Daniel R. Collins Apr 28 '16 at 17:34\n• See: youtube.com/… – Daniel R. Collins Apr 28 '16 at 17:38\n• You're not really addressing why, specifically, we use real numbers instead of some other extension of the rationals, which is, I think, the most important aspect of the question. – Kyle Strand Apr 29 '16 at 21:36\n• @Kyle: Fair enough. I was musing on why the natural, integer, and rational numbers arise naturally for humans, given the OP's clarification, \"Note that I am not asking why $\\mathbb{Q}$ is insufficient as a number system; this has been asked and answered on this site and elsewhere. Rather, why, in a more deep sense, are $\\mathbb{N} \\subset \\mathbb{Z} \\subset \\mathbb{Q} \\subset \\mathbb{R}$ so crucial to mathematics? Would we be able to construct a meaningful study of mathematics with absolutely no reference to these sets, or are they fundamentally imperative?\" – Andrew D. Hwang Apr 29 '16 at 23:17\n• @Daniel: Interesting; thank you for the video link. Again, I was deliberately being a bit provocative in saying humans invented the natural numbers, but although Peano's axioms for the natural numbers are simple and close to everyday experience, I don't see that the natural numbers exist in a manner independent of human minds. Practically speaking (e.g., in terms of classical computation), I see no way to distinguish the infinite set of naturals from a sufficiently long interval $\\{0, 1, 2, ..., N\\}$. (Maybe this is a semantic issue with the meaning of \"exist\".) – Andrew D. Hwang Apr 29 '16 at 23:46\n\nI've often asked myself the same thing, and this is what I tell myself. $\\mathbb R$ is (up to order-preserving field isomorphism) the only totally ordered, complete field. This is pretty big news, because these two nice structures lead to so many others we find useful to study in math. $\\mathbb R$ (and more generally $\\mathbb R^n$) is so great because a plethora of these fundamental \"structures\" studied in math are present in (at least some subset of) $\\mathbb R$. When we learn of new concepts, it's natural (crucial) to seek examples, and we often find solace in the usual first stop -- $\\mathbb R^n$.\n\nHere's a poor-at-best survery of some of the aforementioned structures that $\\mathbb R$ has.\n\nAlgebra\n\n• Group -- we can combine elements, i.e., $a + b$, invert them, i.e., $a^{-1}$.\n• Field -- we get more ways to combine elements, $+, -, \\times, \\div$.\n• Ordered field -- we get to do things like transitivity, i.e., $a < b \\wedge b < c \\implies a < c$, and \"add inequalities\", i.e., $a \\leq b \\wedge c \\leq d \\implies a + c \\leq b + d$.\n• Vector space -- linear algebra's pretty important. Arrow-like addition is very physical.\n\nAnalysis\n\n• Completeness -- analysts love sequences... to converge. This allows for a lot of \"take a sequence...\" arguments which start with a probably-desired sequence that ends up being Cauchy.\n• Compactness -- we always want to exploit compactness in analysis, and $\\mathbb R^n$ has a particularly nice characerization of it.\n• Hilbert Space -- we all love Hilbert space. Orthogonality is a useful tool. So is the spectral theorem.\n• Measure space -- measuring is very physical, and crucial for integrating! $\\mathbb R$ is the natural setting for the famous Lebesgue measure, and all measures map into the \"subset\" $[0,\\infty]$ of $\\mathbb R$. For Riemann integration, the (Darboux) definitions hinge on the least upper bound property of $\\mathbb R$.\n\nGeometry\n\n• Metric Space -- we can measure distances $d(p,q)$ between points. This is very physical. The triangle inequality is here too, which is even more useful in normed spaces, where it reads $\\\|u + v\\\| \\leq \\\|u\\\| + \\\|v\\\|$, because it leads to many useful estimates in analysis.\n• Manifolds -- things that by definition locally look like $\\mathbb R^n$. Many \"objects\" that we deal with early in math are manifolds (we just didn't know it at the time).\n• All of the separation axioms (Hausdorff, regular, normal, ...).\n• All of the countability axioms (separable, Lindelof, ...).\n\nNot shown (for the sake of space and the inevitable lack of completeness) is the interrelatedness between many of these properties for $\\mathbb R$, which is another indispensable virtue of $\\mathbb R$.\n\n• This similar question has a very good answer. – Jon Warneke Apr 28 '16 at 5:38\n• Absolutely. Manifolds are basically defined as \"let's keep all the nice properties of $\\mathbb{R}^n$\". – Turion Apr 28 '16 at 11:19\n• Isn't $\\mathbb R$ the only totally-ordered complete Archimedean field? – Ben Millwood Apr 28 '16 at 21:56\n• Given a totally-ordered field, completeness is equivalent to the least-upper-bound property, and the Archimedean property follows from the least-upper-bound property. – Jon Warneke Apr 28 '16 at 22:57\n• Note that it's possible to define a somewhat-analogous concept of \"computable completeness\" that does apply to the computable reals, even though the traditional Dedekind (or equivalent) definition clearly does not apply: math.stackexchange.com/a/418011/52057 – Kyle Strand Apr 29 '16 at 21:45\n\nA nice property of real numbers is that they are complete: every Cauchy sequence converges.\n\nIn analysis, mathematicians like to study spaces that are complete. People study Banach spaces rather than ordinary normed spaces; study Hilbert spaces rather than ordinary inner product spaces. A space that is not complete does not have as nice properties as complete spaces.\n\nEver since Pythagoras' disciple discovered that $\\sqrt 2$ is irrational, that was the beginning that signals that the rational numbers $\\mathbb{Q}$ is not enough to represent all quantities.\n\nHowever, there are some \"problems\" with the real numbers, and I know at least one professor (not to mention his name here) who does not believe in the real numbers. His reason, if I remember correctly, is that once you go deeper, a real number is not just a string of decimals: it is a equivalence class of Cauchy sequences. Not only is each Cauchy sequence infinite, each equivalence class is infinite (uncountable I think). This is the price to pay for dealing with real numbers.\n\nIt's also worth noting that this representative of is a strong bias towards analysis. An algebraist generally cares minimally about $\\mathbb{R}$, generally field extensions of $\\mathbb{Q}$ and $\\mathbb{F}_p$ are far more interesting. The badly named so-called Real Numbers are very much unimportant in Combinatorics and Number Theory compared to other sets, but mathematicians spend much more time talking about the Real Numbers because there is a strong bias, both in collegiate education, and younger education, towards analytic topics over algebraic or topological ones.\n\n• I'm not sure I agree with this. Even if the thing I'm really interested in is algebraic extensions of $\\mathbb{Q}$, I will end up studying completions like $\\mathbb{C}$ and $\\mathbb{Q}_p$. If I think I'm a combinatorialist, I might still end up using generating functions and doing countour integrals for asymptotics (e.g. to prove Stirling's approximation). Perhaps the reals are over-emphasized in the standard curriculum, but they don't cleanly belong to \"analysis\", and I would think that the vast majority of mathematicians care about them in some form or other. – Slade Apr 28 '16 at 19:42\n• I mean, yes it comes up (I'm not saying it doesn't), but the idea of R as foundational is basically an analysis thing. – Stella Biderman Apr 28 '16 at 19:49\n\nAn amazingly large part of Riemann-Stieltjes integration theory, as given in Apostol's Mathematical Analysis, can be developed in the setting of rational number system. Completeness property of real number system is needed, only when we try to prove that every continuous function is integrable.\n\nIt's the existential proofs of various theorems in Mathematics, that are the things of real challenge. They take 90% energy of a mathematician. Applied mathematicians and theoretical physicists don't take bother in proving existential theorems, and thus, they save their 90% energy.\n\nThe whole edifice of Dedekind's construction of real number system is meant to prove: There exists a complete ordered field." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9460911,"math_prob":0.95631284,"size":3484,"snap":"2020-34-2020-40","text_gpt3_token_len":768,"char_repetition_ratio":0.106609195,"word_repetition_ratio":0.0035587188,"special_character_ratio":0.22531573,"punctuation_ratio":0.14307004,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9916279,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-08-07T18:47:39Z\",\"WARC-Record-ID\":\"<urn:uuid:b3340d1f-bf20-455f-af2e-461149a18e8a>\",\"Content-Length\":\"210283\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:44f2c6ff-63b8-4599-b2b8-b0e99d926ab3>\",\"WARC-Concurrent-To\":\"<urn:uuid:62d046a6-d21c-4fb3-8587-673d09f33c1e>\",\"WARC-IP-Address\":\"151.101.65.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/1762147/why-do-we-study-real-numbers\",\"WARC-Payload-Digest\":\"sha1:TYUVH2EBNFE5NNVOZJYNWSSFD32K67WH\",\"WARC-Block-Digest\":\"sha1:V3ZSDMFRJVGK2AUYUVC46HJMO5QXYCNH\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-34/CC-MAIN-2020-34_segments_1596439737206.16_warc_CC-MAIN-20200807172851-20200807202851-00039.warc.gz\"}"}
https://socratic.org/questions/how-do-you-solve-9-2x-le-3-or-3x-10-le-6-x#621040	[ "# How do you solve 9-2x \\le 3 or 3x+10 \\le 6-x?\n\nMay 27, 2018\n\nSee explanation\n\n#### Explanation:\n\nWe have two conditions that are combined to define the limit of values that may be assigned to $x$\n\nCondition 1: $9 - 2 x \\le 3$\nCondition 2: $2 x + 10 \\le 6 - x$\n\nConsider condition 1\n\nAdd $2 x$ to both sides\n\n$9 \\textcolor{w h i t e}{\\text{dd}} \\underbrace{- 2 x + 2 x} \\le 3 + 2 x$\n\n$9 \\textcolor{w h i t e}{\\text{ddd\")+ 0color(white)(\"dddd}} \\le 2 x + 3$\n\nSubtract 3 from both sides\n\n$9 - 3 \\le 2 x + 0$\n\nDivide both sides by 2\n\n$\\frac{9 - 3}{2} \\le \\frac{2}{2} \\textcolor{w h i t e}{.} x$\n\n$3 \\le x$\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nConsider condition 2\n\nAdd $x$ to both sides\n\n$3 x + 10 \\le 6$\n\nSubtract 10 from both sides\n\n$3 x \\le - 4$\n\nDivide both sides by 3\n\n$x \\le - \\frac{4}{3}$\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nCombining these we have:\n\n$- \\frac{4}{3} \\ge x \\ge 3$\n\nIn other words $x$ does not take on any values between and excluding $- \\frac{4}{3} \\mathmr{and} 3$", null, "" ]	[ null, "https://useruploads.socratic.org/cHZnKU4ET1CwsYCpquSQ_The%20inequality%201.bmp", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7067378,"math_prob":0.9998784,"size":613,"snap":"2022-05-2022-21","text_gpt3_token_len":156,"char_repetition_ratio":0.21839081,"word_repetition_ratio":0.06382979,"special_character_ratio":0.3425775,"punctuation_ratio":0.057142857,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9994411,"pos_list":[0,1,2],"im_url_duplicate_count":[null,5,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-05-23T23:46:06Z\",\"WARC-Record-ID\":\"<urn:uuid:14b147cb-2970-4678-9f2f-454638f19a90>\",\"Content-Length\":\"34630\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9c470776-7864-4f87-880e-fdfacfbd5476>\",\"WARC-Concurrent-To\":\"<urn:uuid:bef23bc2-2867-42f7-9ab4-5dcfbe63c34a>\",\"WARC-IP-Address\":\"216.239.34.21\",\"WARC-Target-URI\":\"https://socratic.org/questions/how-do-you-solve-9-2x-le-3-or-3x-10-le-6-x#621040\",\"WARC-Payload-Digest\":\"sha1:H77YM5FM3DKF5N3MZ4W4MAXRDF4V3T22\",\"WARC-Block-Digest\":\"sha1:QVU7FDOHVMHL5BZHOZSUPB7K2E2XTO5O\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-21/CC-MAIN-2022-21_segments_1652662562106.58_warc_CC-MAIN-20220523224456-20220524014456-00230.warc.gz\"}"}
https://math.stackexchange.com/questions/1743385/epsilon-delta-proof	[ "# Epsilon-delta proof\n\nSuppose $\\epsilon> 0$ and $\\displaystyle\\delta = \\min \\left\\{1,\\frac{\\epsilon}{10}\\right\\}$\n\nIf $0 < \|x -1 \|< \\delta$, does this imply $\|x^{2}-1\|<\\epsilon$?\n\nI have verified that for $\\epsilon=20$, $\\delta=1$ and $x=1.9$, it is true.\n\nI have also verified that for $\\epsilon=5$, $\\displaystyle\\delta=\\frac{1}{2}$ and $x=1.2$, it is true.\n\nI am having problems proving it.\n\nNow $\|x^{2}-1\|<\\epsilon$ is $\|x-1\|\\cdot\|x+1\|<\\epsilon$\n\nAnd I know $\\displaystyle2<\|x+1\|<\\frac{\\epsilon}{10}+2$, or\n\n$2<\|x+1\|<3$\n\nBut not sure where to go next.\n\nto solve this type of problem you have to find relationship between $\\delta$ and $\\epsilon$ in worst case. in other word you have to find an upper bound for $\\delta$ that satisfies expression related to $\\epsilon$.\n\nfor example we have to start from $\|f(x)-L\|<\\epsilon$ and conclude something like $\|x-a\|\|g(x)\|<\\epsilon$. then by finding an upper bound on $\|g(x)\|$ like $M$ we can conclude that $\\delta<min\\{\\frac{\\epsilon}{M},\\delta_1\\}$ where $\\delta_1$ appears in some cases that function $f(x)$ is not defined in the specified interval. so $\\delta_1$ is value that $f(x)$ is well defined at $(a-\\delta_1,a+\\delta_1)$\n\nto reach this goal we can write: $$\|x^{2}-1\|<\\epsilon \\Rightarrow \|x-1\|\\cdot\|x+1\|<\\epsilon$$ we see that since $x \\to 1$ therefore an upper bound on $\|x+1\|$ is $3$. so it can be concluded that $\|x-1\|\\cdot\|x+1\|<3\\epsilon$. so in order to ensure $\|x-1\|<\\delta$ yields $\|x^2-1\|<\\epsilon$ we have $\\delta<\\frac{\\epsilon}{3}$.\n\nso we can conclude that $\\delta = \\min\\{1,\\frac{\\epsilon}{10}\\}$ gives us $\|x^{2}-1\|<\\epsilon$ according to transitive property of Real numbers.\n\n$0\\leq \|x-1\|\|x+1\|<\\delta(\\delta+2)$ because if $x\\geq 1$ then $x-1<\\delta$ implies $x+1<2+\\delta$ and if $x<1$ then $x+1<2$ and so $\|x+1\|<2+\\delta$ for $x\\geq-1$. If $x<-1$ then $\|x-1\|=1-x<\\delta$ implies $\|x+1\|=-x-1<\\delta-2<\\delta+2$. If $\\epsilon\\leq 10$ then $\\delta=\\frac{\\epsilon}{10}$ and thus\n\n$0\\leq\|x-1\|\|x+1\|<\\delta(2+\\delta)=\\frac{\\epsilon^2}{100}+\\frac{\\epsilon}{5}\\leq\\epsilon$\n\nwhere the last inequality holds for $\\epsilon\\leq 80$.\n\nFinally if $\\epsilon>10$ then $0\\leq \|x-1\|\|x+1\|<\\delta(\\delta+2)=3<\\epsilon$." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.808997,"math_prob":1.000007,"size":525,"snap":"2020-10-2020-16","text_gpt3_token_len":205,"char_repetition_ratio":0.15163147,"word_repetition_ratio":0.0,"special_character_ratio":0.42095238,"punctuation_ratio":0.12173913,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000095,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-04-01T15:39:41Z\",\"WARC-Record-ID\":\"<urn:uuid:1fb56ef3-b176-47b8-92d7-9299f8445420>\",\"Content-Length\":\"148915\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0434d5b8-cca5-4102-9cad-176c8002355c>\",\"WARC-Concurrent-To\":\"<urn:uuid:3cecebf2-438c-430f-9f09-0d3961b05419>\",\"WARC-IP-Address\":\"151.101.129.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/1743385/epsilon-delta-proof\",\"WARC-Payload-Digest\":\"sha1:CAUL5BALHFR4YUUTJAENUVFEWGQTAHBM\",\"WARC-Block-Digest\":\"sha1:7D6RMI6H6QBYODYHTNCO6JTZPMSEUESJ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-16/CC-MAIN-2020-16_segments_1585370505731.37_warc_CC-MAIN-20200401130837-20200401160837-00543.warc.gz\"}"}
http://www.stylejonction.ca/pages/Air-Jordan-3-Toronto	[ "You are here: Home - Air Jordan 3 Toronto A novel algorithm is proposed for\n\n# Air Jordan 3 Toronto A novel algorithm is proposed for", null, "A novel algorithm is proposed for the independent component analysis (ICA). In the algorithm, ICA is accomplished by geometrical transformation of a Air Jordan Retro 8 scatter diagram. We evaluate the performance of the algorithm by separating original independent images from some mixed images. Our algorithm fulfills considerable reduction of processing time while maintaining the same level of accuracy as conventional algorithms. For some groups of species, extinction rates are orders of magnitude higher than expected background rates — many species now last nearer a fateful second than their destined hour. We consider greedy algorithms that allow partial regret. As an example we consider a variant of the cheapest insertion algorithm for the TSP. Our numerical study indicates that in most cases it significantly reduces the relative error, and the added computational time is quite small. The dynamical J-T problem is reduced to a system of Air Jordan 3 Toronto two ordinary linear first-order differential equations. An ansatz for the ground state is made which contains Judd's isolated exact solutions as special cases. The ground state energy is calculated as function of the interaction constant for different values of the angular momentum j. Excellent agreement with numerical calculations is found for all values of j." ]	[ null, "http://www.stylejonction.ca/pages/images/UYGre/AJ/air-jordan-3-toronto-A-novel-algorithm-is-proposed-for.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9189537,"math_prob":0.9765555,"size":1391,"snap":"2020-24-2020-29","text_gpt3_token_len":246,"char_repetition_ratio":0.11175198,"word_repetition_ratio":0.018957347,"special_character_ratio":0.17109993,"punctuation_ratio":0.064935066,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95604306,"pos_list":[0,1,2],"im_url_duplicate_count":[null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-05-26T19:20:35Z\",\"WARC-Record-ID\":\"<urn:uuid:bc719c11-7fff-4a67-920a-d1bc0fa3fed3>\",\"Content-Length\":\"7870\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7f400eae-f8b0-4908-8b66-3d2809801fa5>\",\"WARC-Concurrent-To\":\"<urn:uuid:297d54f3-200f-43a5-83c1-33d794c16a2c>\",\"WARC-IP-Address\":\"209.134.27.242\",\"WARC-Target-URI\":\"http://www.stylejonction.ca/pages/Air-Jordan-3-Toronto\",\"WARC-Payload-Digest\":\"sha1:ZXDPLECZ7UJVZ63MCNKZD3CD2FB3PXYE\",\"WARC-Block-Digest\":\"sha1:FDTLNBF6VSVBWVGZBWWZ7O52JYVASU2N\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-24/CC-MAIN-2020-24_segments_1590347391309.4_warc_CC-MAIN-20200526191453-20200526221453-00502.warc.gz\"}"}
https://docs.w3cub.com/tensorflow~2.3/compat/v1/keras/losses	[ "/TensorFlow 2.3\n\n# Module: tf.compat.v1.keras.losses\n\nBuilt-in loss functions.\n\n## Classes\n\n`class BinaryCrossentropy`: Computes the cross-entropy loss between true labels and predicted labels.\n\n`class CategoricalCrossentropy`: Computes the crossentropy loss between the labels and predictions.\n\n`class CategoricalHinge`: Computes the categorical hinge loss between `y_true` and `y_pred`.\n\n`class CosineSimilarity`: Computes the cosine similarity between labels and predictions.\n\n`class Hinge`: Computes the hinge loss between `y_true` and `y_pred`.\n\n`class Huber`: Computes the Huber loss between `y_true` and `y_pred`.\n\n`class KLDivergence`: Computes Kullback-Leibler divergence loss between `y_true` and `y_pred`.\n\n`class LogCosh`: Computes the logarithm of the hyperbolic cosine of the prediction error.\n\n`class Loss`: Loss base class.\n\n`class MeanAbsoluteError`: Computes the mean of absolute difference between labels and predictions.\n\n`class MeanAbsolutePercentageError`: Computes the mean absolute percentage error between `y_true` and `y_pred`.\n\n`class MeanSquaredError`: Computes the mean of squares of errors between labels and predictions.\n\n`class MeanSquaredLogarithmicError`: Computes the mean squared logarithmic error between `y_true` and `y_pred`.\n\n`class Poisson`: Computes the Poisson loss between `y_true` and `y_pred`.\n\n`class SparseCategoricalCrossentropy`: Computes the crossentropy loss between the labels and predictions.\n\n`class SquaredHinge`: Computes the squared hinge loss between `y_true` and `y_pred`.\n\n## Functions\n\n`KLD(...)`: Computes Kullback-Leibler divergence loss between `y_true` and `y_pred`.\n\n`MAE(...)`: Computes the mean absolute error between labels and predictions.\n\n`MAPE(...)`: Computes the mean absolute percentage error between `y_true` and `y_pred`.\n\n`MSE(...)`: Computes the mean squared error between labels and predictions.\n\n`MSLE(...)`: Computes the mean squared logarithmic error between `y_true` and `y_pred`.\n\n`binary_crossentropy(...)`: Computes the binary crossentropy loss.\n\n`categorical_crossentropy(...)`: Computes the categorical crossentropy loss.\n\n`categorical_hinge(...)`: Computes the categorical hinge loss between `y_true` and `y_pred`.\n\n`cosine(...)`: Computes the cosine similarity between labels and predictions.\n\n`cosine_proximity(...)`: Computes the cosine similarity between labels and predictions.\n\n`cosine_similarity(...)`: Computes the cosine similarity between labels and predictions.\n\n`deserialize(...)`: Deserializes a serialized loss class/function instance.\n\n`get(...)`: Retrieves a Keras loss as a `function`/`Loss` class instance.\n\n`hinge(...)`: Computes the hinge loss between `y_true` and `y_pred`.\n\n`kl_divergence(...)`: Computes Kullback-Leibler divergence loss between `y_true` and `y_pred`.\n\n`kld(...)`: Computes Kullback-Leibler divergence loss between `y_true` and `y_pred`.\n\n`kullback_leibler_divergence(...)`: Computes Kullback-Leibler divergence loss between `y_true` and `y_pred`.\n\n`log_cosh(...)`: Logarithm of the hyperbolic cosine of the prediction error.\n\n`logcosh(...)`: Logarithm of the hyperbolic cosine of the prediction error.\n\n`mae(...)`: Computes the mean absolute error between labels and predictions.\n\n`mape(...)`: Computes the mean absolute percentage error between `y_true` and `y_pred`.\n\n`mean_absolute_error(...)`: Computes the mean absolute error between labels and predictions.\n\n`mean_absolute_percentage_error(...)`: Computes the mean absolute percentage error between `y_true` and `y_pred`.\n\n`mean_squared_error(...)`: Computes the mean squared error between labels and predictions.\n\n`mean_squared_logarithmic_error(...)`: Computes the mean squared logarithmic error between `y_true` and `y_pred`.\n\n`mse(...)`: Computes the mean squared error between labels and predictions.\n\n`msle(...)`: Computes the mean squared logarithmic error between `y_true` and `y_pred`.\n\n`poisson(...)`: Computes the Poisson loss between y_true and y_pred.\n\n`serialize(...)`: Serializes loss function or `Loss` instance.\n\n`sparse_categorical_crossentropy(...)`: Computes the sparse categorical crossentropy loss.\n\n`squared_hinge(...)`: Computes the squared hinge loss between `y_true` and `y_pred`." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6607383,"math_prob":0.8892002,"size":4153,"snap":"2021-21-2021-25","text_gpt3_token_len":869,"char_repetition_ratio":0.25572428,"word_repetition_ratio":0.4019608,"special_character_ratio":0.2099687,"punctuation_ratio":0.27808988,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99952483,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-05-14T20:03:46Z\",\"WARC-Record-ID\":\"<urn:uuid:d800cb00-5a40-4c57-99f8-818fd542f4b8>\",\"Content-Length\":\"15996\",\"Content-Type\":\"application/http; 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