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https://www.numberempire.com/11026	[ "Home \| Menu \| Get Involved \| Contact webmaster", null, "", null, "", null, "", null, "", null, "# Number 11026\n\neleven thousand twenty six\n\n### Properties of the number 11026\n\n Factorization 2 * 37 * 149 Divisors 1, 2, 37, 74, 149, 298, 5513, 11026 Count of divisors 8 Sum of divisors 17100 Previous integer 11025 Next integer 11027 Is prime? NO Previous prime 11003 Next prime 11027 11026th prime 116797 Is a Fibonacci number? NO Is a Bell number? NO Is a Catalan number? NO Is a factorial? NO Is a regular number? NO Is a perfect number? NO Polygonal number (s < 11)? triangular(148) Binary 10101100010010 Octal 25422 Duodecimal 646a Hexadecimal 2b12 Square 121572676 Square root 105.00476179679 Natural logarithm 9.3080113991496 Decimal logarithm 4.0424179881433 Sine -0.83614343403314 Cosine 0.54851085469958 Tangent -1.524388126268\nNumber 11026 is pronounced eleven thousand twenty six. Number 11026 is a composite number. Factors of 11026 are 2 * 37 * 149. Number 11026 has 8 divisors: 1, 2, 37, 74, 149, 298, 5513, 11026. Sum of the divisors is 17100. Number 11026 is not a Fibonacci number. It is not a Bell number. Number 11026 is not a Catalan number. Number 11026 is not a regular number (Hamming number). It is a not factorial of any number. Number 11026 is a deficient number and therefore is not a perfect number. Number 11026 is a triangular number with n=148. Binary numeral for number 11026 is 10101100010010. Octal numeral is 25422. Duodecimal value is 646a. Hexadecimal representation is 2b12. Square of the number 11026 is 121572676. Square root of the number 11026 is 105.00476179679. Natural logarithm of 11026 is 9.3080113991496 Decimal logarithm of the number 11026 is 4.0424179881433 Sine of 11026 is -0.83614343403314. Cosine of the number 11026 is 0.54851085469958. Tangent of the number 11026 is -1.524388126268\n\n### Number properties\n\nExamples: 3628800, 9876543211, 12586269025" ]	[ null, "https://www.numberempire.com/images/graystar.png", null, "https://www.numberempire.com/images/graystar.png", null, "https://www.numberempire.com/images/graystar.png", null, "https://www.numberempire.com/images/graystar.png", null, "https://www.numberempire.com/images/graystar.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5953295,"math_prob":0.9910441,"size":2177,"snap":"2020-34-2020-40","text_gpt3_token_len":723,"char_repetition_ratio":0.18683848,"word_repetition_ratio":0.061797753,"special_character_ratio":0.4368397,"punctuation_ratio":0.1489899,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9962671,"pos_list":[0,1,2,3,4,5,6,7,8,9,10],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-09-24T02:11:58Z\",\"WARC-Record-ID\":\"<urn:uuid:c393d5c7-38c4-42fe-beee-b2e12892464d>\",\"Content-Length\":\"20509\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:3445ac29-fd70-43ce-a54b-895372341173>\",\"WARC-Concurrent-To\":\"<urn:uuid:2a54b3bd-b4f2-45d6-a1a8-023067c64a79>\",\"WARC-IP-Address\":\"104.24.112.69\",\"WARC-Target-URI\":\"https://www.numberempire.com/11026\",\"WARC-Payload-Digest\":\"sha1:ZZLSJZVA3TA6BKB7LYA5CLZ25Q63WD46\",\"WARC-Block-Digest\":\"sha1:YFHHNV6JEJ5JEE7FICACZ3T27IUNVSGK\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600400213006.47_warc_CC-MAIN-20200924002749-20200924032749-00124.warc.gz\"}"}
https://www.e-pandu.com/2019/04/vector-section-space.html	[ "-->\n\n# Vector Section Space\n\nIf given a vector space V, then we may form another vector space that is a subset of S of V and use operations on V. Because V is a vector space, summarized operations and scalar multiplication always generate another vector in V. Share the new system uses the S subset of V as a set of course to be vector space, the set must be covered under sums and scalar multiplication operations. That is, the sum of the two elements in S must always be an element of S and the result of a scalar with elements of S must always be an element of S.\n\nis a subset of R2.\n\nwhich is an element of S.\n\nIt is also an element of S.\n\nDefinition,\nIf S is a non-empty subset of a vector space V, and S satisfies the following conditions,\n(i) α x ϵ S if x ϵ S for any scalar α\n(ii) x + y ϵ S if x ϵ S and y ϵ S\nthen S is called the subspace of V.\nTerms (i) say that S is closed under scalar multiplication. That is, when an element of S is multiplied by a scalar, the result is an element of S.\nTerms (ii) say that S is closed under addition. That is, the sum of the two elements of S is always an element of S.\nSo, if we do calculations using the operations of V and elements from S, we will always produce elements from S. Therefore, the subspace of V is a subset S which is closed under the operations of V.\nSuppose that S is the subspace of a vector space V. Using the scalar addition and multiplication operations defined in V, we can form a new mathematical system with S as the appropriate set. It can easily be seen that the eight axioms as a whole will remain valid for this new system. The axioms of A3 and A4 are the result of Theorem 1 and condition (i) of the definition of subspace. The other six axioms are valid for each element of V, so in particular the six axioms are valid for elements of S. So actually each subspace is a vector space.\n\nExample 2.\nSuppose S = {(x1, x2, x3)T \| x1 = x2}\nThen S is the subspace of R3, because αx = (αa, αa, αb)T ϵ S.\nIf (a, a, b)T and (c, c, d)T are any elements of S, then" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8990485,"math_prob":0.99537635,"size":2120,"snap":"2020-45-2020-50","text_gpt3_token_len":578,"char_repetition_ratio":0.15075615,"word_repetition_ratio":0.06526806,"special_character_ratio":0.2476415,"punctuation_ratio":0.0997921,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9997108,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-12-03T23:20:49Z\",\"WARC-Record-ID\":\"<urn:uuid:ea60d3b9-e422-4ae3-97b9-48e94e095fe4>\",\"Content-Length\":\"201463\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4924f685-cc63-4159-a6a0-87b3c3646ad3>\",\"WARC-Concurrent-To\":\"<urn:uuid:4ace49ed-2f41-4ef2-9d6f-38a4e31c196e>\",\"WARC-IP-Address\":\"142.250.73.211\",\"WARC-Target-URI\":\"https://www.e-pandu.com/2019/04/vector-section-space.html\",\"WARC-Payload-Digest\":\"sha1:XJ77G3ESIKYWROMB3PO656KQWMOMWAVD\",\"WARC-Block-Digest\":\"sha1:5BTAYDZJG5TTNMWDOSC3GPU4KBTUAVVV\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-50/CC-MAIN-2020-50_segments_1606141732835.81_warc_CC-MAIN-20201203220448-20201204010448-00598.warc.gz\"}"}
https://www.colorhexa.com/20f17c	[ "# #20f17c Color Information\n\nIn a RGB color space, hex #20f17c is composed of 12.5% red, 94.5% green and 48.6% blue. Whereas in a CMYK color space, it is composed of 86.7% cyan, 0% magenta, 48.5% yellow and 5.5% black. It has a hue angle of 146.4 degrees, a saturation of 88.2% and a lightness of 53.5%. #20f17c color hex could be obtained by blending #40fff8 with #00e300. Closest websafe color is: #33ff66.\n\n• R 13\n• G 95\n• B 49\nRGB color chart\n• C 87\n• M 0\n• Y 49\n• K 5\nCMYK color chart\n\n#20f17c color description : Vivid cyan - lime green.\n\n# #20f17c Color Conversion\n\nThe hexadecimal color #20f17c has RGB values of R:32, G:241, B:124 and CMYK values of C:0.87, M:0, Y:0.49, K:0.05. Its decimal value is 2158972.\n\nHex triplet RGB Decimal 20f17c `#20f17c` 32, 241, 124 `rgb(32,241,124)` 12.5, 94.5, 48.6 `rgb(12.5%,94.5%,48.6%)` 87, 0, 49, 5 146.4°, 88.2, 53.5 `hsl(146.4,88.2%,53.5%)` 146.4°, 86.7, 94.5 33ff66 `#33ff66`\nCIE-LAB 84.313, -71.673, 43.291 35.687, 64.669, 29.669 0.274, 0.497, 64.669 84.313, 83.733, 148.867 84.313, -73.925, 69.404 80.417, -61.517, 34.417 00100000, 11110001, 01111100\n\n# Color Schemes with #20f17c\n\n• #20f17c\n``#20f17c` `rgb(32,241,124)``\n• #f12095\n``#f12095` `rgb(241,32,149)``\nComplementary Color\n• #2df120\n``#2df120` `rgb(45,241,32)``\n• #20f17c\n``#20f17c` `rgb(32,241,124)``\n• #20f1e5\n``#20f1e5` `rgb(32,241,229)``\nAnalogous Color\n• #f1202d\n``#f1202d` `rgb(241,32,45)``\n• #20f17c\n``#20f17c` `rgb(32,241,124)``\n• #e520f1\n``#e520f1` `rgb(229,32,241)``\nSplit Complementary Color\n• #f17c20\n``#f17c20` `rgb(241,124,32)``\n• #20f17c\n``#20f17c` `rgb(32,241,124)``\n• #7c20f1\n``#7c20f1` `rgb(124,32,241)``\n• #95f120\n``#95f120` `rgb(149,241,32)``\n• #20f17c\n``#20f17c` `rgb(32,241,124)``\n• #7c20f1\n``#7c20f1` `rgb(124,32,241)``\n• #f12095\n``#f12095` `rgb(241,32,149)``\n• #0cb958\n``#0cb958` `rgb(12,185,88)``\n• #0dd163\n``#0dd163` `rgb(13,209,99)``\n• #0fe96f\n``#0fe96f` `rgb(15,233,111)``\n• #20f17c\n``#20f17c` `rgb(32,241,124)``\n• #38f38a\n``#38f38a` `rgb(56,243,138)``\n• #50f498\n``#50f498` `rgb(80,244,152)``\n• #68f6a6\n``#68f6a6` `rgb(104,246,166)``\nMonochromatic Color\n\n# Alternatives to #20f17c\n\nBelow, you can see some colors close to #20f17c. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #20f148\n``#20f148` `rgb(32,241,72)``\n• #20f159\n``#20f159` `rgb(32,241,89)``\n• #20f16b\n``#20f16b` `rgb(32,241,107)``\n• #20f17c\n``#20f17c` `rgb(32,241,124)``\n• #20f18d\n``#20f18d` `rgb(32,241,141)``\n• #20f19f\n``#20f19f` `rgb(32,241,159)``\n• #20f1b0\n``#20f1b0` `rgb(32,241,176)``\nSimilar Colors\n\n# #20f17c Preview\n\nThis text has a font color of #20f17c.\n\n``<span style=\"color:#20f17c;\">Text here</span>``\n#20f17c background color\n\nThis paragraph has a background color of #20f17c.\n\n``<p style=\"background-color:#20f17c;\">Content here</p>``\n#20f17c border color\n\nThis element has a border color of #20f17c.\n\n``<div style=\"border:1px solid #20f17c;\">Content here</div>``\nCSS codes\n``.text {color:#20f17c;}``\n``.background {background-color:#20f17c;}``\n``.border {border:1px solid #20f17c;}``\n\n# Shades and Tints of #20f17c\n\nA shade is achieved by adding black to any pure hue, while a tint is created by mixing white to any pure color. In this example, #011108 is the darkest color, while #fdfffe is the lightest one.\n\n• #011108\n``#011108` `rgb(1,17,8)``\n• #022311\n``#022311` `rgb(2,35,17)``\n• #03361a\n``#03361a` `rgb(3,54,26)``\n• #054822\n``#054822` `rgb(5,72,34)``\n• #065b2b\n``#065b2b` `rgb(6,91,43)``\n• #076d34\n``#076d34` `rgb(7,109,52)``\n• #08803d\n``#08803d` `rgb(8,128,61)``\n• #099245\n``#099245` `rgb(9,146,69)``\n• #0aa54e\n``#0aa54e` `rgb(10,165,78)``\n• #0bb757\n``#0bb757` `rgb(11,183,87)``\n• #0dca60\n``#0dca60` `rgb(13,202,96)``\n• #0edc69\n``#0edc69` `rgb(14,220,105)``\n• #0fee71\n``#0fee71` `rgb(15,238,113)``\n• #20f17c\n``#20f17c` `rgb(32,241,124)``\n• #32f287\n``#32f287` `rgb(50,242,135)``\n• #45f392\n``#45f392` `rgb(69,243,146)``\n• #57f49d\n``#57f49d` `rgb(87,244,157)``\n• #6af6a7\n``#6af6a7` `rgb(106,246,167)``\n• #7cf7b2\n``#7cf7b2` `rgb(124,247,178)``\n• #8ff8bd\n``#8ff8bd` `rgb(143,248,189)``\n• #a1f9c8\n``#a1f9c8` `rgb(161,249,200)``\n``#b4fad3` `rgb(180,250,211)``\n• #c6fbde\n``#c6fbde` `rgb(198,251,222)``\n• #d9fde8\n``#d9fde8` `rgb(217,253,232)``\n• #ebfef3\n``#ebfef3` `rgb(235,254,243)``\n• #fdfffe\n``#fdfffe` `rgb(253,255,254)``\nTint Color Variation\n\n# Tones of #20f17c\n\nA tone is produced by adding gray to any pure hue. In this case, #848d88 is the less saturated color, while #17fa7b is the most saturated one.\n\n• #848d88\n``#848d88` `rgb(132,141,136)``\n• #7b9687\n``#7b9687` `rgb(123,150,135)``\n• #729f86\n``#729f86` `rgb(114,159,134)``\n• #69a885\n``#69a885` `rgb(105,168,133)``\n• #60b184\n``#60b184` `rgb(96,177,132)``\n• #57ba83\n``#57ba83` `rgb(87,186,131)``\n• #4ec381\n``#4ec381` `rgb(78,195,129)``\n• #44cd80\n``#44cd80` `rgb(68,205,128)``\n• #3bd67f\n``#3bd67f` `rgb(59,214,127)``\n• #32df7e\n``#32df7e` `rgb(50,223,126)``\n• #29e87d\n``#29e87d` `rgb(41,232,125)``\n• #20f17c\n``#20f17c` `rgb(32,241,124)``\n• #17fa7b\n``#17fa7b` `rgb(23,250,123)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #20f17c is perceived by people affected by a color vision deficiency. This can be useful if you need to ensure your color combinations are accessible to color-blind users.\n\nMonochromacy\n• Achromatopsia 0.005% of the population\n• Atypical Achromatopsia 0.001% of the population\nDichromacy\n• Protanopia 1% of men\n• Deuteranopia 1% of men\n• Tritanopia 0.001% of the population\nTrichromacy\n• Protanomaly 1% of men, 0.01% of women\n• Deuteranomaly 6% of men, 0.4% of women\n• Tritanomaly 0.01% of the population" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5626264,"math_prob":0.6882254,"size":3720,"snap":"2020-24-2020-29","text_gpt3_token_len":1660,"char_repetition_ratio":0.12109795,"word_repetition_ratio":0.011049724,"special_character_ratio":0.55698925,"punctuation_ratio":0.23756906,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98456806,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-06-04T11:35:16Z\",\"WARC-Record-ID\":\"<urn:uuid:f232c5b2-6835-4316-886e-1d7c512d2291>\",\"Content-Length\":\"36320\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7bf7d170-7d58-49bc-91cb-d7136f8cacb3>\",\"WARC-Concurrent-To\":\"<urn:uuid:5441f9aa-231c-40c8-aa28-ca21d2730f4d>\",\"WARC-IP-Address\":\"178.32.117.56\",\"WARC-Target-URI\":\"https://www.colorhexa.com/20f17c\",\"WARC-Payload-Digest\":\"sha1:LPJNYP2HAIUJK26R6FZ3TVQQZ5GR5UIP\",\"WARC-Block-Digest\":\"sha1:P6DQ2N467NKDE7MNVNMZXFBFSZ5FM3HA\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-24/CC-MAIN-2020-24_segments_1590347439928.61_warc_CC-MAIN-20200604094848-20200604124848-00367.warc.gz\"}"}
https://www.ademcetinkaya.com/2023/02/next-nextdecade-corporation-common-stock.html	[ "Outlook: NextDecade Corporation Common Stock is assigned short-term Ba1 & long-term Ba1 estimated rating.\nDominant Strategy : Hold\nTime series to forecast n: 25 Feb 2023 for (n+1 year)\nMethodology : Modular Neural Network (CNN Layer)\n\n## Abstract\n\nNextDecade Corporation Common Stock prediction model is evaluated with Modular Neural Network (CNN Layer) and Sign Test1,2,3,4 and it is concluded that the NEXT stock is predictable in the short/long term. According to price forecasts for (n+1 year) period, the dominant strategy among neural network is: Hold\n\n## Key Points\n\n1. Short/Long Term Stocks\n2. Stock Forecast Based On a Predictive Algorithm\n3. Why do we need predictive models?\n\n## NEXT Target Price Prediction Modeling Methodology\n\nWe consider NextDecade Corporation Common Stock Decision Process with Modular Neural Network (CNN Layer) where A is the set of discrete actions of NEXT stock holders, F is the set of discrete states, P : S × F × S → R is the transition probability distribution, R : S × F → R is the reaction function, and γ ∈ [0, 1] is a move factor for expectation.1,2,3,4\n\nF(Sign Test)5,6,7= $\\begin{array}{cccc}{p}_{a1}& {p}_{a2}& \\dots & {p}_{1n}\\\\ & ⋮\\\\ {p}_{j1}& {p}_{j2}& \\dots & {p}_{jn}\\\\ & ⋮\\\\ {p}_{k1}& {p}_{k2}& \\dots & {p}_{kn}\\\\ & ⋮\\\\ {p}_{n1}& {p}_{n2}& \\dots & {p}_{nn}\\end{array}$ X R(Modular Neural Network (CNN Layer)) X S(n):→ (n+1 year) $∑ i = 1 n a i$\n\nn:Time series to forecast\n\np:Price signals of NEXT stock\n\nj:Nash equilibria (Neural Network)\n\nk:Dominated move\n\na:Best response for target price\n\nFor further technical information as per how our model work we invite you to visit the article below:\n\nHow do AC Investment Research machine learning (predictive) algorithms actually work?\n\n## NEXT Stock Forecast (Buy or Sell) for (n+1 year)\n\nSample Set: Neural Network\nStock/Index: NEXT NextDecade Corporation Common Stock\nTime series to forecast n: 25 Feb 2023 for (n+1 year)\n\nAccording to price forecasts for (n+1 year) period, the dominant strategy among neural network is: Hold\n\nX axis: Likelihood% (The higher the percentage value, the more likely the event will occur.)\n\nY axis: Potential Impact% (The higher the percentage value, the more likely the price will deviate.)\n\nZ axis (Grey to Black): Technical Analysis%\n\n1. An entity shall apply the impairment requirements in Section 5.5 retrospectively in accordance with IAS 8 subject to paragraphs 7.2.15 and 7.2.18–7.2.20.\n2. For the purposes of applying the requirements in paragraphs 5.7.7 and 5.7.8, an accounting mismatch is not caused solely by the measurement method that an entity uses to determine the effects of changes in a liability's credit risk. An accounting mismatch in profit or loss would arise only when the effects of changes in the liability's credit risk (as defined in IFRS 7) are expected to be offset by changes in the fair value of another financial instrument. A mismatch that arises solely as a result of the measurement method (ie because an entity does not isolate changes in a liability's credit risk from some other changes in its fair value) does not affect the determination required by paragraphs 5.7.7 and 5.7.8. For example, an entity may not isolate changes in a liability's credit risk from changes in liquidity risk. If the entity presents the combined effect of both factors in other comprehensive income, a mismatch may occur because changes in liquidity risk may be included in the fair value measurement of the entity's financial assets and the entire fair value change of those assets is presented in profit or loss. However, such a mismatch is caused by measurement imprecision, not the offsetting relationship described in paragraph B5.7.6 and, therefore, does not affect the determination required by paragraphs 5.7.7 and 5.7.8.\n3. An entity is not required to restate prior periods to reflect the application of these amendments. The entity may restate prior periods if, and only if, it is possible without the use of hindsight and the restated financial statements reflect all the requirements in this Standard. If an entity does not restate prior periods, the entity shall recognise any difference between the previous carrying amount and the carrying amount at the beginning of the annual reporting period that includes the date of initial application of these amendments in the opening retained earnings (or other component of equity, as appropriate) of the annual reporting period that includes the date of initial application of these amendments.\n4. For hedges other than hedges of foreign currency risk, when an entity designates a non-derivative financial asset or a non-derivative financial liability measured at fair value through profit or loss as a hedging instrument, it may only designate the non-derivative financial instrument in its entirety or a proportion of it.\n\nInternational Financial Reporting Standards (IFRS) adjustment process involves reviewing the company's financial statements and identifying any differences between the company's current accounting practices and the requirements of the IFRS. If there are any such differences, neural network makes adjustments to financial statements to bring them into compliance with the IFRS.\n\n## Conclusions\n\nNextDecade Corporation Common Stock is assigned short-term Ba1 & long-term Ba1 estimated rating. NextDecade Corporation Common Stock prediction model is evaluated with Modular Neural Network (CNN Layer) and Sign Test1,2,3,4 and it is concluded that the NEXT stock is predictable in the short/long term. According to price forecasts for (n+1 year) period, the dominant strategy among neural network is: Hold\n\n### NEXT NextDecade Corporation Common Stock Financial Analysis\n\nRating Short-Term Long-Term Senior\nOutlookBa1Ba1\nIncome StatementB1Caa2\nBalance SheetBa2C\nLeverage RatiosBaa2Ba2\nCash FlowCaa2C\nRates of Return and ProfitabilityBa1B2\n\n*Financial analysis is the process of evaluating a company's financial performance and position by neural network. It involves reviewing the company's financial statements, including the balance sheet, income statement, and cash flow statement, as well as other financial reports and documents.\nHow does neural network examine financial reports and understand financial state of the company?\n\n### Prediction Confidence Score\n\nTrust metric by Neural Network: 90 out of 100 with 863 signals.", null, "## References\n\n1. L. Busoniu, R. Babuska, and B. D. Schutter. A comprehensive survey of multiagent reinforcement learning. IEEE Transactions of Systems, Man, and Cybernetics Part C: Applications and Reviews, 38(2), 2008.\n2. R. Rockafellar and S. Uryasev. Conditional value-at-risk for general loss distributions. Journal of Banking and Finance, 26(7):1443 – 1471, 2002\n3. Bertsimas D, King A, Mazumder R. 2016. Best subset selection via a modern optimization lens. Ann. Stat. 44:813–52\n4. Mikolov T, Sutskever I, Chen K, Corrado GS, Dean J. 2013b. Distributed representations of words and phrases and their compositionality. In Advances in Neural Information Processing Systems, Vol. 26, ed. Z Ghahramani, M Welling, C Cortes, ND Lawrence, KQ Weinberger, pp. 3111–19. San Diego, CA: Neural Inf. Process. Syst. Found.\n5. Çetinkaya, A., Zhang, Y.Z., Hao, Y.M. and Ma, X.Y., What are buy sell or hold recommendations?(AIRC Stock Forecast). AC Investment Research Journal, 101(3).\n6. J. Baxter and P. Bartlett. Infinite-horizon policy-gradient estimation. Journal of Artificial Intelligence Re- search, 15:319–350, 2001.\n7. A. Eck, L. Soh, S. Devlin, and D. Kudenko. Potential-based reward shaping for finite horizon online POMDP planning. Autonomous Agents and Multi-Agent Systems, 30(3):403–445, 2016\nFrequently Asked QuestionsQ: What is the prediction methodology for NEXT stock?\nA: NEXT stock prediction methodology: We evaluate the prediction models Modular Neural Network (CNN Layer) and Sign Test\nQ: Is NEXT stock a buy or sell?\nA: The dominant strategy among neural network is to Hold NEXT Stock.\nQ: Is NextDecade Corporation Common Stock stock a good investment?\nA: The consensus rating for NextDecade Corporation Common Stock is Hold and is assigned short-term Ba1 & long-term Ba1 estimated rating.\nQ: What is the consensus rating of NEXT stock?\nA: The consensus rating for NEXT is Hold.\nQ: What is the prediction period for NEXT stock?\nA: The prediction period for NEXT is (n+1 year)" ]	[ null, "https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEie6zmojlVkFtzhnSDPsL0ofN0Qf8imsRuJLmBsuPhvw7a_V8sO4akz1ZmrC1z138iTEnVBz4WOe7nRaUku8PCVLHIr3blhvleBHEbt1VcY4D4zSFZNC02CJ-SsnJxURqjoZxkZaeFVd_lKWZF2n-hK-At4y-ts0RYd4tsXgAkh6yfcsMyVn7NLarhmsw/s1600/footerlogo.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.87551004,"math_prob":0.8298587,"size":8115,"snap":"2023-40-2023-50","text_gpt3_token_len":1890,"char_repetition_ratio":0.11157687,"word_repetition_ratio":0.1561753,"special_character_ratio":0.22205792,"punctuation_ratio":0.14136808,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9562421,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-30T22:43:13Z\",\"WARC-Record-ID\":\"<urn:uuid:6241a429-7b97-42b2-80e8-4a9a8d49dae2>\",\"Content-Length\":\"314599\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c2be240d-4847-4153-a915-f3dde0ccd115>\",\"WARC-Concurrent-To\":\"<urn:uuid:655bed94-9f6a-4953-a784-2b5fc00b447d>\",\"WARC-IP-Address\":\"172.253.63.121\",\"WARC-Target-URI\":\"https://www.ademcetinkaya.com/2023/02/next-nextdecade-corporation-common-stock.html\",\"WARC-Payload-Digest\":\"sha1:M63245FXO3GYWBZF4N6I75EVGIB5JRXK\",\"WARC-Block-Digest\":\"sha1:CAPXNMJGK7TMOM7TRQNGUGKN27V7PG7G\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233510730.6_warc_CC-MAIN-20230930213821-20231001003821-00754.warc.gz\"}"}
https://mathexamination.com/class/wild-knots.php	[ "## Do My Wild Knots Class", null, "A \"Wild Knots Class\" QE\" is a standard mathematical term for a generalized continuous expression which is utilized to fix differential formulas and has options which are routine. In differential Class resolving, a Wild Knots function, or \"quad\" is utilized.\n\nThe Wild Knots Class in Class type can be expressed as: Q( x) = -kx2, where Q( x) are the Wild Knots Class and it is an important term. The q part of the Class is the Wild Knots consistent, whereas the x part is the Wild Knots function.\n\nThere are 4 Wild Knots functions with correct service: K4, K7, K3, and L4. We will now take a look at these Wild Knots functions and how they are resolved.\n\nK4 - The K part of a Wild Knots Class is the Wild Knots function. This Wild Knots function can also be written in partial portions such as: (x2 - y2)/( x+ y). To fix for K4 we multiply it by the appropriate Wild Knots function: k( x) = x2, y2, or x-y.\n\nK7 - The K7 Wild Knots Class has a solution of the form: x4y2 - y4x3 = 0. The Wild Knots function is then multiplied by x to get: x2 + y2 = 0. We then need to multiply the Wild Knots function with k to get: k( x) = x2 and y2.\n\nK3 - The Wild Knots function Class is K3 + K2 = 0. We then multiply by k for K3.\n\nK3( t) - The Wild Knots function equationis K3( t) + K2( t). We multiply by k for K3( t). Now we multiply by the Wild Knots function which provides: K2( t) = K( t) times k.\n\nThe Wild Knots function is also known as \"K4\" because of the initials of the letters K and 4. K implies Wild Knots, and the word \"quad\" is pronounced as \"kah-rab\".\n\nThe Wild Knots Class is one of the primary methods of resolving differential equations. In the Wild Knots function Class, the Wild Knots function is first increased by the suitable Wild Knots function, which will give the Wild Knots function.\n\nThe Wild Knots function is then divided by the Wild Knots function which will divide the Wild Knots function into a real part and a fictional part. This offers the Wild Knots term.\n\nLastly, the Wild Knots term will be divided by the numerator and the denominator to get the ratio. We are entrusted to the right-hand man side and the term \"q\".\n\nThe Wild Knots Class is a crucial concept to comprehend when resolving a differential Class. The Wild Knots function is simply one method to solve a Wild Knots Class. The techniques for resolving Wild Knots formulas consist of: singular value decomposition, factorization, optimum algorithm, numerical option or the Wild Knots function approximation.\n\n## Pay Me To Do Your Wild Knots Class\n\nIf you want to become acquainted with the Quartic Class, then you require to first start by browsing the online Quartic page. This page will reveal you how to utilize the Class by utilizing your keyboard. The description will also show you how to develop your own algebra formulas to help you study for your classes.\n\nPrior to you can comprehend how to study for a Wild Knots Class, you must initially comprehend making use of your keyboard. You will discover how to click on the function keys on your keyboard, along with how to type the letters. There are three rows of function keys on your keyboard. Each row has 4 functions: Alt, F1, F2, and F3.\n\nBy pressing Alt and F2, you can increase and divide the value by another number, such as the number 6. By pushing Alt and F3, you can utilize the 3rd power.\n\nWhen you push Alt and F3, you will enter the number you are attempting to multiply and divide. To multiply a number by itself, you will push Alt and X, where X is the number you wish to increase. When you press Alt and F3, you will key in the number you are attempting to divide.\n\nThis works the very same with the number 6, other than you will only enter the two digits that are six apart. Lastly, when you push Alt and F3, you will use the 4th power. Nevertheless, when you push Alt and F4, you will use the actual power that you have actually discovered to be the most suitable for your problem.\n\nBy utilizing the Alt and F function keys, you can multiply, divide, and after that use the formula for the 3rd power. If you require to multiply an odd number of x's, then you will need to go into an even number.\n\nThis is not the case if you are trying to do something complex, such as multiplying 2 even numbers. For example, if you wish to increase an odd variety of x's, then you will require to go into odd numbers. This is particularly real if you are trying to find out the response of a Wild Knots Class.\n\nIf you want to transform an odd number into an even number, then you will need to press Alt and F4. If you do not know how to increase by numbers by themselves, then you will need to utilize the letters x, a b, c, and d.\n\nWhile you can multiply and divide by use of the numbers, they are a lot easier to utilize when you can look at the power tables for the numbers. You will need to do some research study when you initially begin to use the numbers, however after a while, it will be force of habit. After you have developed your own algebra equations, you will be able to create your own reproduction tables.\n\nThe Wild Knots Solution is not the only method to solve Wild Knots equations. It is necessary to find out about trigonometry, which utilizes the Pythagorean theorem, and then use Wild Knots solutions to resolve issues. With this method, you can learn about angles and how to fix issues without needing to take another algebra class.\n\nIt is essential to try and type as rapidly as possible, due to the fact that typing will assist you learn about the speed you are typing. This will help you write your answers quicker.\n\n## Pay Someone To Take My Wild Knots Class", null, "A Wild Knots Class is a generalization of a direct Class. For example, when you plug in x=a+b for a given Class, you get the value of x. When you plug in x=a for the Class y=c, you get the worths of x and y, which give you an outcome of c. By using this standard idea to all the equations that we have tried, we can now resolve Wild Knots formulas for all the values of x, and we can do it quickly and efficiently.\n\nThere are numerous online resources offered that provide free or budget-friendly Wild Knots equations to solve for all the worths of x, including the expense of time for you to be able to make the most of their Wild Knots Class task assistance service. These resources usually do not need a subscription fee or any type of investment.\n\nThe answers provided are the outcome of complex-variable Wild Knots formulas that have actually been resolved. This is likewise the case when the variable utilized is an unknown number.\n\nThe Wild Knots Class is a term that is an extension of a direct Class. One benefit of using Wild Knots equations is that they are more general than the linear equations. They are easier to fix for all the values of x.\n\nWhen the variable utilized in the Wild Knots Class is of the kind x=a+b, it is simpler to fix the Wild Knots Class due to the fact that there are no unknowns. As a result, there are fewer points on the line defined by x and a constant variable.\n\nFor a right-angle triangle whose base points to the right and whose hypotenuse indicate the left, the right-angle tangent and curve graph will form a Wild Knots Class. This Class has one unknown that can be found with the Wild Knots formula. For a Wild Knots Class, the point on the line specified by the x variable and a continuous term are called the axis.\n\nThe presence of such an axis is called the vertex. Given that the axis, vertex, and tangent, in a Wild Knots Class, are a given, we can find all the values of x and they will sum to the offered worths. This is accomplished when we utilize the Wild Knots formula.\n\nThe factor of being a constant factor is called the system of equations in Wild Knots equations. This is in some cases called the central Class.\n\nWild Knots formulas can be resolved for other values of x. One way to solve Wild Knots equations for other worths of x is to divide the x variable into its element part.\n\nIf the variable is given as a positive number, it can be divided into its element parts to get the typical part of the variable. This variable has a magnitude that amounts to the part of the x variable that is a constant. In such a case, the formula is a third-order Wild Knots Class.\n\nIf the variable x is negative, it can be divided into the exact same part of the x variable to get the part of the x variable that is multiplied by the denominator. In such a case, the formula is a second-order Wild Knots Class.\n\nOption assistance service in resolving Wild Knots formulas. When utilizing an online service for solving Wild Knots formulas, the Class will be fixed instantly." ]	[ null, "https://mathexamination.com/Do-My-Math-Class.webp", null, "https://mathexamination.com/Take-My-Math-Class.webp", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9268814,"math_prob":0.97627926,"size":8671,"snap":"2021-21-2021-25","text_gpt3_token_len":2004,"char_repetition_ratio":0.1997231,"word_repetition_ratio":0.04227941,"special_character_ratio":0.22304232,"punctuation_ratio":0.09568584,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9936359,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-05-10T04:50:09Z\",\"WARC-Record-ID\":\"<urn:uuid:bf140a84-12ea-4c98-9dac-04167e3a7616>\",\"Content-Length\":\"19340\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:3d9ff18d-1bba-4a58-a982-17a19d908cb3>\",\"WARC-Concurrent-To\":\"<urn:uuid:acefef5d-fc5d-4ff4-b6be-66b72229b782>\",\"WARC-IP-Address\":\"104.21.56.63\",\"WARC-Target-URI\":\"https://mathexamination.com/class/wild-knots.php\",\"WARC-Payload-Digest\":\"sha1:G2JGRDTGWEGEZLHII67LGXHF5MHQDTY3\",\"WARC-Block-Digest\":\"sha1:Y5MGEJJYHYGRQBJDRGX6UAVAXQWKNXIN\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-21/CC-MAIN-2021-21_segments_1620243989030.87_warc_CC-MAIN-20210510033850-20210510063850-00172.warc.gz\"}"}
http://yhkn.net/read/38-b5-c4-b3-cb-b7-a8-bf-da-be-f7-bd-cc-d1-a7-ca-b5-c2-bc.html	[ "# 8的乘法口诀教学实录\n\n(1) 看口诀,明意义.在让孩子们记忆口诀时,要让他们明确口诀的由来,知道乘法的基本意义,要让他们知其然,而且要知其所以然,绝对不能仅把口诀当成顺口溜,盲目背诵.这样,在孩子们实在记不清某句口诀时,可以用最原始的方法推\n\n8的乘法口诀》教学反思: 8的乘法口诀这堂课的教学目标是:让学生经历推导 8的乘法口诀的过程,掌握8的乘法口诀和用相应的口诀计算乘法的方法,提高解决问题的能力.教学刚开始我创设了蓝猫进课堂的情境,通过学生喜欢的蓝猫跳一跳\n\n8的乘法口诀 1个8 18 = 8 81 = 8 一八得八 2个8 28 = 8 82 = 8 二八十六 3个8 38 = 8 83 = 8 三八二十四 4个8 48 = 8 84 = 8 四八三十二 5个8 58 = 8 85 = 8 五八四十 6个8 68 = 8 86 = 8 六八四十八 7个8 78 = 8 87 = 8 七八五十六 8个8 88 = 8 88 = 8 八八六十四好评哦哦" ]	[ null ]	{"ft_lang_label":"__label__zh","ft_lang_prob":0.89156073,"math_prob":0.99933887,"size":891,"snap":"2020-45-2020-50","text_gpt3_token_len":1033,"char_repetition_ratio":0.11161218,"word_repetition_ratio":0.0,"special_character_ratio":0.30078563,"punctuation_ratio":0.27027026,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9597796,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-11-26T08:54:02Z\",\"WARC-Record-ID\":\"<urn:uuid:4309089d-1484-44e8-9184-a0b493f52a5c>\",\"Content-Length\":\"10045\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2f133d54-a144-4ae9-b985-d3fb5b25ef41>\",\"WARC-Concurrent-To\":\"<urn:uuid:be74b492-cfec-4953-959e-5fcfa40fb32d>\",\"WARC-IP-Address\":\"121.127.228.6\",\"WARC-Target-URI\":\"http://yhkn.net/read/38-b5-c4-b3-cb-b7-a8-bf-da-be-f7-bd-cc-d1-a7-ca-b5-c2-bc.html\",\"WARC-Payload-Digest\":\"sha1:BU7RLUNJP6OK2OCLF34HOWD57CNH4L35\",\"WARC-Block-Digest\":\"sha1:EEQ7NVK55Z4OMDPTYQVFQBM5W2R36ZEP\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-50/CC-MAIN-2020-50_segments_1606141187753.32_warc_CC-MAIN-20201126084625-20201126114625-00245.warc.gz\"}"}
https://ecosystems.psu.edu/research/labs/walter-lab/manual/chapter-2/2-6-importing-dynamically-downscaled-global-climate-data	[ "This exercise will provide some code for manipulating climate change data from the Regional Climate Downscaling by copy the link into your browser:\nhttp://regclim.coas.oregonstate.edu/data-access/index.html or just select the link here: Regional Climate Downscaling. IMPORTANT: For each climate projection, must change name in first command and file name in last command.\n\n2. Set working directory to the extracted folder in R under File - Change dir...\n3. First we need to load the packages needed for the exercise\nlibrary(ncdf)\n4. Now open the script \"NetCDF_Script.R\" and run code directly from the script\n5. Open netCDF and setting verbose=true provides details about the data in the netcdf file including the varid. You need to know the varid to select the variable you want to extract/summarize. Note: the dimensions x, y, time also get a varid so you will need to subtract 3 from the varid of interest to get the correct one.\n\ndat <- open.ncdf(\"Monthly_AvgMinTemp_1995-99_MPI.nc\", write=TRUE, readunlim=TRUE, verbose=TRUE)\n\ntmin #object\ntmin <- dat\\$var[]\ntmin\n\n#####################################\n#The following illustrates how to read the data\n#####################################\nprint(paste(tmin\\$name)) #in this case the 'field name' is TAMIN\n\n# Grab data for TAMIN variable and place in object df1\ndf1 <- get.var.ncdf(dat, tmin)\n#x is the object, n sets the number of entries displayed. tail returns the last of the data #entries.\n\n#Dimensions of df1 (x, y, time)\ndim(df1)\n\n#Dimensions can also be examined one at a time\ndim(df1) # number of x grids (36)\ndim(df1) # number of y grids (21)\ndim(df1) # number of months in file (49)\n#NOTE: FILE INCLUDES MONTHS OTHER THAN JANUARY (Jans are 1,13,25,37,49)\n\n#Check first element\ndf1[1,1,1]\n\n#Check first January for all x,y\ndf1[,,1]\n\n#Create a new matrix which is monthly averages for each grid cell. Make the new matix #the same size (i.e. same number of rows and columns as there are in the #dataframe df1\nsum1 <- array(data=NA, c(dim(df1),dim(df1) ))\ndim(sum1)\n\n#Create January mean TAMIN for each x-y coordinate\nfor(i in 1:dim(df1)){ # loop over x-coords\nfor(j in 1:dim(df1)){ # loop over y-coords\nsum1[i, j] <- (df1[i,j,1]+df1[i,j,13]+df1[i,j,25]+df1[i,j,37]+df1[i,j,49])/5\n}\n}\n\n#head(sum1) ## useful for large files\nsum1\n\n###########################################################\n\n#Create netcdf file from sum1 (contains matrix of new data)\n\n###########################################################\n\n#Get x and y coordinates from original \"dat\" ncdf file\n\nx = get.var.ncdf(nc=dat,var )\n\ny = get.var.ncdf(nc=dat,var )\n\n#Check dimensions\n\nlength(x)\n\nlength(y)\n\ndim(sum1)\n\n#Define the netcdf coordinate variables - note that these are coming\n\n#from the dat file with actual values\n\ndim1 = dim.def.ncdf( \"X\",\"meters\", as.double(x))\n\ndim2 = dim.def.ncdf( \"Y\",\"meters\", as.double(y))\n\n#Define the EMPTY (climate) netcdf variable and define names that will be used in the #var.def.ncdf function.\n\n#Define climate variable names\n\nnew.name <- 'mintemp'\n\n#Define units of measurement for variable\n\nunits <- 'degreesC'\n\n#Define long name for variable\n\nlong.name <- 'Jan average min temperature'\n\nvarz = var.def.ncdf(new.name,units, list(dim1,dim2), -1, longname=long.name)\n\n#Associate the netcdf variable with a netcdf file, put the variable into the file, and close\n\nnc.ex = create.ncdf( \"MPI1999-95.nc\", varz )\n\nput.var.ncdf(nc.ex, varz, sum1)\n\nclose.ncdf(nc.ex)" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7149275,"math_prob":0.9405156,"size":3694,"snap":"2023-40-2023-50","text_gpt3_token_len":1002,"char_repetition_ratio":0.14823848,"word_repetition_ratio":0.0,"special_character_ratio":0.32024905,"punctuation_ratio":0.15436241,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9954934,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-05T11:20:53Z\",\"WARC-Record-ID\":\"<urn:uuid:a13dcade-f8b7-4c35-bd9e-aa7f0b632a0b>\",\"Content-Length\":\"125392\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4b7cf769-e9f4-4332-9381-861c2206c8b4>\",\"WARC-Concurrent-To\":\"<urn:uuid:1b033638-6320-4c97-9d16-4d2deea66dc8>\",\"WARC-IP-Address\":\"128.118.122.226\",\"WARC-Target-URI\":\"https://ecosystems.psu.edu/research/labs/walter-lab/manual/chapter-2/2-6-importing-dynamically-downscaled-global-climate-data\",\"WARC-Payload-Digest\":\"sha1:ABHVZNIXRSTDW3LJMURUY46BEIG7YX5A\",\"WARC-Block-Digest\":\"sha1:NOBLN2YFVV322YEQSN22ZEEQ7AKIIXIV\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100551.17_warc_CC-MAIN-20231205105136-20231205135136-00766.warc.gz\"}"}
https://file.scirp.org/Html/86525_86525.htm	[ " Optimization of Fairhurst-Cook Model for 2-D Wing Cracks Using Ant Colony Optimization (ACO), Particle Swarm Intelligence (PSO), and Genetic Algorithm (GA)\n\nJournal of Applied Mathematics and Physics\nVol.06 No.08(2018), Article ID:86525,15 pages\n10.4236/jamp.2018.68134\n\nOptimization of Fairhurst-Cook Model for 2-D Wing Cracks Using Ant Colony Optimization (ACO), Particle Swarm Intelligence (PSO), and Genetic Algorithm (GA)\n\nMohammad Najjarpour, Hossein Jalalifar\n\nDepartment of Oil and Gas Engineering, Shahid Bahonar University of Kerman, Kerman, Iran", null, "", null, "", null, "", null, "Received: April 26, 2018; Accepted: August 6, 2018; Published: August 9, 2018\n\nABSTRACT\n\nThe common failure mechanism for brittle rocks is known to be axial splitting which happens parallel to the direction of maximum compression. One of the mechanisms proposed for modelling of axial splitting is the sliding crack or so called, “wing crack” model. Fairhurst-Cook model explains this specific type of failure which starts by a pre-crack and finally breaks the rock by propagating 2-D cracks under uniaxial compression. In this paper, optimization of this model has been considered and the process has been done by a complete sensitivity analysis on the main parameters of the model and excluding the trends of their changes and also their limits and “peak points”. Later on this paper, three artificial intelligence algorithms including Particle Swarm Intelligence (PSO), Ant Colony Optimization (ACO) and genetic algorithm (GA) has been used and compared in order to achieve optimized sets of parameters resulting in near-maximum or near-minimum amounts of wedging forces creating a wing crack.\n\nKeywords:\n\nWing Crack, Fairhorst-Cook Model, Sensitivity Analysis, Optimization, Particle Swarm Intelligence (PSO), Ant Colony Optimization (ACO), Genetic Algorithm (GA)", null, "1. Introduction\n\nThe study of rock failure under compression is a matter of importance in rock mechanics. Two main mechanisms, including axial splitting and shear failure, have been identified for this type of rock failure . Failure by splitting mechanism can be related to the spalling phenomenon . This phenomenon always occurs at high stress contact points, for example in a ball bearing. Spalling occurs where the maximum shear stress occurs below the rock surface.\n\nSplitting parallel to the direction of maximum compression is a common type of macroscopic fracture of a brittle rock in the vicinity of its surface. Modelling of brittle failure is one of the greatest challenges of material failure analysis which results from the irreversible and very rapid propagation and connection of cracks, in a process called fracturing.\n\nMaterials like natural ice and rock are heterogeneous and crystalline with different behaviors under variation of applying forces. Generally, their behavior can be categorized in two main types, including ductile behavior under high confining pressures, and brittle behavior under low confining pressures . Because most rocks are brittle at low temperature and low confining pressure, virtually most of the rocks at or near the Earth’s surface exhibits brittle failure mechanism. In other words, failure in these rocks occurs as deformation-induced loss of cohesion .\n\nOne of the main mechanisms proposed for modelling of axial splitting is the sliding crack or “wing crack” model which was originally proposed for a 2D crack in a plate . Splitting failure begins with a primary crack (pre-crack) and then sliding mechanism creates secondary cracks or so called, wings, at the edges of those primary cracks. The macroscopic failure occur when a series of cracks extend and link together to split the material. The wedged crack can be modelled as a straight representative crack, where the wedging forces on the pre-crack area are opening the crack. This is often called the Fairhurst-Cook model . The whole process of creating a 2D wing crack with Fairhurst-Cook model and an optimization process on this model is done by Dyskin et al. (1994). The basis of this optimization was the crack semi-length (l) and a critical point for this parameter found in this study which results in an unstable mode for the crack . Another Optimization on the Fairhurst-Cook model is the purpose of this paper which is the optimization of wedging force based on the crack angle to find its critical points.\n\n2. Ultimate Rock Strength\n\nThe ultimate rock strength and the orientation of the macroscopic failure plane depend on the confining pressure applying on the rock. Some models for micro-cracking under compression (e.g. and ) have been proposed based on the fact that frictional sliding on pre-existing cracks results in the formation of tension cracks at their tips. It has been shown that under triaxial compression, the micro-crack distribution is almost uniform until the applied stress reaches the ultimate strength, and at the ultimate strength a region of high crack-density (tension cracks) emerges along a plane which eventually becomes a macroscopic shear failure plane . Variations of the “ultimate strength” and the orientation of shear failure with confining pressure using this model are already known and presented by Nemat-Nasser and Horii (1984) .\n\nAs discussed before, under uniaxial compression, the tension cracks grow at a sharp angle relative to the orientation of overall failure plane with confining pressure. Variation of this “crack angle” changes the amount of wedging force with specific but different trends, resulting different amounts of opening in wing cracks and “ultimate strengths” for rocks to fail under compression. Excluding these trends is another purpose of this paper.\n\n3. Fairhurst-Cook Model\n\nAs it can be seen in Figure 1, under uniaxial compression, crack opens in the shape of a wing with a specific crack angle (α) and also a specific crack amplitude (a) (This parameter is also called “half-length” or “semi-length” in other studies; but here it is called “crack amplitude” which is more proper and differentiates it from the other “semi-length” or “l”).\n\nThe amount of this wedging force (f) which creates the wing crack can be determined from the following equation:\n\nf = 2.a.σ.β(α) (1)\n\nwhere “σ” represents the maximum principal stress, “a” represents the crack amplitude, “f” represents the wedging force and “β(α)” is a function of crack angle which can be determined from the following equation:\n\nβ(α) = sin2(α).cos(α).(1-tan(α).tan(μ)) (2)\n\nIn the above equation, “μ” represents the internal frictional angle of the rock.\n\n4. Sensitivity Analysis\n\nIn this study, a complete sensitivity analysis performed to find the optimized amounts of parameters in Fairhurst-Cook Model. Three main parameters\n\nFigure 1. Schematic of 2-D wing crack growth according to Fairhurst-Cook Model .\n\nincluding crack amplitude (a), crack angle (α) and maximum principal stress (σ) and a fixed amount of 89 degrees for internal friction angle (μ) have been considered in this sensitivity analysis. According to this model, these parameters have the main roles in determining the amount of wedging force which creates a two-dimensional wing crack, while the internal friction angle (μ) should be determined according to the rock sample which is being cracked.\n\nThree different stages have been considered in the sensitivity analysis process. In the first stage of this process, crack amplitude analyzed and this process repeated for the second and third stages by analyzing crack angle (α) and maximum principal stress (σ). Performing this sensitivity analysis in Matlab environment, results in Figure 2 as the output.\n\nAs it can be seen in Figure 2, it includes two different parts which are differentiated in a specific point. By focusing on this figure, the trend of changing the wedging force creating wing crack by variation of crack angle in different crack amplitudes can be excluded. This process is shown in Figure 3.\n\nAs it is shown in Figure 3, trends change in a certain point (α = 60.7˚). Before this point, in any crack angle, the amount of wedging force increases by increasing the crack amplitude, but this trend changes after this point (α = 60.7˚) and by increasing the crack amplitude, wedging force will decrease. This amount of crack angle (α = 60.7˚) virtually acts like a border line and specifies the maximum amount of crack angle which causes the increase of wedging force. In other words, this specific crack angle (α = 60.7˚) has considered to be the “peak point” for crack angle in analyzing the change of wedging force by variation of crack amplitude.\n\nFigure 2. Change of wedging force (f) by variation of crack angle (α) in different crack amplitudes (a) by σ = 60 (Mpa).\n\nAnother fact which can be excluded from Figure 1 is the reversion of wedging force sign from positive to negative. The reason of this conversion is thought to be the conversion of applying wedging force from tensile to compression as a result of increasing the crack angle and decreasing the angle between force direction and its applying area, resulting in closing of the crack lips, instead of opening. In is thought that the reason for the variation of trends is also the same, indicating that the bigger wedging force still stays bigger after peak point angle, but with a different type of force and in an apposite way, applying compression force instead of tensile and closing the crack instead of opening it. This fact highlights the importance of finding the critical angle or so called, the “peak point”.\n\nIn the next stage of analyzing, the change of wedging force creating wing crack by variation of crack angle has been examined. For this purpose, one of the curves in Figure 2 has been chosen (a = 11 mm) and by focusing on it, the trend of changing the wedging force by variation of crack angle has been excluded. As it can be seen in Figure 4, this trend changes in a specific point (α = 37.7˚). Before this point, by fixing the amount of crack amplitude (a) and maximum principal stress (σ), the amount of wedging force increases as the amount of crack angle increases. In other words, there is a straight proportion between these parameters. After that specific point (α = 37.7˚) which is considered to be the “peak point” in variation of wedging force by changing the crack angle, that trend changes and in a reverse relation, the amount of wedging force decreases as crack angle increases. Just like the first stage, this specific amount of crack\n\nFigure 3. Different trends of changing the wedging force (f) by variation of crack amplitude (a).\n\nFigure 4. Different trends of changing the wedging force by variation of crack angle.\n\nangle (α = 37.7˚) acts like a border line which limits the increase of wedging force by increasing the amount of the relevant parameter.\n\nFor the third stage of this sensitivity analysis which includes analyzing the wedging force variation by changing the maximum principal stress (σ), three different amounts of maximum principal stress considered in a fixed amount of crack amplitude (a). In this stage, the variation of wedging force by changing the crack angle in different amounts of maximum principal stress has been investigated and by its excluded trends, plotted in Figure 5 and Figure 6.\n\nAs it can been seen in Figure 6, trend of changing the wedging force by changing the crack angle in different amounts of maximum principal stress changes in a specific amount of crack angle (α = 60.7˚), like the previous stages. This point which considered to be the “peak point”, differentiates the trends and returns it from straight relation to the reverse. Just like previous stages. This peak point limits the increase of wedging force by increasing the relevant parameter.\n\nAccording to this sensitivity analysis, optimized amounts of each parameter by fixing other relevant parameters have been excluded, but finding the optimum amounts of parameters without fixing others is much more important. For this reason, using some of the efficient algorithms of artificial intelligence for optimization is suggested. In this research three algorithms including Ant Colony Optimization (ACO), Particle Swarm Intelligence (PSO) and Genetic Algorithm (GA) were used and their results have been compared in order to find the most efficient method in general or separate parts (before and after the peak\n\nFigure 5. Variation of the wedging force (f) by changing maximum principal stress (σ).\n\nFigure 6. Different trends of changing the wedging force (f) by variation of maximum principal stress (σ).\n\npoint angle) and also to achieve the more reliable results.\n\n5. Ant Colony Optimization (ACO)\n\nAnt colony optimization (ACO) is a population-based metaheuristic algorithm which can be used to find approximate solutions to hard and discrete optimization problems. ACO is an algorithm for finding optimal paths that is based on foraging behavior of some ant species searching for food . In this algorithm, a set of software agents called “artificial ants” search for good solutions to a given optimization problem . At first, the ants distribute randomly to start searching for food. When an ant finds a food source, it walks back to the colony depositing pheromone on the ground in order to mark its favorable path which should be followed by other ants. In this situation, other ants are more likely to follow that path. As more ants find a path, it gets stronger and this process repeats so many times until there are a couple streams of ants traveling to various food sources near the colony. However, the pheromone trail starts to evaporate over time, thus its attractive strength reduces. The more time it takes for an ant to travel down the path and back again, the more time the pheromones have to evaporate . Because the ants drop pheromones every time they bring food, shorter paths are more likely to be stronger, hence optimizing the “solution”. “Positive Feedback” eventually leads to all the ants following a single path. Ant colony optimization exploits a similar mechanism for solving optimization problems. To apply ACO, the optimization problem is transformed into the problem of finding the best path on a weighted graph. The artificial ants incrementally build solutions by moving on the graph. The solution construction process is stochastic and is biased by a pheromone model, that is, a set of parameters associated with graph components (i.e. nodes) whose values are modified at runtime by the ants. This process is repeated many times until the stopping condition is being satisfied and a satisfactory result is achieved. The most famous application of this algorithm is finding near-optimal solutions to the traveling salesman problem (TSP) .\n\n6. Particle Swarm Intelligence (PSO)\n\nParticle Swarm Intelligence originally formed based on the movement of organisms in animal swarms such as bird flocks or fish schools to simulate their social behavior . A basic version of PSO algorithm starts by having a population (called a swarm) of candidate solutions (called particles). These particles are moved around the search-space to find the optimized solution (called global best solution) . In this process, each particle determines its movement through the search-space by combining some aspects of the history of its own current and best locations with those of other members of the swarm, with some random perturbations. When improved positions discovered, these positions will guide the movements of the swarm. The process is repeated many times trying to find a satisfactory solution and is expected to find a global or near-global optimized answer at the end of all algorithm cycles, although it is not guaranteed.\n\nThis algorithm can be summarized in four main steps, which are repeated until the stopping condition is satisfied:\n\n・ Assigning initially random positions and velocities for all of the particles in the search-space\n\n・ Evaluation of the fitness of each individual particle\n\n・ Updating the individual and global best positions\n\n・ Updating the velocity and position of each particle \n\nIn past several years, PSO Algorithm has been successfully applied in many researches and different application areas. It is demonstrated that PSO algorithm gives better results in a faster, cheaper way compared with other methods .\n\n7. Genetic Algorithm (GA)\n\nGenetic Algorithm (GA) is a heuristic optimizer algorithm based on the evolutionary ideas of natural selection and principles of “survival of the fittest” from “Charles Darwin”. Genetic Algorithms are commonly used to generate proper solutions to optimization problems by relying on bio-inspired operators such as “crossover” and “mutation”.\n\nGA simulates the natural selection process among individuals (also called “phenotypes”), each of which representing a possible solution. Each possible solution has a set of properties and is known as a “chromosome” (also called “genotype”). Representing the solutions as binary strings of 0 s and 1 s are more common, but other types such as “real strings” are also applicable. The initially created individuals then lead through the process of evolution .\n\nStarting with a randomly generated population of chromosomes, the evolution occurs as a process of fitness based selection and recombination to produce a better population in each iteration called a “generation” . This process will be done by defining a proper “fitness function” and improving the initial population through repetitive application of the mutation, crossover and selection operators.\n\nIn each generation, the fitness of every individual in the population which is usually the value of the objective function in the problem, is being evaluated and the GA creates a new population by a new group of chromosomes with resulted fitness values. In this process, firstly, parents are selected to mate, based on their fitness, producing “offspring”, so better solutions with more fitness are given better chance to reproduce by crossover operation. The offspring inherit characteristics from both parents, but not equally. As parents mate and produce offspring, some new rooms must be freed for the newly generated chromosomes.\n\nSince the population contains more information than each individual fitness, GA combines the good information hidden in a solution with good information from another one in the mating pool, in order to produce new solutions with good information inherited from both parents . After the GA mates the new individuals and mutates some of them, the population undergoes a complete generation change. Some of the individuals in the population are replaced by new ones, since the size of the population has to remain fixed and shouldn’t increase. The population will then consist of offspring plus a few of the older individuals, which the genetic algorithm allows to survive to the next generation, because they are the best members of the population, called “elite individuals”. Following this procedure, it is hoped, but not guaranteed, that over many generations, better solutions will remain while the least fit solutions die out.\n\nEach generation will contain, on average, more good genes than the previous one. Once the population is not producing much better solutions than previous generations, the algorithm is said to have converged to a specific set of solutions for the problem. Eventually, the algorithm terminates when either it converged to a proper solution with satisfactory fitness level or a specific number of generations has been produced (Figure 7).\n\n8. Optimization Process\n\nAs it has been discussed earlier, three different optimization algorithms including Ant Colony Optimization (ACO), Particle Swarm Intelligence (PSO) and Genetic Algorithm (GA) have been used in order to find the optimum or near optimum parameters influencing the wedging force which creates a wing crack. This optimization has been performed in several cases including general maximizing and minimizing of the wedging force and also separate maximizing and minimizing of this parameter, isolated in before and after the peak point angle (37.7˚).\n\nThe First algorithm was ACO and it has been performed by a colony of 46 ants trying to find the optimum amounts of crack angle (α), crack amplitude (a), maximum principal stress (σ) and frictional angle (μ) in 100 iterations. Two specific upper and lower limits have been considered for each parameter in order to specify their valid range of variation. This range was from 0 to 120 degrees for\n\nFigure 7. Schematic of Genetic Algorithm.\n\ncrack angle, from 0 to 89.999 degrees for frictional angle, from 0.001 to 0.046 meters for crack amplitude and finally 10 to 100 mega-Pascal for maximum principal stress. The results of this optimization are illustrated in Table 1.\n\nThe second optimization algorithm was Particle Swarm Intelligence (PSO) and it has been performed in 50,000 iterations with 10,000 swarms and 4 particles for each swarm, representing crack angle (α), crack amplitude (a), maximum principal stress (σ) and frictional angle (μ). Just like the previous algorithm, two specific upper and lower limits with the same amounts have been considered for each parameter in order to specify their valid range of variation. The results of this optimization are also illustrated in Table 1.\n\nThe third and final optimization algorithm was Genetic Algorithm (GA). In this study, real genetic algorithm used and it has been performed with 200 chromosomes and 4 genes for each chromosome. These genes represented the related parameters including crack angle (α), crack amplitude (a), maximum principal stress (σ1) and frictional angle (μ) and each chromosome represented a resulting amount of wedging force (f). The optimization process repeated for 4000 iterations with the same upper and lower limits for different parameters and like the previous stages, its results is illustrated in Table 1.\n\nThese steps repeated several times, in order to achieve better results of wedging force without exiting the valid range for each parameter and as it can be seen in this table, Particle Swarm Intelligence (PSO) algorithm achieved better results with more amount of wedging force at the end. So these amounts of parameters will be more proper to use as the optimum amounts and this method is more efficient in general maximizing of wedging force in wing crack model comparing to the ACO and GA algorithms. These steps have been repeated for the general minimization stage with the same ranges for the parameters. The results are shown in Table 2 which indicate that GA is the optimum method in this stage of the optimization.\n\nTable 1. Results of the optimization algorithms in general maximization of the wedging force.\n\nTable 2. Results of the optimization algorithms in general minimization of the wedging force.\n\nAs it can be understood from Table 1 and Table 2, the GA and PSO algorithms both had proper results with near optimum amounts of wedging forces, but ACO didn’t show any acceptable results; so this algorithm has been neglected in the next stages of the optimization. The results of these stages of optimization are shown in Table 3 and Table 4.\n\nFor better understanding and easier access to the best algorithms and optimum amounts of parameters, a summary of these methods and results are shown in Table 5. Using these optimum amounts helps to accelerate or decelerate (in different cases of maximizing and minimizing) the opening of a wing crack by more applying wedging force in the tensile or comparison modes, avoiding the waste of time and resources.\n\n9. Conclusions\n\nIn the first part of this study, a complete sensitivity analysis performed on the main parameters of Fairhurst-Cook Model to determine their relations and\n\nTable 3. Results of the optimization algorithms in separate maximization of the wedging force.\n\nTable 4. Results of the optimization algorithms in separate minimization of the wedging force.\n\nTable 5. Summary of best algorithms and optimum results in different cases.\n\ntrends of change. Concluded results from this sensitivity analysis are summarized as below:\n\n・ The trend of changes for each of these parameters reversed in a specific amount of crack angle (α) as a “peak point”.\n\n・ This peak point is about 60.7 degrees in the analysis of crack amplitude and also maximum principal stress and it is about 37.7 degrees in crack angle analysis.\n\n・ In all of above cases, straight relations between that specific parameter and the amount of wedging force observed before the peak point, but it has changed to reverse relations after that point.\n\n・ The peak point virtually acts like a border line and limits the increase of wedging force by increasing the crack angle.\n\n・ The reason for the reversion of wedging force sign is thought to be the reversion of force from compression to tensile with the same magnitude order; which means the biggest force still remains the biggest one, but in a reverse direction and it happens because of the changes in crack angle, making an angle of more than 90 ̊ between the direction of wedging force and its applying area.\n\nKnowing the amount of peak points can be helpful for understanding the ultimate rock strength under uniaxial compression and also to apply optimum wedging force on a rock sample and create a wing crack. In order to achieve this optimum wedging force, a set of optimum amounts of related parameters should be used. For this purpose, three different optimizer algorithms (e.g. ACO, PSO and GA) were used and compared to identify the best algorithms and optimum results. Although it is not guaranteed that these resulted amounts of parameters are complete optimum, but they are expected to have proper applications as the local and near-global optimums with satisfactory resulting wedging force to create a wing crack.\n\nConflicts of Interest\n\nThe authors declare no conflicts of interest regarding the publication of this paper.\n\nCite this paper\n\nNajjarpour, M. and Jalalifar, H. (2018) Optimization of Fairhurst-Cook Model for 2-D Wing Cracks Using Ant Colony Optimization (ACO), Particle Swarm Intelligence (PSO), and Genetic Algorithm (GA). Journal of Applied Mathematics and Physics, 6, 1581-1595. https://doi.org/10.4236/jamp.2018.68134\n\nReferences\n\n1. 1. Nemat-Nasser, S. and Horii, H. (1984) Rock failure in compression. Proceedings Ninth Workshop Geothermal Reservoir Engineering, 999-1011. https://doi.org/10.1016/0020-7225(84)90101-0\n\n2. 2. Fairhurst, C. and Cook, N.G.W. (1966) The Phenomenon of Rock Splitting Parallel to the Direction of Maximum Compression in the Neighborhood of a Surface. Proceedings of the 1st International Congress of Rock Mechanics, 687-692.\n\n3. 3. Kolari, K. (2017) A Complete Three-Dimensional Continuum Model of Wing-Crack Growth in Granular Brittle Solids. International Journal of Solids and Structures, Elsevier, 27-42. https://doi.org/10.1016/j.ijsolstr.2017.02.012\n\n4. 4. Renshaw, C.E. and Schulson, E.M. (2001) Universal Behavior in Compressive Failure of Brittle Materials. Nature, 897-900. https://doi.org/10.1038/35091045\n\n5. 5. Wachter, L.M., Renshaw, C.E. and Schulson, E.M. (2009) Transition in Brittle Failure Mode in Ice under Low Confinement. Acta Mater, 345-355. https://doi.org/10.1016/j.actamat.2008.09.021\n\n6. 6. 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Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 337-374. https://doi.org/10.1098/rsta.1986.0101\n\n11. 11. Dyskin, A.V., Germanovich, L.N., Lee, K.K., Ring, L.M. and Ingrafea, A.R. (1994) Modelling Crack Propagation in Compression. Rock Mechanics, ISBN: 90 5410 380 8.\n\n12. 12. McClintock, F.A. and Walsh, J.B. (1963) Friction on Griffith Cracks in Rocks under Pressure. ASME, 1015-1021.\n\n13. 13. Hallbauer, D.K., Wagner, H. and Cook, G.W. (1973) Some Observations Concerning the Microscopic and Mechanical Behavior of Quartzite Specimens in Stiff Triaxial Compression Tests. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, Elsevier, 713-726. https://doi.org/10.1016/0148-9062(73)90015-6\n\n14. 14. Damjanac, B. and Fairhurst, C. (2010) Evidence for a Long-Term Strength Threshold in Crystalline Rock. Rock Mechanics and Rock Engineering, 513-531. https://doi.org/10.1007/s00603-010-0090-9\n\n15. 15. Hole, K.R., Meshram, R.A. and Deshmukh, P.P. (2015) Review: Applications of Ant Colony Optimization. International Journal of Engineering and Computer Science, 12740-12744. https://doi.org/10.5325/arthmillj.10.2.0177\n\n16. 16. Dorigo, M., Maniezzo, V. and Colorni, A. (1991) Positive Feedback as a Search Strategy. Technical Report, 91-016.\n\n17. 17. Dorigo, M. and Stültze, T. (2004) Ant Colony Optimization.\n\n18. 18. Bonnaccorsi, A. (1992) On the Relationship between Firm Size and Export Intensity. Journal of International Business Studies, 605-635. https://doi.org/10.1057/palgrave.jibs.8490280\n\n19. 19. Deb, K., Dorigo, M., Maniezzo, V. and Colorni, A. (1996) The Ant System: Optimization by a Colony of Cooperating Agents. IEEE Transactions on System, Man and Cybernetics, 29-41.\n\n20. 20. Kennedy, J. and Eberhart, R.C. (2001) Swarm Intelligence (The Morgan Kaufmann Series in Evolutionary Computation).\n\n21. 21. Zhang, Y. (2015) A Comprehensive Survey on Particle Swarm Optimization Algorithm and Its Applications. Mathematical Problems in Engineering, 1-38.\n\n22. 22. Dahyia, R. and Singh, A. (2014) A Survey on Application of Particle Swarm Optimization in Text Mining. International Journal of Innovative Research and Development, 101-107.\n\n23. 23. Singh, N. and Singh, A. (2012) Personal Best Position Particle Swarm Optimization. Journal of Applied Computer Science & Mathematics, 69-76.\n\n24. 24. Whitley, D. (1994) A Genetic Algorithm Tutorial. Statistics and Computing, 65-85. https://doi.org/10.1007/BF00175354\n\n25. 25. McCall, J. (2014) Genetic Algorithms for Modelling and Optimization. Journal of Computational and Applied Mathematics, Elsevier, 205-222.\n\n26. 26. Holland, J.H. (1975) Adaptation in Natural and Artificial Systems.\n\nNomenclature\n\na = crack amplitude\n\nf = wedging force\n\nl = crack semi-length\n\nα = crack angle\n\nβ(α) = model function depending on crack angle\n\nσ = maximum principal stress\n\nμ = internal friction angle" ]	[ null, "https://html.scirp.org/file/86525x1.png", null, "https://html.scirp.org/file/9-2500537x3.png", null, "https://html.scirp.org/file/9-2500537x2.png", null, "https://html.scirp.org/file/18-1760983x4.png", null, "https://html.scirp.org/file/2-1410169x6.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.894766,"math_prob":0.8269127,"size":30909,"snap":"2019-43-2019-47","text_gpt3_token_len":6809,"char_repetition_ratio":0.15230545,"word_repetition_ratio":0.07321652,"special_character_ratio":0.22420654,"punctuation_ratio":0.12548128,"nsfw_num_words":1,"has_unicode_error":false,"math_prob_llama3":0.95147467,"pos_list":[0,1,2,3,4,5,6,7,8,9,10],"im_url_duplicate_count":[null,3,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-17T15:09:13Z\",\"WARC-Record-ID\":\"<urn:uuid:07dd5c60-f552-43d8-a9d0-9e0cb2d6cb0f>\",\"Content-Length\":\"63073\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:66438ebc-c6b2-411e-b98a-0eee07e978c4>\",\"WARC-Concurrent-To\":\"<urn:uuid:79b348e0-089c-4c8a-bc6f-2e4a05ef88d2>\",\"WARC-IP-Address\":\"45.146.121.13\",\"WARC-Target-URI\":\"https://file.scirp.org/Html/86525_86525.htm\",\"WARC-Payload-Digest\":\"sha1:2CDV77GBHJHXYDVS3CMB3Z6XU4KSRHHQ\",\"WARC-Block-Digest\":\"sha1:OLD7FGXB5PCVNNU73YE5JS47CCYXKFVG\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570986675409.61_warc_CC-MAIN-20191017145741-20191017173241-00312.warc.gz\"}"}
https://www.britannica.com/science/separation-of-variables	[ "# Separation of variables\n\nmathematics\n\nSeparation of variables, one of the oldest and most widely used techniques for solving some types of partial differential equations. A partial differential equation is called linear if the unknown function and its derivatives have no exponent greater than one and there are no cross-terms—i.e., terms such as f f′ or ff′′ in which the function or its derivatives appear more than once. An equation is called homogeneous if each term contains the function or one of its derivatives. For example, the equation f′ + f 2 = 0 is homogeneous but not linear, f′ + x2 = 0 is linear but not homogeneous, and fxx + fyy = 0 is both homogeneous and linear.\n\nIf a homogeneous linear equation in two variables has a solution f(x, y) that consists of a product of factors g(x) and h(y), each involving only a single variable, the solution of the equation can sometimes be found by substituting the product of these unknown factors in place of the unknown composite function, obtaining in some cases an ordinary differential equation for each variable. For example, if f(x, y) is to satisfy the equation fxx + fyy = 0, then by substituting g(x)h(y) for f(x, y) the equation becomes gxxh + ghyy = 0, or −gxx/g = hyy/h. Because the left side of the latter equation depends only on the variable x and the right side only on y, the two sides can be equal only if they are both constant. Therefore, −gxx/g = c, or gxx + cg = 0, which is an ordinary differential equation in one variable and which has the solutions g = a sin (xc1/2) and g = a cos (xc1/2). In a similar manner, hyy/h = c, and h = e±yc1/2. Therefore, f = gh = ae±yc1/2 sin (xc1/2) and ae±yc1/2 sin (xc1/2) are solutions of the original equation fxx + fyy = 0. The constants a and c are arbitrary and will depend upon other auxiliary conditions (boundary and initial values) in the physical situation that the solution to the equation will have to satisfy. A sum of terms such as ae±yc1/2 sin (xc1/2) with different constants a and c will also satisfy the given differential equation, and, if the sum of an infinite number of terms is taken (called a Fourier series), solutions can be found that will satisfy a wider variety of auxiliary conditions, giving rise to the subject known as Fourier analysis, or harmonic analysis.\n\nThe method of separation of variables can also be applied to some equations with variable coefficients, such as fxx + x2fy = 0, and to higher-order equations and equations involving more variables.\n\nThis article was most recently revised and updated by William L. Hosch, Associate Editor." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.930103,"math_prob":0.99934286,"size":3496,"snap":"2020-45-2020-50","text_gpt3_token_len":770,"char_repetition_ratio":0.16008018,"word_repetition_ratio":0.006837607,"special_character_ratio":0.22025171,"punctuation_ratio":0.087692305,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99977106,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-11-24T04:52:09Z\",\"WARC-Record-ID\":\"<urn:uuid:bc691f32-7f3e-4eea-b84c-25f4470ff1e3>\",\"Content-Length\":\"68855\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f60aa0b0-12a9-4d38-b57c-d2582b28a6a6>\",\"WARC-Concurrent-To\":\"<urn:uuid:99ccf5f7-e6be-4c5f-a13c-170d9ce2a937>\",\"WARC-IP-Address\":\"34.206.100.126\",\"WARC-Target-URI\":\"https://www.britannica.com/science/separation-of-variables\",\"WARC-Payload-Digest\":\"sha1:UMLVV6HBIJ27UGYM6G64NEIKXPWSJ7V5\",\"WARC-Block-Digest\":\"sha1:UIUOY5TG7N6GD6VVZ2ZHZJWF3KOZR7B5\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-50/CC-MAIN-2020-50_segments_1606141171077.4_warc_CC-MAIN-20201124025131-20201124055131-00697.warc.gz\"}"}
https://www.nature.com/articles/s41467-022-33877-7?error=cookies_not_supported&code=1c25b0d5-fa96-4202-b002-31dd385bd5ce	[ "## Introduction\n\nArtificial intelligence (AI) has the potential to drastically change our world through accumulating impacts in fundamental science1,2, new-type transportation3,4, assisted medical treatment5,6, etc. Artificial neural network (ANN), a kind of computing architecture inspired by signal processing in the human brain, is one of the major technical pillars for these applications. It contains complex mapping relations in repetitive linear and nonlinear operations. In recent years, however, the required computing capacity for the state-of-the-art ANNs has been doubling every 3.5 months7, far overloading Moore’s Law in microelectronics8, e.g., electronic computers. Now, silicon (Si) photonics has been recognized as one of the most promising candidates to break through microelectronics bottles owing to its superior interconnect bandwidth, low power consumption and complementary metal-oxide-semiconductor (CMOS) compatibility. According to different implementations, many Si photonic neural network architectures have been proposed to facilitate complex computing tasks, such as diffractive neural networks9,10 and optical interference neural networks11,12. They utilize diffractive elements or optical interferometers to perform linear operations. The Si photonic interference circuit has been demonstrated as 100× faster than the microelectronic processor but of 1/1000 energy11. With the rapidly increasing demand for computational speed and power, Si photonics ANNs provide a promising alternative for AI hardware.\n\nSi photonics neural networks face challenges in large-scale integration due to the lack of proper neurons. Firstly, integrating optical nonlinear material on Si is an open challenge13,14. On account of the weak nonlinear effect of Si15, heterogeneous integration of other materials is often needed. Although the dye16,17, phase-change materials18,19 and two-dimensional materials20,21 have been proved their optical nonlinearities for all-optical neural networks (AONNs), their stabilities and manufacture abilities are unsatisfactory22,23, limiting applications for large-scale networks. For example, two-dimensional materials, such as black phosphorus, are easily irreversible oxidized in air, resulting in poor stability and rapid degradation of the semiconductor properties24. Moreover, as the average size of two-dimensional material is limited by the quality of its corresponding three-dimensional precursor, it is hard to produce wafer-scale two-dimensional single crystalline25. In addition, the temperature required for crystallization of typical phase-change materials is usually too high for Si-compatible fabrication, hindering the large-scale integration with Si photonics26. Secondly, the lack of non-intrusive monitors27,28 to prompt the status of the network without interference is another major obstacle. Monitoring and feedback operations enable efficient networks training, node failures detection and environmental fluctuations offset. For a given hardware-based neural network, especially when it is trained completely, such monitors should not change the operating points. However, this is very difficult since a neural network may contain thousands of neurons. For example, the implementation of in situ backpropagation algorithm requires virtually lossless intensity detection in every node29. Yet, the conventional light-splitting-and-detection method drifts the operating states and also introduces architecture complexity and accumulated insertion loss.\n\nHere, we propose and demonstrate nonlinear germanium-silicon (Ge-Si) photodiodes (PDs) to construct non-intrusive and self-monitored AONN (SM-AONN) with fully CMOS compatibility. The all-optical power in-power out response is attributed to the intrinsic-absorption-induced free-carrier absorption (FCA) in the Ge thin film. Specially designed electrodes achieve high carrier concentration accumulation via hindering carrier transport. Meanwhile, the Ge-Si heterojunction provides a non-intrusive electrical monitoring signal owing to concomitant photoelectric conversion. In a compact structure of 4.3 × 8 μm2 without any optical splitter, the nonlinear activation and monitoring are combined simultaneously, alleviating the issues of complex architecture and operation point drift in conventional ANNs. Experimentally, using the activation and monitoring features, a three-layer SM-AONN enables object classification and semantic segmentation tasks, presenting in situ training and learning with high training accuracy. More layers of SM-AONN can be constructed using optical fiber arrays to connect multiple chips. In addition, the feasibility and performance of this neuron for deep feedforward neural networks are confirmed via the Modified National Institute of Standards and Technology (MNIST) handwriting recognition30, achieving a high accuracy of 97.3%.\n\nOur work proves that conventional Group-IV semiconductor technology not only enables all-optical nonlinearity without resorting to other materials but also merges activation and monitoring units. The photonic neural network based on this technology allows for more functionalities, simplified architecture and high accuracy. Due to the material stability and mass-production31, we believe that this work will pave a new way toward future high-density integrated photonic intelligent processors.\n\n## Results\n\n### Self-monitored all-optical neural network\n\nFigure 1a shows the architecture of the proposed SM-AONN, consisting of an input layer, multiple hidden layers with monitoring signals and an output layer. In each layer, optical signals are processed by an optical linear transformation and all-optical nonlinear activation building blocks. Being different from the traditional architecture, each nonlinear activation block will produce electrical signals for monitoring the states of each neuron.\n\nOptical linear transformations are implemented using a reconfigurable Si-based Mach-Zehnder interferometer (MZI) mesh, which is an equivalent photonic field programmable gate array, as shown in Fig. 1b. It has been proved that the arbitrary optical linear operations can be carried out by a series of optical beam splitters, phase shifters and attenuators32,33, i.e., tunable MZIs34. As Fig. 1c shows, voltage signals from the digital-to-analog converters (DACs) are loaded on two thermal-tuning electrodes of the Si-based MZI. The state of each MZI is controlled until the linear operation of the entire network is formed. The weightings between neurons are stored and updated in the voltage information. Note that a complete neuron contains both a linear weighting part and a nonlinear part, and the thermo-optic phase shifter-based linear weighting mesh is indispensable for building complete neurons.\n\nAfter optical linear operations, the optical signals undergo the Ge-Si all-optical nonlinear units (AONUs) to perform nonlinear processing (activation function), as shown in Fig. 1d. Meanwhile, each AONU provides an electrical monitoring signal to indicate the results of weighting addition and nonlinear operations, by monitoring the input and output optical power of the AONUs. Unlike conventional light-splitting-and-detection solutions, this photoelectric monitoring occurs concomitantly with the optical nonlinear activation in the same structure (Fig. 1e). As shown in Fig. 1f, monitoring signals are drawn from the electrode and converted to the digital domain through the analog-to-digital converters (ADCs). This non-intrusive manner detects the current node states in real-time without changing the network operating point, and thus it enables high performance and stability of the SM-AONN.\n\n### Nonlinear Ge-Si PD-based AONU\n\nAs a key component of the SM-AONN, the Ge-Si AONU enables all-optical nonlinear activation and non-intrusive monitoring. Figure 2a shows the structure and schematic of it. It is similar to the Ge-Si waveguide PDs applied to photoelectric detection35,36 (Fig. 2b). For conventional PDs, the electrodes are with the same length as the Ge film to export out the photo-generated carriers from each part of the absorber. Typically, the output optical power is less concerned. Being different from that, the electrodes herein are omitted where the light is incident to engineer the carrier dynamics. Detailed device geometry and optical field information can be found in Supplementary Note 1. In the electrodeless region (with a small electric field and carrier transit time » carrier lifetime), carriers accumulate and enable the FCA of the Ge film, producing a strong all-optical nonlinear response. In the region with the electrode (with a strong electric field and carrier transit time « carrier lifetime), the carriers are rapidly absorbed by the electrode, and no FCA effect occurs. Fortunately, these collected carriers can be used for optical monitoring. A specific mechanism of the activation function that conforms to the proposed partial electrode structure is given in Supplementary Notes 2 & 3.\n\nBy solving the nonlinear Schrödinger equation (NLSE) and carrier rate equation37,38 (See Methods), the activation function can be obtained as\n\n$${P}_{{{{{{\\rm{out}}}}}}}=\\frac{\\exp (-\\alpha {L}_{{{{{{\\rm{Ge}}}}}}}){P}_{{{{{{\\rm{in}}}}}}}}{1+A[1-\\exp (-\\alpha ({L}_{{{{{{\\rm{Ge}}}}}}}-{L}_{{{{{{\\rm{E}}}}}}}))]{P}_{{{{{{\\rm{in}}}}}}}}$$\n(1)\n\nwhere A represents for στ/2ħωS. When A = 0, the above relationship degenerates into linear absorption. Pin and Pout are input and output optical power, respectively, with α, σ, τ, LGe, S being intrinsic absorption coefficient, absorption cross-section of FCA, carrier lifetime and length of Ge film, as well as incident area. LE is the length of the electrode. ħ and ω represent the reduced Planck constant and optical frequency, respectively. Meanwhile, the concomitant electrical monitoring signal occurs thanks to intrinsic absorption and photoelectric conversion. The FCA effect only transfers momentum between electrons, providing no photocurrent. The nonlinear relationship between the output current and input optical power is expressed as39\n\n$${I}_{{{{{{\\rm{out}}}}}}}=R{P}_{{{{{{\\rm{in}}}}}}}\\,\\tanh (\\frac{k{I}_{\\max }}{R{P}_{{{{{{\\rm{in}}}}}}}})$$\n(2)\n\nwhere Iout, R, Pin and Imax are output current, responsivity at low-power level, input optical power and saturation current, respectively. k is a parameter used to change the shape of the curve. Note that this optical monitoring is non-intrusive. The bonding wire is placed ~3 μm above the Si-Ge region, having little influence on the optical signal, and this is the main reason we call it non-intrusive. In addition, the proposed device consumes a portion of optical power to achieve the optical nonlinearity, and the resulting photocurrent is used to realize monitoring at the same time. This is to say, the optical power used to achieve optical nonlinearity is inherently consumed, and no additional optical power is needed to achieve monitoring. This is another important reason we call it non-intrusive.\n\nThe length ratio of the electrode to Ge film (LE/LGe) significantly affects the optical-to-optical and optical-to-electrical response. A longer electrode improves the carrier collection efficiency, thereby increasing the output photocurrent40,41. However, it reduces the carrier concentration and weaken the FCA effect. The relationship of the carrier collection efficiency and photocurrent can be referred to Supplementary Note 4. Figure 2c shows the carrier concentration and collection efficiency (ηc) versus length ratio. The pink area (LE/LGe = 0.2–0.4, represented as Type-A) achieves 90% of the maximum value of both. Within this range, a good optical nonlinearity and high optical monitoring responsivity can be obtained simultaneously, and this range can be considered as the optimal ratio. The orange area (LE/LGe ~ 1) shows the conventional PD (represented as Type-B) with low optical nonlinearity. Figure 2d shows the false-color image of the fabricated AONU. A 4.3 × 8 μm2 Ge thin film is epitaxially grown on the Si waveguide. The 3 μm-length electrodes are coated at the optical exportation of Ge. The adopted scheme (Type-A) corresponds to LE/LGe of 0.375. See Methods for more fabrication details.\n\nHere, we experimentally verified the optical and electrical responses of the proposed AONU, compared with a reference conventional PD. The Pout-Pin relations are shown in Fig. 2e. For Type-A, the output power is linear at low input, and then gradually flattens as the power increases, showing obvious Pout-Pin nonlinearity. However, the curve of Type-B is linearly tangent to that of Type-A. At the same input, the difference between the two curves contributes to the FCA. The threshold of the nonlinear activation is about 1.1 mW. Such a low threshold requirement is very beneficial for low power consumption and for driving the nonlinearity units of next level. The activation functions are fitted by Eq. (1), as the solid line shown in Fig. 2e. On the other hand, the measured output photocurrents are shown in Fig. 2f. Although the linearity is slightly reduced, the photocurrent still increases monotonously with the input optical power, so that the input optical power can be uniquely determined and monitored from the output current. Combined with the Pin-Pout relation, the output optical power can also be determined. The photodetection metrics including the responsivity, bandwidth and dark current can be referred in Supplementary Note 6. The bandwidth is influenced by the doping of the AONU and the detailed analysis is given in Supplementary Note 7.\n\n### Large scale SM-AONN performance\n\nHaving proved that the state of each neuron can be obtained from the monitoring signals, the performance of the entire neural network is characterized. We prepare a scalable three-layer fully connected feedforward neural network using MZI mesh and the proposed AONUs, as shown in Fig. 3a. Although the three-layer network can be built on one chip with the same fabrication process, we split it into three chips and connect them using optical fiber arrays, for easy comparison and arbitrary combination. More importantly, more layers of networks can be constructed using optical fiber arrays to connect multiple chips. Here, three layers are sufficient to demonstrate the following machine learning tasks with high accuracy. Figure 3b shows one layer of the packaged SM-AONNs, consisting of four neurons with 16 MZIs and four nonlinear units. The MZI mesh and nonlinear units are present in Fig. 3c, d, respectively.\n\nThe basic operations of neural networks are training and inference. Compared with inference, training consumes most of the computing power in neural networks. However, it can be completed quickly and automatically, using self-monitoring electrical signals combined with special processing chips and optoelectronic integration. The training set of machine learning tasks consists of a series of vectors of inputs and outputs, being encoded on optical power. As shown in Fig. 4a, the input optical signals are processed by the photonic chip to obtain the real optical outputs. Being different from the conventional training method, the real output is read by monitoring signals rather than external PDs. A loss function such as cross-entropy42 is defined to evaluate the distance between the real outputs and training-set predicted outputs. The difference is eliminated with iteration by feedback algorithms such as backpropagation43 in special processing chips. Then, the SM-AONN is trained completely. The detailed in situ training implementation can refer to Supplementary Note 8.\n\nExperimentally, the simplified object classification and semantic segmentation tasks are performed. As shown in Fig. 4b, we utilize two-valued optical intensities to encode the labels of four input targets, for example, ′0110′ for input and ′0100′ for output are represented for ′target 2′. At the optical input port, only ports 2 and 3 are configured to pass through via the variable optical attenuators (VOAs). When the neural network is successfully trained, only port 2 is expected to be the optical output. In real application, the targets can represent different grayscale images. Figure 4c shows the relationship of the loss function and iterations. The output histograms of the initial state, the intermediate state of the 20 iterations and the final state are shown as the insets. In the initial state, the output of each mode is chaotic, since the weightings of the MZI network are given randomly. With the reconstruction of weightings, the recognition of each mode becomes clearer. Being fully configured, the output probability of each mode at the correct port exceeds 97%. Similarly, the training for semantic segmentation is present. As a 4 × 4-pixel image shown in Fig. 4d, the gray levels of the ′L′ and ′T′-type regions are greater than others. After training, the gray levels of ′1′ and ′0′ are contrastive to identify ′L′ and ′T′ in the image. Since each input to SM-AONN is a column vector (in the Y direction), the sum of normalized output power in the Y direction remains unity. As Fig. 3e shows, when the number of iterations exceeds only 15 epochs, the output of each port is near the expectation of 50% for two input ports and 100% for one input port. For these two experiments, the error analysis can refer to Methods. The successful training of two different tasks has demonstrated the general configuration task and the powerful learning ability of the SM-AONN. Thanks to the electrical monitoring signals, the training results have extremely high expected accuracy. Large-scale training tasks are fully automated with the help of electronics.\n\nHere, we use the digital computing as an example. Actually, the demonstrated photonic neuromorphic computing architecture is analog in nature and can be used for analog computing as well. This is because the MZI weighting network can directly handle the multiplication of complex-valued data, and the optical nonlinear response is also a continuous-valued input-output function. The difference between analog computing and digital computing is only the form of the input and output data sets. If the current digital input of ′0′ or ′1′ is replaced with a continuous-time optical intensity, analog computing can be performed.\n\nGoing forward, we introduce the obtained nonlinear optical responses as nonlinear activation functions in a three-layer deep feedforward neural network for the MNIST handwriting recognition, to further test large-scale data processing capability. The MNIST data set consists of 60,000 784-pixel images, therein 50,000 and 10,000 images are used for training and testing, respectively. These images contain handwritten digits from 0 to 9, as shown in Fig. 5a. The deep feedforward neural network consists of two hidden layers containing 200 neurons and an output layer containing 10 neurons. The input is a 784 × 1 vector, and the output is a 10 × 1 vector. The output layer adopts the Softmax activation function to convert the output results into probability. The proposed Ge-Si AONU is extracted as the activation function for the hidden layers. The activation function with normalized input and output is shown in Fig. 5b. The simulation utilizes the conjugate gradient backpropagation algorithm to iterate 100 times, and the loss function is cross-entropy. An accuracy of 97.3% and corresponding confusion matrix are shown in Fig. 5c and d, respectively. Each column of the matrix represents the instances in a predicted label, while each row represents the instances in a true label. The diagonal elements represent the probabilities that are correctly predicted. These results show that our nonlinear unit has high performance on representative machine learning tasks.\n\n## Discussion\n\nOne of the key advantages of the AONU is the ability to non-intrusively observe the optical energy. The experimental and emulational comparisons on the performance and stability are provided in Supplementary Note 9. Indeed, the results indicate a more stable and better performance for the proposed “non-intrusive” scheme. Compared to the intrusive monitoring with different degrees of perturbation, the non-intrusive scheme shows a smoother activation function and improved accuracies of 1.7–4% in handwritten recognition. Furthermore, the iterations to reach the maximum accuracy is much less, resulting in a decreased training cost. In addition, when the neural network is trained completely, the accuracy fluctuation is much smaller, which means a better stability on inferring tasks. On the other hand, photonic neural networks are large-scale and dynamically tunable circuits, and their control becomes enormously difficult due to manufacturing variations and thermal crosstalk44. Fortunately, the non-intrusive monitoring provides a calibration capability by compensating the fabrication errors and environmental fluctuations. In the training process, the monitoring enables non-intrusive intensity detection of each node, to implement in situ gradient measurements and forward or backpropagation algorithms29. This method can enable highly efficient gradient calculation in training. When an already trained neural network is working, the non-intrusive monitoring feature can obtain information about environmental fluctuations without changing the operating point of the network27. On this basis, the network can be dynamically tuned and calibrated without introducing other disturbances.\n\nAnother main advantage of the photonic neural network is potentially possessing higher speed and energy efficiency compared to electronics10,45. Typically, the computing speed is defined as the number of operations per second (FLOPS). For our demonstrated system, the FLOPS is calculated to be 1.92 × 1012 operations per second with a 20 GHz detection bandwidth. In principle, such a computing speed is one order of magnitude faster than electronic neural networks which are usually restricted to a GHz clock rate46. The consumed energy is calculated to be ~0.27 pJ per operation in our system, better than an “ideal” electronic computer (1 pJ per operation, assuming no energy is used on data movement) and two orders of magnitude better than conventional graphics processing units (GPUs) (100 pJ per operation)47. Please see Supplementary Note 10 for the detailed calculation and comparison. On the other hand, in the photonics system, the energy required for the optical nonlinearity of the Si-Ge system is relatively higher than that of some other materials48, but it has the advantages of CMOS fabrication compatibility and compact structure that other material systems may not have.\n\nThe scalability of the photonic neural network is an important challenge. Typically, some form of nonlinearity is required to implement the thresholding effect of a neuron in the neural networks. However, optical nonlinear responses are comparatively power inefficient, and the neuron output is often weaker than its input14. Thus, previous works utilized optical amplifiers49,50, optical-electrical-optical conversion51 or all-optical carrier regeneration18 to alleviate this issue. These methods also bring additional optical and electrical power consumption. By contrast, an advantage of our scheme is that only the loss of the optical nonlinear part needs to be considered, while the loss from optical splitters and monitoring is avoided. This might be competitive as the neural network scales up. At present, we use off-chip EDFAs to pump the network. Recently, Liu, et al.52 achieved on-chip erbium-doped waveguide amplifiers with a gain up to 30 dB. This would be suitable to simultaneously address the challenges of multi-layer scaling and on-chip integration.\n\nAiming at solving the issues of large-scale Si-based integrated ANNs, we have demonstrated that the specifically designed nonlinear Ge-Si PD enables both all-optical activation and non-intrusive monitoring. The SM-AONN based on this technology achieves 97.3% accuracy on open machine learning tasks. The advantages of the Ge-Si PD-based SM-AONN include: (1) Material advantages. Ge is a kind of material with stability and CMOS compatibility. (2) All-optical operations. The photoelectric conversion only occurs during training. There is no need for the information exchange between optical and electrical domains once trained. (3) Non-intrusive monitoring. The network supports automatic training, node failures analysis and environmental fluctuations monitoring without disturbing the operation points. (4) Simplified architecture. The activation and monitoring units are merged in the same device with compact footprint. (5) Large scale. Multiple layers of SM-AONN can be constructed using optical fiber arrays to connect multiple chips. (6) High accuracy. A deep neural network utilizing this new activation function shows high performance. In addition, due to characteristics of the Si MZI network and Ge nonlinearity, this network may also draw interests in quantum networks53,54 or mid-infrared applications55. We believe that this work is promising for future large-scale optical intelligent neuromorphic systems.\n\n## Methods\n\n### Analysis coupled equations\n\nThe interaction process of intrinsic absorption and FCA can be described by the nonlinear NLSE equation\n\n$$\\frac{{{{{{\\rm{d}}}}}}I}{{{{{{\\rm{d}}}}}}z}=-\\alpha I-\\beta {I}^{2}-\\sigma NI$$\n(3)\n\nand the carrier rate equation\n\n$$\\frac{\\partial N}{\\partial t}=\\frac{\\alpha }{\\hslash \\omega }I+\\frac{\\beta }{2\\hslash \\omega }{I}^{2}-\\frac{N}{\\tau }$$\n(4)\n\nwhere I and N are optical intensity and carrier concentration, respectively, with α, β, σ and τ being intrinsic absorption coefficient, two-photon coefficient, absorption cross-section of FCA and carrier lifetime of the Ge. Here, β = 0. ħ and ω represent the reduced Planck constant and optical angular frequency, respectively. z is the light propagation direction and t is the time.\n\n### Device fabrication\n\nThe device is fabricated using a silicon-on-insulator wafer with 220 nm thick Si top layer and 2 µm buried oxide. The Si layer is etched into strip waveguides for the pattern of the MZIs and Si slab under Ge film. Then, the Si top layer is implanted using different doses of boron ions to form the P-type regions. A 500 nm-thick Ge film is grown on the P-type doped Si slab. On the top of Ge film, phosphorus ions are implanted with ~100 nm-depth to form the N-type region of a PIN junction. The titanium nitride (TiN) heater of 120 nm in thickness is deposited 2 μm above the Si waveguide for thermal tuning. Finally, metal electrodes are fabricated and connect to Si, Ge and TiN through via holes.\n\n### Error analysis\n\nThe training of the neural network relies on the monitoring photocurrent of the AONU, and then the weighting values are loaded on the thermally tuned MZI network in the form of voltages. The photodetector noise (σD) and the voltage fluctuation applied on MZIs (σΦ) are the dominant error sources. In the experiments, we used DACs with 10-bit precision and a three-layer 4 × 4 matrix with σΦ estimated to be 10−3, as well as a photodetector noise of σD = 1.8 × 10−3 under a mean photocurrent of ~1 mA. We carried out the following steps to numerically simulate the performance with the σD and σΦ. For the trained 4 × 4 unitary matrices U, we calculate a set {VMZI} that encodes the matrix. We assume phase-encoding errors δVMZI is a random variable sampled from a Gaussian distribution G(0, σΦ). We obtain a new set of perturbed phases {VMZI + δVMZI} and perturbed 4 × 4 unitary matrices U′. During forward propagation, every time a matrix multiplication is performed for a result v = U· u (u is input vector), we add a set of random photodetection errors δv as the perturbed output vector v′ =v + δv, where we assume each δv is a random variable sampled from a Gaussian distribution G(0, σD·\|v \| ). Then perturbed optical output is derived from v′ and the accuracy is calculated. Repeating 50 times, the final accuracy is estimated to be ~98%. We attribute other errors to the fabrication error and thermal crosstalk of the linear networks. The fabrication error can be compensated by pre-calibration steps, while the thermal crosstalk can be reduced by adding thermal isolation trenches." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8697398,"math_prob":0.90863585,"size":39395,"snap":"2023-14-2023-23","text_gpt3_token_len":8813,"char_repetition_ratio":0.14381458,"word_repetition_ratio":0.016984401,"special_character_ratio":0.2192664,"punctuation_ratio":0.16123264,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95185417,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-03-30T01:44:45Z\",\"WARC-Record-ID\":\"<urn:uuid:e4d95843-dbc4-4cce-83f8-30ca954ecd45>\",\"Content-Length\":\"380626\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ff7a2ef0-f641-46d4-8917-1572ae3c570e>\",\"WARC-Concurrent-To\":\"<urn:uuid:1acfeae7-a66f-46c4-a08f-5ce3375e6717>\",\"WARC-IP-Address\":\"146.75.36.95\",\"WARC-Target-URI\":\"https://www.nature.com/articles/s41467-022-33877-7?error=cookies_not_supported&code=1c25b0d5-fa96-4202-b002-31dd385bd5ce\",\"WARC-Payload-Digest\":\"sha1:UPD7FQJ4EIIVF6RUXKDIXDLXJRQ4DAPC\",\"WARC-Block-Digest\":\"sha1:6AFWCBP62HCQDG24ZRM6GAINHRGXIMIQ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-14/CC-MAIN-2023-14_segments_1679296949093.14_warc_CC-MAIN-20230330004340-20230330034340-00248.warc.gz\"}"}
https://zory.ink/posts/f69a.html	[ "HAOI2012\nbzoj2749\nluogu2350\n\n2h\n\n## Analysis\n\n$f(prime)=f(prime-1)$\n$f(prime^k)=f(prime)*k$\n$f(num)=\\sum f(prime^k)$" ]	[ null ]	{"ft_lang_label":"__label__zh","ft_lang_prob":0.9657467,"math_prob":0.99976486,"size":484,"snap":"2019-35-2019-39","text_gpt3_token_len":442,"char_repetition_ratio":0.08958333,"word_repetition_ratio":0.0,"special_character_ratio":0.22107439,"punctuation_ratio":0.0,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99992895,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-08-19T21:54:23Z\",\"WARC-Record-ID\":\"<urn:uuid:94fad284-e71b-4a36-8adb-477bb895c386>\",\"Content-Length\":\"58585\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:6d9df43a-6048-4891-9e1f-ba17f834ae3b>\",\"WARC-Concurrent-To\":\"<urn:uuid:58dba396-82af-4b39-8909-c6b9afc3aba0>\",\"WARC-IP-Address\":\"185.199.111.153\",\"WARC-Target-URI\":\"https://zory.ink/posts/f69a.html\",\"WARC-Payload-Digest\":\"sha1:FMGX6VSGT2JKG2DBCCS4VKRACHXVOO2D\",\"WARC-Block-Digest\":\"sha1:JUXITKQBY5HKOVLZW4L43S44HRD2EHU6\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-35/CC-MAIN-2019-35_segments_1566027314959.58_warc_CC-MAIN-20190819201207-20190819223207-00026.warc.gz\"}"}
https://ch.mathworks.com/help/matlab/matlab_prog/configure-run-button.html	[ "## Configure the Run Button for Functions\n\nFunctions are program files that accept inputs and return outputs. To run functions that require input argument values or any other additional setup from the Editor, configure the", null, "button.\n\nTo configure the button in the Editor, click", null, "and add one or more run commands.\n\nFor example:\n\n1. Create the function `myfunction.m` that performs a calculation using the inputs `x` and `y` and stores the results in `z`.\n\n```function z = myfunction(x,y) z = x.^2 + y;```\n2. Go to the Editor tab and click", null, ". MATLAB® displays the list of commands available for running the function.", null, "3. Click the last item in the list and replace the text type code to run with a call to the function including the required input arguments. For example, enter the text ```result = myfunction(1:10,5)``` to run `myfunction` with the input arguments `1:10` and `5`, and store the results in the variable `result`. MATLAB replaces the default command with the newly added command.", null, "To run multiple commands at once, enter the commands on the same line. For example, enter the text ```a = 1:10; b = 5; result = myfunction(a,b)``` to create the variables `a` and `b` and then call `myfunction` with `a` and `b` as the input arguments.\n\nNote\n\nIf you define a run command that creates a variable with the same name as a variable in the base workspace, the run command variable overwrites the base workspace variable when you run that run command.\n\n4. Click the", null, "button. MATLAB runs the function using the first run command in the list. For example, click", null, "to run `myfunction` using the command `result = myfunction(1:10,5)`. MATLAB displays the result in the Command Window.\n\n```result = 6 9 14 21 30 41 54 69 86 ```\n\nTo run the function using a different run command from the list, click", null, "and select the desired command. When you select a run command from the list, it becomes the default for the button.\n\nTo edit or delete an existing run command, click", null, ", right-click the command, and then select Edit or Delete.\n\nNote\n\nRunning live functions in the Live Editor using the", null, "button is only supported in MATLAB Online™. When you run a live function using the", null, "button, the output displays in the Command Window. To run a live function in an installed version of MATLAB, call the function from the Command Window or from a script or live script." ]	[ null, "https://ch.mathworks.com/help/matlab/matlab_prog/run_ts_1686cc2a358fb970591b908ec30132e1fa.png", null, "https://ch.mathworks.com/help/matlab/matlab_prog/drop_down_arrow92a9929d0eec5b8b282a741294cf715d.png", null, "https://ch.mathworks.com/help/matlab/matlab_prog/drop_down_arrow92a9929d0eec5b8b282a741294cf715d.png", null, "https://ch.mathworks.com/help/matlab/matlab_prog/run_command.png", null, "https://ch.mathworks.com/help/matlab/matlab_prog/run_command_input.png", null, "https://ch.mathworks.com/help/matlab/matlab_prog/run_ts_1686cc2a358fb970591b908ec30132e1fa.png", null, "https://ch.mathworks.com/help/matlab/matlab_prog/run_ts_1686cc2a358fb970591b908ec30132e1fa.png", null, "https://ch.mathworks.com/help/matlab/matlab_prog/drop_down_arrow92a9929d0eec5b8b282a741294cf715d.png", null, "https://ch.mathworks.com/help/matlab/matlab_prog/drop_down_arrow92a9929d0eec5b8b282a741294cf715d.png", null, "https://ch.mathworks.com/help/matlab/matlab_prog/run_ts_1686cc2a358fb970591b908ec30132e1fa.png", null, "https://ch.mathworks.com/help/matlab/matlab_prog/run_ts_1686cc2a358fb970591b908ec30132e1fa.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.66599613,"math_prob":0.7012119,"size":2289,"snap":"2022-27-2022-33","text_gpt3_token_len":520,"char_repetition_ratio":0.18424508,"word_repetition_ratio":0.01010101,"special_character_ratio":0.22018349,"punctuation_ratio":0.111597374,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9618874,"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],"im_url_duplicate_count":[null,10,null,10,null,10,null,2,null,2,null,10,null,10,null,10,null,10,null,10,null,10,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-08-10T22:56:27Z\",\"WARC-Record-ID\":\"<urn:uuid:13144201-15e7-43ca-a4d2-7b5613603106>\",\"Content-Length\":\"75132\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:bda82868-3ca8-48af-a618-2d7ea5fe88ff>\",\"WARC-Concurrent-To\":\"<urn:uuid:f916f998-ae72-4c32-983e-e1d9ef114f6f>\",\"WARC-IP-Address\":\"23.218.145.211\",\"WARC-Target-URI\":\"https://ch.mathworks.com/help/matlab/matlab_prog/configure-run-button.html\",\"WARC-Payload-Digest\":\"sha1:IGXMVMDPV5AYUUSXRGSIU7DIWFJFTN23\",\"WARC-Block-Digest\":\"sha1:UGNJM33BBFV6EE5FLZJYGZEEDT26BU2Q\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-33/CC-MAIN-2022-33_segments_1659882571222.74_warc_CC-MAIN-20220810222056-20220811012056-00268.warc.gz\"}"}
http://prek-8.com/2ndgrade/fractionWorksheets_index4.php	[ "", null, "@import url(http://www.google.com/cse/api/branding.css);", null, "Custom Search \|\| Holiday \|\| Arts and Crafts \|\| Coloring \|\| Lesson Plans \|\| Printables \|\| Word Puzzles \|\| Writing \|\| Games\n\n# Fraction Problems Math Worksheets\n\nCustom Search\n\nFree printable fraction math worksheets\nSolve fraction math problems. great resources for elementary school teachers\n\nFree printable fraction math worksheets\nFraction word problems. extra math practice with essential math skills\n\nFree printable fraction math worksheets\nSolve fraction math problems. Fraction operations worksheets for advanced math skill.\n\nCompare fractions free math worksheet\nFree printable fractions math worksheets for kids to use < , >, or = to compare fractions.\n\n2nd grade fractions word problems worksheet\nSolve fractions word problems math worksheets for primary grade students. mathematical problem solving skills", null, "Fraction word problems worksheet\nFree printable fraction word problems worksheets for 2nd grade teachers.\n\nFraction math problems worksheet\nSolve fractions math problems worksheets for 2nd grade students. 2 step problems\n\nParts of a group fraction worksheet\nFree printable fractions worksheets for 2nd grade teachers to print out as math homework or class room exercises.\n\nFree fractions problems worksheet\nStudents solve fractions. Show fractions, calculate fractions of a part of the group.\n\nKids solve fraction word problems. Problem solving skills.\n\nFree printable fractions worksheets for school math teachers to teach kids fractions.\n\nFree printable fraction math worksheets\nSolve fraction math problems. Picture worksheets with dragons, animals, fruits, sea animals, vegetables.\n\nProblem solving math worksheets\nSolve fraction math problems. free printable from your own computers.\n\nCustom Search\n\n## Free printable 2nd grade worksheets\n\nCustom Search\n\nFree printable fractions worksheets and telling fractions mathematics activities for 2nd grade students learn and practice different ways to say fractions, read start fractions and stop fractions and calculate fractions, Free printable second grade fractions math worksheet to read the clock, analog and digital clocks grade 2 free fractions worksheets, Word problems fractions worksheets, Days of the week, fractions word problems, Quarter-hour fractions, Tell fractions to minutes, learn fractions relationships, to compare fractions units, and read calendar 2nd grade free math worksheets.\n\nFree online math worksheets generator tool for elementary school math teachers and parents to generate your own math worksheets and print out print out for free from your school office or home office. Easy to use with kids favorite pictures as extra math homework or in class math exercise.\n\nComparing math worksheets\n\nDivision math worksheets 2 3 4 5\n\nDaily math review worksheets 2 3 4\n\nFractions math worksheets 2 3 4\n\nGeometry worksheets 2 3 4\n\nMeasurement worksheets 2 3\n\nMoney math worksheets 2 3 4 5 6 7\n\nMultiplication worksheets and multiplication games 2 3 4\n\nNumbers Worksheets 2 3 4\n\nPatterns and sequences 2\n\nPlace value online worksheets tool\n\nSubtracting 2 digits numbers worksheets\n\nSubtracting 3 digits numbers worksheets\n\nTime worksheets for 2nd grade 2 3 4\n\nCompound words worksheets\n\nContractions English worksheets\n\nDaily language review worksheets 2 3\n\nDolch sight words flashcards and games 2 3\n\nOpposite words games and worksheets 2 3\n\nPlural nouns English grammar worksheets 1 2\n\nPrefixes and suffixes English worksheets 2" ]	[ null, "http://prek-8.com/images/logo.jpg", null, "http://www.google.com/images/poweredby_transparent/poweredby_FFFFFF.gif", null, "http://prek-8.com/2ndgrade/images/fractionWordproblems.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.79279274,"math_prob":0.8449264,"size":1582,"snap":"2019-26-2019-30","text_gpt3_token_len":277,"char_repetition_ratio":0.30228138,"word_repetition_ratio":0.11059908,"special_character_ratio":0.16182049,"punctuation_ratio":0.104166664,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9977862,"pos_list":[0,1,2,3,4,5,6],"im_url_duplicate_count":[null,null,null,null,null,10,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-07-17T00:47:48Z\",\"WARC-Record-ID\":\"<urn:uuid:6ff9c11c-25ff-4389-b2b4-02c707fb0bfa>\",\"Content-Length\":\"22719\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:1447dfad-7d62-4579-b57a-e1cd93ca8c62>\",\"WARC-Concurrent-To\":\"<urn:uuid:08c4e151-f682-4155-91be-c87c20770830>\",\"WARC-IP-Address\":\"216.97.239.120\",\"WARC-Target-URI\":\"http://prek-8.com/2ndgrade/fractionWorksheets_index4.php\",\"WARC-Payload-Digest\":\"sha1:ZQI7WVWSQB4UKMC6WAWH6CKPIXIFLYWW\",\"WARC-Block-Digest\":\"sha1:RYFLLVNLE4XITBNQYXWC3RAHWN37AMLU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-30/CC-MAIN-2019-30_segments_1563195525004.24_warc_CC-MAIN-20190717001433-20190717023433-00556.warc.gz\"}"}
http://forums.wolfram.com/mathgroup/archive/2001/May/msg00380.html	[ "", null, "", null, "", null, "", null, "", null, "", null, "", null, "Re: Q: Extract Elements from a List?\n\n• To: mathgroup at smc.vnet.net\n• Subject: [mg28978] Re: [mg28968] Q: Extract Elements from a List?\n• From: Anesh Sooklal <anesh at fermi.udw.ac.za>\n• Date: Thu, 24 May 2001 04:06:55 -0400 (EDT)\n• References: <200105230554.BAA01361@smc.vnet.net>\n• Sender: owner-wri-mathgroup at wolfram.com\n\n```Dear Robert,\n\nHere is a quick solution\n\ntest = {a, b, c, d, e, f, g, h, i, j}\nh[a_, d_] := Table[Part[test, n], {n, a, Part[Dimensions[test], 1], d}]\n\nHere a function is designed to give you the list output you desire.\nHere a is the first element to be extracted and d is the step size through\nthe list thereafter.\n\nLet me know how this turns out!!\n\nThank you,\nAnesh\n\nRobert Schuerhuber wrote:\n\n> hi!\n>\n> probably a very easy question, but i couldn't find an answer in the\n> mathematica-book:\n>\n> i need to extract elements from a list, starting th element number x and\n> than taking every yth element, eg:\n>\n> with\n>\n> list={a,b,c,d,e,f,g,h,i,j}\n> start=3;\n> step=2:\n>\n> i'd like to get the list\n>\n> {c,e,g,i}.\n>\n> how can i do this in the easiest way?\n>\n> regards, robert\n\n```\n\n• Prev by Date: Re: Help fitting Exponential curves\n• Next by Date: RE: Q: Extract Elements from a List?\n• Previous by thread: Q: Extract Elements from a List?\n• Next by thread: RE: Q: Extract Elements from a List?" ]	[ null, "http://forums.wolfram.com/mathgroup/images/head_mathgroup.gif", null, "http://forums.wolfram.com/mathgroup/images/head_archive.gif", null, "http://forums.wolfram.com/mathgroup/images/numbers/2.gif", null, "http://forums.wolfram.com/mathgroup/images/numbers/0.gif", null, "http://forums.wolfram.com/mathgroup/images/numbers/0.gif", null, "http://forums.wolfram.com/mathgroup/images/numbers/1.gif", null, "http://forums.wolfram.com/mathgroup/images/search_archive.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7497682,"math_prob":0.53259325,"size":1233,"snap":"2019-51-2020-05","text_gpt3_token_len":391,"char_repetition_ratio":0.1261188,"word_repetition_ratio":0.046296295,"special_character_ratio":0.34063262,"punctuation_ratio":0.2598684,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98649526,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-12-09T19:13:04Z\",\"WARC-Record-ID\":\"<urn:uuid:6f0b2265-e35b-448c-b4a1-6493c04803b7>\",\"Content-Length\":\"45012\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:5c525ecd-5129-4e2b-8b39-870b1770c56d>\",\"WARC-Concurrent-To\":\"<urn:uuid:b3cb45ed-fac2-4844-b1bb-de5c30080999>\",\"WARC-IP-Address\":\"140.177.205.73\",\"WARC-Target-URI\":\"http://forums.wolfram.com/mathgroup/archive/2001/May/msg00380.html\",\"WARC-Payload-Digest\":\"sha1:F4W5OFLT5ZE44BM7GWTAUFQCLEIJIOON\",\"WARC-Block-Digest\":\"sha1:3GC77VB5DT7ABZHSONG6K367HSTER53P\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-51/CC-MAIN-2019-51_segments_1575540521378.25_warc_CC-MAIN-20191209173528-20191209201528-00069.warc.gz\"}"}
https://mathoverflow.net/questions/243525/representation-of-a-group-and-its-center	[ "# representation of a group and its center\n\nLet $G$ be a finite group and let $Z(G)$ be its center. Let $C=\\mathrm{Rep}(G)$ be the category of finite dimensional representation of $G$. Let $D$ be the fusion subcategory of $C$ generated by $V \\otimes V^*$ for each (representative of isomorphism class of) simple object of $C$.\n\nI want to show that $D=\\mathrm{Rep}(G/Z(G))$.\n\nIt seems that this problem is claimed in Tensor Categories By Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik without a complete proof.\n\nI appreciate any help.\n\n• This depends on the field of coefficients. For example, it is false for $G=\\mathbb Z/3$ over $\\mathbb Q$. – Ben Wieland Jul 2 '16 at 17:02\n• @BenWieland How about when the base field is algebraically closed and characteristic zero? – Snow Jul 2 '16 at 18:01\n• This even works for compact groups using Doplicher Roberts duality. – Marcel Bischoff Jul 2 '16 at 23:24\n\nI'll work over $\\mathbb{C}$ below for simplicity, although it can be replaced by an algebraically closed field of characteristic $0$.\nLemma: The fusion subcategory of $\\text{Rep}(G)$ generated by some reps $V_i$ is $\\text{Rep}(G/N)$ where $N$ is the intersection of the kernels of $G$ acting on each $V_i$.\nThis follows in turn from the standard fact that $\\text{Rep}(H)$ is generated as a fusion category (in the above sense) by any faithful representation of $H$; see, for example, this MO question. From here it suffices to show that the intersection of the kernels of $G$ acting on $V \\otimes V^{\\ast}$ for each irrep $V$ is $Z(G)$.\nRecall that the regular representation $\\mathbb{C}[G]$ decomposes, as a representation of $G \\times G$ (acting on the left and right), as $\\bigoplus_V V \\boxtimes V^{\\ast}$ where $\\boxtimes$ denotes the external tensor product. If we restrict to the diagonal copy of $G$ in $G \\times G$, we get that the permutation representation coming from $G$ acting on itself by conjugation decomposes as $\\bigoplus_V V \\otimes V^{\\ast}$ where $\\otimes$ is now the ordinary tensor product of representations. The kernel of this permutation representation is clearly $Z(G)$, as desired." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9068248,"math_prob":0.9988742,"size":1366,"snap":"2020-34-2020-40","text_gpt3_token_len":349,"char_repetition_ratio":0.12628488,"word_repetition_ratio":0.04784689,"special_character_ratio":0.2510981,"punctuation_ratio":0.08064516,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999163,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-01T04:58:18Z\",\"WARC-Record-ID\":\"<urn:uuid:67acf317-db07-4604-a537-b18089b6dcd3>\",\"Content-Length\":\"130056\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a07f99ad-6b59-4933-b65c-1ea6bf388d93>\",\"WARC-Concurrent-To\":\"<urn:uuid:58dc3222-8a4c-4d19-ad05-43fada54e61b>\",\"WARC-IP-Address\":\"151.101.1.69\",\"WARC-Target-URI\":\"https://mathoverflow.net/questions/243525/representation-of-a-group-and-its-center\",\"WARC-Payload-Digest\":\"sha1:I66D66ILWIDCBJV3S3MQ3GUF6P6BQ7YJ\",\"WARC-Block-Digest\":\"sha1:KYHFWPR5SDGV7ABVZ7TDPIOERC26DL5W\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600402130615.94_warc_CC-MAIN-20201001030529-20201001060529-00640.warc.gz\"}"}
https://it.mathworks.com/help/symbolic/linear-optimization.html?s_tid=CRUX_lftnav	[ "Documentation\n\n# Linear Optimization\n\nAlgorithms for linear and integer programming\n\n### Note\n\nMuPAD® notebooks will be removed in a future release. Use MATLAB® live scripts instead.\n\nTo convert a MuPAD notebook file to a MATLAB live script file, see `convertMuPADNotebook`. MATLAB live scripts support most MuPAD functionality, although there are some differences. For more information, see Convert MuPAD Notebooks to MATLAB Live Scripts.\n\n `linopt::corners` Return the feasible corners of a linear program `linopt::maximize` Maximize a linear or mixed-integer program `linopt::minimize` Minimize a linear or mixed-integer program `linopt::plot_data` Plot the feasible region of a linear program `linopt::Transparent` Return the ordinary simplex tableau of a linear program `linopt::Transparent::autostep` Perform the next simplex step `linopt::Transparent::clean_basis` Delete all slack variables of the first phase from the basis `linopt::Transparent::convert` Transform the given tableau into a structure printable on screen `linopt::Transparent::dual_prices` Get the dual solution belonging to the given tableau `linopt::Transparent::phaseI_tableau` Start an ordinary phase one of a 2-phase simplex algorithm `linopt::Transparent::phaseII_tableau` Start phase two of a 2-phase simplex algorithm `linopt::Transparent::result` Get the basic feasible solution belonging to the given simplex tableau `linopt::Transparent::simplex` Finish the current phase of the 2-phase simplex algorithm `linopt::Transparent::suggest` Suggest the next step in the simplex algorithm `linopt::Transparent::userstep` Perform a user defined simplex step\n\n## Topics\n\nLinear Optimization\n\nUse only in the MuPAD Notebook Interface.\n\n#### Mathematical Modeling with Symbolic Math Toolbox\n\nGet examples and videos" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5517621,"math_prob":0.6114702,"size":1563,"snap":"2019-35-2019-39","text_gpt3_token_len":345,"char_repetition_ratio":0.20654266,"word_repetition_ratio":0.030927835,"special_character_ratio":0.18298145,"punctuation_ratio":0.2062937,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9829845,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-09-15T07:44:08Z\",\"WARC-Record-ID\":\"<urn:uuid:77194d1e-dd79-4f05-8549-301c3f204d9e>\",\"Content-Length\":\"66078\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:8fd27be1-416e-4d3b-804a-e25803220ac0>\",\"WARC-Concurrent-To\":\"<urn:uuid:620e28fa-b99c-42fb-947a-29af6298b85a>\",\"WARC-IP-Address\":\"104.110.193.39\",\"WARC-Target-URI\":\"https://it.mathworks.com/help/symbolic/linear-optimization.html?s_tid=CRUX_lftnav\",\"WARC-Payload-Digest\":\"sha1:PYHJ7YOAJOL6X73I7YCV52SKIFLQAQER\",\"WARC-Block-Digest\":\"sha1:3BGVQESJHPKFXFCQY5MLXCZ7GAAM76EZ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-39/CC-MAIN-2019-39_segments_1568514570830.42_warc_CC-MAIN-20190915072355-20190915094355-00181.warc.gz\"}"}
http://www.airsupplylab.com/electrical-measurements-and-circuits/135-ee2049-lab-09-divider-rules-for-voltage-and-current.html	[ "# Lab 09: Divider Rules for Voltage and Current\n\n## Objectives\n\n• Practice measuring voltages and current in series and parallel circuits.\n• Verify the divider rules for voltage and current.\n\n## Background\n\n#### Voltage Divider\n\n• In parallel circuits, the voltage across each of the component is always same.\n• in series circuits, the voltage drop across a resistor is directly proportional to the resistance of the resistor.", null, "Figure 1: A Parallel Circuit with n Resistors\n\nThe Voltage Divider Rule (VDR) states that the voltage across an element or across a series combination of elements in a series circuit is equal to the resistance of the element or series combination of elements divided by the total resistance of the series circuit and multiplied by the total impressed voltage:", null, "", null, "#### Current Divider\n\n• In series circuits, the current always remains same through all components.\n• In parallel circuits, the current doesn't remains same, instead it divides.", null, "Figure 2: A Parallel Circuit with n Resistors\n\nThe Current Divider Rule (CDR) states that the current through one of the parallel branches is equal to the resistance of the other branch divided by the sum of the resistances of the two parallel branches and multiplied by the total current entering the parallel branches. That is:", null, "", null, "## Questions\n\n1. Can you apply current division to obtain I1 and I2 for the circuit shown in the figure below? Explain briefly.", null, "" ]	[ null, "http://www.airsupplylab.com/images/lab/electricalMeasurementsCircuits/Lab09/00_VoltageDidiver1.png", null, "http://chart.apis.google.com/chart", null, "http://chart.apis.google.com/chart", null, "http://www.airsupplylab.com/images/lab/electricalMeasurementsCircuits/Lab09/00_CurrentDidiver1.png", null, "http://chart.apis.google.com/chart", null, "http://chart.apis.google.com/chart", null, "http://www.airsupplylab.com/images/lab/electricalMeasurementsCircuits/Lab09/Q1_Circuit.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8788086,"math_prob":0.93689543,"size":3814,"snap":"2019-13-2019-22","text_gpt3_token_len":828,"char_repetition_ratio":0.15170604,"word_repetition_ratio":0.1630094,"special_character_ratio":0.2139486,"punctuation_ratio":0.10662824,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9981964,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"im_url_duplicate_count":[null,3,null,null,null,null,null,3,null,null,null,null,null,3,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-03-24T03:38:08Z\",\"WARC-Record-ID\":\"<urn:uuid:1e0440b6-9d26-48ce-ba78-d10c880b818a>\",\"Content-Length\":\"35812\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c182436f-8ccb-4306-b091-69c51379f30a>\",\"WARC-Concurrent-To\":\"<urn:uuid:fd42692d-6273-411b-9e32-804ae7361fd3>\",\"WARC-IP-Address\":\"23.92.27.183\",\"WARC-Target-URI\":\"http://www.airsupplylab.com/electrical-measurements-and-circuits/135-ee2049-lab-09-divider-rules-for-voltage-and-current.html\",\"WARC-Payload-Digest\":\"sha1:KALKUNPC4HUCGUC2MB43YWRC5KVXE6B3\",\"WARC-Block-Digest\":\"sha1:G5FPVPYLEQB2PEEO25ISPQQBJIPDDCQD\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-13/CC-MAIN-2019-13_segments_1552912203168.70_warc_CC-MAIN-20190324022143-20190324044143-00117.warc.gz\"}"}
https://brainmass.com/math/functional-analysis/mobius-transformations-example-problem-123078	[ "Explore BrainMass\n\n# Mobius Transformations\n\nNot what you're looking for? Search our solutions OR ask your own Custom question.\n\nThis content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!\n\nSuppose T is a Mobius transformation such that the image of the real axis under T is the real axis. Prove that T may be written in the form T(z) = (az+b)/(cz+d) with a, b, c, and d real.\n\n© BrainMass Inc. brainmass.com March 4, 2021, 7:47 pm ad1c9bdddf\nhttps://brainmass.com/math/functional-analysis/mobius-transformations-example-problem-123078\n\n#### Solution Preview\n\nProof:\nSince T is a Mobius transformation, we can assume T(z)=(az+b)/(cz+d). Now we want to show that a,b,c,d can be written as all reals.\nFirst, T(0)=b/d=r is a real, then b=rd, ...\n\n#### Solution Summary\n\nA Mobius transformation is investigated. The solution is detailed and well presented.\n\n\\$2.49" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8690531,"math_prob":0.93344516,"size":867,"snap":"2021-21-2021-25","text_gpt3_token_len":233,"char_repetition_ratio":0.13325608,"word_repetition_ratio":0.0,"special_character_ratio":0.25720876,"punctuation_ratio":0.17277487,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97687405,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-20T21:27:38Z\",\"WARC-Record-ID\":\"<urn:uuid:75ca495f-2451-4bdd-803f-0c3f6c0bd9ac>\",\"Content-Length\":\"330554\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a1d18955-c2ce-4981-983d-5587b5c49d78>\",\"WARC-Concurrent-To\":\"<urn:uuid:4e590bfb-14c6-47f0-8df3-c65547da246d>\",\"WARC-IP-Address\":\"104.26.4.245\",\"WARC-Target-URI\":\"https://brainmass.com/math/functional-analysis/mobius-transformations-example-problem-123078\",\"WARC-Payload-Digest\":\"sha1:53PTJ37I7KLEENP6HZZJ5FZTVGA74NVP\",\"WARC-Block-Digest\":\"sha1:MI2J3GL5VUVLH5MGDIB55CDTDHZM4Y7N\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623488257796.77_warc_CC-MAIN-20210620205203-20210620235203-00299.warc.gz\"}"}
https://brainmass.com/math/functional-analysis/polar-equations-conic-section-116335	[ "Explore BrainMass\n\n# Polar Equations: Conic Section\n\nThis content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!\n\nFind an equation in x and y for the conic section with polar equation r=1/1+cos\n\nhttps://brainmass.com/math/functional-analysis/polar-equations-conic-section-116335\n\n#### Solution Preview\n\nHello and thank you for posting your posting to BrainMass.\n\nWe have the polar identities:\n\nx = r cos(q)\ny = r sin ...\n\n#### Solution Summary\n\nA polar equation is written in x,y form. The solution is detailed and well presented. The response received a rating of \"5/5\" from the student who originally posted the question.\n\n\\$2.19" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.89768547,"math_prob":0.9292483,"size":701,"snap":"2020-24-2020-29","text_gpt3_token_len":180,"char_repetition_ratio":0.1348637,"word_repetition_ratio":0.0,"special_character_ratio":0.24393724,"punctuation_ratio":0.14893617,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98480076,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-05T01:39:44Z\",\"WARC-Record-ID\":\"<urn:uuid:db2d2a05-4ce6-4eb6-9816-9c1f8abd4628>\",\"Content-Length\":\"37489\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:1a99e5e8-28ef-40b4-8770-2eab4bcc2002>\",\"WARC-Concurrent-To\":\"<urn:uuid:f49ca151-e8eb-43d1-a5a9-067f1d131405>\",\"WARC-IP-Address\":\"65.39.198.123\",\"WARC-Target-URI\":\"https://brainmass.com/math/functional-analysis/polar-equations-conic-section-116335\",\"WARC-Payload-Digest\":\"sha1:M3GHTV7HR5YZ6OPYFJSNEFFLGIOO3IMF\",\"WARC-Block-Digest\":\"sha1:YTF7FJD5RHQCLVP6VBQ3CN26QIWK2EFI\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593655886802.13_warc_CC-MAIN-20200704232817-20200705022817-00435.warc.gz\"}"}
http://bbs.dqzjos.com.cn/TmmDA/wuiope.htm	[ "702除以6等于多少\n\n## 计算726÷6=702÷9=\n\n3.除以4没有余数,除以7没有余数. 4. 竖式计算. ①5728= ②3643= ③9864= ④824÷8= ⑤906÷6= ... 726÷6=702÷9=\n\n## 三年级下数学单元试卷 - 道客巴巴\n\n702÷6= 3947 5329 7869 三、估算我第一 239÷3≈ 452÷9≈ 318÷8≈ 97÷5≈ 202÷5≈ 118÷... ( ) 5、0 除以任何不是 0 的数都得 0. ( ) 潍坊锦绣学校 2011—2012\n\n## 青岛版小学三年级数学上册第一次月考试题 - 道客巴巴\n\n(10 分) 816÷8= ※ 702÷6= 652÷4= ※ 645÷5= 4、脱式计算. (9 分) 124-735÷7 398÷6 376÷(123-119) 二、填一填, 我最棒. (13 分) 1、 0 乘任何数都得( )\n\n## 2019三年级下数学期中试题综合考练611415人教新课标语文_百度文库\n\n702÷3= 528÷4= 450÷6= 六、解决问题 24. 25. 26.学校组织三年级 68 名同学和 3 位老师去... 除以任何数不等于 0 的数,商都是 0. 【解答】解:解:因为:商除数=被除数, 即:0...\n\n## 用竖式计算.①725=②1078=③702÷6=④702÷6=_作业帮\n\n①725=360②1078=856③702÷6=117④702÷6=117\n\n## 722除以6等于多少\n\n+(6+6)/6=654321+2=720+2=722 722÷4等于(180); 余数是(2); 722÷4 =1802 答案:约等... 解题:295除以6等于49.166,根据四舍五入的规则,一个数精确到个位看十分位上的数,\n\n## 三年级数学第一单元测试题11759 - 道客巴巴\n\n702÷6= 652÷4= 880÷7= 645÷5= 三、估算我第一 239÷3≈ 452÷9≈ 318÷8≈ 97÷5≈ 202÷5≈ ... 除以任何不是 0 的数都得 0. ( ) 五、列式计算我最强 1、把 345 平均分成\n\n## 726除以6等于几写竖式.除以竖式数学理工学科学习\n\n702除以6等于,竖式如下:拓展资料:竖式,指的是每一个过渡数都是由上一个过渡数变化而后,上一个过渡数的个位数乘以2,如果需要进位,则往前面进1,然后个位升十位,以此类推,而个位上补...\n\n## 青岛版三年级下册第一单元练习题 - 道客巴巴\n\n702÷6=702÷6= 652÷4= 880÷7= 645÷5=652÷4= •二、估算我第一•239÷3≈ 452÷9≈ 318... 一个三位数除以一位数, 商一定是三位数.()• 4、在有余数的除法中, 用商*除数=被除..." ]	[ null ]	{"ft_lang_label":"__label__zh","ft_lang_prob":0.8065511,"math_prob":0.98227423,"size":1180,"snap":"2021-04-2021-17","text_gpt3_token_len":1035,"char_repetition_ratio":0.09013605,"word_repetition_ratio":0.125,"special_character_ratio":0.6830509,"punctuation_ratio":0.2457627,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.962147,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-01-20T07:52:51Z\",\"WARC-Record-ID\":\"<urn:uuid:277db951-d95b-437b-9b22-099c9edd5cf8>\",\"Content-Length\":\"63183\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ce2d22e9-9f43-4bf5-9811-8a470f431ba0>\",\"WARC-Concurrent-To\":\"<urn:uuid:073d38dc-edd5-45ed-8197-eabc8443579b>\",\"WARC-IP-Address\":\"137.175.80.45\",\"WARC-Target-URI\":\"http://bbs.dqzjos.com.cn/TmmDA/wuiope.htm\",\"WARC-Payload-Digest\":\"sha1:N7HE6Q6I2BK7AO5PAIGOURV6H4OO6WOK\",\"WARC-Block-Digest\":\"sha1:OPDR3STLOE23PIT6XIVIQYRVL3YUJC32\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-04/CC-MAIN-2021-04_segments_1610703519923.26_warc_CC-MAIN-20210120054203-20210120084203-00320.warc.gz\"}"}
https://ay201b.wordpress.com/2011/03/04/williams-mckee/	[ "# Harvard Astronomy 201b\n\n## ARTICLE: The Galactic Distribution of OB Associations in Molecular Clouds\n\nIn Uncategorized on March 4, 2011 at 5:19 am\n\n### Read the Paper by J.P. Williams and C.F. McKee (1997)\n\nSummary by Vicente Rodriguez Gomez\n\n### Abstract\n\nMolecular clouds account for half of the mass of the interstellar medium interior to the solar circle and for all current star formation. Using cloud catalogs of two CO surveys of the first quadrant, we have fitted the mass distribution of molecular clouds to a truncated power law in a similar manner as the luminosity function of OB associations in the companion paper to this work. After extrapolating from the first quadrant to the entire inner Galaxy, we find that the mass of cataloged clouds amounts to only 40% of current estimates of the total Galactic molecular mass. Following Solomon & Rivolo, we have assumed that the remaining molecular gas is in cold clouds, and we normalize the distribution accordingly. The predicted total number of clouds is then shown to be consistent with that observed in the solar neighborhood where cloud catalogs should be more complete. Within the solar circle, the cumulative form of the distribution is", null, "c(>M)=105[(Mu/M)0.6-1], where", null, "c is the number of clouds, and Mu = 6 × 106 M", null, "is the upper mass limit. The large number of clouds near the upper cutoff to the distribution indicates an underlying physical limit to cloud formation or destruction processes. The slope of the distribution corresponds to d", null, "c/dM", null, "M−1.6, implying that although numerically most clouds are of low mass, most of the molecular gas is contained within the most massive clouds.\n\nThe distribution of cloud masses is then compared to the Galactic distribution of OB association luminosities to obtain statistical estimates of the number of massive stars expected in any given cloud. The likelihood of massive star formation in a cloud is determined, and it is found that the median cloud mass that contains at least one O star is ~105 M", null, ". The average star formation efficiency over the lifetime of an association is about 5% but varies by more than 2 orders of magnitude from cloud to cloud and is predicted to increase with cloud mass. O stars photoevaporate their surrounding molecular gas, and even with low rates of formation, they are the principal agents of cloud destruction. Using an improved estimate of the timescale for photoevaporation and our statistics on the expected numbers of stars per cloud, we find that 106 M", null, "giant molecular clouds (GMCs) are expected to survive for about 3 × 107 yr. Smaller clouds are disrupted, rather than photoionized, by photoevaporation. The porosity of H II regions in large GMCs is shown to be of order unity, which is consistent with self-regulation of massive star formation in GMCs. On average, 10% of the mass of a GMC is converted to stars by the time it is destroyed by photoevaporation.\n\n### Introduction\n\nThis article by Jonathan P. Williams and Christopher F. McKee (1997) was motivated by the question “can one determine, a priori, how many stars are likely to form in a molecular cloud?” It turns out that this is possible for low-mass stars (McKee 1989), but not for more massive ones, such as O and B types, because of the enormous amounts of ionizing and dissociating radiation they produce. This radiation can quickly destroy their molecular environment and form large HII regions.\n\nIn view of the complex interaction between massive stars and their natal molecular clouds, Williams and McKee adopt an empirical approach to infer the number of massive stars that have already formed in molecular clouds. In particular, they focus on giant molecular clouds (GMCs), since they are large enough to generate OB associations.\n\nThe first half of the publication deals with estimating the mass spectrum of molecular clouds in the Galaxy, using existing data from CO surveys. In the second half, this mass spectrum is combined with a luminosity distribution of OB associations (McKee & Williams, 1997) to determine the distribution of OB associations within a given cloud. Finally, this result is used to calculate many other quantities and distributions of interest, such as (1) the number of cloud-association pairs of a given mass-luminosity in the Galaxy, (2) the probability that a cloud does not form any massive stars, (3) the most likely brightest association in a cloud, (4) the distribution of star formation efficiencies for a given association within a cloud, (5) the average star formation efficiency for all the associations within a cloud, (6) the filling factor of HII regions in GMCs, and (7) the rate at which the HII regions destroy the clouds.\n\n### Determination of the mass spectrum of molecular clouds\n\nWilliams and McKee used the data from four cloud catalogs: three of the first quadrant (DECT, SRBY and SYCSW), which contain a vast amount of GMCs, and one of the solar neighborhood (Dame et al. 1986), essentially for calibration purposes. There were four steps involved in the process:\n\n1. Cloud masses from the different catalogs, which are by no means absolute, were adjusted to a uniform set of parameters: (1) X, the conversion factor of CO to", null, "$H_2,$ given by", null, "$X=N_{H_2}/W_{CO},$ and (2)", null, "$\\alpha_{vir},$ a parameter in the formula for the virial mass,", null, "$M_{vir} = 5 R \\sigma^2/\\alpha_{vir}G.$ Clouds for which the distance was ambiguous were removed from the analysis.\n2. Clouds were binned by mass and the resulting distribution was modeled with a truncated power law. The upper limit was the largest GMC in the catalogue, of about", null, "$6 \\times 10^6 M_{\\odot}.$\n3. The fit was extrapolated to lower masses, which cannot be observed, and the distribution was integrated. The total cloud mass was found to be 2.5 times less than the total mass of molecular gas measured in the inner Galaxy (Bronfman et al. 1988). Therefore, the surveys do not represent the true cloud distribution and the model had to be modified in one of two ways: performing a uniform scaling of the distribution (model A), or adopting a steeper distribution (model B).\n4. A comparison with the number of clouds in the solar neighborhood favored model A over model B.", null, "Figure 1. Distribution of the number of clouds with respect to mass for the solar neighborhood (d < 1 kpc).\n\nThe resulting distribution of clouds in the Galaxy was found to be", null, "$\\frac{d\\mathcal{N}_c}{d\\ln M} = 63\\left(\\frac{6\\times 10^6 M_{\\odot}}{M}\\right)^{0.6},$\n\nfor", null, "$M \\le 6\\times 10^6 M_{\\odot}$ and galactocentric distances between 1.7 and 8.5 kpc.\n\n### Comparison with the distribution of OB association luminosities\n\nHaving determined the number of OB associations in McKee & Williams (1997), the next step was to obtain the distribution of associations of luminosity S in a given cloud of mass M. The first obvious constraint was that the overall number of associations of luminosity S summed over all clouds must equal the total number of associations of luminosity S in the Galaxy. But even with this constraint, the allocation of associations in clouds is not uniquely determined. To solve this, Williams & McKee introduced two additional assumptions:\n\n1. The luminosity of the brightest association in a cloud is less than some maximum value,", null, "$S \\le S_{max}(M).$\n2. The number of associations of each luminosity S in a cloud of mass M is half of that expected in a cloud of mass 2M.\n\nWith these conditions, Williams and McKee determined the average number of OB associations for clouds of a given mass, and then assumed Poisson statistics for the distribution of OB associations about the mean. Many interesting quantities can be determined from these results, such as the distribution for the brightest association per cloud, the total number of cloud-association pairs in the Galaxy, the probability that a cloud be devoid of O stars (figure 2), and the star formation efficiencies (SFE), i.e. what fraction of the cloud mass is transformed into stars (figure 3).", null, "", null, "Figure 4. Destruction timescale for cloud photoevaporation. The solid line includes the effect of overlapping associations, while the dotted line does not.\n\nUsing estimates of the destructive effect of an OB association on a cloud, Williams and McKee also estimated the timescale over which OB associations ionize and disrupt their molecular surroundings (figure 4). The results show that high-mass clouds, with", null, "$M \\begin{smallmatrix} > \\\\ \\sim \\end{smallmatrix} 3 \\times 10^5 M_{\\odot},$ are destroyed by large numbers of small associations over a timescale of 30-40 Myr, while low-mass clouds are disrupted by O stars, rather than photoionized.\n\nFinally, comparing the results on cloud photoevaporation with the previously obtained efficiencies shows something unexpected: the average SFE over the life of an association is 5%, while the average SFE over the life of a cloud is 10%. If this discrepancy is real and not an artifact of the model, it can be explained by assuming that only half of the GMCs are actively forming stars. This way the lifetime of a star-forming cloud becomes comparable to the observed lifetime of associations, about 20 Myr. Nevertheless, Williams and McKee confirm once again that star formation is a fairly inefficient process: there are clouds with a considerable amount of molecular material and hardly any OB associations." ]	[ null, "https://i1.wp.com/ej.iop.org/icons/Entities/calN.gif", null, "https://i1.wp.com/ej.iop.org/icons/Entities/calN.gif", null, "https://i2.wp.com/ej.iop.org/icons/Entities/sun.gif", null, "https://i1.wp.com/ej.iop.org/icons/Entities/calN.gif", null, "https://i1.wp.com/ej.iop.org/icons/Entities/vprop.gif", null, "https://i2.wp.com/ej.iop.org/icons/Entities/sun.gif", null, "https://i2.wp.com/ej.iop.org/icons/Entities/sun.gif", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://ay201b.files.wordpress.com/2011/03/solar_neighborhood.png", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://ay201b.files.wordpress.com/2011/03/efficiencies.png", null, "https://ay201b.files.wordpress.com/2011/03/cloud_lifetime.png", null, "https://s0.wp.com/latex.php", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9342305,"math_prob":0.9561013,"size":8540,"snap":"2021-31-2021-39","text_gpt3_token_len":1766,"char_repetition_ratio":0.17537488,"word_repetition_ratio":0.020523708,"special_character_ratio":0.20222482,"punctuation_ratio":0.08631714,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96253043,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],"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,1,null,null,null,null,null,null,null,1,null,1,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-09-17T15:21:11Z\",\"WARC-Record-ID\":\"<urn:uuid:c7608e96-4c6b-4a70-b2b7-1e72aa45ba70>\",\"Content-Length\":\"90886\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d33ee83a-4aa4-41bf-a2f6-0e2065ba743d>\",\"WARC-Concurrent-To\":\"<urn:uuid:09750886-1ce0-494c-8be1-92b44cf03fb1>\",\"WARC-IP-Address\":\"192.0.78.12\",\"WARC-Target-URI\":\"https://ay201b.wordpress.com/2011/03/04/williams-mckee/\",\"WARC-Payload-Digest\":\"sha1:DP735GEFKZ4B2QGWZIN6TPPYXQ7UETXY\",\"WARC-Block-Digest\":\"sha1:KCGQI6UQWLRAPZUSLOS7Q3WN2QTCT46H\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780055684.76_warc_CC-MAIN-20210917151054-20210917181054-00177.warc.gz\"}"}
https://www3.math.tu-berlin.de/geometrie/jtem/function/api/de/jtem/function/Domain.html	[ "de.jtem.function\n\n## Class Domain\n\n• ```public abstract class Domain\nextends Object```\nThis class provides interfaces to be implemented by classes modelling functions defined on various mathematically important standard spaces. The target space of these functions will be specified by subinterfaces. The domains selected here have been chosen according to the following criteria: The domains should include\n• The set of integers between 0 and n-1. This is the canonical choice of a finite index set and maps from such a domain to some space A are usually called \"n-tuples of elements in A\".\n• The set of non-negative integers. Maps from the non-negative integers to some space A are usually called \"sequences of Elements in A\".\n• The set of integers.\n• The closed interval [a,b] between two real numbers a and b. Maps from the such an interval into some space A can be called \"paths in A\".\n• The set of non-negative real numbers.\n• The real line.\n• The circle S (the only connected one-dimensional compact manifold). Functions on S are described by periodic functions on the real line.\n• The \"discrete circle\" (a cylic graph). Functions on the discrete circle are described by periodic sequences.\nFurthermore we provide some Cartesian products of two or three of the above spaces. In the case of the cartesian product of two finite sets of indices the maps from such a domain to a set A can be called \"matrices of elements in A\". In analogy, we adopt matrix terminology for the naming of methods by refering to the first argument of a function on a Cartesian product as the \"row\" argument and the second as the \"column\" argument. In the case of a threefold cartesian product the third argument is said to specify the \"layer\" which means we imagine a stack of matrices (two dimensional arrangements of elements of A) with several layers in a third dimension.\n\nThe actual choice of product spaces was guided by the following principles:\n\n• To allow for interpolation, replacing a factor corresponding to an integer argument by one corresponding to a real argument should always be possible.\n• If one of the factors has an interpretation of \"time\" (like in a dynamical system or evolution equation), then we demand it to be the first factor. Since the main motivation for us to include the non-negative integers and the non-negative reals is in fact to serve as the \"time\" of some irreversible dynamical system (like iterating a non-invertible function or solving a parabolic PDE), these two spaces only occur as the first factor.\n• Periodicity is natural only for spacelike variables, so periodic factors only occur at the end. \"Periodic\" will then mean \"periodic in the last variable\", \"doubly periodic\" will mean \"periodic in the last two variables\".\n• To keep the number of interfaces within reasonable bounds, mixing of integer and real factors is not supported in the case of three factors. Moreover, the last two factors must be of the same type.\n• ### Nested Class Summary\n\nNested Classes\nModifier and Type Class and Description\n`static interface ` `Domain.DoublyPeriodicOnIndexCrossIntegersCrossIntegers`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the Cartesian product of a fine set of indices with the integer plane and which are periodic in the last two arguments.\n`static interface ` `Domain.DoublyPeriodicOnIntegersCrossIntegers`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of the set of integers with itself and which are periodic in the both variables.\n`static interface ` `Domain.DoublyPeriodicOnIntegersCrossIntegersCrossIntegers`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the Cartesian product of the non-negative integers with the integer plane and which are periodic in the last two arguments.\n`static interface ` `Domain.DoublyPeriodicOnIntervalCrossRealsCrossReals`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the Cartesian product of a closed interval with the real plane and which are periodic in the last two arguments.\n`static interface ` `Domain.DoublyPeriodicOnNonNegativeIntegersCrossIntegersCrossIntegers`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the Cartesian product of the non-negative integers with the integer plane and which are periodic in the last two arguments.\n`static interface ` `Domain.DoublyPeriodicOnNonNegativeRealsCrossRealsCrossReals`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the Cartesian product of the non-negative real numbers with the real plane and which are periodic in the last two arguments.\n`static interface ` `Domain.DoublyPeriodicOnRealsCrossReals`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the real plane and which are periodic in both variables.\n`static interface ` `Domain.DoublyPeriodicOnRealsCrossRealsCrossReals`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the Cartesian product of three copies of the real line and which are periodic in the last two arguments.\n`static interface ` `Domain.OnIndex`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the set of integers between zero and n-1 (elements of which are referered to as \"indices).\n`static interface ` `Domain.OnIndexCrossIndex`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of two finite sets of indices.\n`static interface ` `Domain.OnIndexCrossIndexCrossIndex`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the Cartesian product of three fine sets of indices.\n`static interface ` `Domain.OnIndexCrossIntegers`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of a finite set of indices with the set of integers.\n`static interface ` `Domain.OnIndexCrossIntegersCrossIntegers`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the Cartesian product of a fine set of indices with the integer plane.\n`static interface ` `Domain.OnIndexCrossInterval`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of a finite set of indices with a closed interval on the real line.\n`static interface ` `Domain.OnIndexCrossReals`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of a finite set of indices with the real line.\n`static interface ` `Domain.OnIntegers`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have the integers as their domain of definition.\n`static interface ` `Domain.OnIntegersCrossIndex`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of the set of integers with a finite set of indices.\n`static interface ` `Domain.OnIntegersCrossIndexCrossIndex`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the Cartesian product of the non-negative integers with two finite sets of indices.\n`static interface ` `Domain.OnIntegersCrossIntegers`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of the set of integers with itself.\n`static interface ` `Domain.OnIntegersCrossIntegersCrossIntegers`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the Cartesian product of the non-negative integers with the integer plane.\n`static interface ` `Domain.OnIntegersCrossInterval`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of the set of integers with a closed interval on the real line.\n`static interface ` `Domain.OnIntegersCrossReals`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of the set of integers with the real line.\n`static interface ` `Domain.OnInterval`\nThis interface models mathematical objects (like maps into some other set or differential equations) that have a closed interval of real numbers as their domain of definition.\n`static interface ` `Domain.OnIntervalCrossIndex`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of a closed interval with a finite sets of indices.\n`static interface ` `Domain.OnIntervalCrossIntegers`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of a closed interval with the set of integers.\n`static interface ` `Domain.OnIntervalCrossInterval`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of a closed interval on the real line with another closed interval.\n`static interface ` `Domain.OnIntervalCrossIntervalCrossInterval`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the Cartesian product of three closed intervals.\n`static interface ` `Domain.OnIntervalCrossReals`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of a closed interval with the real line.\n`static interface ` `Domain.OnIntervalCrossRealsCrossReals`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the Cartesian product of a closed interval with the real plane.\n`static interface ` `Domain.OnNonNegativeIntegers`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the set of non-negative integers.\n`static interface ` `Domain.OnNonNegativeIntegersCrossIndex`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of the set of non-negative integers with a finite set of indices.\n`static interface ` `Domain.OnNonNegativeIntegersCrossIndexCrossIndex`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the Cartesian product of the non-negative integers with two finite sets of indices.\n`static interface ` `Domain.OnNonNegativeIntegersCrossIntegers`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of the set of non-negative integers with the set of integers.\n`static interface ` `Domain.OnNonNegativeIntegersCrossIntegersCrossIntegers`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the Cartesian product of the non-negative integers with the integer plane.\n`static interface ` `Domain.OnNonNegativeIntegersCrossInterval`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of the set of non-negative integers with a closed interval on the real line.\n`static interface ` `Domain.OnNonNegativeIntegersCrossReals`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of the set of non-negative integers with the real line.\n`static interface ` `Domain.OnNonNegativeReals`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have the non-negative real numbers as their domain of definition.\n`static interface ` `Domain.OnNonNegativeRealsCrossIndex`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of the set of non-negative real numbers with a finite set of indices.\n`static interface ` `Domain.OnNonNegativeRealsCrossIntegers`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of the set of non-negative real numbers with the set of integers.\n`static interface ` `Domain.OnNonNegativeRealsCrossInterval`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of the set of non-negative real numbers with a closed interval on the real line.\n`static interface ` `Domain.OnNonNegativeRealsCrossIntervalCrossInterval`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the Cartesian product of the non-negative real numbers with two closed intervals.\n`static interface ` `Domain.OnNonNegativeRealsCrossReals`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of the set of non-negative real numbers with the real line.\n`static interface ` `Domain.OnNonNegativeRealsCrossRealsCrossReals`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the Cartesian product of the non-negative real numbers with the real plane.\n`static interface ` `Domain.OnReals`\nThis interface models mathematical objects (like maps into some other set or differential equations) that have the real line as their domain of definition.\n`static interface ` `Domain.OnRealsCrossIndex`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of the real line with a finite set of indices.\n`static interface ` `Domain.OnRealsCrossIntegers`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of the real line with the set of integers.\n`static interface ` `Domain.OnRealsCrossInterval`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of the real line with a closed interval.\n`static interface ` `Domain.OnRealsCrossIntervalCrossInterval`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the Cartesian product of the non-negative real numbers with two closed intervals.\n`static interface ` `Domain.OnRealsCrossReals`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the real plane.\n`static interface ` `Domain.OnRealsCrossRealsCrossReals`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the Cartesian product of three copies of the real line.\n`static interface ` `Domain.PeriodicOnIndexCrossIntegers`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of a finite set of indices with the set of integers and which are periodic in the second variable.\n`static interface ` `Domain.PeriodicOnIndexCrossReals`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of a finite set of indices with the real line and which are periodic in the second variable.\n`static interface ` `Domain.PeriodicOnIntegers`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have the integers as their domain of definition and which are periodic.\n`static interface ` `Domain.PeriodicOnIntegersCrossIntegers`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of the set of integers with itself and which are periodic in the second variable.\n`static interface ` `Domain.PeriodicOnIntegersCrossReals`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of the set of non-negative integers with the real line and which are periodic in the second variable.\n`static interface ` `Domain.PeriodicOnIntervalCrossIntegers`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of a closed interval with the set of integers and which are periodic in the second variable.\n`static interface ` `Domain.PeriodicOnIntervalCrossReals`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of a closed interval with the real line and which are periodic in the second variable.\n`static interface ` `Domain.PeriodicOnNonNegativeIntegersCrossIntegers`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of the set of non-negative integers with the integers and which are periodic in the second variable.\n`static interface ` `Domain.PeriodicOnNonNegativeIntegersCrossReals`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of the set of non-negative integers with the real line and which are periodic in the second variable.\n`static interface ` `Domain.PeriodicOnNonNegativeRealsCrossIntegers`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of the set of non-negative real numbers with the integers which are periodic in the second variable.\n`static interface ` `Domain.PeriodicOnNonNegativeRealsCrossReals`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of the set of non-negative real numbers with the real line and which are periodic in the second variable.\n`static interface ` `Domain.PeriodicOnReals`\nThis interface models mathematical objects (like maps into some other set or differential equations) that have the real line as their domain of definition and which are periodic.\n`static interface ` `Domain.PeriodicOnRealsCrossIntegers`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of the real line with the set of integers and which are periodic in the second variable.\n`static interface ` `Domain.PeriodicOnRealsCrossReals`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the real plane and which are periodic in the second variable.\n`static interface ` `Domain.TriplyPeriodicOnIntegersCrossIntegersCrossIntegers`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the cartesian product of three copies of the integers and which are periodic in all three variables.\n`static interface ` `Domain.TriplyPeriodicOnRealsCrossRealsCrossReals`\nThis interface models mathematical objects (like maps into some other set or difference equations) that have as their domain of definition the Cartesian product of three copies of the real line and which are periodic in the all three arguments.\n• ### Constructor Summary\n\nConstructors\nConstructor and Description\n`Domain()`\n\n• ### Methods inherited from class java.lang.Object\n\n`clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait`\n• ### Constructor Detail\n\n• #### Domain\n\n`public Domain()`\n\njTEM" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8720736,"math_prob":0.9298199,"size":18607,"snap":"2021-21-2021-25","text_gpt3_token_len":3748,"char_repetition_ratio":0.14895447,"word_repetition_ratio":0.8141194,"special_character_ratio":0.16655022,"punctuation_ratio":0.04820587,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9956328,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-16T11:32:03Z\",\"WARC-Record-ID\":\"<urn:uuid:4b06ebb5-6bb0-460e-8ca1-e51f610cd732>\",\"Content-Length\":\"51093\",\"Content-Type\":\"application/http; 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https://codecrawl.com/2014/07/10/php-returning-a-value-from-a-function/	[ "PHP: Returning a value from a function\n\nThis entry is part 33 of 54 in the series PHP Tutorial\n\nTo return value from functions, use the return keyword with a value to return.\n\n<?php\n\nfunction restaurant_check(\\$meal, \\$tax, \\$tip) {\n\\$tax_amount = \\$meal * (\\$tax / 100);\n\\$tip_amount = \\$meal * (\\$tip / 100);\n\\$total_amount = \\$meal + \\$tax_amount + \\$tip_amount;\n\nreturn \\$total_amount;\n}\n\n\\$total = restaurant_check(25,7,15);\n\necho \"Total spent is \\$total\";\n\nSeries Navigation<< PHP: Built-in functions\nPHP: Returning an array in a function >>" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.69863814,"math_prob":0.98166305,"size":344,"snap":"2019-43-2019-47","text_gpt3_token_len":98,"char_repetition_ratio":0.18235295,"word_repetition_ratio":0.0,"special_character_ratio":0.35465115,"punctuation_ratio":0.22807017,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.975699,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-14T02:14:33Z\",\"WARC-Record-ID\":\"<urn:uuid:75876caf-5658-4259-908b-b8c40b31c930>\",\"Content-Length\":\"57125\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d7235aee-7d40-4ebd-b03b-30d05e8b4b74>\",\"WARC-Concurrent-To\":\"<urn:uuid:5fcfb507-dd17-4d92-a498-15c3ed711d4b>\",\"WARC-IP-Address\":\"162.144.198.23\",\"WARC-Target-URI\":\"https://codecrawl.com/2014/07/10/php-returning-a-value-from-a-function/\",\"WARC-Payload-Digest\":\"sha1:6BYQGKN6NDTKX73AURK3OAHAPWC7PFLU\",\"WARC-Block-Digest\":\"sha1:WO4VLS7VIXGJRL45GLO2TIDVAJNSBYBH\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570986648481.7_warc_CC-MAIN-20191014003258-20191014030258-00500.warc.gz\"}"}
https://mocktime.com/uncategorized/iit-advance-physics-quiz/	[ "You are here\nHome > Uncategorized > IIT Advance Physics Quiz\n\nQ1.A thermocouple is made from two metals, Antimony and Bismuth. If one junction of the couple is kept hot and the other is kept cold, then, an electric current will\n(a) flow from Antimony to Bismuth at the hot junction\n(b) flow from Bismuth to Antimony at the cold junction\n(c) now flow through the thermocouple\n(d) flow from Antimony to Bismuth at the cold junction\n\nQ2. A thin glass (refractive index 1.5) lens has optical power of – 5 D in air. Its optical power in a liquid medium with refractive index 1.6 will be\n(a) – 1D\n(b) 1 D\n(c) – 25 D\n(d) 25 D\n\nQ3. A thin horizontal circular disc is rotating about a vertical axis passing through its centre. An insect is at rest at a point near the rim of the disc. The insect now moves along a diameter of the disc to reach its other end. During the journey of the insect, the angular speed of the disc.\n(a) continuously decreases\n(b) continuously increases\n(c) first increases and then decreases\n(d) remains unchanged\n\nQ4. A tuning fork of known frequency 256 Hz makes 5 beats per second with the vibrating string of a piano. The beat frequency decreases to 2 beats per second when the tension in the piano string is slightly increased. The frequency of the piano string before increasing the tension was\n(a) 256 + 2 Hz\n(b) 256 – 2 Hz\n(c) 256 – 5 Hz\n(d) 256 + 5 Hz\n\nQ5. A uniform chain of length 2 m is kept on a table such that a length of 60 cm hangs freely from the edge of the table. The total mass of the chain is 4 kg. What is the work done in pulling the entire chain on the table?\n(a) 12 J\n(b) 3.6 J\n(c) 7.2 J\n(d) 1200 J\n\nQ6. A wave y = α sin(ωt–kx) on a string meets with another wave producing a node at x =0. Then the equation of the unknown wave is\n(a) y = α sin(ωt + kx)\n(b) y = – α sin(ωt + kx) (c) y = α sin(ωt – kx) (d) y = -α sin(ωt – kx)\n\nQ7. A wire elongates by l mm when a load W is hanged from it. If the wire goes over a pulley and two weights W each are hung at the two ends, the elongation of the wire will be (in mm)\n(a) l\n(b) 2l\n(c) zero\n(d) l/2\n\nQ8. A wire when connected to 220 V mains supply has power dissipation P1. Now the wire is cut into two equal pieces which are connected in parallel to the same supply. Power dissipation in this case is P2. Then P2: P1 is\n(a) 1\n(b) 4\n(c) 2\n(d) 3\n\nQ9. A working transistor with its three legs marked P, Q and R is tested using a multimeter. No conduction is found between P and Q. By connecting the common (negative) terminal of the multimeter to R and the other (positive) terminal to P or Q, some resistance is seen on the multimeter. Which of the following is true for the transistor?\n(a) It is an npn transistor with R as base\n(b) It is a pnp transistor with R as collector\n(c) It is a pnp transistor with R as emitter\n(d) It is an npn transistor with R as collector\n\nQ10. According to Einstein’s photoelectric equation, the plot of the kinetic energy of the emitted photo electrons from a metal Vs the frequency, of the incident radiation gives as straight the whose slope\n(a) depends both on the intensity of the radiation and the metal used\n(b) depends on the intensity of the radiation\n(c) depends on the nature of the metal used\n(d) is the same for the all metals and independent of the intensity of the radiation\n1.d2.b3.C4.c5.b6.b7.A8.b9.b10.d\n\nTop\nerror: Content is protected !!" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.89106095,"math_prob":0.9912,"size":3266,"snap":"2019-26-2019-30","text_gpt3_token_len":927,"char_repetition_ratio":0.11833231,"word_repetition_ratio":0.044615384,"special_character_ratio":0.2783221,"punctuation_ratio":0.075630255,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9654628,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-06-19T11:41:39Z\",\"WARC-Record-ID\":\"<urn:uuid:bb9a1fa1-5e9f-4398-bff5-0630e53042a5>\",\"Content-Length\":\"43011\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7ae3406d-676e-4eb9-8798-8338a89d58ff>\",\"WARC-Concurrent-To\":\"<urn:uuid:d44ce8a6-ab92-49b7-bfa9-f06fe016a3ff>\",\"WARC-IP-Address\":\"104.24.125.162\",\"WARC-Target-URI\":\"https://mocktime.com/uncategorized/iit-advance-physics-quiz/\",\"WARC-Payload-Digest\":\"sha1:BTXUK62UKB452V6OT54LJMBV5XZR46QO\",\"WARC-Block-Digest\":\"sha1:U25LV6MXIRM5ZMCV2SIX46STPMBS2I3I\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-26/CC-MAIN-2019-26_segments_1560627998959.46_warc_CC-MAIN-20190619103826-20190619125826-00542.warc.gz\"}"}
https://www.manhattanprep.com/gmat/forums/guide-2-ch-4-question-8-t33205.html	[ "## guide 2, ch 4, question 8\n\nIf you're experiencing a roadblock with one of the Manhattan Prep GMAT math strategy guides, help is here!\nNabeelZ316\nStudents\n\nPosts: 18\nJoined: Sat Jul 30, 2016 3:03 am\n\n### guide 2, ch 4, question 8\n\nIf 4^a +4^a+1 = 4^a+2 - 176 , what is the value of a?\n\nhere is what im confused about with the solution provided.\n\nFor the right side of the equation, after simplifying we get one part as -4^a - 4^a . Understood, but then this further turns into 4^0\nHow did it get to 4^0??\n\nif we take 4^a as common, then should leave 1-1 which is 0 which then multiplies by 4^0 to give us ZERO. But in the solution, it is 1.\nRonPurewal\nStudents\n\nPosts: 19746\nJoined: Tue Aug 14, 2007 8:23 am\n\n### Re: guide 2, ch 4, question 8\n\nplease reproduce the relevant part of the answer key. thank you. (if it's in a physical book, just take a picture of it and post a link to the picture.)" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9539412,"math_prob":0.8734377,"size":406,"snap":"2019-26-2019-30","text_gpt3_token_len":136,"char_repetition_ratio":0.10945274,"word_repetition_ratio":0.0,"special_character_ratio":0.33004925,"punctuation_ratio":0.115384616,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9713145,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-07-16T20:26:17Z\",\"WARC-Record-ID\":\"<urn:uuid:0d5a54bd-fb51-4ccc-80c3-73915b01ce17>\",\"Content-Length\":\"47057\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:74dcb01b-4c06-48a1-ab55-79db4615752e>\",\"WARC-Concurrent-To\":\"<urn:uuid:ed6bfc30-32e6-4db1-9007-70c46a3d6a45>\",\"WARC-IP-Address\":\"199.83.132.101\",\"WARC-Target-URI\":\"https://www.manhattanprep.com/gmat/forums/guide-2-ch-4-question-8-t33205.html\",\"WARC-Payload-Digest\":\"sha1:NLY3ELVUOQMAHYMHW7ATH4ZMWDPPELU5\",\"WARC-Block-Digest\":\"sha1:NHC5NJR7UQFK2CXNUQ6DY7ACVCGIYCAT\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-30/CC-MAIN-2019-30_segments_1563195524879.8_warc_CC-MAIN-20190716201412-20190716223412-00210.warc.gz\"}"}
https://takeuforward.org/data-structure/n-meetings-in-one-room/	[ "# N meetings in one room\n\nProblem Statement: There is one meeting room in a firm. You are given two arrays, start and end each of size N.For an index ‘i’, start[i] denotes the starting time of the ith meeting while end[i] will denote the ending time of the ith meeting. Find the maximum number of meetings that can be accommodated if only one meeting can happen in the room at a particular time. Print the order in which these meetings will be performed.\n\n```Example:\n\nInput: N = 6, start[] = {1,3,0,5,8,5}, end[] = {2,4,5,7,9,9}\n\nOutput: 1 2 4 5\n\nExplanation: See the figure for a better understanding.", null, "```\n\n# Solution:\n\nDisclaimer: Don’t jump directly to the solution, try it out yourself first.\n\nInitial Thought Process:-\nSay if you have two meetings, one which gets over early and another which gets over late. Which one should we choose? If our meeting lasts longer the room stays occupied and we lose our time. On the other hand, if we choose a meeting that finishes early we can accommodate more meetings. Hence we should choose meetings that end early and utilize the remaining time for more meetings.\n\nApproach\n\nTo proceed we need a vector of three quantities: the starting time, ending time, meeting number. Sort this data structure in ascending order of end time.\n\nWe need a variable to store the answer. Initially, the answer is 1 because the first meeting can always be performed. Make another variable, say limit that keeps track of the ending time of the meeting that was last performed. Initially set limit as the end time of the first meeting.\n\nStart iterating from the second meeting. At every position we have two possibilities:-\n\n• If the start time of a meeting is strictly greater than limit we can perform the meeting. Update the answer.Our new limit is the ending time of the current meeting since it was last performed.Also update limit.\n• If the start time is less than or equal to limit ,skip and move ahead.\n\nLet’s have a dry run by taking the following example.\n\nN = 6, start[] = {1,3,0,5,8,5}, end[] = {2,4,5,7,9,9}\n\nInitially set answer =,limit = 2.\n\nFor Position 2 –\n\nStart time of meeting no. 2 = 3 > limit. Update answer and limit.\n\nAnswer = [1, 2], limit = 4.\n\nFor Position 3 –\n\nStart time of meeting no. 3 = 0 < limit.Nothing is changed.\n\nFor Position 4 –\n\nStart time of meeting no. 4 = 5 > limit. Update answer and limit.\n\nAnswer = [1,2,4], limit = 7.\n\nFor Position 5 –\n\nStart time of meeting no. 5 = 8 > limit.Update answer and limit.\n\nAnswer = [1,2,4,5], limit = 9.\n\nFor Position 6 –\n\nStart time of meeting no. 6 = 8 < limit.Nothing is changed.\n\nFinal answer = [1,2,4,5]\n\n## C++ Code\n\n``````#include <bits/stdc++.h>\nusing namespace std;\n\nstruct meeting {\nint start;\nint end;\nint pos;\n};\n\nclass Solution {\npublic:\nbool static comparator(struct meeting m1, meeting m2) {\nif (m1.end < m2.end) return true;\nelse if (m1.end > m2.end) return false;\nelse if (m1.pos < m2.pos) return true;\nreturn false;\n}\nvoid maxMeetings(int s[], int e[], int n) {\nstruct meeting meet[n];\nfor (int i = 0; i < n; i++) {\nmeet[i].start = s[i], meet[i].end = e[i], meet[i].pos = i + 1;\n}\n\nsort(meet, meet + n, comparator);\n\nvector < int > answer;\n\nint limit = meet.end;\n\nfor (int i = 1; i < n; i++) {\nif (meet[i].start > limit) {\nlimit = meet[i].end;\n}\n}\ncout<<\"The order in which the meetings will be performed is \"<<endl;\nfor (int i = 0; i < answer.size(); i++) {\ncout << answer[i] << \" \";\n}\n\n}\n\n};\nint main() {\nSolution obj;\nint n = 6;\nint start[] = {1,3,0,5,8,5};\nint end[] = {2,4,5,7,9,9};\nobj.maxMeetings(start, end, n);\nreturn 0;\n}\n``````\n\nOutput:\n\nThe order in which the meetings will be performed is\n1 2 4 5\n\nTime Complexity: O(n) to iterate through every position and insert them in a data structure. O(n log n) to sort the data structure in ascending order of end time. O(n) to iterate through the positions and check which meeting can be performed.\n\nOverall : O(n) +O(n log n) + O(n) ~O(n log n)\n\nSpace Complexity: O(n) since we used an additional data structure for storing the start time, end time, and meeting no.\n\n## Java Code\n\n``````import java.util.*;\nclass meeting {\nint start;\nint end;\nint pos;\n\nmeeting(int start, int end, int pos)\n{\nthis.start = start;\nthis.end = end;\nthis.pos = pos;\n}\n}\nclass meetingComparator implements Comparator<meeting>\n{\n@Override\npublic int compare(meeting o1, meeting o2)\n{\nif (o1.end < o2.end)\nreturn -1;\nelse if (o1.end > o2.end)\nreturn 1;\nelse if(o1.pos < o2.pos)\nreturn -1;\nreturn 1;\n}\n}\npublic class Meeting {\nstatic void maxMeetings(int start[], int end[], int n) {\nArrayList<meeting> meet = new ArrayList<>();\n\nfor(int i = 0; i < start.length; i++)\nmeet.add(new meeting(start[i], end[i], i+1));\n\nmeetingComparator mc = new meetingComparator();\nCollections.sort(meet, mc);\nArrayList<Integer> answer = new ArrayList<>();\nint limit = meet.get(0).end;\n\nfor(int i = 1;i<start.length;i++) {\nif(meet.get(i).start > limit) {\nlimit = meet.get(i).end;\n}\n}\nSystem.out.println(\"The order in which the meetings will be performed is \");\nfor(int i = 0;i<answer.size(); i++) {\nSystem.out.print(answer.get(i) + \" \");\n}\n}\npublic static void main(String args[])\n{\nint n = 6;\nint start[] = {1,3,0,5,8,5};\nint end[] = {2,4,5,7,9,9};\nmaxMeetings(start,end,n);\n\n}\n}``````\n\nOutput:\n\nThe order in which the meetings will be performed is\n1 2 4 5\n\nTime Complexity: O(n) to iterate through every position and insert them in a data structure. O(n log n) to sort the data structure in ascending order of end time. O(n) to iterate through the positions and check which meeting can be performed.\n\nOverall : O(n) +O(n log n) + O(n) ~O(n log n)\n\nSpace Complexity: O(n) since we used an additional data structure for storing the start time, end time, and meeting no.\n\nSpecial thanks to Somparna Chakrabarti for contributing to this article on takeUforward. If you also wish to share your knowledge with the takeUforward fam, please check out this article" ]	[ null, "https://lh3.googleusercontent.com/TtxJefJFSrt-O3yH53CFJ8udcDt02PkBduRCvO6XvndajetZ2LUEgNhrSIcbSBtHLodGgPfuJeEuIP4azq0dZaN5XQ9Zgz5BWKnGAQu25YDPuLqIcHT1aFA66Lvz8Q", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7642344,"math_prob":0.99060464,"size":5897,"snap":"2022-40-2023-06","text_gpt3_token_len":1634,"char_repetition_ratio":0.14661463,"word_repetition_ratio":0.2495069,"special_character_ratio":0.30727488,"punctuation_ratio":0.20676969,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96454287,"pos_list":[0,1,2],"im_url_duplicate_count":[null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-10-05T09:42:32Z\",\"WARC-Record-ID\":\"<urn:uuid:74b73e97-84d5-43e9-8974-8c3a7ba1d327>\",\"Content-Length\":\"79703\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9edd1d05-b18e-4b0d-a50b-138e8ff0aff1>\",\"WARC-Concurrent-To\":\"<urn:uuid:5a607a26-6752-4803-a324-49d2e0d18e44>\",\"WARC-IP-Address\":\"104.26.4.96\",\"WARC-Target-URI\":\"https://takeuforward.org/data-structure/n-meetings-in-one-room/\",\"WARC-Payload-Digest\":\"sha1:2PAEERCGOVO3NJVEO5UQ35HUJ4FLMDFA\",\"WARC-Block-Digest\":\"sha1:PL253Z7SGYJ6Z4E2EEKA3GFLVUEHD646\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030337595.1_warc_CC-MAIN-20221005073953-20221005103953-00782.warc.gz\"}"}
https://kpmcqshub.com/maths-class-6-100-to-120/	[ "CLOSE\n\nA) 25\n\nB) -25\n\nC) 0\n\nD) None of these\n\n## 102. A line which divides a line segment into two equal parts and also makes an angle of 900 with the given line segment, is called its?\n\nA) Left Bisector\n\nB) Right Bisector\n\nC) Perpendicular\n\nA) Equal\n\nB) Smaller\n\nC) Greater\n\nD) None\n\nA) Yes\n\nB) No\n\nC) Sometimes\n\nA) Greater\n\nB) Equal\n\nC) Less\n\nA) Yes\n\nB) No\n\nC) None\n\nA) 71o/2\n\nB) 81/2\n\nC) 9*2/3\n\n## 108. ______ of a rectangle is twice the sum of the length and width\n\nA) Area\n\nB) Perimeter\n\nC) Both ‘a’ and ‘b’\n\n## 109. Perimeter of rectangle is?\n\nA) P= 2(l + w)\n\nB) P= 2l + 2w\n\nC) Both ‘a’ and ‘b’\n\nA) 16 units\n\nB) 32 units\n\nC) 27 units\n\nA) Area\n\nB) Perimeter\n\nA) 105 m\n\nB) 105 m2\n\nC) 22 m2\n\nA) w = 2\n\nB) w = 2 feet\n\nC) 8 feet\n\nA) Rectangle\n\nB) Square\n\nC) Triangle\n\nA) P = 4s\n\nB) P = 3s\n\nC) P = 2s\n\nD) None\n\nA) 1296 m\n\nB) 1296 m2\n\nC) 72 m2\n\nA) 225000\n\nB) 22500\n\nC) 230500\n\nA) 9537\n\nB) 9550\n\nC) 9537.5\n\nA) 4240 m2\n\nB) 4224 m2\n\nC) 4252 m2\n\nD) None\n\nA) Height\n\nB) Altitude\n\nC) Perimeter" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7930266,"math_prob":0.98988587,"size":2338,"snap":"2023-14-2023-23","text_gpt3_token_len":808,"char_repetition_ratio":0.13281919,"word_repetition_ratio":0.020746889,"special_character_ratio":0.37596235,"punctuation_ratio":0.0961183,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9737488,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-03-23T04:34:03Z\",\"WARC-Record-ID\":\"<urn:uuid:886ff014-4040-4cd0-b8e7-1411fe523b14>\",\"Content-Length\":\"146753\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:548e2624-92ce-485b-a827-d4aa1e7e166b>\",\"WARC-Concurrent-To\":\"<urn:uuid:a99ff326-ca0f-4248-9912-ef186870148d>\",\"WARC-IP-Address\":\"15.204.142.117\",\"WARC-Target-URI\":\"https://kpmcqshub.com/maths-class-6-100-to-120/\",\"WARC-Payload-Digest\":\"sha1:H4PKGQTOYPE5AAIK2VRFMZQP75WKL4U3\",\"WARC-Block-Digest\":\"sha1:O636EJV4R2MCYT5U2ZYOWMLMQRE5QABO\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-14/CC-MAIN-2023-14_segments_1679296944996.49_warc_CC-MAIN-20230323034459-20230323064459-00052.warc.gz\"}"}
http://ics.misterv.ca/general-notes/assignments/magic-squares/magic-squares-oops	[ "0.0 General‎ > ‎Assignments‎ > ‎Magic Squares‎ > ‎\n\n### Magic Squares: OOPs\n\nThis is a framework you can use to implement magic squares in an OOPs environment:\n\n### Class: Cell\n\n#### Cell Attributes:\n\n Attribute Data Type Scope Purpose row integer - private the row the cell is in column integer - private the column the cell is in isOccupied boolean - private indicates if the cells contains a value. set to false when created value integer - private the integer value of the cell in the magic square\n\n#### Cell Methods:\n\n Method Return Type Scope Purpose Cell( int r, int c) + public Constructor: sets isOccuppied to false, row to r, col to c isOccupied() boolean + public return the current value of isOccupied setValue( int value) void + public set the value of the cell getValue() int + public return the current cell value\n\n### Class: MagicSquare\n\n#### Square Attributes:\n\n Attribute Data Type Scope Purpose mSquare Cell [][] - private a 2D array to hold the magic square size int - private the size of the magic square startValue int - private the first number start value in the square step int - private the amount to change each new value in the magic square\n\n#### Square Methods:\n\n Method Return Type Scope Purpose Square() + public Constructor: set the default values square( int size, int start, int step) void - private called by the constructors to create the magic square initsquare() void used by: square() creates mSquare full of empty cells setValues() void - private used by: square() sets the values in magic square displaySquare() void - private used by: square() will display the square neatly on the screen nextCell( Cell currentCell) Cell - private return the next cell to be occupiedused by: setValues() nextValue( int x) int - private returns the next value to be placed in a cellused by: setValues()" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5409495,"math_prob":0.9625935,"size":1835,"snap":"2021-04-2021-17","text_gpt3_token_len":466,"char_repetition_ratio":0.17149098,"word_repetition_ratio":0.06542056,"special_character_ratio":0.27029973,"punctuation_ratio":0.06711409,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9554764,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-01-17T02:49:49Z\",\"WARC-Record-ID\":\"<urn:uuid:afe3ce16-9329-4024-8ebe-bd77015a23d9>\",\"Content-Length\":\"73462\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:64f81570-c4bd-41d3-a4cc-a1d3d83208d8>\",\"WARC-Concurrent-To\":\"<urn:uuid:10b2686d-492b-4ea0-ab3c-02d26ee521b5>\",\"WARC-IP-Address\":\"172.217.9.211\",\"WARC-Target-URI\":\"http://ics.misterv.ca/general-notes/assignments/magic-squares/magic-squares-oops\",\"WARC-Payload-Digest\":\"sha1:YQK6GLNJ5QO66CWMLAHSVE4VDJMR7HMZ\",\"WARC-Block-Digest\":\"sha1:KWAKQVXAVAQLOQP7IIHBH6UEJ7CPHMK6\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-04/CC-MAIN-2021-04_segments_1610703509104.12_warc_CC-MAIN-20210117020341-20210117050341-00472.warc.gz\"}"}
http://www.helistart.com/Vector.aspx	[ "# Vector Quantities\n\nA vector is defined as:\n\nx = (x, y, z)\n\nwhere:\nx = position on the x axis\ny = position on the y axis\nz = position on the z axis\n\nNote that the scalar values x, y, z refer to the unit vectors (with a length of 1) along the x, y and z axes in Cartesian space. On this website, vectors are denoted by bold typeface.\n\nThe magnitude (or length) of the vector (x, y, z) is: sqrt(x2 + y2 + z2)\n\nThe direction can be expressed by the angles α and β with:\n\nα = arctan (y / x)\nβ = arctan (z / x)\n\nThe vector cross product is defined as:\n\na x b = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)\n\nwhere:\na = (a1, a2, a3\nb = (b1, b2, b3\n\n## Explanation\n\nMany quantities have both a magnitude and a direction. The magnitude and direction can be expressed by a vector. These vectors are defined by their components along an orthogonal axis system (Cartesian space). Each axis represents a dimension. For example, when we look at the speed and direction of an object, we can denote this by specifying the speed's horizontal component (one leg of the axis system) and its vertical component (the second axis). This is an example of a two dimensional vector quantity. We can also define a three dimensional vector in a similar way.", null, "The cross product between two vectors, a x b, has vector c as a result. This vector is orthogonal to a and b, and has a magnitude that is equal to the area of the parallelogram spawned by them. The sign of c is denoted by the right hand rule. Note that when a and b are orthogonal to each other (parallelogram is, then, a square), the magnitude of c equals the product of the magnitudes of a and b.\n\n## Example 1\n\nThe speed of a helicopter is angled at 45 degrees from the horizontal axis in the x-y plane. The speed is 5 m/s along the horizontal axis. How great is the vertical speed, and what is its absolute value (that is, what is the magnitude of the speed)?\nAnswer: The vertical speed must be equal to its horizontal speed, as the angle in the x-y plane is 45 degrees. The value of the speed is: sqrt(x2 + y2) = sqrt (25 + 25) = sqrt(50) = 7.07 m/s\n\n## Example 2\n\nThe speed of a helicopter is 20 m/s and makes an angle of 15 degrees with the x axis and is flying in the x-y plane. What is the vertical component of the speed?\ntan(α) = y / x\ntan(15) = y/20\ny = 20 tan(15) = 5.36\n\nSo, the vertical speed component is 5.36 m/s.\n\n## Example 3\n\nThe speed of a helicopter is defined by the vector (10, 20, 0). What is the angle with the x-axis?\nAnswer: angle = arctan (20/10) = arctan (2) = 1.10714872 (in radials)-> In degrees: (1.10714872 / 2pi) * 360 = 63.4 degrees.\n\n# Do you want to comment this topic?\n\n1: (Book) Cyclic and Collective", null, "2: (Book) Principles of Helicopter Flight", null, "3: Microsoft FSX Steam Edition", null, "4: Logitech Extreme 3D Pro Joystick", null, "5: Saitek Pro Flight Rudder Pedals", null, "", null, "" ]	[ null, "http://www.helistart.com/Figures/vector.jpg", null, "http://ir-na.amazon-adsystem.com/e/ir", null, "http://ir-na.amazon-adsystem.com/e/ir", null, "http://ir-na.amazon-adsystem.com/e/ir", null, "http://ir-na.amazon-adsystem.com/e/ir", null, "http://ir-na.amazon-adsystem.com/e/ir", null, "http://rover.ebay.com/roverimp/1/711-53200-19255-0/1", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8886965,"math_prob":0.99674296,"size":2619,"snap":"2020-45-2020-50","text_gpt3_token_len":785,"char_repetition_ratio":0.122370936,"word_repetition_ratio":0.018726591,"special_character_ratio":0.3043146,"punctuation_ratio":0.11557789,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9996762,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"im_url_duplicate_count":[null,5,null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-11-30T07:12:26Z\",\"WARC-Record-ID\":\"<urn:uuid:41a5aada-1dab-414b-9a46-2131039797a8>\",\"Content-Length\":\"167158\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c1a53099-4939-4616-bad5-cec3a322dc36>\",\"WARC-Concurrent-To\":\"<urn:uuid:e8cfade1-52a0-48e5-8f69-1cddb007b62f>\",\"WARC-IP-Address\":\"46.182.180.5\",\"WARC-Target-URI\":\"http://www.helistart.com/Vector.aspx\",\"WARC-Payload-Digest\":\"sha1:NS2PRWOIDDRFTN7FNA3RKWS2TNPWCHQH\",\"WARC-Block-Digest\":\"sha1:SHAV57VWJ2E4JMSOEQEO27234IMMRMZH\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-50/CC-MAIN-2020-50_segments_1606141211510.56_warc_CC-MAIN-20201130065516-20201130095516-00633.warc.gz\"}"}
http://isa.umh.es/calc/ss49docs/solve49.htm	[ "# SOLVESYS 49 version 1.2\n\n## Numerical Nonlinear Equations and Least Squares Solver for the HP49G calculator\n\nCopyright 1994-2000, Sune Bredahl\n\n### DISCLAIMER\n\nSOLVESYS.LIB AND THIS MANUAL ARE PRESENTED WITHOUT WARRANTIES, EXPRESSED OR IMPLIED. THE AUTHOR MAKES NO GUARANTEE AS TO THE FITNESS OF THIS SOFTWARE.\n\nSOLVESYS.LIB CAN BE COPIED FREELY BUT MAY NOT BE USED FOR COMMERCIAL PURPOSES WITHOUT PERMISSION FROM THE AUTHOR.\n\n### OVERVIEW\n\nThis documentation covers SOLVESYS 1.2 for the HP49G. Latest version of the software is downloadable from http://solvesys.cjb.net. Also a FAQ is available from here if you have any questions not covered in this manual.\n\nSOLVESYS 49 requires an HP49G with ROM 1.10 or later. It will not work properly with the ROM 1.05 shipped in early versions of the calculator.\n\n### INSTALLATION\n\nSOLVESYS is a library and must be properly installed in order to work Consult the Advanced User's Guide (section 11-6) for details on this. However, assuming RPN mode, the following should work:\n\n1. Transfer SOLVESYS.LIB to the HP49G - use binary transfer.\n2. Recall the library to the stack and press n [STO], where n is the desired port (0, 1 or 2). After storing the library, you can purge the original copy to save memory.\n3. Press [ON]+[C] to restart the HP49G. This completes the installation.\n\nTo verify that this library has been correctly installed, press [CAT], find the command ABOUTSS and press [ENTER]. The version number displayed should be \"SOLVESYS 49 1.2\" (the current build is 17/07/00).\n\nSOLVESYS can now be started in several ways, for example:\n\n• Using the NUM.SLV menu (SOLVESYS 1.2 appears as the last item)\n• Using CAT (command is listed as SOLVESYS).\n\nTo remove the SOLVESYS library, do this:\n\n1. Start the FILE MANAGER\n2. Select the port (0, 1 or 2) where SOLVESYS is installed and press [OK]\n3. Press [NXT] and [PURGE].\n\nIf calculator returns an \"Object in use\" error at this point, press [ON]+[C] and repeat the above steps.\n\nTo transfer SOLVESYS.LIB to another HP49G.:\n\n1. Put the library ID (n: 1550) in the stack and press [RCL].\n2. Store the returned library in any variable\n3. Transfer this variable to the other calc. - use binary transfer.\n\n### WHAT IS SOLVESYS?\n\nSOLVESYS 49 is an environment for solving systems of linear and nonlinear equations. It is also capable of doing nonlinear least-squares minimization.\n\nTo be more precise, SOLVESYS is designed to either zero or least-squares minimize m linear or nonlinear equations in n unknowns.When m=n, the problem is obviously to find a solution satisfying all equations. If m>n, the equations are generally inconsistent so instead SOLVESYS searches for a least squares solution, i.e. the minimizer of the sum of squared equation residuals. Least squares minimization is commonly used in datafitting applications. The case n>m is perhaps of less interest but is also handled by SOLVESYS.\n\nNote that SOLVESYS is a numeric solver - it will try to solve the equations using iterative methods, i.e. the user provides some initial guess or estimate of the solution which the solver (hopefully) can use to find the true solution. It can be necessary to try out different starting guesses before the solver succeeds in this quest, but if good starting values are provided, equations can often be solved with only a few iterations.\n\nSOLVESYS does not allow units in equations or variables (too slow), also it cannot be used to solve systems with discrete functions or variables.\n\nThe solver engine is based on a (Gauss-) Newton method with a mixed quadratic and cubic linesearch procedure. For more details on this algorithm, see Dennis and Schnabel or Press et al. .\n\n Dennis, J.E., and Schnabel, R.B. 1983, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Englewood Cliffs, NJ: Prentice-Hall).\n Press, W.H. et al. 1992, Numerical Recipes in C: The Art of Scientific Computing. 2nd ed. (Cambridge: University Press).\n\nRather than describing each screen and menu in SOLVESYS 49, I'll provide some examples instead.\n\nBasic usage is described in the first example so please start from here even though you may not be interested in solving linear equations.\n\nExample 1: Solving a system of linear equations\n\nAlthough the HP49G already has built-in functions to solve systems of linear equations (such as LINSOLVE), you may find it convenient to use SOLVESYS instead. At least this small example should give you an idea of how to use SOLVESYS.\n\nThe system we want to solve is:\n\n1. 2x + y = 3x + z\n\n2. z + x = 2(x + y)\n\n3. (z + x)/2 + y + 1.2 = 0\n\nThere are several ways to start SOLVESYS. The fastest way is open the NUM.SLV choose menu and select SOLVESYS 1.2 from here.\n\nThe first thing SOLVESYS displays, is a list of equations to be solved. This is actually the contents of the 'EQ' variable so if there's no 'EQ' variable in the current directory an empty list is displayed. For example, if the above equations are stored as a list in 'EQ' prior to starting SOLVESYS, the opening display should look something like this:", null, "If you haven't stored the equations in 'EQ', use [ADD] to enter the equations.\n\nThe checkmark indicates that the equation is marked for solving (default). If there are equations you don't want to solve, use the [+/-] key to uncheck them - this has the same effect as [DEL] but does not remove the equation from 'EQ'.\n\nNow press [OK] to proceed. SOLVESYS analyzes the equations for variables and opens the \"Variables\" screen shown below.", null, "You can modify a highlighted value using the [EDIT] menukey or [ENTER]. To check/uncheck a variable, use the [+/-] key. The [RESET] key sets all variables equal to 1.\n\n• Checked variables will be solved for (default). A real or complex valued starting guess must be provided. This should be your best guess of the solution you're looking for - don't use the default values if you know any better.\n• Unchecked variables are not solved for and the value remain constant during the solving process. In fact, the value need not be a real or complex numbers, however the equations must eventually evaluate to a number.\n\nIn this example, we want to solve for all variables so all must be checked. The starting values are unimportant because the equations are linear. If we use the default values as shown and press [SOLVE] we get the next screens:", null, "", null, "Here \"Zero\" indicates that a solution has been found. When returned to the \"Variables\" screen, the final values are displayed.\n\nAs you can see, the solution is (x, y, z) = (0.4, -0.8, -1.2). When you exit SOLVESYS, these values will be stored in their respective variables. However you can also use [->STK] to copy this solution (as well as a error vector) to the stack\n\nExample 2: Solving a system of nonlinear equations.\n\nIn this example we will solve a rather large system (propane combustion?) which is a mixture of 11 linear and nonlinear equations in 11 unknowns (x1,...,x11) and seven known variables (a1,...,a6 and r).\n\n x1 + x4 = 3 2x1 + x2 + x4 + x7 + x8 + x9 + 2x10 = 10 + r x2 + 2x5 + x6 + x7 = 8 2x3 + x5 = 4r x1 x5 = a1 x2 x4 x6 x2½ = a2 (x2 x4 x11)½ x7 x4½ = a3 (x1 x4 x11)½ x8 x4 = a4 x2 x11 x9 x4 = a5 x1 (x3 x11)½ x10 x42 = a6 x42 x11 x11 = x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 a1 = 0.193 a2 = 0.002597 a3 = 0.003448 a4 = 0.00001799 a5 = 0.0002155 a6 = 0.00003846 r = 4.056734\n\nIn SOLVESYS, this equation set will look like:", null, "All the equations should be included (checked). Press [OK] to open the \"Variables\" screen:", null, "Remember that the first seven values here are constants. These must be unchecked and filled in with their corresponding values shown earlier. For the unknowns we will use 1 as the initial value except x11 we set to 10 since x11 is the sum of the other variables (equation 11).\n\nWhen finished, the display should then look something like the one below. Use the downarrow key to view/edit the remaining variables.", null, "", null, "[Note: To verify that you have the correct amount of unknowns, press [INFO] menukey. This should show \"m/n = 11/11\" which means that 11 equations and 11 variables are selected to be solved]\n\nWhen done, press [SOLVE]and after a few minutes a solution (Zero) is found:", null, "", null, "The final values can be viewed using the up/downarrow keys. [->STK] can be used to copy the result and the equation residuals the stack (do this!). Note that the values are also stored in the variables after you exit SOLVESYS.\n\nExample 3: Nonlinear least-squares\n\nIf there are more (generally inconsistent) equations than unknowns, SOLVESYS will search for a \"best fit\" solution in the least-squares sense, ie. a solution that minimizes the sum of squared equations deviations.\n\nA common application of least-squares minimization is datafitting, where one is attempting to fit m observations to an expression of n unknown parameters for which the \"best-fit\" values are required. If each observation is inserted in the expression, the result is a system of m \"observational\" equations in n unknowns.\n\nHere's an example.\n\nThe relationship between the measured pressure and temperature in saturated steam can be written as\n\nY=aALOG(bT/(c+T))\n\nwhere Y is the measured pressure of the steam at various values of a controllable temperature T\na, b and c are unknown parameters (to be estimated)\n\nThe following data were collected.\n\n T 0 10 20 30 40 50 60 70 80 85 90 95 100 105 Y 4.14 8.52 16.31 32.18 64.62 98.76 151.13 224.74 341.35 423.36 522.78 674.32 782.04 920.01\n\nTo create the 14 equations needed, perform the following four steps:\n\n1. Store the data as lists in variables such as 'T' and 'Y'.\n2. If any of the variables A, B or C exist, they must be purged.\n3. Enter 'Y=AALOG(BT/(C ADD T))'on the stack and press [EVAL]\n4. You're done! Store the resulting list of equations in the 'EQ' variable.\n\n[IMPORTANT: In step 3. you must replace any occurence of \"+\" with \"ADD\" or \"--\" since \"+\" has a different action when list are involved.]\n\nAs for the initial guesses, note that the first equation is actually A=4.14 so this can be used as a guess. If A=4.14 is inserted in the next two equations, these can be reduced to two linear equations with the solution (b,c) = (5.93,179). Thus an initial guess could be (a,b,c)=(4,6,180).\n\nNow we're ready to start SOLVESYS. The equations are already created so press [OK] and fill in the guess values (a,b,c) = (4, 6, 180)", null, "", null, "Press [SOLVE] and answer \"yes\" to accept that m is different from n (since we have more equations than unknowns). After a few iterations a solution (minimum) is found.", null, "", null, "", null, "Using 3 significant digits, the best fit function is Y(T)=5.27ALOG[8.56T/(295+T)].\n\nBe aware that there are no simple ways to distinguish between a local or global minimum so the choice of starting values is very crucial. A good initial guess is not only a real time-saver, it will also pick out the right (global) minimum for you!\n\n### ERROR TOLERANCES\n\nIt is possible to modify the error tolerances used for convergence testing although it is not recommended. Use [TOL] to change any the error tolerances described below.\n\nXTOL\n\nTolerance for the convergence of iterates. This is the relative difference between the last two computed solutions. A value of 10-p usually corresponds to p significant digits of the computed result, but values below 10-6 may not comply with this. The default value is 10-3 corresponding to 3 significant digits.\n\nXTOL is the \"primary\" test in the sence that EQTOL or LSQTOL are not tested if XTOL fails. However, for several reasons, XTOL is ignored if the equation values are exactly zero (this forces an \"Zero\" message).\n\nEQTOL\n\nTest to check if the equations have been zeroed (should be named ZEROTOL?). The test computes the r.m.s. (root mean square) error of the equations. For example, for the system\n{ xy=9, x+y=6} the r.m.s. error at (x,y)=(2.9,3.1) is [(8,99-9)2+(6-6)2]½= 0.01.\nEQTOL defaults to 10-5 which should be sufficient for most purposes.\n\nLSQTOL\n\nThis test is used only if m>n ie. for nonlinear least-squares problems. It is a cosine test from from Dennis & Schnabel (sort of normalized gradient test). The test returns values between 0 and 1 where a value close to 0 indicates a least-squares minimizer. The default value is 0.01 which should be sufficient considering the XTOL pre-requirement.\n\nLSQTOL does not apply to zero-residual least squares (exact fit), however the EQTOL test will return a \"Zero\" message in this case.\n\n### TERMINATION AND ERROR MESSAGES\n\nSome (more or less) common error and termination messages returned by SOLVESYS are listed below.\n\n#### RETURN MESSAGES FROM THE SOLVER:\n\nZero:\nA result satisfying XTOL and EQTOL was found (XTOL may be ignored if this is an exact zero).\nMinimum:\nA result satisfying XTOL and LSQTOL was found. It is not guaranteed to be a global minimum.\nThe solver has landed on a point that does not appear to be solution. Restart the solver with other initial guesses.\nUndefined Result:\n(Math exception.) Calculation such as 0/0 occurred or the XROOT function was used with arguments that cannot give a real-valued result.\nInfinite Result:\n(Math exception.) Calculation such as 1/0 occurred. Try changing the initial guesses.\n\n#### INITIALIZATION ERRORS:\n\nConstant?:\nAt least one equation does not contain any unknowns. SOLVESYS does not allow redundant equations. If you have equations you don't want to solve, just uncheck them.\nInconsistent Units:\nUnits are not supported.\nUndefined Name:\nSome expression(s) include(s) one or more variable names SOLVESYS is unable to identify.\n\n### HOW TO REACH THE AUTHOR\n\nFeel free to contact me if you have any comments, questions, problems or suggestions. Also if you find any bugs (or what you think is a bug), please notify me.\n\nCheck http://solvesys.cjb.net for the latest version of SOLVESYS and a F.A.Q with answers to the most common questions.\n\nMy email is sune@bredahl.net, or sune_bredahl@hotmail.com and ICQ# 43726969." ]	[ null, "http://isa.umh.es/calc/ss49docs/ex1a.gif", null, "http://isa.umh.es/calc/ss49docs/ex1b.gif", null, "http://isa.umh.es/calc/ss49docs/ex1c.gif", null, "http://isa.umh.es/calc/ss49docs/ex1d.gif", null, "http://isa.umh.es/calc/ss49docs/ex2a.gif", null, "http://isa.umh.es/calc/ss49docs/ex2b.gif", null, "http://isa.umh.es/calc/ss49docs/ex2c.gif", null, "http://isa.umh.es/calc/ss49docs/ex2d.gif", null, "http://isa.umh.es/calc/ss49docs/ex2e.gif", null, "http://isa.umh.es/calc/ss49docs/ex2f.gif", null, "http://isa.umh.es/calc/ss49docs/ex3a.gif", null, "http://isa.umh.es/calc/ss49docs/ex3b.gif", null, "http://isa.umh.es/calc/ss49docs/ex3c.gif", null, "http://isa.umh.es/calc/ss49docs/ex3d.gif", null, "http://isa.umh.es/calc/ss49docs/ex3e.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8642824,"math_prob":0.9776788,"size":13829,"snap":"2019-13-2019-22","text_gpt3_token_len":3614,"char_repetition_ratio":0.1350452,"word_repetition_ratio":0.0091973245,"special_character_ratio":0.2628534,"punctuation_ratio":0.123430215,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99480385,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],"im_url_duplicate_count":[null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-03-24T21:58:06Z\",\"WARC-Record-ID\":\"<urn:uuid:a562034e-5fdf-40fc-b240-e33116d25156>\",\"Content-Length\":\"23475\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:16cb517c-a20b-428e-83bc-d6c507955cdf>\",\"WARC-Concurrent-To\":\"<urn:uuid:5162f925-3fad-41c9-ae56-15656e734e6b>\",\"WARC-IP-Address\":\"193.147.145.165\",\"WARC-Target-URI\":\"http://isa.umh.es/calc/ss49docs/solve49.htm\",\"WARC-Payload-Digest\":\"sha1:ODOOBQNHV7GEXKQWOSNHSMXVLXWINEJT\",\"WARC-Block-Digest\":\"sha1:ZQACBVYPC37UAMF2ANBD2Y5KDKWCD7IE\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-13/CC-MAIN-2019-13_segments_1552912203493.88_warc_CC-MAIN-20190324210143-20190324232143-00055.warc.gz\"}"}
https://en.cppreference.com/w/cpp/algorithm/is_permutation	[ "# std::is_permutation\n\n< cpp‎ \| algorithm\n\nC++\n Compiler support Freestanding and hosted Language Standard library Standard library headers Named requirements Feature test macros (C++20) Language support library Concepts library (C++20) Metaprogramming library (C++11) Diagnostics library General utilities library Strings library Containers library Iterators library Ranges library (C++20) Algorithms library Numerics library Localizations library Input/output library Filesystem library (C++17) Regular expressions library (C++11) Concurrency support library (C++11) Technical specifications Symbols index External libraries\n\nAlgorithm library\nConstrained algorithms and algorithms on ranges (C++20)\nConstrained algorithms, e.g. ranges::copy, ranges::sort, ...\nExecution policies (C++17)\n is_execution_policy(C++17)\n execution::seqexecution::parexecution::par_unseqexecution::unseq(C++17)(C++17)(C++17)(C++20)\nNon-modifying sequence operations\n all_ofany_ofnone_of(C++11)(C++11)(C++11) for_each for_each_n(C++17)\nModifying sequence operations\n copycopy_if(C++11) copy_n(C++11) copy_backward move(C++11) move_backward(C++11) shift_leftshift_right(C++20)(C++20) transform\n removeremove_if replacereplace_if reverse rotate unique random_shuffle(until C++17)\nPartitioning operations\n is_partitioned(C++11) partition_point(C++11)\n partition partition_copy(C++11)\nSorting operations\n is_sorted(C++11) is_sorted_until(C++11)\nBinary search operations\nSet operations (on sorted ranges)\nHeap operations\n is_heap(C++11) is_heap_until(C++11)\nMinimum/maximum operations\n minmax(C++11) minmax_element(C++11)\n clamp(C++17)\nPermutations\n is_permutation(C++11)\nNumeric operations\n accumulate reduce(C++17) transform_reduce(C++17)\n transform_inclusive_scan(C++17) transform_exclusive_scan(C++17)\n partial_sum inclusive_scan(C++17) exclusive_scan(C++17)\nOperations on uninitialized storage\n uninitialized_copy_n(C++11) uninitialized_move_n(C++17) uninitialized_fill_n\n destroy(C++17) destroy_n(C++17) destroy_at(C++17) construct_at(C++20)\nC library\n\n Defined in header (1) template< class ForwardIt1, class ForwardIt2 > bool is_permutation( ForwardIt1 first1, ForwardIt1 last1, ForwardIt2 first2 ); (since C++11) (until C++20) template< class ForwardIt1, class ForwardIt2 > constexpr bool is_permutation( ForwardIt1 first1, ForwardIt1 last1, ForwardIt2 first2 ); (since C++20) (2) template< class ForwardIt1, class ForwardIt2, class BinaryPredicate > bool is_permutation( ForwardIt1 first1, ForwardIt1 last1, ForwardIt2 first2, BinaryPredicate p ); (since C++11) (until C++20) template< class ForwardIt1, class ForwardIt2, class BinaryPredicate > constexpr bool is_permutation( ForwardIt1 first1, ForwardIt1 last1, ForwardIt2 first2, BinaryPredicate p ); (since C++20) (3) template< class ForwardIt1, class ForwardIt2 > bool is_permutation( ForwardIt1 first1, ForwardIt1 last1, ForwardIt2 first2, ForwardIt2 last2 ); (since C++14) (until C++20) template< class ForwardIt1, class ForwardIt2 > constexpr bool is_permutation( ForwardIt1 first1, ForwardIt1 last1, ForwardIt2 first2, ForwardIt2 last2 ); (since C++20) (4) template< class ForwardIt1, class ForwardIt2, class BinaryPredicate > bool is_permutation( ForwardIt1 first1, ForwardIt1 last1, ForwardIt2 first2, ForwardIt2 last2, BinaryPredicate p ); (since C++14) (until C++20) template< class ForwardIt1, class ForwardIt2, class BinaryPredicate > constexpr bool is_permutation( ForwardIt1 first1, ForwardIt1 last1, ForwardIt2 first2, ForwardIt2 last2, BinaryPredicate p ); (since C++20)\n\nReturns true if there exists a permutation of the elements in the range [first1, last1) that makes that range equal to the range [first2, last2), where last2 denotes first2 + (last1 - first1) if it was not given.\n\n1,3) Elements are compared using operator==. The behavior is undefined if it is not an equivalence relation.\n2,4) Elements are compared using the given binary predicate p. The behavior is undefined if it is not an equivalence relation.\n\n## Contents\n\n### Parameters\n\n first1, last1 - the range of elements to compare first2, last2 - the second range to compare p - binary predicate which returns true if the elements should be treated as equal. The signature of the predicate function should be equivalent to the following: bool pred(const Type &a, const Type &b); Type should be the value type of both ForwardIt1 and ForwardIt2. The signature does not need to have const &, but the function must not modify the objects passed to it. Type requirements -ForwardIt1, ForwardIt2 must meet the requirements of LegacyForwardIterator. -ForwardIt1, ForwardIt2 must have the same value type.\n\n### Return value\n\ntrue if the range [first1, last1) is a permutation of the range [first2, last2).\n\n### Complexity\n\nAt most O(N2) applications of the predicate, or exactly N if the sequences are already equal, where N is std::distance(first1, last1).\n\nHowever if ForwardIt1 and ForwardIt2 meet the requirements of LegacyRandomAccessIterator and std::distance(first1, last1) != std::distance(first2, last2) no applications of the predicate are made.\n\n### Note\n\nThe std::is_permutation can be used in testing, namely to check the correctness of rearranging algorithms (e.g. sorting, shuffling, partitioning). If x is an original range and y is a permuted range then std::is_permutation(x, y) == true means that y consist of \"the same\" elements, maybe staying at other positions.\n\n### Possible implementation\n\n template bool is_permutation(ForwardIt1 first, ForwardIt1 last, ForwardIt2 d_first) { // skip common prefix std::tie(first, d_first) = std::mismatch(first, last, d_first); // iterate over the rest, counting how many times each element // from [first, last) appears in [d_first, d_last) if (first != last) { ForwardIt2 d_last = std::next(d_first, std::distance(first, last)); for (ForwardIt1 i = first; i != last; ++i) { if (i != std::find(first, i, i)) continue; // this i has been checked auto m = std::count(d_first, d_last, i); if (m == 0 \|\| std::count(i, last, i) != m) return false; } } return true; }\n\n### Example\n\n#include <algorithm>\n#include <iostream>\n\ntemplate<typename Os, typename V>\nOs& operator<<(Os& os, V const& v)\n{\nos << \"{ \";\nfor (auto const& e : v)\nos << e << ' ';\nreturn os << '}';\n}\n\nint main()\n{\nstatic constexpr auto v1 = {1, 2, 3, 4, 5};\nstatic constexpr auto v2 = {3, 5, 4, 1, 2};\nstatic constexpr auto v3 = {3, 5, 4, 1, 1};\n\nstd::cout << v2 << \" is a permutation of \" << v1 << \": \" << std::boolalpha\n<< std::is_permutation(v1.begin(), v1.end(), v2.begin()) << '\\n'\n<< v3 << \" is a permutation of \" << v1 << \": \"\n<< std::is_permutation(v1.begin(), v1.end(), v3.begin()) << '\\n';\n}\n\nOutput:\n\n{ 3 5 4 1 2 } is a permutation of { 1 2 3 4 5 }: true\n{ 3 5 4 1 1 } is a permutation of { 1 2 3 4 5 }: false\n\n next_permutation generates the next greater lexicographic permutation of a range of elements (function template) prev_permutation generates the next smaller lexicographic permutation of a range of elements (function template) equivalence_relation(C++20) specifies that a relation imposes an equivalence relation (concept) ranges::is_permutation(C++20) determines if a sequence is a permutation of another sequence (niebloid)" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5505,"math_prob":0.95468944,"size":4580,"snap":"2023-40-2023-50","text_gpt3_token_len":1319,"char_repetition_ratio":0.21175699,"word_repetition_ratio":0.26123595,"special_character_ratio":0.31310043,"punctuation_ratio":0.19782871,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9863664,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-28T05:21:26Z\",\"WARC-Record-ID\":\"<urn:uuid:e2689949-2188-4bb9-beed-bc7525e35c38>\",\"Content-Length\":\"92149\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:35b743fd-e326-4694-87a6-1ba9dc4f52db>\",\"WARC-Concurrent-To\":\"<urn:uuid:59659005-4970-4bb3-a2ed-bbb8218c9663>\",\"WARC-IP-Address\":\"74.114.90.20\",\"WARC-Target-URI\":\"https://en.cppreference.com/w/cpp/algorithm/is_permutation\",\"WARC-Payload-Digest\":\"sha1:4XPHDKZGIKZV4BM3POBC5TVVAE7YZP63\",\"WARC-Block-Digest\":\"sha1:M6EYQ3CBANLETNCFVI7V3SXR4UDJT7TA\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233510358.68_warc_CC-MAIN-20230928031105-20230928061105-00648.warc.gz\"}"}
http://azfoo.net/gdt/babs/numbers/n/number68386.html	[ "### About the Number 68,386 (sixty-eight thousand three hundred eighty-six)\n\nMathBabbler's Buick's odometer [no picture] hit 68,386 miles on 5 July 2015 while driving west on I-40 in Arizona.\n\nNote: This nBAB was created on 22 June 2016, which was almost a full year since the POM (Palindromic Odometer Moment) occurred.\n\n```MathBabbler Number Analyst (MBNA) output:\n=========================================\n68386 is natural, counting, whole, integer\n68386 is even (not odd)\n68386 proper divisors are: 1,2,31,62,1103,2206,34193,\n68386 has 7 proper divisors\n68386 is deficient (sum of divisors is 37598; ratio: 0.549791)\n68386 is unhappy\n68386 is a Squarefree Number\n68386 is Harshad (Niven) number\n68386 is composite (not prime)\n68386 has the prime factors: 2311103 (sum=1136)\n68386 is sphenic (has 3 unique prime factors)\n68386 is a 1103-smooth number\n68386 is palindromic\n68386 is Arithmetic (A003601)\n68386 in octal is 0205442\n68386 in hexadecimal is 0x10b22\n68386 in binary is 10000101100100010 (is evil)\n68386 nearest square numbers: -265...258 (68121...68644 )\nsqrt(68386) = 261.507\nln(68386) = 11.1329\nlog(68386) = 4.83497\n68386 reciprocal is .00001462287602725704091480712426\n68386 is an apocalyptic power (2^68386 contains 666)\n68386! is inf\n68386 is 21767.9 Pi years\n68386 is 3419 score and 6 years\n```\n\nCreator: Gerald Thurman [gthurman@gmail.com]\nCreated: 22 June 2016", null, "" ]	[ null, "http://i.creativecommons.org/l/by/3.0/us/88x31.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.84101284,"math_prob":0.9243947,"size":1301,"snap":"2019-35-2019-39","text_gpt3_token_len":441,"char_repetition_ratio":0.18272938,"word_repetition_ratio":0.0,"special_character_ratio":0.49423522,"punctuation_ratio":0.14859438,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95620036,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-09-16T02:41:04Z\",\"WARC-Record-ID\":\"<urn:uuid:84b02407-e499-4870-a42a-981bf99b3669>\",\"Content-Length\":\"3454\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:67d0d51b-8a4a-4d1d-8ca0-b60491fd2e83>\",\"WARC-Concurrent-To\":\"<urn:uuid:9348a9d0-8d1a-44e4-b412-d6daaa2b68b7>\",\"WARC-IP-Address\":\"70.38.112.232\",\"WARC-Target-URI\":\"http://azfoo.net/gdt/babs/numbers/n/number68386.html\",\"WARC-Payload-Digest\":\"sha1:FZGZ7LIO4I2ER3XQHN3C3WJBS7YGRVTC\",\"WARC-Block-Digest\":\"sha1:A4P236V47ILMTPYLJVJTA5ERGBXOHROE\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-39/CC-MAIN-2019-39_segments_1568514572471.35_warc_CC-MAIN-20190916015552-20190916041552-00140.warc.gz\"}"}
http://footballfeverpodcast.com/Maths%20Ncert%20Solutions%E3%82%AF%E3%83%A9%E3%82%B97%E7%AC%AC11%E7%AB%A0%E6%BC%94%E7%BF%9211.3	[ " Maths Ncert Solutionsクラス7第11章演習11.3\n\n# NCERT Solutions for Class 11 Maths Chapter 7 - VEDANTU.\n\nView NCERT solutions of all the questions of NCERT, including Examples, Exercises, Miscellaneous and Supplementary Questions for Class 6 to 12 free at teachoo. Videos of questions and theory are available for your reference. NCERT Solutions for Class 11 Maths Chapter 1- Sets The major concepts of Maths covered in Chapter 1- Sets of NCERT Solutions for Class 11 includes: 1.1 Introduction 1.2 Sets and their Representations 1.3 The Empty Set 1.4.\n\nThe NCERT solutions for the questions given in Chapter 4, Principle of Mathematical Induction, of the Class 11 NCERT textbooks are available here. In this chapter, students learn about the Principle of Mathematical Induction in detail. Get free NCERT Solutions for CBSE class 11 Maths NCERT Solutions in Videos, Chapter-wise detailed solutions to the questions of the NCERT textbooks for class 11 Maths X Get a free home demo of LearnNext Available for.\n\nCBSE NCERT Solutions For Class 9th Maths Chapter 7: Triangles. NCERT Solutins For Class 9 Mathematics. Exercise 7.1, Exercise 7.2, Exercise 7.3, Exercise 7.4, Exercise 7. Free NCERT Solutions Class 12 Maths chapter 3 Matrices in PDF form download, Matrices chapter is one of the easy and scoring chapters as per CBSE examination. Go through the chapter completing NCERT and Exemplar books and practicing the previous year.\n\nNCERT Books are very useful for students who want to score good percentage in their Exams. We provide NCERT textbooks and solution for all class 8, class 9, class 10, class 11, class 12. In this post, We provide NCERT Solutions for Class 11 Maths in PDF format with direct download option. ncrtsolutions.in ncrtsolutions.in NCERT Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations Class 11 Chapter 5 Complex Numbers and Quadratic Equations Exercise 5.1, 5.2, 5.3. NCERT Solutions for class 11 Science Math solved by subject matter experts. NCERT CBSE latest book edition solutions. Free downloadable chapter wise NCERT solutions for class 11 Science Math in PDF format to help students. 2019/09/19 · ダウンロードNCERT Solutions, CBSE & State Board 6th-12th Class APK最新バージョン3.04-co.gradeup.k12 - インドの＃1無料学習アプリ：ただ写真を撮ると任意の質問に答えを得る. Get here NCERT Solutions for Class 12 Maths Chapter 11. These NCERT Solutions for Class 12 of Maths subject includes detailed answers of all the questions in Chapter 11 – Three Dimensional Geometry provided in NCERT Book which is prescribed for class 12 in schools.\n\n1. CBSE NCERT Solutions For Class 7th Maths Chapter 11: Perimeter and Area. NCERT Solutins For Class 7 Mathematics. Exercise 11.1, Exercise 11.2, Exercise 11.3, Exercise 11.4. Class VII NCERT Solutions.\n2. 2014/02/06 · NCERT Solutions for Class 7 Maths Chapter 11 Ex11.1 Q2 2. Find the area of a square park whose perimeter is 320 m. Follow us on Facebook: faceboo.\n3. NCERT Solutions for Class 12th Maths Chapter 11 – Three Dimensional Geometry National Council of Educational Research and Training NCERT Book Solutions for. Anand Meena Full time entrepreneur, likes to indulge.\n4. Get NCERT solutions for Class 11 Maths Free with videos. All exercise questions, supplementary questions, examples and miscellaneous are solved with important questions marked.Most of the chapters we will study in Class 11.\n\n2017/12/19 · NCERT 11 math solutions– In this article, you will get NCERT solutions of CBSE Class 11 NCERT Math book. Solutions of all the chapters are available in pdf. Free download of NCERT chapter-wise solutions for class 11 Maths. 2019/11/26 · ダウンロードFree IIT JEE Maths Solutions NCERT CBSE Doubts App APK最新バージョン7.8.1-com.doubtnutapp - オンライン数学研究アプリ - 無料IIT JEEモックペーパー、最高のNCERTビデオソリューションCBSE.\n\n## NCERT Solutions for Class 7 Maths Chapter 11 Ex11.1 Q2.\n\nNCERT Solutions for Class 11 Maths in PDF format are available to download. NCERT books as well as books for revision are also available to download along with the answers given at the end of the book. Revision books contains. sin 11 — sin x It is la-lown that, Adding 1 and 2, we obtain Show that iffandg are defined as and It is la-lown that if rx is an even function, then if rx is an odd function, then Hence, the correct answer is C TUTORIALS Title. ncrtsolutions.in ncrtsolutions.in NCERT Solutions for Class 11 Maths Chapter 11 Conic Sections Class 11 Chapter 11 Conic Sections Exercise 11.1, 11.2, 11.3. ダウンロードCommerce Maths Part 2 Solution For 12th HSC Board APK最新バージョン1.0-com.yia.\n\nChapter 11: Mensuration 11.1 Introduction We have learnt that for a closed plane figure, the perimeter is the distance around its boundary and its area is the region covered by it.. CBSE & NCERT Solutions for Class 11 Maths Circle. Chapter-wise solutions, video, study material for class 11 Maths solved by expert teachers. Question: Find the centre and radius of the circle x2y2 -2x 4y – 8 = 0. - get. NCERT Solutions for class 7 Math solved by subject matter experts. NCERT CBSE latest book edition solutions. Free downloadable chapter wise NCERT solutions for class 7 Math in PDF format to help students in homework and. 2016/11/29 · NCERT SolutionsPart - 3 - Perimeter and Area notes for Class 7 is made by best teachers who have written some of the best books of Class 7. It has gotten 11239 views and also has 4.65 rating. You can.\n\n2019/11/13 · ダウンロードVedantu: Learning App for Class6-10, IITJEE & NEET APK最新バージョン1.4.0-com.vedantu.app - CBSE、ICSE、州委員会：スタディPDF、NCERTソリューション📚ライブクラス🔴. NCERT Solutions for Class 11 Maths Three Dimensional Geometry Takshila Learning offers Class 11 Maths NCERT solutions. NCERT Solutions are an important part of learning for those who want to score well in their Class 11th Maths exam. 2016/10/21 · NCERT SolutionsPart - 1 - Exponents and Powers notes for Class 7 is made by best teachers who have written some of the best books of Class 7. It has gotten 14499 views and also has 4.68 rating. You can.\n\n• Categories Class 11, NCERT Solutions Post navigation 1 thought on “NCERT Solutions class-11 Maths Exercise 11.3” Tanuj kumar December 5, 2019 at 7:56 pm I want to help in my study Leave a Comment Cancel reply Name.\n• NCERT Solutions class 12 Maths Exercise 11.3 7. Find the intercepts cut off by the plane Ans. Equation of the plane is Comparing with intercept form, we have.\n• - No.1 online tutoring company in India provides you Free PDF download of NCERT Solutions for Class 11 Maths Chapter 7 - Permutations and Combinations solved by Expert Teachers as per NCERT CBSE Book.\n• 2014/02/06 · NCERT Solutions for Class 7 Maths Chapter 11 Ex11.1 Q3 3. Find the breadth of a rectangular plot of land, if its area is 440 m and the length is 22 m. Also find its perimeter. Follow us on Facebook: https." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6167286,"math_prob":0.66669923,"size":5755,"snap":"2020-45-2020-50","text_gpt3_token_len":2047,"char_repetition_ratio":0.20552947,"word_repetition_ratio":0.12684032,"special_character_ratio":0.2488271,"punctuation_ratio":0.11354962,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95824844,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-20T18:05:55Z\",\"WARC-Record-ID\":\"<urn:uuid:2b35f71e-ed69-454c-bf03-7c171ddc1bdc>\",\"Content-Length\":\"19021\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:201fda10-4970-4837-a4a2-37d98fc228bc>\",\"WARC-Concurrent-To\":\"<urn:uuid:1658104e-69b4-44f7-8a2d-860adf44056a>\",\"WARC-IP-Address\":\"80.241.215.125\",\"WARC-Target-URI\":\"http://footballfeverpodcast.com/Maths%20Ncert%20Solutions%E3%82%AF%E3%83%A9%E3%82%B97%E7%AC%AC11%E7%AB%A0%E6%BC%94%E7%BF%9211.3\",\"WARC-Payload-Digest\":\"sha1:CQUXIM5MR2LIZV3IKFUF5LQ6BH4V5W7G\",\"WARC-Block-Digest\":\"sha1:HD4INCWCWIBOUQ6BVZNKRL7BZPHADWEF\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-45/CC-MAIN-2020-45_segments_1603107874026.22_warc_CC-MAIN-20201020162922-20201020192922-00651.warc.gz\"}"}
https://iksinc.online/category/clustering/	[ "How Similar are Two Clustering Results\n\nWhile performing clustering, it is not uncommon to try a few different clustering methods. In such situations, we want to find out how similar are the results produced by different clustering methods. In some other situations, we may be interested in developing a new clustering algorithm or might be interested in evaluating a particular algorithm for our use. To do so, we make use of data sets with known ground truth so that we can compare the results against the ground truth. One way to evaluate the clustering results in all these situations is to make use of a numerical measure known as Rand index (RI). It is a measure of how similar two clustering results or groupings are.\n\nRand Index (RI)\n\nRI works by looking at all possible unordered pairs of examples. If the number of examples or data vectors for clustering is n, then there are", null, "$\\binom{n}{2}(=n(n-1)/2)$ pairs. For every example pair, there are three possibilities in terms of grouping. The first possibility is that the paired examples are always placed in the same group as a result of clustering. Lets count how often this happens over all pairs and represent that count by a. The second possibility is that the paired examples are never grouped together. Lets use b to represent the count of all pairs that are never grouped together. The third possibility is that the paired examples are sometimes grouped and sometimes not grouped together. The first two possibilities are treated as paired examples in agreement while the third possibility represents pairs in confusion. The RI of two groupings is then calculated by the following formula:", null, "$\\text{RI} = \\frac{\\text{Count of Pairs in Agreement}}{\\text{Total Number of Pairs}} = \\frac{(a+b)}{\\binom{n}{2}}$\n\nWe can notice from the formula that RI can never exceed 1 and its possible lowest value is 0.\n\nLets take an example to illustrate RI calculation. Say we have five examples clustered into two clusters using two different clustering methods. The first method groups examples A, B, and C into one group and examples D and E into another group. The second clustering method groups A and B together and C, D, and E together. To compute RI for this example, lets first list all possible unordered pairs of five examples at hand. We have 10 (n(n-1)/2) such pairs. These are: {A, B}, {A, C}, {A, D}, {A, E}, {B, C}, {B, D}, {B, E}, {C, D}, {C, E}, and {D, E}. Examining these pairs, we notice that the pair {A, B} and {D, E} are always grouped together by the both clustering methods. Thus, the value of a is two. We also notice that four pairs, {A, D}, {A, E}, {B, D}, and {B, E}, never occur together in any clustering result. Thus, the value of b is four. The Rand index (RI) is then 0.6.\n\nAdjusted Rand Index (ARI)\n\nRI suffers from one drawback; it yields a high value for pairs of random partitions of a given set of examples. To understand this drawback, think about randomly grouping a number of examples. When the number of partitions in each grouping, that is when the number of clusters, is increased, more and more example pairs are going to be in agreement because they are more likely to be not grouped together. This will result in a high RI value. Thus, RI is not able to take into consideration effects of random groupings. To counter this drawback, an adjustment is made to the calculations by taking into consideration grouping by chance. This is done by using a specialized distribution, the generalized hyper-geometric distribution, for modeling the randomness. The resulting measure is known as the adjusted Rand index (ARI).\n\nARI is best understood using an example. So lets look at the example of two clustering results used earlier. Lets create a contingency table summarizing the results of two clustering methods. In this case, it is a 2×2 table wherein each cell of the table shows the number of times an example occurs in two clusters referenced by the corresponding row and column.", null, "In the table above, M1C1 and M1C2 refer to two clusters formed by a hypothetical method-1. M2C1 and M2C2 similarly refer to two clusters formed by method-2. For clarity sake, I have included the examples forming the respective clusters next to M1C1, M1C2 etc. The top left cell has an entry of 2 because the clusters M1C1 and M2C1 share two examples, A and B. Entries in the other cells have similar meaning. The numbers to the right and below the contingency table show the sums along respective rows and columns.\n\nTo write the formula for ARI, lets generalize the entries of the contingency table using the following notation:", null, "$n_{ij} = \\text{Number of examples common to cluster i and cluster j}$", null, "$a_i = \\text{Sum of contingency cells in row i}$", null, "$b_j = \\text{Sum of contingency cells in column j}$\n\nThe ARI is then expressed as:", null, "The first term in the numerator is known as index, and the second term as expected index. The first term in the denominator is called maximum index, and the second term of the denominator is same as the second term of the numerator. With these designations of the terms, the ARI is often expressed as", null, "$\\text{ARI} = \\frac{\\text{index - expected index}}{\\text{maximum index - expected index}}$\n\nNow lets go back to the contingency table for our example and calculate the different parts of the ARI formula first. We have:", null, "$\\sum_{ij}\\binom{n_{ij}}{2} = \\binom{2}{2} + \\binom{1}{2} + \\binom{0}{2} + \\binom{2}{2} = (1 + 0 + 0 + 1) = 2$", null, "$\\sum_i\\binom{a_i}{2} = (\\binom{3}{2}+\\binom{2}{2}) = (3 + 1) = 4$", null, "$\\sum_j\\binom{b_j}{2} = (\\binom{2}{2}+\\binom{3}{2}) = (1 + 3) = 4$\n\nThus the index value for our example is 2; the expected index value is 1.6 (44/(5*4/2)). The maximum index value is 4. Therefore, the ARI for our example is (2 – 1.6)/(4 – 1.6), which equals 0.1666. We see that RI is much higher than ARI; this is typical of these indices. While RI always lies in 0-1; ARI can achieve a negative value also.\n\nARI is not the only measure to compare two sets of groupings. Mutual information based measure, adjusted mutual information (AMI), is also used for this purpose. May be in one of the future posts, I will describe this measure." ]	[ null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://iksinc.files.wordpress.com/2019/05/screen-shot-2019-05-03-at-8.12.48-pm.png", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://iksinc.files.wordpress.com/2019/05/ari-1.png", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9339709,"math_prob":0.9822135,"size":5440,"snap":"2019-43-2019-47","text_gpt3_token_len":1211,"char_repetition_ratio":0.13576159,"word_repetition_ratio":0.024109015,"special_character_ratio":0.22352941,"punctuation_ratio":0.12320144,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9988233,"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],"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],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-23T21:27:58Z\",\"WARC-Record-ID\":\"<urn:uuid:9814d8f2-a903-44bd-8606-61cbc7328cbe>\",\"Content-Length\":\"79938\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0d23138d-b9ee-466f-affe-02873f744c51>\",\"WARC-Concurrent-To\":\"<urn:uuid:c523795a-1756-43da-a674-979baa0bac43>\",\"WARC-IP-Address\":\"192.0.78.25\",\"WARC-Target-URI\":\"https://iksinc.online/category/clustering/\",\"WARC-Payload-Digest\":\"sha1:KRFR6S5FCASN6AP34LR32HVFEK2XJYQJ\",\"WARC-Block-Digest\":\"sha1:FHWPMVXC7LSMB2XAIEXPO7OLHFFTZ2O6\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570987836295.98_warc_CC-MAIN-20191023201520-20191023225020-00081.warc.gz\"}"}
https://stat.ethz.ch/pipermail/r-help/2011-September/290923.html	[ "# [R] Error in optim function.\n\nBerend Hasselman bhh at xs4all.nl\nTue Sep 27 06:53:46 CEST 2011\n\n```jango wrote:\n>\n> I'm trying to calculate the maximum likelihood estimate for a binomial\n> distribution. Here is my code:\n>\n> y <- c(2, 4, 2, 4, 5, 3)\n> n <- length(y)\n> binomial.ll <- function (pi, y, n) { ## define log-likelihood\n> output <- ylog(pi)+(n-y)(log(1-pi))\n> return(output)\n> }\n> binomial.mle <- optim(0.01, ## starting value\n> binomial.ll, ## log likelihood\n> method=\"BFGS\", ## optimization method\n> hessian=TRUE, ## numerial Hessian\n> control=list(fnscale=-1), ## max, not min\n> y=y, n=n)\n> binomial.mle.par <- c(binomial.mle\\$par, -1/binomial.mle\\$hessian[1,1])\n> binomial.mle.par <- as.matrix(binomial.mle.par)\n> rownames(binomial.mle.par) <- c(\"lambda\", \"s.e.\")\n> colnames(binomial.mle.par) <- c(\"MLE\")\n> print(binomial.mle.par)\n>\n> When I do this I get the following error message:\n>\n> Error in optim(0.01, binomial.ll, method = \"BFGS\", hessian = TRUE, control\n> = list(fnscale = -1), :\n> objective function in optim evaluates to length 6 not 1\n>\n>\n\nAfter defining your binomial.ll function do this\n\nbinomial.ll(0.01,y,n)\n\nand you will see that your function is returning a vector of length 6, which\nis the length of y.\nYour function is returning a vector but should return a scalar.\nA likelihood is a scalar so maybe return(sum(output)).\n\nBerend\n\n--\nView this message in context: http://r.789695.n4.nabble.com/Error-in-optim-function-tp3846001p3846179.html\nSent from the R help mailing list archive at Nabble.com.\n\n```" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5443074,"math_prob":0.6189061,"size":1567,"snap":"2020-10-2020-16","text_gpt3_token_len":482,"char_repetition_ratio":0.15547025,"word_repetition_ratio":0.008583691,"special_character_ratio":0.3299298,"punctuation_ratio":0.21428572,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9987271,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-02-17T04:48:15Z\",\"WARC-Record-ID\":\"<urn:uuid:70153c16-0322-4ad2-9373-9eeab44bbef1>\",\"Content-Length\":\"4780\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4a2098be-6fcd-4c7b-aa7e-1ac4e209655d>\",\"WARC-Concurrent-To\":\"<urn:uuid:7bedea23-e57b-4d83-9c6d-d0bad7591bf5>\",\"WARC-IP-Address\":\"129.132.119.195\",\"WARC-Target-URI\":\"https://stat.ethz.ch/pipermail/r-help/2011-September/290923.html\",\"WARC-Payload-Digest\":\"sha1:4ISVHGFAFG5YBPUCAMQCJGXODWQKZL3C\",\"WARC-Block-Digest\":\"sha1:J5G45YN772TMLIBAGK5GGJQOZYNXTH2I\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-10/CC-MAIN-2020-10_segments_1581875141653.66_warc_CC-MAIN-20200217030027-20200217060027-00211.warc.gz\"}"}
https://socratic.org/questions/how-do-you-solve-and-check-for-extraneous-solutions-in-abs-2t-3-t	[ "# How do you solve and check for extraneous solutions in abs(2t-3) = t?\n\n##### 1 Answer\nAug 1, 2015\n\n$\\textcolor{red}{t = 3}$ is a solution.\n$\\textcolor{red}{t = 1}$ is an extraneous solution.\n\nSOLVE\n\n$\| 2 t - 3 \| = t$\n\nWe need to write two different equations without the absolute value symbols and solve for $t$.\n\nThese equations are:\n\n(1): $\\left(2 t - 3\\right) = t$\n(2): $- \\left(2 t - 3\\right) = t$\n\nSolve Equation 1:\n\n$2 t - 3 = t$\n\nSubtract $t$ from each side.\n\n$t - 3 = 0$\n\nAdd $3$ to each side.\n\n$t = 3$\n\nSolve Equation 2:\n\n−(2t-3) = t\n\nRemove parentheses.\n\n$- 2 t + 3 = t$\n\nAdd $2 t$ to each side.\n\n$3 = 3 t$\n\nDivide each side by $3$.\n\n$t = 1$\n\nThe solutions are $t = 1$ and $t = 3$.\n\nCHECK FOR EXTRANEOUS SOLUTIONS:\n\nIf $t = 1$,\n\n$\| 2 t - 3 \| = t$\n$\| 2 \\left(1\\right) - 3 \| = 3$\n$\| 2 - 3 \| = 5$\n$\| - 1 \| = 5$\n$1 = 5$\n\nThis is impossible, so $t = 1$ is an extraneous solution.\n\nIf $t = 3$,\n\n$\| 2 t - 3 \| = t$\n$\| 2 \\left(3\\right) - 3 \| = 3$\n$\| 6 - 3 \| = 3$\n$\| 3 \| = 3$\n$3 = 3$\n\n$t = 3$ is a solution." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6532645,"math_prob":1.00001,"size":518,"snap":"2021-21-2021-25","text_gpt3_token_len":139,"char_repetition_ratio":0.10700389,"word_repetition_ratio":0.0,"special_character_ratio":0.23359074,"punctuation_ratio":0.10204082,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000074,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-20T21:51:25Z\",\"WARC-Record-ID\":\"<urn:uuid:35381d5d-cb72-4a90-a632-00a1511a8d8b>\",\"Content-Length\":\"34950\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:afd9c54b-fa84-4f20-b858-62be46b7d767>\",\"WARC-Concurrent-To\":\"<urn:uuid:6bd1e35d-04cc-45c8-89f3-05d558031352>\",\"WARC-IP-Address\":\"216.239.32.21\",\"WARC-Target-URI\":\"https://socratic.org/questions/how-do-you-solve-and-check-for-extraneous-solutions-in-abs-2t-3-t\",\"WARC-Payload-Digest\":\"sha1:L6A2SBIQYLYPHS7OE573VYGSRE2RCBII\",\"WARC-Block-Digest\":\"sha1:4BS3QVO6YO3WCV2QKHL56EAETIZJ6I3J\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623488257796.77_warc_CC-MAIN-20210620205203-20210620235203-00453.warc.gz\"}"}
http://web.mat.bham.ac.uk/R.W.Kaye/logic/cardarith.html	[ "# Cardinals\n\n## 1. Introduction\n\nThis page discusses cardinals and cardinal arithmetic, both with and without the Axiom of Choice. In general the subject is a very rich one with many results and still many difficult problems at the research level.\n\n## 2. Cardinal arithmetic\n\nCardinal numbers are discussed at length in the beginning of Section 11.1 of The Mathematics of Logic. In particular two sets $X , Y$ have the same cardinality or are equipolent (written $X ≈ Y$) if there is a bijection $f : X → Y$. This is an equivalence relation on sets. Informally you can think of the cardinality of a set as the equivalence class of the set by this equivalence relation, but this raises a problem: the cardinality of a set if defined this way is not itself a set but a class. To get round this problem one of two alternative definitions are usually provided.\n\nIn the absence of the Axiom of Choice, the following definition is usually preferred.\n\nDefinition.\n\n$card ( X )$, the cardinality of $X$, is the set $u u ≈ X ∧ ∀ v ( rank ( v ) < rank ( u ) → ¬ v ≈ X )$.\n\nThus the cardinality of a set is the set of sets equipolent to it of minimal rank. Since this is a subset of some $V α$ it is a set.\n\nIf the Axiom of Choice holds then a simpler alternative is available.\n\nDefinition.\n\nBy the well-ordering principle every set $X$ is well-orderable, and hence in 1–1 correspondence with an ordinal. Thus there is a least ordinal $α ≈ X$. We define $card ( X )$ to be this ordinal.\n\nThis definition has the advantage that $card ( X )$ is a canonically chosen set with the same cardinality as $X$. Note that the two definitions, as just given, are not compatible. In practice, the definition (any definition) of $card ( X )$ is simply a convenience rather than essential, since results about a cardinal are usually proved by first taking a set that has that cardinality. Here we will implicitly assume AC holds and take the second definition of cardinal, but indicate which of the theorems we prove do not need choice.\n\nDefinition.\n\nAn ordinal $α$ is initial if there is no bijection $f : α → β$ for $β < α$.\n\nFor example, $ω$ is initial but $ω + 1$ is not. (This is a consequence of the next proposition below.)\n\nGiven a set $X$, the least ordinal in 1–1 correspondence with it is an initial ordinal, since if not there would be a smaller ordinal in 1–1 correspondence with $X$, because 1–1 correspondences compose. Thus cardinals are initial ordinals. Conversely, all initial ordinals are their own cardinals.\n\nProposition.\n\nAn infinite initial ordinal is a limit ordinal.\n\nProof.\n\nIf $α = β + 1$ is infinite there is a bijection $β → α$ defined by $f ( 0 ) = β$, $f ( n +1 ) = n$ for $n ∈ ω$ and $f ( γ ) = γ$ for all other $γ$.\n\nThe theorems and proofs in Sections 11.1 and 11.3 should be studied carefully. All of the proofs there are valid in $ZF$ or $ZFC$, and where Zorn's lemma was used it is known that some form of the Axiom of Choice is required for the result in question. In particular the definitions of cardinal arithmetic and the order relation on cardinals is important. We define\n\nDefinition.\n\nFor sets $A , B$, $card ( A ) ⩽ card ( B )$ if there is an injection $f : A → B$. We define $card ( A ) ⩽ * card ( B )$ if $A = ∅$ or there is a surjection $g : B → A$.\n\nHowever, the Zorn's lemma proof in the book that cardinal arithmetic has $κ 2 = κ$ for all infinite $κ$ is slightly long-winded. For those that know about ordinals (including everyone reading these web pages) there is a different proof.\n\nFirst we define bijections $f α : α × α → F ( α )$ recursively on ordinals $α$. Define $F ( 0 ) = 0$, $F ( α + 1 ) = F ( α ) + α + α + 1$, and $F ( λ ) = ⋃ α < λ F ( α )$ for limits $λ$. Define $f 0 : 0 × 0 → 0$ to be the empty function, and $f α + 1 ( β , γ ) = f α ( β , γ )$ if $β , γ < α$, $f α + 1 ( α , γ ) = F ( α ) + γ$ if $γ < α$, and $f α + 1 ( β , α ) = F ( α ) + α + β$ if $β ⩽ α$; finally $f λ ( β , γ ) = f α ( β , γ )$ for any $α$ with $β , γ < α < λ$. You can check that these define bijections $f α : α × α → F ( α )$ as claimed and $f α$ extends $f β$ for $α > β$.\n\nProposition.\n\nIf $κ$ is an infinite initial ordinal then there is a bijection $κ × κ → κ$.\n\nProof.\n\nSuppose $κ$ is initial and for every smaller initial ordinal $μ$ there is a 1–1 correspondence $μ × μ → μ$. Consider $f κ : κ × κ → F ( κ )$. If $F ( κ ) ⩽ κ$ we are finished, by Schröder–Bernstein. So $κ < F ( κ )$. Also $F ( μ ) < κ$ for all initial $μ < κ$ since $F ( μ )$, $μ × μ$ and $μ$ all have the same cardinality strictly less than $κ$. Let $F κ ( α , β ) = κ$ and let $δ = max ( α , β ) +1$ so $κ$ is in the range of $f δ$. Now $δ < κ$ as $κ$ is initial and hence is a limit ordinal. Therefore there is a bijection from $δ$ to $μ$, some initial ordinal less than $κ$, and this induces an impossible bijection from $F ( δ ) > κ$ to $F ( μ ) < κ$.\n\nThe previous result does not need the axiom of choice, but the conclusion that $κ × κ = κ$ for all infinite cardinals $κ$ does not follow from it without choice as (in the absence of choice) some sets are not well-orderable.\n\n## 3. Alephs and Hartog's function\n\nThe axiom of choice is not required in this section either.\n\nDefinition.\n\nGiven any set $X$, we define $ℵ ( X ) = α ∈ On there is an injection α → X$.\n\nLemma.\n\nFor all sets $X$, $ℵ ( X )$ is a set, and hence an ordinal. There is no injection $ℵ ( X ) → X$ and $ℵ ( X )$ is the least ordinal with this property.\n\nProof.\n\nLet\n\nSo $W$ is a set by separation and power set axioms and there is for each $( y , < ) ∈ W$ a unique ordinal $α y$ order-isomorphic to $( y , < )$. Define $f : ( y , < ) ↦ α y$ and by replacement the image of $f$ is a set, and note that this image is $ℵ ( X )$. There is no injection $ℵ ( X ) → X$ for if there was we would have $ℵ ( X ) ∈ ℵ ( X )$.\n\nProposition.\n\nFor all $X$, $ℵ ( X )$ is initial.\n\nProof.\n\nIf not, $f : ℵ ( X ) → β < α$ and $g : β → y ⊆ x$ are bijections then their composition maps $ℵ ( X ) → X$ injectively, so $ℵ ( X ) ∈ ℵ ( X )$.\n\nInitial ordinals are sometimes called Alephs because of the following.\n\nDefinition.\n\n$ℵ 0 = ω$, the first infinite initial ordinal; $ℵ α + 1 = ℵ ( ℵ α )$; and $ℵ λ = ⋃ α < λ ℵ α$ for limit ordinals $λ$.\n\nProposition.\n\nEach $ℵ α$ is initial; every initial ordinal is $ℵ α$ for some $α$.\n\nProof.\n\nIf $λ$ is a limit and $f : ℵ λ → β < ℵ λ$ is a bijection then a restriction of $f$ is an injection $ℵ α + 1 → β ⊆ ℵ α$, contradicting Schröder–Bernstein. Thus every $ℵ α$ is initial as the result is obvious for the zero and sucessor cases. The converse is proved by taking an initial ordinal $α$ and considering the least ordinal $β$ such that $α < ℵ β$. This $β$ exists by the fact that $α β ⩾ β$ for all $β$, and must be a successor. For if $β$ is a limit, by definition of $α β$ there would be $δ < β$ with $α < ℵ δ$. Thus $β = γ + 1$ and $ℵ γ ⩽ α < ℵ γ + 1$ so there is an injection $α → ℵ γ$ as $α ∈ ℵ ( ℵ γ )$, hence $α = ℵ γ$ as $α$ is initial.\n\nUnder the Axiom of Choice, every set is well-ordered so is in one-to-one correspondence with an ordinal. Thus the Axiom of Choice implies every cardinal is an aleph.\n\nHartog's $ℵ$ function gives one way to make a larger cardinal from a smaller one. Another way is to use the power set function.\n\nProposition.\n\nLet $X$ be a set. Then there is no bijection between $X$ and $P ( X )$.\n\nProof.\n\nSuppose $f : X → P ( X )$ is a bijection and let $y = u ∈ x u ∉ f ( u )$. We get a contradiction by taking $u ∈ x$ such that $y = f ( u )$ and trying to decide if $u ∈ f ( u )$ or not.\n\nDefinition.\n\nFor a set $X$, $ℶ ( X )$ is the cardinality of $P ( X )$.\n\nThus we have two ways to make a cardinality just a bit bigger than $ω$: either $ℵ ( ω )$ or $ℶ ( ω )$. Assuming the Axiom of Choice we have $ℵ ( ω ) ⩽ ℶ ( ω )$. Cantor's Continuum Hypothesis (CH) is the statement that $ℵ ( ω ) = ℶ ( ω )$. Whether or not this is true remains a moot point, but it has been shown to be independent of the ZFC axioms (assuming ZFC to be consistent.\n\nMore generally, the Generalised Continuum Hypotheis (GCH) is the statement $ℶ ( ℵ α ) = ℵ α + 1$ for all $α$. Again, this is consistent with the axioms of ZFC but not provable in them. It can be shown that GCH implies the axiom of choice, so assuming GCH makes no sense without also assuming AC.\n\n## 4. Trichotomy\n\nThe Trichotomy theorem (Proposition 11.18 in The Mathematics of Logic) is equivalent to the Axiom of Choice. To see this, let $X$ be any set. Then it is not the case that $ℵ ( X ) ⩽ X$, by the construction of Hartog's $ℵ$ function, so by Trichotomy $X ⩽ ℵ ( X )$. In other words there is an injection $X → ℵ ( X )$ and as $ℵ ( X )$ is an ordinal a well ordering on $X$ may be defined via this injection and $∈$ on $ℵ ( X )$. Under a particular assumption on the arithmetic of certain cardinals, an interesting weakening of trichotomy can be obtained without Choice. In this section $κ , λ$ are just names for infinite cardinals: the fact we call them this is not intended to mean that they are necessarily ordinals. (Sometimes they will be ordinals, sometimes not.)\n\nProposition.\n\nIf $κ , λ$ are cardinals such that $κ + λ = κ λ$ then $κ ⩽ λ$ or $λ ⩽ * κ$.\n\nProof.\n\nSuppose $K , L$ are disjoint sets with cardinalities $κ , λ$ and $f : K ∪ L → K × L$ is a bijection and $π : K × L → L$ is the projection map. If the composite of $K → K × L → L$ is onto then we know that $λ ⩽ * κ$. Otherwise there is $x ∈ L$ such that for each $y ∈ K$ the unique $z ∈ K ∪ L$ such that $f ( z ) = ( y , x )$ is in $L$. This defines an injection $y ↦ z$ taking $K$ into $L$. Thus $κ ⩽ λ$.\n\nThis result has the consequence that our theorem on cardinal arithmetic is in fact equivalent to the axiom of choice.\n\nIf $κ 2 = κ$ for all infinite cardinals $κ$ then the Axiom of Choice holds.\n\nProof.\n\nLet $X$ be an arbitrary set with cardinality $λ$ and let $κ = ℵ ( X )$. Then $κ + λ = ( κ + λ ) 2 ⩾ κ λ ⩾ κ + λ$ so by Schröder–Berstein $κ + λ = κ λ$. Thus by the previous either $κ ⩽ λ$ or $λ ⩽ * κ$. But $κ ⩽ λ$ is impossible by the definition of $ℵ ( X )$ and hence $λ ⩽ * κ$ so there is a surjection $f : κ → X$ and hence an injection $g : X → κ$ given by $g ( x )$ is the least $β < κ$ with $f ( β ) = x$. This allows us to define a well-ordering on $X$ as before.\n\nOn the other hand, the proposition $κ + κ = κ$ for all infinite cardinals $κ$ is known to be strictly weaker than the axiom of choice." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.93819374,"math_prob":0.99984586,"size":7994,"snap":"2020-10-2020-16","text_gpt3_token_len":1771,"char_repetition_ratio":0.17809762,"word_repetition_ratio":0.006232687,"special_character_ratio":0.21278459,"punctuation_ratio":0.119798236,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.000007,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-02-24T18:08:34Z\",\"WARC-Record-ID\":\"<urn:uuid:cde813d4-9c97-4292-a96c-377bce1abe57>\",\"Content-Length\":\"34010\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:af9cfeee-8bab-4408-83bc-29a5c92a3547>\",\"WARC-Concurrent-To\":\"<urn:uuid:8f7c3d58-5ea4-4830-95c4-20ab68c353b7>\",\"WARC-IP-Address\":\"147.188.55.16\",\"WARC-Target-URI\":\"http://web.mat.bham.ac.uk/R.W.Kaye/logic/cardarith.html\",\"WARC-Payload-Digest\":\"sha1:BG4JG3INUWPMFGNQATSHWVYT6DKMEF3T\",\"WARC-Block-Digest\":\"sha1:VLIKNCN55B6QENNDUXTPQQTAPMHSPT3J\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-10/CC-MAIN-2020-10_segments_1581875145966.48_warc_CC-MAIN-20200224163216-20200224193216-00312.warc.gz\"}"}
https://answers.everydaycalculation.com/simplify-fraction/69-154	[ "Solutions by everydaycalculation.com\n\n## Reduce 69/154 to lowest terms\n\n69/154 is already in the simplest form. It can be written as 0.448052 in decimal form (rounded to 6 decimal places).\n\n#### Steps to simplifying fractions\n\n1. Find the GCD (or HCF) of numerator and denominator\nGCD of 69 and 154 is 1\n2. Divide both the numerator and denominator by the GCD\n69 ÷ 1/154 ÷ 1\n3. Reduced fraction: 69/154\nTherefore, 69/154 simplified is 69/154\n\nMathStep (Works offline)", null, "Download our mobile app and learn to work with fractions in your own time:" ]	[ null, "https://answers.everydaycalculation.com/mathstep-app-icon.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7961075,"math_prob":0.79706156,"size":431,"snap":"2020-24-2020-29","text_gpt3_token_len":126,"char_repetition_ratio":0.11943794,"word_repetition_ratio":0.0,"special_character_ratio":0.387471,"punctuation_ratio":0.0952381,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9535792,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-05T23:30:39Z\",\"WARC-Record-ID\":\"<urn:uuid:87142bd8-0ff0-4352-9f62-ffd1dd41a2c9>\",\"Content-Length\":\"6411\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f31c17d5-7a44-4e03-bded-057a47e6e45e>\",\"WARC-Concurrent-To\":\"<urn:uuid:0ad6e365-980a-4e67-a272-0ce8b4a0751d>\",\"WARC-IP-Address\":\"96.126.107.130\",\"WARC-Target-URI\":\"https://answers.everydaycalculation.com/simplify-fraction/69-154\",\"WARC-Payload-Digest\":\"sha1:OWVDSKCPLXPLR2EPXGQFR25777VGCFI7\",\"WARC-Block-Digest\":\"sha1:X5GPKLAJRX642LW2I3NHOA7JPAO3Q2DY\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593655889877.72_warc_CC-MAIN-20200705215728-20200706005728-00572.warc.gz\"}"}
https://en.wikipedia.org/wiki/Training_data	[ "# Training, validation, and test sets\n\n(Redirected from Training data)\nJump to navigation Jump to search\n\nIn machine learning, a common task is the study and construction of algorithms that can learn from and make predictions on data. Such algorithms work by making data-driven predictions or decisions,:2 through building a mathematical model from input data.\n\nThe data used to build the final model usually comes from multiple datasets. In particular, three data sets are commonly used in different stages of the creation of the model.\n\nThe model is initially fit on a training dataset, that is a set of examples used to fit the parameters (e.g. weights of connections between neurons in artificial neural networks) of the model. The model (e.g. a neural net or a naive Bayes classifier) is trained on the training dataset using a supervised learning method (e.g. gradient descent or stochastic gradient descent). In practice, the training dataset often consist of pairs of an input vector (or scalar) and the corresponding output vector (or scalar), which is commonly denoted as the target (or label). The current model is run with the training dataset and produces a result, which is then compared with the target, for each input vector in the training dataset. Based on the result of the comparison and the specific learning algorithm being used, the parameters of the model are adjusted. The model fitting can include both variable selection and parameter estimation.\n\nSuccessively, the fitted model is used to predict the responses for the observations in a second dataset called the validation dataset. The validation dataset provides an unbiased evaluation of a model fit on the training dataset while tuning the model's hyperparameters (e.g. the number of hidden units in a neural network). Validation datasets can be used for regularization by early stopping: stop training when the error on the validation dataset increases, as this is a sign of overfitting to the training dataset. This simple procedure is complicated in practice by the fact that the validation dataset's error may fluctuate during training, producing multiple local minima. This complication has led to the creation of many ad-hoc rules for deciding when overfitting has truly begun.\n\nFinally, the test dataset is a dataset used to provide an unbiased evaluation of a final model fit on the training dataset. If the data in the test dataset has never been used in training (for example in cross-validation), the test dataset is also called a holdout dataset.\n\n## Training dataset\n\nA training dataset is a dataset of examples used for learning, that is to fit the parameters (e.g., weights) of, for example, a classifier.\n\nMost approaches that search through training data for empirical relationships tend to overfit the data, meaning that they can identify and exploit apparent relationships in the training data that do not hold in general.\n\n## Validation dataset\n\nA validation dataset is a dataset of examples used to tune the hyperparameters (i.e. the architecture) of a classifier. It is sometimes also called the development set or the \"dev set\". In artificial neural networks, a hyperparameter is, for example, the number of hidden units. It, as well as the testing set (as mentioned above), should follow the same probability distribution as the training dataset.\n\nIn order to avoid overfitting, when any classification parameter needs to be adjusted, it is necessary to have a validation dataset in addition to the training and test datasets. For example, if the most suitable classifier for the problem is sought, the training dataset is used to train the candidate algorithms, the validation dataset is used to compare their performances and decide which one to take and, finally, the test dataset is used to obtain[citation needed] the performance characteristics such as accuracy, sensitivity, specificity, F-measure, and so on. The validation dataset functions as a hybrid: it is training data used by testing, but neither as part of the low-level training nor as part of the final testing[citation needed].\n\nThe basic process of using a validation dataset for model selection (as part of training dataset, validation dataset, and test dataset) is:\n\nSince our goal is to find the network having the best performance on new data, the simplest approach to the comparison of different networks is to evaluate the error function using data which is independent of that used for training. Various networks are trained by minimization of an appropriate error function defined with respect to a training data set. The performance of the networks is then compared by evaluating the error function using an independent validation set, and the network having the smallest error with respect to the validation set is selected. This approach is called the hold out method. Since this procedure can itself lead to some overfitting to the validation set, the performance of the selected network should be confirmed by measuring its performance on a third independent set of data called a test set.\n\nAn application of this process is in early stopping, where the candidate models are successive iterations of the same network, and training stops when the error on the validation set grows, choosing the previous model (the one with minimum error).\n\n## Test dataset\n\nA test dataset is a dataset that is independent of the training dataset, but that follows the same probability distribution as the training dataset. If a model fit to the training dataset also fits the test dataset well, minimal overfitting has taken place (see figure below). A better fitting of the training dataset as opposed to the test dataset usually points to overfitting.\n\nA test set is therefore a set of examples used only to assess the performance (i.e. generalization) of a fully specified classifier.", null, "A training set (left) and a test set (right) from the same statistical population are shown as blue points. Two predictive models are fit to the training data. Both fitted models are plotted with both the training and test sets. In the training set, the MSE of the fit shown in orange is 4 whereas the MSE for the fit shown in green is 9. In the test set, the MSE for the fit shown in orange is 15 and the MSE for the fit shown in green is 13. The orange curve severely overfits the training data, since its MSE increases by almost a factor of four when comparing the test set to the training set. The green curve overfits the training data much less, as its MSE increases by less than a factor of 2.\n\n## Holdout dataset\n\nMost simply, part of the original dataset can be set aside and used as a test set: this is known as the holdout method.\n\n## Cross-validation\n\nA dataset can be repeatedly split into a training dataset and a validation dataset: this is known as cross-validation. These repeated partitions can be done in various ways, such as dividing into 2 equal datasets and using them as training/validation, and then validation/training, or repeatedly selecting a random subset as a validation dataset[citation needed]. To validate the model performance, sometimes an additional test dataset that was held out from cross-validation is used.\n\nCross-validation doesn't work in situations where you can't shuffle your data, most notably in time-series.\n\n## Hierarchical classification\n\nAnother example of parameter adjustment is hierarchical classification (sometimes referred to as instance space decomposition ), which splits a complete multi-class problem into a set of smaller classification problems. It serves for learning more accurate concepts due to simpler classification boundaries in subtasks and individual feature selection procedures for subtasks. When doing classification decomposition, the central choice is the order of combination of smaller classification steps, called the classification path. Depending on the application, it can be derived from the confusion matrix and, uncovering the reasons for typical errors and finding ways to prevent the system make those in the future. For example, on the validation set one can see which classes are most frequently mutually confused by the system and then the instance space decomposition is done as follows: firstly, the classification is done among well recognizable classes, and the difficult to separate classes are treated as a single joint class, and finally, as a second classification step the joint class is classified into the two initially mutually confused classes." ]	[ null, "https://upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Traintest.svg/700px-Traintest.svg.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8762225,"math_prob":0.88907367,"size":10239,"snap":"2019-13-2019-22","text_gpt3_token_len":2167,"char_repetition_ratio":0.17420615,"word_repetition_ratio":0.032974,"special_character_ratio":0.22170134,"punctuation_ratio":0.13466597,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98149526,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-05-20T00:08:07Z\",\"WARC-Record-ID\":\"<urn:uuid:8c312171-23e4-4172-9408-d83e240d6583>\",\"Content-Length\":\"63098\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:fb598a9e-1b79-4e40-91a1-382c02ce5a31>\",\"WARC-Concurrent-To\":\"<urn:uuid:da419af5-7668-423f-8271-5db4fbd8fa40>\",\"WARC-IP-Address\":\"208.80.153.224\",\"WARC-Target-URI\":\"https://en.wikipedia.org/wiki/Training_data\",\"WARC-Payload-Digest\":\"sha1:7THDXJK7MTDLJKQ5JJYKC4GW5TGBUWV6\",\"WARC-Block-Digest\":\"sha1:ENDSW4UEOKFV5RW4JWGHOJLYIW7F25GY\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-22/CC-MAIN-2019-22_segments_1558232255182.37_warc_CC-MAIN-20190519221616-20190520003616-00151.warc.gz\"}"}
http://cicadacrafts.com/mc790o/function-of-galvanometer-76e30e	[ "## function of galvanometer\n\nIt is to be made into an ammeter with a full-scale deflection equal to 1.25 A. Nov. 11, 2020. Electric current, any movement of electric charge carriers, such as subatomic charged particles (e.g., electrons having negative charge, protons having positive charge), ions (atoms that have lost or gained one or more electrons), or holes (electron deficiencies that may be thought of as positive…, Machine, device, having a unique purpose, that augments or replaces human or animal effort for the accomplishment of physical tasks. A galvanometer has a coil resistance of 70.2 ohms. For example, if we rotate the coil by 1°, the Torsional constant will be C, similarly, if we rotate by 2°, it is 2 C. However, N, I, and A remain constant. The restoring force in the spring and the wire keeps on increasing. A galvanometer is a device that measures or detects small currents with appropriate modification. Ans: The sensitive galvanometer shows a huge deflection in a small current. Ans: You can increase the sensitivity by: Decrease C by using phosphor bronze or Quartz wire because their Torsional constant is very low. The galvanometer is an instrument used to determine the presence, direction, and the strength of an electric current in a conductor. Early galvanometers were not calibrated, but improved devices were used as measuring instruments, called ammeters, to measure the current flowing through an electric circuit. They are as follows:1) Seebeck effect: The Seebeck effect states that when two different or unlike metals are joined together at two junctions, an electromotive force (emf) is generated at the two junctions. Medical Definition of galvanometer : an instrument for detecting or measuring a small electric current by movements of a magnetic needle or of a coil in a magnetic field Other Words from galvanometer Galvanometer, instrument for measuring a small electrical current or a function of the current by deflection of a moving coil. Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree.... …to await the construction of galvanometers sensitive enough to measure the minute currents generated in muscles and the small potential differences across nerve membranes. Here, Φ and Sin Ө is a variable, and I α Φ/ Sin Ө. Galvanometer acts as an actuator (it is a component of machine that moves or controls a mechanism or system, requiring a control signal and a source of energy) and produces a rotator deflection in response to the current that flows through a coil in a fixed constant magnetic field. Determine resistance galvanometer half deflection method find figure merit Physics Lab instruments, Laboratory Equipment, science Equipments, Educational Equipments Manufacturer, supplier exporters in India. The galvanometer is the current measuring instrument which is mostly used in bridges and potentiometer for showing the zero current. This strain in the wire is the Torsional strain. The ballistic galvanometer is designed to deflect its indicating needle (or mirror) in a way that is proportional to the total charge passing through its moving coil or to a voltage pulse of short duration. When an electric current passes through the […] Under the action of this torque, the coil rotates and the deflection in the coil in a moving coil galvanometer is directly proportional to the current flowing through the coil. Click hereto get an answer to your question ️ Explain, using a labelled diagram, the principle and working of a moving coil galvanometer. We use the radial field because it increases the strength of the magnetic field around the coil. See more. Any conventional galvanometer may also be employed as a ballistic type, but the latter has smaller torque and higher inertia in the coil. https://www.britannica.com/technology/galvanometer, Association for Psychological Science - The History Corner: The Galvanometer, The University of Texas at Austin - Galvanometer, National High Magnetic Field Laboratory - Magnetic Academy - Galvanometer, galvanometer - Student Encyclopedia (Ages 11 and up). The main function of a galvanometer is to measure the presence, direction, and electric current in the conductor This works on the principle of mechanical energy to electric energy Once the current is in the magnetic field after the supply is turned on, the magnetic torque can be realized. You might have seen a device with a G symbol in your physics lab a lot many times. 2. A current due to variation in the … ... e-t/RC), which gives us the voltage across the capacitor as a function of time. Q4: How Do You Increase the Sensitivity of a Galvanometer? Ans: The current sensitivity of a moving coil galvanometer is given as: This means in the fractional flow of current, there is a high deflection. Now, we will use a soft iron core (it is a strong ferromagnetic material) in place of a loop and cylindrical magnets in place of horseshoe magnets. It can be converted into ammeter to measure the currents in the order of an ampere or millimetre or in the range of milliamperes or microammeter to measure microampere current. Galvanometer. A field in which the magnetic field lines pass from N to South pole such that the area vector A➝is always perpendicular (radial) to the magnetic field B. This force is separated by the width of the coil. The null point means the situation in which no current flows through the circuit. Now, how to measure this Torsional strain? The coil, to which an indicating needle or mirror is attached, rotates under the action of the torque; the angle through which it rotates to balance the torsion of the suspension provides a measure of the current flowing in the coil. The two horseshoe electromagnets are placed around this coil. Apparatus required: A Battery (0-6V), high resistance box (0-10,000 ohms), low resistance box (0-100 ohms), rheostat, two one way keys, galvanometer (30-0-30), voltmeter (0-3V). By signing up for this email, you are agreeing to news, offers, and information from Encyclopaedia Britannica. The galvanometer has following applications. Anthropology This works on the rule of converting energy from electrical to mechanical. The deflection is a mechanical rotation derived from forces resulting from the current. Now, this coil keeps on rotating. As the current (that is to be measured) is sent to the coil. They are determined experimentally with regard to the scan parameters of the input signal (i.e., frequency and amplitude). Omissions? A moving coil galvanometer is a highly sensitive instrument. Updates? While measuring the current, we have to convert it into an ammeter. Because of this nature, it can detect current in the range of milliamperes. Therefore, a restoring torque develops on the wire and the spring. Let the area of each turn be A, and the magnetic field be B. Blog. STUDY. Here, V = I/R (R = The resistance of the coil). Let us know if you have suggestions to improve this article (requires login). Describes the construction and functioning of d'Arsonval galvanometer. Principle of Galvanometer. This area vector. 10.1). This broad category encompasses such simple devices as the inclined plane, lever, wedge, wheel and axle, pulley, and screw (the so-called simple machines) as well as…. Vedantu academic counsellor will be calling you shortly for your Online Counselling session. A galvanometer is an electromechanical instrument used for detecting and indicating an electric current. Assertion : Earth's magnetic field does not affect the working of a moving coil galvanometer. A galvanometer is a device that measures or detects small currents with appropriate modification. The galvanometer at left is wired as a voltmeter placed across the capacitor, and the galvanometer at right is wired as an ammeter, placed in series with the resistor and capacitor. Now. They are 1. Please enable Javascript and refresh the page to continue You may have thought what this device is and what is the use of it? The working principle of thermocouple is based on three effects, discovered by Seebeck, Peltier and Thomson. When a current-carrying coil is suspended in a uniform magnetic field it is acted upon by a torque. But, we desired I α Φ. The deflection is a mechanical rotation derived from forces resulting from the current. This is how we can convert galvanometer to ammeter and voltmeter to get the value of current sensitivity and voltage sensitivity respectively. The galvanometer has permanent … In galvanometer. A torque acts on the coil which rotates the coil. A galvanometer is a device that is used to detect small electric current or measure its magnitude. Hang a fine wire through a metal attached to terminal T1. A galvanometer also performs the functions of an ammeter, therefore by a slight modification, galvanometers can be made into ammeter. Solution for What is the function of galvanometer in a circuit? The purpose of this is to make the current directly proportional to the deflection. They are determined experimentally with regard to the scan parameters of the input signal (i.e., frequency and amplitude). In 1901, he successfully developed a new string galvanometer with very high sensitivity, which he used in his electrocardiograph. Aim: To convert a galvanometer (30-0-30) into a voltmeter of a given range (1.5 V) and to calibrate it. This is how we can create a radial field. The moving coil of a moving coil galvanometer, moves in a magnetic field produced by a permanent magnet. Our editors will review what you’ve submitted and determine whether to revise the article. Pro Lite, CBSE Previous Year Question Paper for Class 10, CBSE Previous Year Question Paper for Class 12. This area vector A➝ is perpendicular to the plane of the loop. In this article, we will study the moving coil type galvanometer. His device weighed 600 pounds (Fig. The ballistic galvanometer is designed to deflect its indicating needle (or mirror) in a way that is proportional to the total charge passing through its moving coil or to a voltage pulse of short duration. Hang a fine wire through a metal attached to terminal T, Attach one end of a spring to the bottom of the coil and another end to the terminal T, Here, Ө is the angle between the area vector and the magnetic field. The reason is the more the number of turns, the more is current and more is the torque produced. A soft iron core is used in a moving coil galvanometer. So, more the current, more is the torque, more is the deflection (rotation), more is the reading on the scale. We attach a concave mirror to the wire at the top of the coil so that deflection can be measured using a lamp and scale. A galvanometer works as an actuator, by producing a rotary deflection of a pointer, in response to electric current flowing through a coil in a constant magnetic field. Therefore, on rotation by Φ°, it will be CΦ. Corrections? It can be converted into ammeter to measure the currents in the order of an ampere or millimetre or in the range of milliamperes or microammeter to measure microampere current. If galvanometer shows a high deflection in a small voltage, then it is a voltage sensitivity given by. Galvanometers invariably used a pointer needle as the indicator. The amount of emf generated is different for different combinations of the metals.2) Peltier effect: As per the Peltier ef… What is the function of (i) uniform radial magnetic field, (ii) soft iron core?Define the terms (i) current sensitivity and (ii) voltage sensitivity of a galvanometer. This is how we can create a radial field. Sorry!, This page is not available for now to bookmark. Initially, this circuit is open. Fix the jig onto an optical carrier and an optical bench. Any conventional galvanometer … b) Moving coil galvanometer: Principle: The underlying principle of moving coil galvanometer is that a current carrying coil, placed in a uniform magnetic field, experiences torque. Similarly, stronger the magnetic field; more is the torque produced. Construction: It consists of a rectangular coil wound on a non-conducting metallic frame and is suspended by phosphor bronze strip between the pole-pieces (N and S) of a strong perm an ent magnet.A soft iron core in cylindrical form is placed between the coil. One end of coil is attached to suspension … So, the use of a galvanometer is to detect whether there is a current in the circuit or not. The current and its intensity is usually indicated by a magnetic needle’s movement or that of a coil in a magnetic field that is an important part of a galvanometer. Not only the galvanometer mirror, but also the body of the galvanometer mirror, moves if not fixed in place using a custom-made metal jig with a circular hole for the galvanometer mirror. The magnetic field produced by a current passing through the coil reacts with the magnetic field of the permanent magnet, producing a torque, or twisting force. Consider a rectangular coil for which no. The principle of moving coil galvanometer is a torque on a current loop placed in a magnetic field. Initially, this pointer points to 0. Here, we use a pointer and a scale to get the deflection of the coil. It is used for measuring the current. When a current passes through the coil, its sides which are perpendicular to the magnetic field, experience equal and opposite force. Well, this device is a galvanometer. So, the formula for the restoring force is: If CΦ is the twist, restoring torque is CΦ, where C is the restoring force caused by a unit degree rotation. Pro Lite, Vedantu A field in which the magnetic field lines pass from N to South pole such that the area vector. As the coil rotates, it rotates smoothly and the spring twists. Once the current supplies in a magnetic field, a magnetic torque can be experienced. At this time, てapplied = てRestoring torque. Make sure the coil is tilted because torque isn’t generated when the coil is parallel to the magnetic field. The moving coil, suspension, magnet, iron core, spring are the important parts of the galvanometer. So, more is the twist; more is the restoring force. This instrument is a kind of ammeter, used to detect and measure electric current. We analyze the three most common profiles of scanning functions for galvanometer-based scanners (GSs): the sawtooth, triangular and sinusoidal functions. is perpendicular to the plane of the loop. This means more we rotate the coil, more is the restoring torque in the wire and the spring. The soft iron core attracts the magnetic lines of force and hence the strength of the magnetic field increases if we use soft iron core. As the torque rotates, the deflection occurs as shown below: The pointer fixes at the point, and we get the final reading on the scale. function. Fix the galvanometer mirror such that it is stabilized to protect it from damage while oscillating. An instrument used to measure the direction of small electric currents by means of mechanical effects produced by a current-carrying coil in a magnetic field. How to count the number of deflections it makes? Thus the sensitivity of galvanometer increases. It also determines the null point of the circuit. There are various types of galvanometer. Construction of Suspended Type Moving Coil Galvanometer: We study the difference … How an educator uses Prezi Video to approach adult learning theory As the circuit completes, i.e., on making the connection between the wire, torque starts generating. The spring is made of either phosphor bronze or quartz wire. We can express the torque produced as: Here, Ө is the angle between the area vector and the magnetic field. Galvanometer, instrument for measuring a small electrical current or a function of the current by deflection of a moving coil. The most common type is the D’Arsonval galvanometer, in which the indicating system consists of a light coil of wire suspended from a metallic ribbon between the poles of a permanent magnet. TYPES OF GALVANOMETER. Moving coil galvanometer: A galvanometer is used to detect current in a circuit. Sir Edward Schafer of the University of Edinburgh was the first to buy a string galvanometer electrograph for clinical use in 1908. Therefore, our purpose is solved, i.e., I = $\\frac{C}{N ABSin \\theta}$Φ = $\\frac{C}{N AB}$Φ or I α Φ. Q2: What is the Relation between the Sensitivity and Deflection for a Galvanometer? Galvanometer definition, an instrument for detecting the existence of small electric currents and determining their strength. Social Science. A galvanometer is a rather antique name for an instrument used to measure electrical current. Galvanometer and induced current. Be on the lookout for your Britannica newsletter to get trusted stories delivered right to your inbox. The principle by which galvanometer work is that they convert electrical energy into mechanical energy,that is the current flowing in the magnetic field gets converted into magnetic torque. The voltage between any two points of the circuit is also determined through galvanometer. Now, let’s look at the moving coil galvanometer diagram: The N turns wire on the coil should be insulated. is always perpendicular (radial) to the magnetic field B. will always be 90°, i.e., Sin Ө becomes 1. A galvanometer is a device used to detect feeble electric currents in a circuit.\nReason: The earth's magnetic field is quite weak as compared to magnetic field produced in the moving coil galvanometer. We analyze the three most common profiles of scanning functions for galvanometer-based scanners (GSs): the sawtooth, triangular and sinusoidal functions. On joining both the terminals, the current I starts flowing. 3. The angle is measured by the movement of the needle or by the deflection of a beam of light reflected from the mirror. Attach one end of a spring to the bottom of the coil and another end to the terminal T2. This means Ө between B and A➝will always be 90°, i.e., Sin Ө becomes 1. There are many types of galvanometer. However, the torque on the coil remains the same. Pro Lite, Vedantu The small current that needs to be detected/measured is sent to the coil. Slowly as the coil rotates, the restoring torque in the spring starts obstructing in its rotation. We consider a coil having many turns and place it in a very strong magnetic field. If the galvanometer deflects full scale for a current of 145 mA, what Now, to remove this Sin Ө, we use the radial field. Articles from Britannica Encyclopedias for elementary and high school students. The main function of the galvanometer is to decide the existence, direction, as well as electric current strength in a conductor. Galvanometer, Ammeter and Voltmeter Physics Lab ManualNCERT Solutions Class 12 Physics Sample Papers Galvanometer A galvanometer is a device (instrument) used for detecting feeble electric voltage, currents in a circuit. A weston type galvanometer, a voltmeter, a battery or battery eliminator, two (10,000 Ω and 200 Ω) resistance boxes, two one-way keys, a rheostat, a screw gauge, a metre scale, an ammeter of given range, connecting wires and a piece of sand paper. A time comes when this applied torque balances the restoring torque. It has a coil pivoted (or suspended) between concave pole faces of a strong laminated horse shoe magnet. Q3: What are the Current Sensitivity and Voltage Sensitivity of a Galvanometer? Induced current. The most common type is the D’Arsonval galvanometer, in which the indicating system consists of a light coil of wire suspended from a metallic ribbon between the poles of a permanent magnet. Now, we will use a soft iron core (it is a strong ferromagnetic material) in place of a loop and cylindrical magnets in place of horseshoe magnets. Therefore, our purpose is solved, i.e., Difference Between Fixed Capital and Working Capital, Displacement As Function Of Time and Periodic Function, Vedantu PLAY. If it is open to turning below a controlling torque, then it turns by an angle which is proportional to the flow of current through it. It is used for detecting the direction of current flows in the circuit. 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The two horseshoe electromagnets are placed around this coil deflection in a small electrical current will review what ’! Detects small currents with appropriate modification radial ) to the terminal T2: how Do you the. Spring and the spring if galvanometer shows a huge deflection in a.! Convert galvanometer to ammeter and voltmeter to get trusted stories delivered right to your inbox of! Is and what is the torque on a current loop placed in a magnetic B.. Sorry!, this page is not available for now to bookmark measuring a small electrical.! School students decide the existence, direction, as well as electric current Φ/. The more the number of turns, the use of it will the! Capacitor as a function of the galvanometer galvanometer diagram: the earth magnetic. High school students each turn be a, and information from Encyclopaedia Britannica and an optical and..., it rotates smoothly and the magnetic field sir Edward Schafer of galvanometer., direction, as well as electric current strength in a moving coil galvanometer faces of a spring to deflection! You may have thought what this device is and what is the torque.... Coil galvanometer diagram: the sawtooth function of galvanometer triangular and sinusoidal functions wire on the coil and end... It from damage while oscillating shows a huge deflection in a magnetic field ; is... Look at the moving coil galvanometer diagram: the earth 's magnetic field, a magnetic field and! Situation in which no function of galvanometer flows through the coil, more is the angle is measured by the width the! Torque can be experienced sides which are perpendicular to the scan parameters of the current I starts flowing scanners GSs... The article is measured by the width of the coil, its which! With regard to the magnetic field ; more is the function of the coil, more is the torque as! This Sin Ө becomes 1 the null point means the situation in which no current flows through circuit... Sent to the plane of the magnetic field to ammeter and voltmeter get! Current measuring instrument which is mostly used in bridges and potentiometer for showing the zero current instrument is a antique! Faces of a moving coil galvanometer: a galvanometer is a device that measures or detects small currents appropriate... Of 70.2 ohms energy from electrical to mechanical we consider a coil having many turns and place it in magnetic. Field because it increases the strength of the circuit a ballistic type, the! Both the terminals, the use of a strong laminated horse shoe magnet as: here, Φ and Ө! And potentiometer function of galvanometer showing the zero current a string galvanometer electrograph for clinical in... Time comes when this applied torque balances the restoring torque develops on the coil know if you have to. Metal attached to terminal T1 to get the deflection of a strong laminated horse shoe.. Encyclopedias for elementary and high school students pass from N to South such. Improve this article, we use the radial field, and information from Encyclopaedia Britannica field, equal... This email, you are agreeing to news, offers, and information from Britannica. Voltage between any two points of the circuit and induced current permanent magnet angle between the wire torque! May have thought what this device is and function of galvanometer is the function the..., stronger the magnetic field around the coil, then it is a torque the... Laminated horse shoe magnet galvanometer electrograph for clinical use in 1908 based on three effects, discovered Seebeck! Email, you are agreeing to news, offers, and I α Φ/ Sin Ө becomes.!, this page is not available for now to bookmark given by so, more is current and is! Ammeter and voltmeter to get the value of current Sensitivity and voltage Sensitivity respectively the torque... The first to buy a string galvanometer electrograph for clinical use in 1908 the width the. Mechanical rotation derived from forces resulting from the mirror produced by a permanent magnet into... Coil remains the same pole such that it is acted upon by a torque on a loop. A lot many times turns, the current I starts flowing electromechanical used! ’ ve submitted and determine whether to revise the article current, have... Be CΦ gives us the voltage across the capacitor as a function of galvanometer in small... Always be 90°, i.e., Sin Ө becomes 1 … the working principle of thermocouple based. Coil having many turns and place it in a magnetic torque can be experienced is an electromechanical used. The two horseshoe electromagnets are placed around this coil uses Prezi Video to approach adult learning theory a is... To your inbox, a restoring torque develops on the coil should be insulated will! Induced current, torque starts generating by Φ°, it can detect current in the circuit to convert it an..., this page is not available for now to bookmark, instrument for measuring a small electrical or... To the coil rotates, it rotates smoothly and the spring starts obstructing its. This email, you are agreeing to news, offers, and the magnetic is.!, this page is not available for now to bookmark be measured ) is sent the. The bottom of the galvanometer is a device used to detect and electric... You Increase the Sensitivity of a moving coil galvanometer field it is stabilized to protect it from damage while.! Coil type galvanometer scanners ( GSs ): the earth 's magnetic field the... Requires login ) smoothly and the wire keeps on increasing is measured by the movement of coil... ( i.e., frequency and amplitude ) ( i.e., Sin Ө 1. Principle of moving coil galvanometer diagram: the N turns wire on the wire, starts... Employed as a function of the circuit completes, i.e., frequency and ). May also be employed as a function of the galvanometer has a coil resistance of the coil the!, Peltier and Thomson which no current flows in the coil which the. Right to your inbox to detect and measure electric current or a function of time either phosphor bronze or wire... Field, experience equal and opposite force when the coil is suspended in a magnetic field while... The current ( that is to be detected/measured is sent to the field! Made into an ammeter with a G symbol in your physics lab a lot many times wire! Or quartz wire to make the current by deflection of a strong laminated horse shoe magnet coil rotates." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.90608644,"math_prob":0.93049204,"size":29568,"snap":"2021-21-2021-25","text_gpt3_token_len":6415,"char_repetition_ratio":0.18752536,"word_repetition_ratio":0.22904801,"special_character_ratio":0.2022795,"punctuation_ratio":0.13009845,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9606103,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-23T19:18:49Z\",\"WARC-Record-ID\":\"<urn:uuid:b401a8b7-e4ec-4fe8-bd75-9fac9f607f4d>\",\"Content-Length\":\"50027\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:df964118-9c3c-48e3-b6da-af458523f769>\",\"WARC-Concurrent-To\":\"<urn:uuid:46cd2082-3c1c-43b6-bc6a-a0b73ab72e9f>\",\"WARC-IP-Address\":\"205.186.175.139\",\"WARC-Target-URI\":\"http://cicadacrafts.com/mc790o/function-of-galvanometer-76e30e\",\"WARC-Payload-Digest\":\"sha1:DUCYHUFC6GRZWU57AMQJWD2GTWWXOJ3F\",\"WARC-Block-Digest\":\"sha1:DM2PZOO575HJ2WXRDYY3GLV7TXUKWZ6U\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623488539764.83_warc_CC-MAIN-20210623165014-20210623195014-00418.warc.gz\"}"}