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https://findthefactors.com/2018/09/13/1213-and-level-3/	[ "# 1213 and Level 3\n\nThis puzzle would be a lot tougher to solve if I didn’t put the clues in the order that I did. Just start at the top of the puzzle and work your way down the puzzle clue by clue until you get to the bottom of the puzzle and have the entire thing solved.", null, "Print the puzzles or type the solution in this excel file: 12 factors 1211-1220\n\nNow I’ll share a few facts about the number 1213:\n\n• 1213 is a prime number.\n• Prime factorization: 1213 is prime.\n• The exponent of prime number 1213 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1213 has exactly 2 factors.\n• Factors of 1213: 1, 1213\n• Factor pairs: 1213 = 1 × 1213\n• 1213 has no square factors that allow its square root to be simplified. √1213 ≈ 34.82815\n\nHow do we know that 1213 is a prime number? If 1213 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1213 ≈ 34.8. Since 1213 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1213 is a prime number.", null, "1213 is the sum of these nine consecutive prime numbers:\n109 + 113 + 127 + 131 + 137 + 139 + 149 + 151 + 157 = 1213\n\n27² + 22² = 1213\n1213 is the hypotenuse of a Pythagorean triple:\n245-1188-1213 calculated from 27² – 22², 2(27)(22), 27² + 22²\n\nHere’s another way we know that 1213 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 27² + 22² = 1213 with 27 and 22 having no common prime factors, 1213 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1213 ≈ 34.8. Since 1213 is not divisible by 5, 13, 17, or 29, we know that 1213 is a prime number.\n\nThis site uses Akismet to reduce spam. Learn how your comment data is processed." ]	[ null, "https://i0.wp.com/findthefactors.com/wp-content/uploads/2018/10/1213-Puzzle.jpg", null, "https://i0.wp.com/findthefactors.com/wp-content/uploads/2018/10/1213-Factor-Pairs.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.90970564,"math_prob":0.99696374,"size":1652,"snap":"2023-14-2023-23","text_gpt3_token_len":538,"char_repetition_ratio":0.17354369,"word_repetition_ratio":0.08529412,"special_character_ratio":0.40133172,"punctuation_ratio":0.11859838,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9971676,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,3,null,3,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-06-08T01:12:54Z\",\"WARC-Record-ID\":\"<urn:uuid:00e02d11-70e7-42d1-a6bd-8d90d236c516>\",\"Content-Length\":\"86645\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:8ce2b34d-edd4-413b-adf1-120d46313c7f>\",\"WARC-Concurrent-To\":\"<urn:uuid:9ce52d44-f482-4074-85cb-c586d2d7ca31>\",\"WARC-IP-Address\":\"192.0.78.222\",\"WARC-Target-URI\":\"https://findthefactors.com/2018/09/13/1213-and-level-3/\",\"WARC-Payload-Digest\":\"sha1:CO3RUYSCIBP7RBKY6J2EKJUVAQ5P7NEF\",\"WARC-Block-Digest\":\"sha1:YQHG7PUKW77HUKU4N2BW3IVK3BVYGNM7\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-23/CC-MAIN-2023-23_segments_1685224654031.92_warc_CC-MAIN-20230608003500-20230608033500-00372.warc.gz\"}"}
http://www.mathspadilla.com/macsII/Unit4-LinearProgramming/systems_of_linear_inequations_with_two_unknowns.html	[ "# Systems of linear inequations with two unknowns\n\nA system of linear inequation with two unknowns is a set of linear inequations. The set of solutions is formed by the solutions that verify all the inequations. It is also called feasible region.", null, "Examples:\n\n1)", null, "2)", null, "", null, "Exercise: solve the following systems:", null, "solutions:\n\na)", null, "b)", null, "" ]	[ null, "http://www.mathspadilla.com/macsII/Unit4-LinearProgramming/program_lineal1.png", null, "http://www.mathspadilla.com/macsII/Unit4-LinearProgramming/program_lineal6.png", null, "http://www.mathspadilla.com/macsII/Unit4-LinearProgramming/eXe_LaTeX_math_2.gif", null, "http://www.mathspadilla.com/macsII/Unit4-LinearProgramming/program_lineal2.png", null, "http://www.mathspadilla.com/macsII/Unit4-LinearProgramming/eXe_LaTeX_math_6.1.gif", null, "http://www.mathspadilla.com/macsII/Unit4-LinearProgramming/program_lineal12.png", null, "http://www.mathspadilla.com/macsII/Unit4-LinearProgramming/program_lineal13.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.92039996,"math_prob":0.9959158,"size":420,"snap":"2022-27-2022-33","text_gpt3_token_len":94,"char_repetition_ratio":0.15384616,"word_repetition_ratio":0.0,"special_character_ratio":0.1904762,"punctuation_ratio":0.10810811,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9961904,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"im_url_duplicate_count":[null,1,null,1,null,1,null,1,null,1,null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-07-06T13:53:27Z\",\"WARC-Record-ID\":\"<urn:uuid:c5a2d35e-e9ff-40c7-852c-a184feb73a06>\",\"Content-Length\":\"4054\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:8919d3a6-b3b7-4b63-a1e5-9be81a21ca6a>\",\"WARC-Concurrent-To\":\"<urn:uuid:9fa94c0b-61bb-4f07-bf4b-ae9d7adb380d>\",\"WARC-IP-Address\":\"31.214.178.24\",\"WARC-Target-URI\":\"http://www.mathspadilla.com/macsII/Unit4-LinearProgramming/systems_of_linear_inequations_with_two_unknowns.html\",\"WARC-Payload-Digest\":\"sha1:EQ2RPTAZVM5TNWCCLCPGIFQUM7Q6A2NT\",\"WARC-Block-Digest\":\"sha1:TBPFGJEQFPWJSFTID7TNJUJSJKACCGEY\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656104672585.89_warc_CC-MAIN-20220706121103-20220706151103-00637.warc.gz\"}"}
https://www.colorhexa.com/02bf8e	[ "# #02bf8e Color Information\n\nIn a RGB color space, hex #02bf8e is composed of 0.8% red, 74.9% green and 55.7% blue. Whereas in a CMYK color space, it is composed of 99% cyan, 0% magenta, 25.7% yellow and 25.1% black. It has a hue angle of 164.4 degrees, a saturation of 97.9% and a lightness of 37.8%. #02bf8e color hex could be obtained by blending #04ffff with #007f1d. Closest websafe color is: #00cc99.\n\n• R 1\n• G 75\n• B 56\nRGB color chart\n• C 99\n• M 0\n• Y 26\n• K 25\nCMYK color chart\n\n#02bf8e color description : Strong cyan - lime green.\n\n# #02bf8e Color Conversion\n\nThe hexadecimal color #02bf8e has RGB values of R:2, G:191, B:142 and CMYK values of C:0.99, M:0, Y:0.26, K:0.25. Its decimal value is 180110.\n\nHex triplet RGB Decimal 02bf8e `#02bf8e` 2, 191, 142 `rgb(2,191,142)` 0.8, 74.9, 55.7 `rgb(0.8%,74.9%,55.7%)` 99, 0, 26, 25 164.4°, 97.9, 37.8 `hsl(164.4,97.9%,37.8%)` 164.4°, 99, 74.9 00cc99 `#00cc99`\nCIE-LAB 68.914, -52.027, 13.542 23.536, 39.225, 31.92 0.249, 0.414, 39.225 68.914, 53.761, 165.411 68.914, -58.057, 27.34 62.63, -42.522, 13.623 00000010, 10111111, 10001110\n\n# Color Schemes with #02bf8e\n\n• #02bf8e\n``#02bf8e` `rgb(2,191,142)``\n• #bf0233\n``#bf0233` `rgb(191,2,51)``\nComplementary Color\n• #02bf30\n``#02bf30` `rgb(2,191,48)``\n• #02bf8e\n``#02bf8e` `rgb(2,191,142)``\n• #0292bf\n``#0292bf` `rgb(2,146,191)``\nAnalogous Color\n• #bf3002\n``#bf3002` `rgb(191,48,2)``\n• #02bf8e\n``#02bf8e` `rgb(2,191,142)``\n• #bf0292\n``#bf0292` `rgb(191,2,146)``\nSplit Complementary Color\n• #bf8e02\n``#bf8e02` `rgb(191,142,2)``\n• #02bf8e\n``#02bf8e` `rgb(2,191,142)``\n• #8e02bf\n``#8e02bf` `rgb(142,2,191)``\n• #33bf02\n``#33bf02` `rgb(51,191,2)``\n• #02bf8e\n``#02bf8e` `rgb(2,191,142)``\n• #8e02bf\n``#8e02bf` `rgb(142,2,191)``\n• #bf0233\n``#bf0233` `rgb(191,2,51)``\n• #017356\n``#017356` `rgb(1,115,86)``\n• #018d68\n``#018d68` `rgb(1,141,104)``\n• #02a67b\n``#02a67b` `rgb(2,166,123)``\n• #02bf8e\n``#02bf8e` `rgb(2,191,142)``\n• #02d8a1\n``#02d8a1` `rgb(2,216,161)``\n• #03f1b4\n``#03f1b4` `rgb(3,241,180)``\n• #11fdbf\n``#11fdbf` `rgb(17,253,191)``\nMonochromatic Color\n\n# Alternatives to #02bf8e\n\nBelow, you can see some colors close to #02bf8e. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #02bf5f\n``#02bf5f` `rgb(2,191,95)``\n• #02bf6f\n``#02bf6f` `rgb(2,191,111)``\n• #02bf7e\n``#02bf7e` `rgb(2,191,126)``\n• #02bf8e\n``#02bf8e` `rgb(2,191,142)``\n• #02bf9e\n``#02bf9e` `rgb(2,191,158)``\n• #02bfae\n``#02bfae` `rgb(2,191,174)``\n• #02bfbd\n``#02bfbd` `rgb(2,191,189)``\nSimilar Colors\n\n# #02bf8e Preview\n\nThis text has a font color of #02bf8e.\n\n``<span style=\"color:#02bf8e;\">Text here</span>``\n#02bf8e background color\n\nThis paragraph has a background color of #02bf8e.\n\n``<p style=\"background-color:#02bf8e;\">Content here</p>``\n#02bf8e border color\n\nThis element has a border color of #02bf8e.\n\n``<div style=\"border:1px solid #02bf8e;\">Content here</div>``\nCSS codes\n``.text {color:#02bf8e;}``\n``.background {background-color:#02bf8e;}``\n``.border {border:1px solid #02bf8e;}``\n\n# Shades and Tints of #02bf8e\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, #00100c is the darkest color, while #fcfffe is the lightest one.\n\n• #00100c\n``#00100c` `rgb(0,16,12)``\n• #00241b\n``#00241b` `rgb(0,36,27)``\n• #013729\n``#013729` `rgb(1,55,41)``\n• #014b37\n``#014b37` `rgb(1,75,55)``\n• #015e46\n``#015e46` `rgb(1,94,70)``\n• #017154\n``#017154` `rgb(1,113,84)``\n• #018563\n``#018563` `rgb(1,133,99)``\n• #029871\n``#029871` `rgb(2,152,113)``\n• #02ac80\n``#02ac80` `rgb(2,172,128)``\n• #02bf8e\n``#02bf8e` `rgb(2,191,142)``\n• #02d29c\n``#02d29c` `rgb(2,210,156)``\n• #02e6ab\n``#02e6ab` `rgb(2,230,171)``\n• #03f9b9\n``#03f9b9` `rgb(3,249,185)``\n• #13fdc0\n``#13fdc0` `rgb(19,253,192)``\n• #26fdc5\n``#26fdc5` `rgb(38,253,197)``\n• #3afdca\n``#3afdca` `rgb(58,253,202)``\n• #4dfdd0\n``#4dfdd0` `rgb(77,253,208)``\n• #61fdd5\n``#61fdd5` `rgb(97,253,213)``\n• #74feda\n``#74feda` `rgb(116,254,218)``\n• #87fedf\n``#87fedf` `rgb(135,254,223)``\n• #9bfee4\n``#9bfee4` `rgb(155,254,228)``\n• #aefee9\n``#aefee9` `rgb(174,254,233)``\n• #c2feef\n``#c2feef` `rgb(194,254,239)``\n• #d5fff4\n``#d5fff4` `rgb(213,255,244)``\n• #e8fff9\n``#e8fff9` `rgb(232,255,249)``\n• #fcfffe\n``#fcfffe` `rgb(252,255,254)``\nTint Color Variation\n\n# Tones of #02bf8e\n\nA tone is produced by adding gray to any pure hue. In this case, #5b6663 is the less saturated color, while #02bf8e is the most saturated one.\n\n• #5b6663\n``#5b6663` `rgb(91,102,99)``\n• #546d67\n``#546d67` `rgb(84,109,103)``\n• #4c756a\n``#4c756a` `rgb(76,117,106)``\n• #457c6e\n``#457c6e` `rgb(69,124,110)``\n• #3d8471\n``#3d8471` `rgb(61,132,113)``\n• #368b75\n``#368b75` `rgb(54,139,117)``\n• #2f9279\n``#2f9279` `rgb(47,146,121)``\n• #279a7c\n``#279a7c` `rgb(39,154,124)``\n• #20a180\n``#20a180` `rgb(32,161,128)``\n• #18a983\n``#18a983` `rgb(24,169,131)``\n• #11b087\n``#11b087` `rgb(17,176,135)``\n• #09b88a\n``#09b88a` `rgb(9,184,138)``\n• #02bf8e\n``#02bf8e` `rgb(2,191,142)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #02bf8e 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":0.52199996,"math_prob":0.7245757,"size":3701,"snap":"2019-51-2020-05","text_gpt3_token_len":1648,"char_repetition_ratio":0.12740059,"word_repetition_ratio":0.011049724,"special_character_ratio":0.54552823,"punctuation_ratio":0.236404,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98381245,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-12-13T08:17:11Z\",\"WARC-Record-ID\":\"<urn:uuid:68ff9de2-3ecd-434f-b85f-f9b825f5f328>\",\"Content-Length\":\"36271\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b99cf887-88f1-401e-a469-057679429ec0>\",\"WARC-Concurrent-To\":\"<urn:uuid:b5839b4b-c32a-46cf-8ead-95d0614d3754>\",\"WARC-IP-Address\":\"178.32.117.56\",\"WARC-Target-URI\":\"https://www.colorhexa.com/02bf8e\",\"WARC-Payload-Digest\":\"sha1:IH6WSIYGPPWDKINJ3BXS4ZWS37GGRF7H\",\"WARC-Block-Digest\":\"sha1:LJ4FOFSUXMZR45K5DLCWRBXKCUEMAWN3\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-51/CC-MAIN-2019-51_segments_1575540551267.14_warc_CC-MAIN-20191213071155-20191213095155-00109.warc.gz\"}"}
https://swi-prolog.discourse.group/t/checking-for-membership-in-list-without-instantiating-anonymous-variables/1542	[ "", null, "# Checking for membership in list without instantiating anonymous variables\n\nHi!\nBelow i want to check if `p(a)` and `p(b)` are members of the list. But after the fist call to `member/2`,` L` is instantiated to `p(a)` so the second call to `member/2` fails. Is there a way to avoid this? Ie checking for membership without changing the list that involves anonymous variables?\n\n``````myList([p(_)]).\n\nmain:-\nmyList(L),\nmember(p(a), L),\nmember(p(b), L).\n``````\n\nCheers/JCR\n\nThis works, but I know Jan will most likely show something that is better that I am not aware. It uses copy_term/2\n\n``````other :-\nmyList(L),\ncopy_term(L,L1),\nmember(p(a), L),\nmember(p(b), L1).\n``````\n``````?- main.\nfalse.\n\n?- other.\ntrue.\n``````\n1 Like\n\nDouble negation?\n\n``````main:-\nmyList(L),\n\\+ \\+ member(p(a), L),\n\\+ \\+ member(p(b), L).``````\n1 Like\n\nTo this day I know that is important to know but have no clue as how or the reasoning behind it. Care to explain.\n\n1 Like\n\nNegation never binds variables in the negated goal. Thus, by using double negation you can verify that a goal succeeds without binding any of its variables.\n\n4 Likes\n\nThanks @EricGT.\n\nI should have explained more clearly in my original post that i don’t know how many times `member/2` will be called and what the first argument of the compound `p` will be. So what i have is something like\n\n``````myList([p(_)]).\n\nmain:-\nmyList(L),\nmember(p(a), L),\nmember(p(b), L),\n...,\nmember(p(n), L).\n``````\n\nWhere there is an unknown number of such `member/2` predicates each with `p(unknownAtom)`.\n\nCheers/JC\n\n1 Like\n\nThanks! This solved my problem.\n\nThe predicate forall/2 is implemented as `\\+ ( Cond, \\+ Action)` , i.e., There is no instantiation of Cond for which Action is false. . The use of double negation implies that forall/2 does not change any variable bindings . It proves a relation. The forall/2 control structure can be used for its side-effects." ]	[ null, "https://aws1.discourse-cdn.com/free1/uploads/swiprolog/original/1X/73ebf150a3746e8f54f93423fb28d18c434847c9.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8279352,"math_prob":0.9182764,"size":366,"snap":"2019-51-2020-05","text_gpt3_token_len":102,"char_repetition_ratio":0.12707183,"word_repetition_ratio":0.0,"special_character_ratio":0.28142077,"punctuation_ratio":0.14942528,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95015085,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-12-12T19:48:29Z\",\"WARC-Record-ID\":\"<urn:uuid:c4b1ff6c-06c9-471e-85ac-3b97e6fc0b63>\",\"Content-Length\":\"23828\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:1e596919-e6e9-45d5-9e3f-75e62033da68>\",\"WARC-Concurrent-To\":\"<urn:uuid:e34e9c3c-1f21-40f2-ac50-0e55bb00af6f>\",\"WARC-IP-Address\":\"66.220.12.139\",\"WARC-Target-URI\":\"https://swi-prolog.discourse.group/t/checking-for-membership-in-list-without-instantiating-anonymous-variables/1542\",\"WARC-Payload-Digest\":\"sha1:I56KTCLTBDVLVMZLGMLZRHGXG2QZQU5I\",\"WARC-Block-Digest\":\"sha1:SXQOUGIQBCGLQPBRGKWQL6INPNQ4V2GV\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-51/CC-MAIN-2019-51_segments_1575540545146.75_warc_CC-MAIN-20191212181310-20191212205310-00219.warc.gz\"}"}
https://dlmf.nist.gov/search/search?q=cylinder%20functions	[ "# cylinder functions\n\n(0.013 seconds)\n\n## 1—10 of 83 matching pages\n\n##### 1: 12.1 Special Notation\nUnless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values. The main functions treated in this chapter are the parabolic cylinder functions (PCFs), also known as Weber parabolic cylinder functions: $U\\left(a,z\\right)$, $V\\left(a,z\\right)$, $\\overline{U}\\left(a,z\\right)$, and $W\\left(a,z\\right)$. …An older notation, due to Whittaker (1902), for $U\\left(a,z\\right)$ is $D_{\\nu}\\left(z\\right)$. …\n##### 3: 10.29 Recurrence Relations and Derivatives\nWith $\\mathscr{Z}_{\\nu}\\left(z\\right)$ defined as in §10.25(ii),\n$\\mathscr{Z}_{\\nu-1}\\left(z\\right)-\\mathscr{Z}_{\\nu+1}\\left(z\\right)=(2\\nu/z)% \\mathscr{Z}_{\\nu}\\left(z\\right),$\n$\\mathscr{Z}_{\\nu-1}\\left(z\\right)+\\mathscr{Z}_{\\nu+1}\\left(z\\right)=2\\mathscr{% Z}_{\\nu}'\\left(z\\right).$\n$\\mathscr{Z}_{\\nu}'\\left(z\\right)=\\mathscr{Z}_{\\nu-1}\\left(z\\right)-(\\nu/z)% \\mathscr{Z}_{\\nu}\\left(z\\right),$\n$\\mathscr{Z}_{\\nu}'\\left(z\\right)=\\mathscr{Z}_{\\nu+1}\\left(z\\right)+(\\nu/z)% \\mathscr{Z}_{\\nu}\\left(z\\right).$\n##### 4: 10.36 Other Differential Equations\nThe quantity $\\lambda^{2}$ in (10.13.1)–(10.13.6) and (10.13.8) can be replaced by $-\\lambda^{2}$ if at the same time the symbol $\\mathscr{C}$ in the given solutions is replaced by $\\mathscr{Z}$. …\n10.36.1 $z^{2}(z^{2}+\\nu^{2})w^{\\prime\\prime}+z(z^{2}+3\\nu^{2})w^{\\prime}-\\left((z^{2}+% \\nu^{2})^{2}+z^{2}-\\nu^{2}\\right)w=0,$ $w=\\mathscr{Z}_{\\nu}'\\left(z\\right)$,\n10.36.2 ${z^{2}w^{\\prime\\prime}+z(1\\pm 2z)w^{\\prime}+(\\pm z-\\nu^{2})w=0},$ $w=e^{\\mp z}\\mathscr{Z}_{\\nu}\\left(z\\right)$.\n##### 5: 12.16 Mathematical Applications\n###### §12.16 Mathematical Applications\nFor examples see §§13.20(iii), 13.20(iv), 14.15(v), and 14.26. …\n##### 6: 10.13 Other Differential Equations\n###### §10.13 Other Differential Equations\n10.13.1 $w^{\\prime\\prime}+\\left(\\lambda^{2}-\\frac{\\nu^{2}-\\tfrac{1}{4}}{z^{2}}\\right)w=0,$ $w=z^{\\frac{1}{2}}\\mathscr{C}_{\\nu}\\left(\\lambda z\\right)$,\nIn (10.13.9)–(10.13.11) $\\mathscr{C}_{\\nu}\\left(z\\right)$, $\\mathscr{D}_{\\mu}(z)$ are any cylinder functions of orders $\\nu,\\mu$, respectively, and $\\vartheta=z(\\!\\ifrac{\\mathrm{d}}{\\mathrm{d}z})$.\n10.13.9 ${z^{2}w^{\\prime\\prime\\prime}+3zw^{\\prime\\prime}+(4z^{2}+1-4\\nu^{2})w^{\\prime}+% 4zw=0},$ $w=\\mathscr{C}_{\\nu}\\left(z\\right)\\mathscr{D}_{\\nu}(z)$,\n10.13.10 ${z^{3}w^{\\prime\\prime\\prime}+z(4z^{2}+1-4\\nu^{2})w^{\\prime}+(4\\nu^{2}-1)w=0},$ $w=z\\mathscr{C}_{\\nu}\\left(z\\right)\\mathscr{D}_{\\nu}(z)$,\n##### 7: 10.6 Recurrence Relations and Derivatives\nWith $\\mathscr{C}_{\\nu}\\left(z\\right)$ defined as in §10.2(ii),\n$\\mathscr{C}_{\\nu-1}\\left(z\\right)+\\mathscr{C}_{\\nu+1}\\left(z\\right)=(2\\nu/z)% \\mathscr{C}_{\\nu}\\left(z\\right),$\n$\\mathscr{C}_{\\nu-1}\\left(z\\right)-\\mathscr{C}_{\\nu+1}\\left(z\\right)=2\\mathscr{% C}_{\\nu}'\\left(z\\right).$\n$\\mathscr{C}_{\\nu}'\\left(z\\right)=\\mathscr{C}_{\\nu-1}\\left(z\\right)-(\\nu/z)% \\mathscr{C}_{\\nu}\\left(z\\right),$\n$\\mathscr{C}_{\\nu}'\\left(z\\right)=-\\mathscr{C}_{\\nu+1}\\left(z\\right)+(\\nu/z)% \\mathscr{C}_{\\nu}\\left(z\\right).$" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6758551,"math_prob":1.0000076,"size":1149,"snap":"2020-24-2020-29","text_gpt3_token_len":371,"char_repetition_ratio":0.139738,"word_repetition_ratio":0.0,"special_character_ratio":0.39251524,"punctuation_ratio":0.2605364,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000077,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-15T01:41:44Z\",\"WARC-Record-ID\":\"<urn:uuid:cfd0304a-9158-4b83-a483-b92e3964fa4a>\",\"Content-Length\":\"60415\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:59e20c03-4b37-482a-be06-420ebbc6986c>\",\"WARC-Concurrent-To\":\"<urn:uuid:2a6cce21-d494-4b1c-9019-ea0b2c32164f>\",\"WARC-IP-Address\":\"129.6.24.27\",\"WARC-Target-URI\":\"https://dlmf.nist.gov/search/search?q=cylinder%20functions\",\"WARC-Payload-Digest\":\"sha1:X7EXBJ54GRWBK3N7XO4HQBP5HXTKD6D6\",\"WARC-Block-Digest\":\"sha1:7BRH3DKM2N2TJMPSKTJLDA4FJWDVT63M\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593657154789.95_warc_CC-MAIN-20200715003838-20200715033838-00480.warc.gz\"}"}
https://scholarworks.umass.edu/physics_faculty_pubs/1121/	[ "## Physics Department Faculty Publication Series\n\n2003\n\n#### Abstract\n\nIn a recent Letter (see also ) the authors presented numerical evidence supporting an idea of a direct transition between the superfluid (SF) and Mott insulating (MI) phases in the disordered Bosonic system, and even studied non-trivial properties of the multicritical line where SF, MI and the Bose Glass (BG) phases meet. The results were obtained from Monte Carlo simulations of the (2+1)-dimensional classical loop-current model with the lattice action S = 1 2K ÞE ~ J=0 XrƒÑ \u0014~ J2(r, ƒÑ) . 2(ƒÊ + v(r)) ~ JƒÑ (r, ƒÑ)\u0015 . (1) where r, ƒÑ are spatial and imaginary time coordinates, and ~ J(r, ƒÑ) are integer current vectors with zero divergence. The spatial disorder potential v(r) is uniformly distributed on the interval (.\u0001,\u0001)." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.85620475,"math_prob":0.9052351,"size":1105,"snap":"2020-24-2020-29","text_gpt3_token_len":304,"char_repetition_ratio":0.08446866,"word_repetition_ratio":0.0,"special_character_ratio":0.24886878,"punctuation_ratio":0.12,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9539341,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-05-28T16:02:41Z\",\"WARC-Record-ID\":\"<urn:uuid:01b6754d-3e4e-43f2-bc53-35acfbf93b1f>\",\"Content-Length\":\"33865\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9871fb46-5a7d-477c-8305-c30b26646e22>\",\"WARC-Concurrent-To\":\"<urn:uuid:bde3b43a-e8a9-47df-a483-0e487db4ccb2>\",\"WARC-IP-Address\":\"13.57.92.51\",\"WARC-Target-URI\":\"https://scholarworks.umass.edu/physics_faculty_pubs/1121/\",\"WARC-Payload-Digest\":\"sha1:ELERQDOO4ZXDTTECLAU4L6RTEDS6WNEF\",\"WARC-Block-Digest\":\"sha1:7F5BS7SWTLCJP737QIAPGVZGABZHTWEZ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-24/CC-MAIN-2020-24_segments_1590347399820.9_warc_CC-MAIN-20200528135528-20200528165528-00242.warc.gz\"}"}
https://dwarffortresswiki.org/index.php/40d:Peat	[ "# 40d:Peat\n\n `░` `░` `░` `░` `░` `░` `░` `.` `=` `=` `=` `░` `░` `░` `.` `.` `=` `=` `=` `░` `░` `.` `.` `.` `.` `=` `=` `░` `.` `.` `.` `.` `.` `.` `=`" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7934366,"math_prob":0.7271297,"size":335,"snap":"2021-31-2021-39","text_gpt3_token_len":103,"char_repetition_ratio":0.054380666,"word_repetition_ratio":0.0,"special_character_ratio":0.2716418,"punctuation_ratio":0.1641791,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000082,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-07-27T21:36:58Z\",\"WARC-Record-ID\":\"<urn:uuid:8dc3910d-0d42-4108-a570-aec5e38477b1>\",\"Content-Length\":\"36612\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:dbd61d80-f00c-45ee-aad2-8865457fca35>\",\"WARC-Concurrent-To\":\"<urn:uuid:8e81a1dd-ca6b-44fc-b856-3118d1cd1a4f>\",\"WARC-IP-Address\":\"38.143.66.236\",\"WARC-Target-URI\":\"https://dwarffortresswiki.org/index.php/40d:Peat\",\"WARC-Payload-Digest\":\"sha1:CX45M5HQLSEZPYC7WHVUEVU2Y2XGGP4G\",\"WARC-Block-Digest\":\"sha1:2SIOXEE7XFRSXLJZSYAMCUQYQOLR2LGE\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-31/CC-MAIN-2021-31_segments_1627046153491.18_warc_CC-MAIN-20210727202227-20210727232227-00278.warc.gz\"}"}
https://help.scilab.org/docs/5.4.1/ja_JP/m2sci_find.html	[ "Scilab Home page \| Wiki \| Bug tracker \| Forge \| Mailing list archives \| ATOMS \| File exchange\nChange language to: English - Français - Português - Русский\n\nSee the recommended documentation of this function\n\nScilab help >> Matlab to Scilab Conversion Tips > Matlab-Scilab equivalents > F > find (Matlab function)\n\n# find (Matlab function)\n\nFind indices and values of nonzero elements\n\n### Matlab/Scilab equivalent\n\n Matlab Scilab `find` `find`\n\n### Particular cases\n\nMatlab function can work with complex values what is not possible in Scilab, however, using abs it is very easy to have the same results.\n\nNote that Scilab function can only return two output values and Matlab one can return a third value that can be computed according to the first two output matrices as explained in Matlab help.\n\nFor example, in [i,j,v]=find(X), v is equal to: X(i+(j-1))*size(X,1).\n\nAnother great difference between Scilab and Matlab is that Matlab returns column vectors of indices when X is a column vector or a matrix but Scilab always returns row vectors. For this kind of input, use matrix to get the same output value...what is done mtlb_find()" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.80236745,"math_prob":0.7489526,"size":1859,"snap":"2020-24-2020-29","text_gpt3_token_len":482,"char_repetition_ratio":0.37196764,"word_repetition_ratio":0.0,"special_character_ratio":0.23614846,"punctuation_ratio":0.074074075,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.999707,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-05T13:29:54Z\",\"WARC-Record-ID\":\"<urn:uuid:2f9bb161-eee8-4851-a56c-e2e9b510c16a>\",\"Content-Length\":\"23461\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:63b2d8db-c26f-4e86-ad37-6c85b6fe827f>\",\"WARC-Concurrent-To\":\"<urn:uuid:09a5b3c7-0bb4-4db5-8c22-57d60993c011>\",\"WARC-IP-Address\":\"176.9.3.186\",\"WARC-Target-URI\":\"https://help.scilab.org/docs/5.4.1/ja_JP/m2sci_find.html\",\"WARC-Payload-Digest\":\"sha1:VE7BE6LIHXVKIPSUNBTIV2LJONDEGES4\",\"WARC-Block-Digest\":\"sha1:54YDARV75ZR52HVWQE2AIDRZTSNEDHRR\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593655887360.60_warc_CC-MAIN-20200705121829-20200705151829-00495.warc.gz\"}"}
http://blog.carriesegal.com/index.php/2010/06/	[ "# Monthly Archives: June 2010\n\n## Schrodinger’s kit: Tools that are in two places at once – physics-math – 28 June 2010 – New Scientist\n\nFiled under Articles\n\nFiled under Side Projects\n\n## Mirrors, concave and convex.\n\nIn class today we were going over how mirrors and lens work. It reminded me of this very neat toy I saw, a Parabolic Mirror Wok, that projects a real image on the items in side up outward.", null, "I wonder what would happen if you had a little screen in the bottom projecting an image?\n\nFiled under Side Projects\n\n## Quama, Laser TV and Liquid Helium\n\nSo I downloaded this new tool to let me blog remotely more easily. Hopefully it will be very helpful! To test it out here is a list of recent articles which have caught my eye.\n\nLaser Phosphor Display (LPD) television – it’s all done with mirrors\n\n(PhysOrg.com) — Californian company Prysm has unveiled a high definition television with a “laser phosphor display” based on their patented method of using lasers reflected off a bank of mirrors to excite pixels on the television screen in a similar way to cathode ray tubes.\nThis is the first time I’ve heard of a TV using lasers. The video link in the article is very cool\n\nThis is a neat video on Liquid Helium. We were studying Temperature in class, the book mentioned superfluids and this is a neat video I found on superfluids.\n\nAnd to see how Quama handles images, here is a recent photo of me from the webcam in my Airie.", null, "Oh Yeah, I’ve also figured out I can learn Chemistry through MIT Opencourseware. So I’ve been working through 5.111 Principles of Chemical Science As taught in: Fall 2008. There is a full video lecture series and the professor teaching it is awesome! I’m very happy to be learning from her.\n\nI would like to see if my LaTeX editor can pick up things. Lets give it a try…\n\n[math]!(a+b)^2[/math]\n\nIf it works I’ve got to track down my lost post on how to write LaTeX again.\n\nFiled under Side Projects\n\n## Quantum Cascade Surface-Emitting Photonic Crystal Laser\n\nWe combine photonic and electronic band structure engineering to create a surface-emitting quantum cascade microcavity laser. A high-index contrast two-dimensional photonic crystal is used to form a micro-resonator that simultaneously provides feedback for laser action and diffracts light vertically from the surface of the semiconductor surface. A top metallic contact allows electrical current injection and provides vertical optical confinement through\na bound surface plasmon wave. The miniaturization and tailorable emission properties of this design are potentially important for sensing applications, while electrical pumping can allow new studies of photonic crystal and surface plasmon structures in nonlinear and near-field optics.\n\nhttp://iqse.harvard.edu/research/docs/science_2003-302-1374.pdf\n\nFiled under Articles\n\n## Entangled photons available on tap – physics-math – 02 June 2010 – New Scientist\n\nFiled under Side Projects\n\n## Bursting bubbles beget tiny copies of themselves – physics-math – 09 June 2010 – New Scientist\n\nFiled under Side Projects\n\n## Bursting bubbles beget tiny copies of themselves – physics-math – 09 June 2010 – New Scientist\n\nFiled under Uncategorized\n\n## Fluids – Chapter Notes\n\nThe Density (rho) is mass per unit volume, the unit is kg/m3.\n[math]displaystyle rho = frac{m}{V}[/math]\nrho equals m over V\n[math] m = rho V[/math]\nm equals rho V\n\nSpecific Gravity is ratio of the density of the substance to the density of water at 4C.\nPressure is force per unit area, when the force is magnitude acting perpendicular to the surface area A.\n[math]displaystyle P = frac {F}{A}[/Math]\nPressure is a scalar with the unit name Pascal, which is N/m2.\nPressure due to liquid is , remember the funny looking p is “rho” aka DENSITY.\nAs the pressure increases, density increases as well though so for cases with gas, pressure is\n[math] displaystyle frac{dP}{dy} = -rho g[/math].\nAnother way to express is equation is\n[math] displaystyle P_2 – P_1 = -int_{y_1}^{y_2} rho g dy [/Math]\nIf you can ignore variations in density this can be expressed as:\n[math] P_2 – P_1 = – rho g(y_2 – y_1) [/math]\nFor an open contain of liquid, you simply add the pressure from the atmosphere\n[Math] P = P_0 + rho gh[/math]\nPressure head: Sometimes the h in Rho g h is called the pressure head.\nAtmospheric Pressure is 101.3 kPa for 14.7lb/in2. sometimes we use bars, which are 10,000 N/m2.\nPressure gauges register gauge pressure. This does not include the atmospheric pressure, so we need to add it in, by adding the pressure of the atmosphere to the gauge.\n\nPascal’s Principle states “If an external pressure is applied to a confined fluid, the pressure at every point within the fluid increases by that amount”\n[Math] P_{out} = P_{in}[/math]\n[math] displaystyle frac{F_{out}}{A_{out}} = frac{F_{In}}{A_{In}}[/math]\n[math] displaystyle frac{F_{out}}{F_{in}} = frac{A_{out}}{A_{In}}[/math]\nThis ratio gives mechanical advantage, Fout/Fin.\nThe buoyant force occur’s because pressure on the bottom surface of a submerged object is greater than the downward pressure on the top of the object. It is equal to the weight of fluid displaced by the object.\n[math]F_b = m’g = rho Vg[/math] where m’g is the weight of the body of fluid equal to the volume of the original submerged object. When an objects float Fb=mg in general. When an object is partially submerged.\n[math] displaystyle frac{V_{displaced}}{V_0} = frac {rho_0}{rho_{final}}[/math]\n\nTools\nHydrometer – measured specific gravity of a liquid\n\nFluid Dynamics (Hydrodynamics)\nStreamline (laminar flow)\nTurbulent flow\nMass flow rate equals change in mass over change in time.\n[math]rho_1A_1v_1 = rho_2A_2v_2[/math] This is the equation of continuity. It reads “initial Density times Area times Velocity equals final Density times Area times Velocity”.\nIf density is constant, The equation of continuity can be written as\n[math]A_1v_1 = A_2v_2[/math]\nBernoulli’s Principle “Where the velocity of a fluid is high, the pressure is low, and where the velocity is low, the pressure is high” Bernoulli’s Equation:\n[math] displaystyle P_1 + frac{1}{2}rho v_1^2 + rho gy_1 = P_2 + frac{1}{2} rho v_2^2+ rho gy_2[/math]\nor in other words\n[math] displaystyle P + frac{1}{2}rho v^2 + rho gy[/math] is constant\nand for both, y is the height of the center of the tube above a fixed reference level.\nLiquid leaves a spigot with the same speed a freely falling object would attain if dropped from the same height.\n[math] v_1 = sqrt{2g(y_2 – y_1)}[/math]\n\nAnd if you want to use Bernoulli’s principle in the case where there is no significant change in height:\n[math]displaystyle P_1 + frac{1}{2} rho v_1^2 = P_2 + frac{1}{2} rho v_2^2 [/math]\n\nFiled under PHY 126\n\n## How to use LaTeX\n\nGreek letters\npi for lowercase\nPi for uppercase\nNo command for \\$Alpha\\$ – just use A\n[Math]pi[/Math]\n[Math]pi[/Math]\n[Math]A[/Math]\n\nFractions\nfrac{numerator}{denominator}\n[Math]displaystyle frac{F}{a}[/Math]\n\nSuperscript\nx^2\n[Math]x^2[/math]\n\nSubscript\nx_2\n[Math]x_2[/Math]\n\nUsing Curly Braces to make groups\nx_{F_2}\nx_{min}\n[Math]x_{F_2}[/Math]\n[Math]x_{min}[/Math]\n\nSuper and Sub\nx_F^3\n[Math]x_F^3[/math]\n\nLimits and Integrals\ndisplaystyle lim_{x to infty} 3x\ndisplaystyle int_0^2 x dx\n[Math]displaystyle lim_{x to infty} 3x [/Math]\n\n[Math] displaystyle int_0^2 x dx [/Math]" ]	[ null, "http://blog.carriesegal.com/wp-content/uploads/2010/06/mirage.jpg", null, "http://blog.carriesegal.com/wp-content/uploads/2010/06/Photo-on-2010-06-26-at-17.46.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.80952036,"math_prob":0.93040234,"size":7320,"snap":"2022-27-2022-33","text_gpt3_token_len":1959,"char_repetition_ratio":0.12028431,"word_repetition_ratio":0.057971016,"special_character_ratio":0.25437158,"punctuation_ratio":0.06486881,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9968577,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,2,null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-08-18T22:51:54Z\",\"WARC-Record-ID\":\"<urn:uuid:dc34b780-324d-461f-89f8-cba85f3cd1b6>\",\"Content-Length\":\"45093\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:1eff94f8-df83-4643-80b2-22bc8614966d>\",\"WARC-Concurrent-To\":\"<urn:uuid:57bca877-d616-4c62-92bc-47c37668454d>\",\"WARC-IP-Address\":\"66.33.223.31\",\"WARC-Target-URI\":\"http://blog.carriesegal.com/index.php/2010/06/\",\"WARC-Payload-Digest\":\"sha1:LQSHHO7ZACFNSW5CGJE2MZV24JRYPEUP\",\"WARC-Block-Digest\":\"sha1:KIDVCSHZFXLSDHGC5YVLOCSFR6X7WAID\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-33/CC-MAIN-2022-33_segments_1659882573533.87_warc_CC-MAIN-20220818215509-20220819005509-00155.warc.gz\"}"}
https://www.lacasadeitigli.it/82936/1592921212.html	[ "•", null, "## C6X Series Jaw Crusher\n\nC6X Jaw Crusher is new equipment used for crushing hard or abrasiveness stones. It is possess of detachable frame without welding structure, double wedge adjusting device, elastic limit damping device and integrated motor seat, which will make C6X Series Jaw Crusher popular in the market.\n\n•", null, "## HPT Hydraulic Cone Crusher\n\nHPT Series High-Efficiency Hydraulic Cone Crusher combines crushing stroke with crushing chamber perfectly, which makes capacity improved and efficiency increased. Besides, hydraulic lubrication control system helps bearing of hydraulic cone crusher get double protection.\n\n•", null, "## VSI6X Series Vertical Crusher\n\nBased on more than 30 years of experiences, VSI6X Series Vertical Shaft Impact Crusher carrying many patents amazes the market greatly. This machine is possess of four openings impeller, special sealing structure and sealed cartridge bearing structure, which makes production convenient and efficient.\n\n# Calculating depreciation of mining equipment\n\n•", null, "#### Calculating Depreciation Of Mining Equipment\n\nmining equipment depreciation lives. Calculating depreciation of mining equipment lives is used, the BEA depreciation profiles are for an entire cohort of assets of afor production-type equipment . Get Price. Depreciation Bonus information clearinghouse.\n\n•", null, "#### Calculating Depreciation Of Mining Equipment\n\ncalculating depreciation of mining equipment taxtreatmentof etsallowances European Commission. Dec 31, 2010 Calculated welfare losses due to differences in national taxation . .. set, an asset type that is typically allowed a linear depreciation of its value benchmark is a firm's investment in new equipment that reduces future f a mine, an\n\n•", null, "#### calculating depreciation of mining equipment\n\ncalculating depreciation of mining equipment taxtreatmentof etsallowances European Commission. Calculated welfare losses due to differences in national taxation . .. set, an asset type that is typically allowed a linear depreciation of its value benchmark is a firm's investment in new equipment that reduces future f a mine, an oil or gas well\n\n•", null, "#### Calculating Depreciation Of Mining Equipment\n\nCalculating Depreciation Of Mining Equipment. Depreciation of ore gold mining machine depreciation rates for gold mining equipmentepreciation rates for gold mining equipmentold price touches 6 month low as currency war talk heats up, depreciation rates for gold mining equipment,fiat currency depreciation increases golds allure as a hedge, frik is editor and writer for mining frik has\n\n•", null, "#### How to Calculate Depreciation on Equipment Bizfluent\n\nSep 26, 2017· Depreciation is an accounting term that refers to the allocation of cost over the period in which an asset is used. In a business, the cost of equipment is generally allocated as depreciation expense over a period of time known as the useful life of the equipment.\n\n•", null, "#### 3. CALCULATION OF MACHINE RATES\n\nCommonly included in fixed costs are equipment depreciation, interest on investment, taxes, and storage, and insurance. 3.2.2 Operating Costs. Operating costs vary directly with the rate of work (Figure 3.1). These costs include the costs of fuel, lubricants, tires, equipment maintenance and repairs. Figure 3.1 Equipment Cost Model. 3.2.3 Labor\n\n•", null, "#### Straight Line Depreciation Calculator\n\nInputsStraight-Line Depreciation FormulaStraight-Line Depreciation ExampleMicrosoft® Excel® Functions equivalent: SlnAsset Cost1. the original value of your asset or the depreciable cost; the necessary amount expended to get an asset ready for its intended useSalvage Value1. the value of the asset at the end of its useful life; also known as residual value or scrap valueUseful Life1. the expected time that the asset will be productive for its expected purposeSee more on calculatorsoup\n•", null, "#### Units of Production Depreciation: How to Calculate & Formula\n\nJul 02, 2019· To calculate units of production depreciation expense, you will apply an average cost per unit rate to the total units the machinery or equipment produces each year. This rate will be the ratio of the total cost of the asset less its salvage value to the estimated number of units it is expected to produce during its useful life.\n\n•", null, "#### 4 Ways to Calculate Depreciation on Fixed Assets wikiHow\n\nMar 29, 2019· How to Calculate Depreciation on Fixed Assets. Depreciation is the method of calculating the cost of an asset over its lifespan. Calculating the depreciation of a fixed asset is simple once you know the formula. === Using Straight Line.\n\n• Views: 1.9M\n•", null, "#### Depreciation Calculator\n\nDepreciationMethods of DepreciationPartial Year DepreciationSalvage ValueConceptually, depreciation is the reduction in value of an asset over time, due to elements such as wear and tear. For instance, a widget-making machine is said to \\\"depreciate\\\" when it produces less widgets one year compared to the year before it, or a car is said to \\\"depreciate\\\" in value after a fender bender or the discovery of a faulty transmission.For accounting in particular, depreciation concerns allocating the cost of an asset over a period of time, usually its useful life. When a comp.\n•", null, "#### Calculating Depreciation Of Mining Equipment\n\nCalculating Depreciation Of Mining Equipment. Depreciation of ore gold mining machine depreciation rates for gold mining equipmentepreciation rates for gold mining equipmentold price touches 6 month low as currency war talk heats up, depreciation rates for gold mining equipment,fiat currency depreciation increases golds allure as a hedge, frik is editor and writer for mining frik has\n\n•", null, "#### calculating depreciation of mining equipment\n\ncalculating depreciation of mining equipment taxtreatmentof etsallowancesEuropean Commission. Dec 31 2010 Calculated welfare losses due to differences in national taxation . .. set an asset type that is typically allowed a linear depreciation of its value benchmark is a firm s investment in new equipment that reduces future f a mine an.\n\n•", null, "#### Calculating Depreciation Of Mining Equipment\n\nSoftware for calculating rates of mining equipment.2014 mttr, mtbr, failure rate, availability and reliability.This is the time-line of a particular equipment where u is operating time uptime in hrs, d is repair time downtime in hrs.Calculating the depreciation of a fixed asset is\n\n•", null, "#### Calculating Depreciation Of Mining Equipment\n\nCalculating depreciation of mining equipment youtube jun 15, 2015 calculating depreciation of mining equipment machine for grinding barite equipment for gold mining crusher belt conveyor conveyor belts for more details get price. click to chat; Mine Mill Equipment Costs Estimators Guide .\n\n•", null, "#### calculating depreciation of mining equipment\n\ncalculating depreciation of mining equipment calculating depreciation of mining equipment taxtreatmentof etsallowances European Commission Dec 31, 2010 Calculated welfare losses due to differences in national taxation set, an asset type that is typically allowed a linear depreciation of its value benchmark is a firm's investment in new\n\n•", null, "#### Depreciation Cost of Construction Equipment\n\nThe equipment life used in calculating depreciation should correspond to the equipment’s expected economic or useful life. Among many depreciation methods, the straight-line method,double-declining balance method,and sum-of-years’-digits method are the most commonly used in the construction equipment industry.\n\n•", null, "#### Depreciation Calculation Methods\n\nMay 22, 2020· How to Calculate Depreciation . Depreciation is calculated by taking the useful life of the asset (available in tables, based on the type of asset, though you may need an accountant for this), less the salvage value of the asset at the end of its useful life (also determined by a table), divided by the cost of the asset (including all costs for acquiring the asset like transportation, set-up\n\n•", null, "#### Units of Production Depreciation: How to Calculate & Formula\n\nJul 02, 2019· To calculate units of production depreciation expense, you will apply an average cost per unit rate to the total units the machinery or equipment produces each year. This rate will be the ratio of the total cost of the asset less its salvage value to the estimated number of units it is expected to produce during its useful life.\n\n•", null, "#### What Is Depreciation Types, Formula & Calculation\n\nAccumulated depreciation is the total depreciation of the fixed asset accumulated up to a specified time. Example: On April 1, 2012, company X purchased an equipment for Rs. 100,000.This is expected to have 5 useful life years.\n\n•", null, "#### MACRS Asset Life table\n\nThe MACRS Asset Life table is derived from Revenue Procedure 87-56 1987-2 CB 674. The table specifies asset lives for property subject to depreciation under the general depreciation system provided in section 168(a) of the IRC or the alternative depreciation system provided in section 168(g).\n\n•", null, "#### 4 Ways to Depreciate Equipment wikiHow\n\nMar 05, 2020· Calculate annual depreciation using the double declining balance depreciation method. First, calculate the straight-line depreciation rate using the cost, salvage value and useful life of the asset. In the first year, apply the double depreciation rate to the cost of the asset to calculate the depreciation expense.\n\n• Views: 161K\n•", null, "#### Depreciation Rate (Formula, Examples) How to Calculate?\n\nThus depreciation rate during the useful life of vehicles would be 20% per year. Example #2. A company purchases 40 units of storage tanks worth \\$1,00,000/- per unit. Tanks have a useful life of 10 years and a scrap value of \\$11000/-. The company uses a Double declining method of depreciation for calculating the depreciation expense for the tanks.\n\n•", null, "#### How to Calculate Depreciation? FreshBooks\n\nFor example, the annual depreciation on an equipment with a useful life of 20 years, a salvage value of \\$2000 and a cost of \\$100,000 is \\$4,900 ((\\$100,000-\\$2,000)/20). The asset must be placed in service (set up and used) in the first year that depreciation is calculated, for accounting and tax purposes.\n\n•", null, "#### Straight Line Depreciation Formula & Guide to Calculate\n\nOther Methods of Depreciation. In addition to straight line depreciation, there are also other methods of calculating depreciation Depreciation Methods The most common types of depreciation methods include straight-line, double declining balance, units of production, and sum of years digits. There are various formulas for calculating depreciation of an asset.\n\n•", null, "#### Calculating Depreciation Of Mining Equipment\n\nSoftware for calculating rates of mining equipment.2014 mttr, mtbr, failure rate, availability and reliability.This is the time-line of a particular equipment where u is operating time uptime in hrs, d is repair time downtime in hrs.Calculating the depreciation of a fixed asset is simple once you know the formula.Using.\n\n•", null, "#### calculating depreciation of mining equipment\n\ncalculating depreciation of mining equipment calculating depreciation of mining equipment taxtreatmentof etsallowances European Commission Dec 31, 2010 Calculated welfare losses due to differences in national taxation set, an asset type that is typically allowed a linear depreciation of its value benchmark is a firm's investment in new\n\n•", null, "#### Calculating Depreciation Of Mining Equipment\n\nCalculating depreciation of mining equipment youtube jun 15, 2015 calculating depreciation of mining equipment machine for grinding barite equipment for gold mining crusher belt conveyor conveyor belts for more details get price. click to chat; Mine Mill Equipment Costs Estimators Guide .\n\n•", null, "#### Calculating Depreciation Of Mining Equipment\n\nCalculating Depreciation Of Mining Equipment. Depreciation of ore gold mining machine depreciation rates for gold mining equipmentepreciation rates for gold mining equipmentold price touches 6 month low as currency war talk heats up, depreciation rates for gold mining equipment,fiat currency depreciation increases golds allure as a hedge, frik is editor and writer for mining\n\n•", null, "#### How to Calculate Depreciation? FreshBooks\n\nFor example, the annual depreciation on an equipment with a useful life of 20 years, a salvage value of \\$2000 and a cost of \\$100,000 is \\$4,900 ((\\$100,000-\\$2,000)/20). The asset must be placed in service (set up and used) in the first year that depreciation is calculated, for accounting and tax purposes.\n\n•", null, "#### Depreciation Calculation Methods\n\nMay 22, 2020· How to Calculate Depreciation . Depreciation is calculated by taking the useful life of the asset (available in tables, based on the type of asset, though you may need an accountant for this), less the salvage value of the asset at the end of its useful life (also determined by a table), divided by the cost of the asset (including all costs for acquiring the\n\n•", null, "#### A Guide to Depreciation for Small Businesses (2020) The\n\nAug 18, 2020· Unlike double declining depreciation, sum-of-the-years depreciation does consider salvage value when calculating depreciation, so your first year depreciation calculation would be: (10 ÷ 55) x\n\n•", null, "#### 4 Ways to Depreciate Equipment wikiHow\n\nMar 05, 2020· Calculate annual depreciation using the double declining balance depreciation method. First, calculate the straight-line depreciation rate using the cost, salvage value and useful life of the asset. In the first year, apply the double depreciation rate to the cost of the asset to calculate the depreciation expense.\n\n• Views: 161K\n•", null, "#### Straight Line Depreciation Formula & Guide to Calculate\n\nOther Methods of Depreciation. In addition to straight line depreciation, there are also other methods of calculating depreciation Depreciation Methods The most common types of depreciation methods include straight-line, double declining balance, units of production, and sum of years digits. There are various formulas for calculating depreciation of an asset.\n\n•", null, "#### Salvage Value Learn How to Calculate an Asset's Salvage\n\nSalvage value is the estimated amount that an asset is worth at the end of its useful life. Salvage value is also known as scrap value or residual value, and is used in calculating depreciation expense. The value depends on how long the company expects to use the asset and how hard the asset is used. For example, if a\n\n•", null, "#### How to calculate depreciation on computer hardware: A\n\nMar 29, 2017· Consider this example: Company A buys a new piece of equipment, the Widget, for \\$100,000. The Widget has a useful life of 10 years. Without depreciation, Company A would show \\$100,000 in expenses\n\n•", null, "#### Depreciation For Mining Operations BMT Insider\n\nNov 20, 2019· The millions of dollars saved through depreciation can be used to invest in new mining equipment, to pay for outstanding business costs or even to expand operations. To find out more about depreciation for commercial property, Request a Quote or contact our expert staff on 1300 728 726.\n\n•", null, "#### Accumulated Depreciation (Definition, Formula) How to\n\nThe equipment is not expected to have any salvage value at the end of its useful life. The equipment is to be depreciated on a straight-line method. Determine the accumulated depreciation at the end of 1 st year and 3 rd year. Below is data for calculation of the accumulated depreciation at the end of 1 st year and 3 rd year.\n\n•", null, "#### Calculating Depreciation Of Mining Equipment\n\nDepreciation On Coal Handling Equipment. Calculating depreciation of mining equipmentccounting treatment of the depletion and depreciation the units ofproduction method is used to calculate depreciation 2 discuss the accounting treatment of the depletion and depreciation on the mine and mining equipmentet priceepreciation rates machinery." ]	[ null, "https://www.lacasadeitigli.it/themes/mininger/col_b1.png", null, "https://www.lacasadeitigli.it/themes/mininger/col_b3.png", null, "https://www.lacasadeitigli.it/themes/mininger/col_b2.png", null, "https://www.lacasadeitigli.it/gallery/95.jpg", null, "https://www.lacasadeitigli.it/gallery/16.jpg", null, "https://www.lacasadeitigli.it/gallery/165.jpg", null, "https://www.lacasadeitigli.it/gallery/171.jpg", null, "https://www.lacasadeitigli.it/gallery/62.jpg", null, "https://www.lacasadeitigli.it/gallery/100.jpg", null, "https://www.lacasadeitigli.it/gallery/37.jpg", null, "https://www.lacasadeitigli.it/gallery/25.jpg", null, "https://www.lacasadeitigli.it/gallery/16.jpg", null, "https://www.lacasadeitigli.it/gallery/110.jpg", null, "https://www.lacasadeitigli.it/gallery/69.jpg", null, "https://www.lacasadeitigli.it/gallery/46.jpg", null, "https://www.lacasadeitigli.it/gallery/50.jpg", null, "https://www.lacasadeitigli.it/gallery/92.jpg", null, "https://www.lacasadeitigli.it/gallery/68.jpg", null, "https://www.lacasadeitigli.it/gallery/142.jpg", null, "https://www.lacasadeitigli.it/gallery/208.jpg", null, "https://www.lacasadeitigli.it/gallery/45.jpg", null, "https://www.lacasadeitigli.it/gallery/208.jpg", null, "https://www.lacasadeitigli.it/gallery/124.jpg", null, "https://www.lacasadeitigli.it/gallery/161.jpg", null, "https://www.lacasadeitigli.it/gallery/139.jpg", null, "https://www.lacasadeitigli.it/gallery/14.jpg", null, "https://www.lacasadeitigli.it/gallery/153.jpg", null, "https://www.lacasadeitigli.it/gallery/218.jpg", null, "https://www.lacasadeitigli.it/gallery/139.jpg", null, "https://www.lacasadeitigli.it/gallery/185.jpg", null, "https://www.lacasadeitigli.it/gallery/13.jpg", null, "https://www.lacasadeitigli.it/gallery/120.jpg", null, "https://www.lacasadeitigli.it/gallery/168.jpg", null, "https://www.lacasadeitigli.it/gallery/210.jpg", null, "https://www.lacasadeitigli.it/gallery/146.jpg", null, "https://www.lacasadeitigli.it/gallery/87.jpg", null, "https://www.lacasadeitigli.it/gallery/213.jpg", null, "https://www.lacasadeitigli.it/gallery/42.jpg", null, "https://www.lacasadeitigli.it/gallery/71.jpg", null, "https://www.lacasadeitigli.it/gallery/141.jpg", null, "https://www.lacasadeitigli.it/gallery/140.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.88501334,"math_prob":0.94884294,"size":15583,"snap":"2020-34-2020-40","text_gpt3_token_len":3171,"char_repetition_ratio":0.2509789,"word_repetition_ratio":0.51829785,"special_character_ratio":0.19360842,"punctuation_ratio":0.09736842,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98457617,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82],"im_url_duplicate_count":[null,2,null,2,null,2,null,2,null,4,null,1,null,1,null,1,null,1,null,1,null,1,null,4,null,1,null,1,null,1,null,1,null,1,null,1,null,2,null,2,null,1,null,2,null,1,null,2,null,3,null,1,null,1,null,1,null,3,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-09-21T03:10:13Z\",\"WARC-Record-ID\":\"<urn:uuid:693f3733-caea-4c26-9f5f-f212d9c69231>\",\"Content-Length\":\"26152\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:315f58cb-9203-4168-8e10-815bed2ebb2c>\",\"WARC-Concurrent-To\":\"<urn:uuid:6c949d33-28b4-46ba-bda1-4fffd69888e0>\",\"WARC-IP-Address\":\"104.24.126.81\",\"WARC-Target-URI\":\"https://www.lacasadeitigli.it/82936/1592921212.html\",\"WARC-Payload-Digest\":\"sha1:LN5M2UC7VJIC5OX7HFEW5DFH2YG2VZI6\",\"WARC-Block-Digest\":\"sha1:JNLCHLZ3EAHXGXLEW6CDCIO6H33CA75Y\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600400198887.3_warc_CC-MAIN-20200921014923-20200921044923-00015.warc.gz\"}"}
https://cs.stackexchange.com/questions/12943/what-is-a-good-algorithm-for-generating-random-dfas/12949	[ "# What is a good algorithm for generating random DFAs?\n\nI am generating random DFAs to test a DFA reduction algorithm on them.\n\nThe algorithm that I'm using right now is as follows: for each state $q$, for each symbol in the alphabet $c$, add $\\delta (q, c)$ to some random state. Each state has the same probability of becoming a final state.\n\nIs this a good method of generating unbiased DFAs? Also, this algorithm doesn't generate a trim DFA (a DFA with no obsolete states) so I'm wondering if there is a better way of generating random DFAs that can somehow ensure that it is trim?\n\n• There's no natural distribution of random DFAs, so it's up to you. What do you want to accomplish? – Yuval Filmus Jun 28 '13 at 5:29\n• I am generating random DFAs to test a DFA reduction algorithm on them. – Duncan Jun 28 '13 at 5:32\n• Perhaps you want to start from a small set of minimal DFAs and blow them up? That way you know that the minimization actually arrives at the right result. – adrianN Jun 28 '13 at 9:30\n• wondering, are you testing a nonoptimal reduction algorithm or is it finding the unique minimal optimum DFA? – vzn Jun 28 '13 at 17:25\n• If you're generating data structures for testing, unbiased isn't important. What's important is to have a good chance of triggering interesting behaviors. To take an analogy, when you test a geometric algorithm, you need to ensure that some of your tests will have three points aligned and other things that would never occur in a random distribution. – Gilles 'SO- stop being evil' Jun 29 '13 at 8:43\n\nCheck out and the discussion in Section 4, Random Automata Generation. The paper benchmarks different DFA minimization algorithms. A uniform random generator is used that produces canonical string representations of complete DFAs with $n$ states and $k$ symbols. They also discuss other methods.\n\n• What does \"canonical string representations\" mean? – Duncan Jun 29 '13 at 3:01\n• @drowse The canonical order is defined over the set of states by traversing the automaton in a breadth-first way choosing at each node the outgoing using the (total) order of $\\Sigma$. Check out the references in the paper. – Juho Jun 29 '13 at 16:54\n\nYou should look at Cyril Nicaud's homepage. In particular, the following references are relevant to your question:\n\nF. Bassino, J. David and C. Nicaud, Enumeration and random generation of possibly incomplete deterministic automata, Pure Mathematics and Applications 19 (2-3) (2009) 1-16.\n\nF. Bassino and C. Nicaud. Enumeration and Random Generation of Accessible Automata. Theor. Comp. Sc.. 381 (2007) 86-104.\n\nThere are algorithms to randomly generate DFAs up to a permutation http://paranthoen.thomas.free.fr/PAPERS/RandDFAToAppearInTCS.ps.gz.\n\nBut, it is also mentionned in the above paper that almost all DFAs are already minimal. Non minimal DFAs are like prime numbers ... there are just few of them. And if you use this algorithm to test minimization algorithm it will be like if you were testing an algorithm on prime number with a simple random number generator. In order to have more non minimal DFAs, you may alter the algorithm by adding a sink state, and redirect an important percentage of the transitions to this sink state.\n\nBut from my point of view, if you want to test the speediness of your implementation, check it against what you want to use it for: with random word sets or random REGEX, create a NFA or a DFA, and then minimize the resulting DFA.\n\none natural strategy is to regard the DFA as a graph and then there are many \"natural\" and highly studied random distributions of graphs, the simplest is probably Erdos-Renyi. in that case you treat all states of the DFA as nodes of the graph and some fixed percent of all possible edges (DFA transitions) are chosen. more sophisticated distributions that are studied much in the more recent era are small world graphs. for the strategy you mention in your question you are apparently choosing the special case $p=1/n$ where $n$ is the number of nodes in the graph. however your strategy, nor Erdos-Renyi, does not guarantee that all the states in the DFA are connected [a natural constraint to add].\n\n• – Gilles 'SO- stop being evil' Aug 18 '13 at 8:17\n• re the edit, another way to see DFAs is as a set of triples defining edges $(v_1,b,v_2)$ where $v_1$ is vertex 1, $v_2$ vertex 2, and $b$ is the transition symbol and any subset of these can be chosen to determine a DFA. – vzn Sep 15 '13 at 17:08" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9292792,"math_prob":0.94680345,"size":527,"snap":"2019-43-2019-47","text_gpt3_token_len":120,"char_repetition_ratio":0.13001911,"word_repetition_ratio":0.0,"special_character_ratio":0.22770399,"punctuation_ratio":0.093457945,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96931064,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-18T16:46:18Z\",\"WARC-Record-ID\":\"<urn:uuid:b1c7e943-4840-424e-ab7b-afc226c8c6a0>\",\"Content-Length\":\"166283\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:76bf39dc-d968-4fa8-b569-ce3876afafe7>\",\"WARC-Concurrent-To\":\"<urn:uuid:8bd7d780-76a7-409a-9269-3034da38b219>\",\"WARC-IP-Address\":\"151.101.193.69\",\"WARC-Target-URI\":\"https://cs.stackexchange.com/questions/12943/what-is-a-good-algorithm-for-generating-random-dfas/12949\",\"WARC-Payload-Digest\":\"sha1:V3B5YYBARKXUY67SB6ZFAKRLMVZEKICA\",\"WARC-Block-Digest\":\"sha1:RMNBU6TBLA6JDEFWJ3HXO7Q45UPIHMAY\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496669809.82_warc_CC-MAIN-20191118154801-20191118182801-00372.warc.gz\"}"}
https://www.osapublishing.org/ol/fulltext.cfm?uri=ol-42-2-290&id=357170	[ "## Abstract\n\nA distributed optical fiber dynamic strain sensor with high spatial and frequency resolution is demonstrated. The sensor, which uses the $ϕ$-OTDR interrogation technique, exhibited a higher sensitivity thanks to an improved optical arrangement and a new signal processing procedure. The proposed sensing system is capable of fully quantifying multiple dynamic perturbations along a 5 km long sensing fiber with a frequency and spatial resolution of 5 Hz and 50 cm, respectively. The strain resolution of the sensor was measured to be 40 nε.\n\nPublished by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.\n\nThe concept of using an optical fiber to map dynamic perturbations along the sensing fiber was first proposed by Taylor and Lee and later demonstrated by their team in 2005 . The proposed sensor was only capable of detecting the dynamic perturbations along the sensing fiber by monitoring the changes in the pattern of the backscattered coherent Rayleigh noise. The volume of research since then has proliferated and has focused on developing distributed optical fiber dynamic strain sensors that are capable of fully quantifying the characteristics of perturbations, such as frequency and amplitude along the sensing fiber. The first distributed optical fiber sensor capable of full characterization of dynamic strains was demonstrated by Song and Hotate . The proposed sensor used Brillouin optical correlation domain analysis (BOCDA) to interrogate 20 m long sensing fiber. Despite its high spatial resolution, the bandwidth of the sensor was limited to 200 Hz mainly due to the slow sensing procedure intrinsic to the BOCDA sensing technique. In addition, the relatively low strain sensitivity of the sensor made it unsuitable for mapping acoustic fields.\n\nInitially, distributed optical fiber dynamic strain sensors had limited dynamic and/or frequency range mainly due to their slow interrogation techniques. In 2013, Masoudi et al. demonstrated a long-range and high-bandwidth distributed dynamic strain sensor capable of measuring dynamic vibrations . The proposed sensing technique was based on phase optical time-domain reflectometry ($ϕ$-OTDR), a technique which determines the induced strain using the phase of the Rayleigh backscattered light. A detailed review of the past and present measurement techniques capable of fully quantifying the dynamic perturbations and a comparison between their key parameters can be found in .\n\nOne of the drawbacks of the current sensing systems for mapping acoustic fields has been their relatively high noise floor [4,6]. The main source of noise in these systems is the amplified spontaneous emission (ASE) noise from the two erbium-doped fiber amplifiers (EDFAs) used to amplify the probe pulse and the backscattered light. The ASE noise include ASE-ASE beat noise, signal-ASE beat noise, and ASE-shot beat noise where the noise level in each case depends on the bandwidth of the ASE spectrum. In the previous publications [4,6] two matched fiber Bragg grating (FBG) filters were used to reduce the ASE. The first was placed after the power amplifier used to amplify the probe pulse. The second was placed after the pre-amplifier to amplify the weak backscattered Rayleigh signal. By reducing the linewidth of these two FBGs the ASE noise can be reduced. The practical limit is, however, governed by the stability of the reflection wavelength of the FBGs. For the two FBGs to remain matched the linewidth of both should be larger than the anticipated wavelength drift arising from changes in the temperature of their environment.\n\nIn this Letter, a novel optical arrangement is introduced which uses a single narrow FBG to perform both functions, i.e., to reduce the ASE noise from both the power amplifier and pre-amplifier. Unlike the sensing systems with two FBGs, the bandwidth of the FBG used in the new arrangement can be very narrow thereby enhancing the rejection of ASE-generated noise. This new experimental arrangement is combined with an improved signal processing procedure that is less susceptible to noise. Much enhanced results, including a fourfold improvement in spatial resolution, a twofold improvement in strain resolution, and an eighteenfold improvement in frequency discrimination, are demonstrated.\n\nThis Letter begins with a brief description of the underlying sensing principles of the sensor followed by a detailed description of the new signal processing procedure and the experimental setup. Next, the experimental results are presented followed by an analysis of the results and conclusion.\n\nThe strain distribution along the sensing fiber is mapped by examining the changes in the phase of the backscattered Rayleigh light . For any given section of the fiber, the phase difference between the backscattered light from the two ends of that section, $Δφ$, is a function of the length of that section, $ℓ$. For any unperturbed section of the sensing fiber, the length and, consequently, the phase difference remain unchanged. Any perturbation that induces a strain $ε$ on the fiber will change the phase difference. The changes in the phase difference as a function of length is given by \n\n$Δφ(ℓ)=ϵℓ[β−12βn2[(1−μ)p12−μp11]],$\nwhere $β$ is the propagation constant of light in the fiber, $n$ is the refractive index of the fiber, $μ$ is the Poisson’s ratio, and $p11$ and $p12$ are strain-optic coefficients. Replacing the values of refractive index ($n=1.456$), Poisson’s ratio ($μ=0.17$), and strain-optic coefficients ($p11=0.121$, $p12=0.27$) in Eq. (1) gives\n$Δφ(ℓ)=ϵℓβ×0.78.$\n\nEquation (2) illustrates that 22% of the phase change is due to the strain-induced refractive index change in the fiber. Therefore, to accurately calculate the strain rate, $Δφ$ should be divided by 0.78. In order to measure the phase difference between the two end points of each section, an imbalanced Mach–Zehnder interferometer (MZI) is used. A symmetrical $3×3$ coupler is employed as the interferometer output coupler to avoid the signal fading problem . It can be shown that the intensity of the light at the three output arms of the MZI is given by \n\n$I1=I0[M+N cos(Δφ(ℓ))],I2=I0[M+N cos(Δφ(ℓ)+2π3)],I3=I0[M+N cos(Δφ(ℓ)−2π3)],$\nwhere $I0$ is the intensity of the input signal and $M$ and $N$ are constant. Hitherto, a differentiate and cross-multiplying (DXM) demodulator was used to extract the phase information from the outputs of the three detectors [4,6]. The main drawback of the DXM demodulator is the use of a differentiator in the demodulation process, which is inherently sensitive to noise. In order to avoid the differentiator, a new signal processing procedure is devised to retrieve the phase information using an arc-tangent function and for phase variation beyond $±π/2$, the new procedure used a fringe-counting algorithm. To retrieve the phase information, the DC components of the three terms in Eq. (3) need to be eliminated first. The DC term can be calculated as follows:\n$S=I1+I2+I33=I0M.$\n\nSubtracting Eq. (4) from the three terms in Eq. (3) gives\n\n$I˙1=I0N cos(Δφ(ℓ)),I˙2=I0N cos(Δφ(ℓ)+2π3),I˙3=I0N cos(Δφ(ℓ)−2π3).$\n\nFinally, using trigonometric identities, the phase information can be recovered as follows:\n\n$Δφ=arctan(I˙2−I˙3I˙1).$\n\nA fringe-counting algorithm is implemented by looking at the transition of $Δφ$ from $+π/2$ to $−π/2$ and vice versa.\n\nThe experimental setup is shown in Fig. 1. A 1550 nm distributed feedback laser diode was modulated with an electro-optic modulator to generate a 5 ns pulse with a repetition rate of 50 μs. The pulse was then amplified by an erbium-doped fiber amplifier (EDFA1) to raise the peak power to 4 W. The amplified pulse was passed through an acousto-optic modulator (AOM1) ($insertion loss=3 dB$; $extinction ratio=50 dB$) followed by an FBG filter ($λB=1550.52 nm$; $Δλ=0.2 nm$; $reflectivity=99.9%$) to remove the ASE from the EDFA1 via circulator C1. The probe pulse with peak power of 2 W was then launched into the sensing fiber via circulator C2. The sensing fiber consisted of a 4.9 m and a 2.4 m long single-mode fiber wrapped around two piezoelectric ceramics (PZTs) (PZT1: a 155 mm diameter ring PZT; PZT2: 115 mm diameter ring PZT) separated by a 100 m unstrained–unheated fiber. A 4.5 km and a 100 m long unstrained–unheated fiber were used to separate the PZTs from the front end and far end of the sensing fiber, respectively. The backscattered Rayleigh signal was collected by the circulator C2 and amplified by the second optical amplifier (EDFA2). The ASE from the second optical amplifier was filter by the same FBG used to filter the ASE of the first optical amplifier via circulator C3. For use in environments with temperature variations greater than (10°C), the FBG should be temperature stabilized.", null, "Fig. 1. Experimental setup. DFB, distributed feedback; PC, polarization controller; EOM, electro-optic modulator; EDFA, erbium-doped fiber amplifier; AOM, acousto-optic modulator; C, circulator; FBG, fiber Bragg grating; PD, photodetector; DAQ, data acquisition system.\n\nThe experimental procedure went through two phases during each interrogation cycle: the pulse-formation phase and the signal-detection phase. The control signals to the two AOMs were adjusted accordingly to direct or block the light during each phase. During the pulse-formation phase, AOM1 was switched on to allow the probe pulse to reach the sensing fiber. During this period, AOM2 remained switched off to block the ASE of EDFA1 from saturating the detectors. If AOM2 does not block the light, the entire ASE spectrum from EDFA1 less 0.2 nm (i.e., FBG bandwidth) passes through the FBG and falls on the detectors. During the signal-detection phase, AOM1 was switched off to isolate the pulse-formation arm of the sensor and AOM2 was turned on to allow the amplified backscattered light through to the MZI after it had been filtered by the FBG to remove ASE generated by EDFA2.\n\nAfter passing through AOM2, the filtered backscattered signal was fed into the thermally insulated MZI with path imbalance of 1 m using a 50/50 coupler. Three photoreceivers (40 V/mA transimpedance; 125 MHz bandwidth) were used to detect the light from the $3×3$ coupler at the output of the MZI. The photoreceivers were sampled using a 250 MHz oscilloscope at a sampling rate of 625 MSa/s.\n\nThe 3D diagram of Fig. 2 illustrates the frequency components present in a 300 m section of the sensing fiber between 4500 and 4800 m. The plot represents the post-processed data obtained from the setup while the large and small PZTs were modulated with 1500 and 2000 Hz sinusoidal signals, respectively. The amplitude of the voltage applied to the large PZT was $7 VPP$ and that of the smaller PZT was set to $10 VPP$.", null, "Fig. 2. Frequency components present along a 300 m stretch of the sensing fiber between 4500 and 4800 m.\n\nThe phase shift and strain level of the larger PZT as a function of the PZT voltage is shown in the diagram of Fig. 3. This diagram demonstrates the PZT response to 1 kHz sinusoidal signals with voltages varying from $1 VPP$ to $8 VPP$. The data points on this figure represent the peak of the strain measured by the sensor.", null, "Fig. 3. PZT input voltage versus sensor output for 1 kHz sinusoidal signal.\n\nFigure 4 represents the frequency response of the larger PZT measured by both the distributed sensor (red data points) and MZI (solid line). The solid line is obtained by characterizing the large PZT using cw light and a conventional MZI. The data was collected by applying a fixed voltage ($7 VPP$) to the PZT while changing the frequency from 100 Hz to 3 kHz.", null, "Fig. 4. Frequency response of the large PZT measured by the distributed sensor (red data points) and MZI (solid line).\n\nTo assess the spatial and frequency resolution of the sensor, the 3D diagram illustrating the output of the sensor is presented by two 2D diagrams in Fig. 5. Figure 5(a) shows the spatial distribution of strain along the sensing fiber at a fixed frequency while Fig. 5(b) shows the frequency spectrum of the dynamic fluctuations at a single point on the sensing fiber. The trace representing the spatial distribution of strain in Fig. 5(a) was obtained by averaging the data acquired from 10 separate measurements. The frequency and amplitude of the input voltage to PZT2 was kept constant at 1 kHz and $8 VPP$, respectively, for the duration of the 10 measurements. To measure the frequency resolution, the frequency of the input signal to PZT1 was changed in 1 Hz steps from 1500 to 1505 Hz while monitoring the frequency spectrum of a fixed point on the sensing fiber. Figure 5(b) shows the data acquired for input frequencies of 1500 and 1505 Hz.", null, "Fig. 5. 2D representation of the 3D diagram depicting the output of the sensor. (a) Spatial distribution of strain along the sensing fiber at 1 kHz. (b) Frequency spectrum of the dynamic fluctuations at 4578 m for 1500 Hz signal (blue trace) and 1505 Hz signal (red trace).\n\nThe two peaks in Fig. 2 correspond to the strains applied by the two PZTs at 4578 and 4681 m. The 3D plot demonstrates that the sensor can accurately measure the location, frequency, and amplitude of dynamic perturbations along the fiber.\n\nFigure 3 shows a linear relationship between the amplitude of the input voltage to the PZT and the measured phase shift for 1 kHz sinusoidal signal. Equation (2) was used to calculate the strain level shown on the right vertical axis of this diagram:\n\n$ϵ=Δφ0.78ℓβ=Δφ0.78ℓ·λ2πn,$\nwhere $λ=1550 nm$, $n=1.456$, and $ℓ$ is determined by the path imbalanced of the MZI and is equal to $ℓ=ΔL/2=0.5 m$. Similar experiments at other frequencies (i.e., 500, 1500, and 2000 Hz) exhibit the same linear characteristic. The minimum detectable strain at 1 kHz was measured to be 40 nε, which shows a factor of 2 improvement from the previous work. Figure 4 shows a good correlation between the data collected with the sensor and the frequency response measured by the MZI. The experimental result also shows that the system can detect frequencies as high as 5 kHz. Based on the observation from Figs. 3 and 4, it can be concluded that the modified sensing arrangement is capable of quantifying the frequency, amplitude, and location of dynamic perturbations anywhere along the sensing fiber with an accuracy of 40 nε. The strain sensitivity of the sensor can be further improved through averaging in expense of a more limited frequency range.\n\nThe other advantage of the modified optical arrangement is that it facilitates the development of a wavelength-division multiplexing sensing system that offers improved frequency and/or sensing range. Such a system can be realized by using a multi-wavelength source and only one FBG for each wavelength in the link between circulator C1 and C3.\n\nThe analysis of Fig. 5(a) demonstrates 10/90% spatial resolution of 50 cm, a factor of 4 improvement on what was achieved previously. Furthermore, Fig. 5(b) shows that two frequencies as close as 5 Hz can be clearly distinguished from one another. A 5 Hz frequency resolution indicates a factor of 18 improvement from the previous results. The improvement in the frequency resolution was achieved by increasing the acquisition time from 10 to 200 ms.\n\nIn summary, the modified experimental setup demonstrated a factor of 4 improvement in gauge length and a factor of 2 improvement in minimum detectable strain. The higher sensitivity was achieved by (1) using a single narrow-bandwidth FBG to filter the ASE from both EDFAs and, (2) eliminating the differentiator from the phase demodulator, which made it inherently less sensitive to noise. The new signal processing procedure was not only less sensitive to noise but also simpler to implement. As a result, the signal processing run time of the new demodulator was 10 times as fast as the DXM demodulation algorithm. The frequency resolution and sensing range of the new setup were measured to be 5 Hz and 5 km, respectively, which indicates a factor of 18 improvement in the frequency resolution and a factor of 5 improvement in the sensing range.\n\nAll data supporting this study are openly available from the University of Southampton repository in Dataset 1, Ref. .\n\n## Funding\n\nEngineering and Physical Sciences Research Council (EPSRC) (EP/N00437X/1).\n\n1. H. F. Taylor and C. E. Lee, “Apparatus and method for fiber optic intrusion sensing,” U.S. patent 5,194,847 (March 16, 1993).\n\n2. J. C. Juarez, E. W. Maier, K. N. Choi, and H. F. Taylor, J. Lightwave Technol. 23, 2081 (2005). [CrossRef]\n\n3. K. Y. Song and K. Hotate, IEEE Photon. Technol. Lett. 19, 1928 (2007). [CrossRef]\n\n4. A. Masoudi, M. Belal, and T. P. Newson, Meas. Sci. Technol. 24, 085204 (2013). [CrossRef]\n\n5. A. Masoudi and T. P. Newson, Rev. Sci. Instrum. 87, 011501 (2016). [CrossRef]\n\n6. C. Wang, C. Wang, Y. Shang, X. Liu, and G. Peng, Opt. Commun. 346, 172 (2015). [CrossRef]\n\n7. K. De Souza and T. P. Newson, Opt. Express 12, 2656 (2004). [CrossRef]\n\n8. G. B. Hocker, Appl. Opt. 18, 1445 (1979). [CrossRef]\n\n### References\n\n• View by:\n\n1. H. F. Taylor and C. E. Lee, “Apparatus and method for fiber optic intrusion sensing,” U.S. patent5,194,847 (March16, 1993).\n2. J. C. Juarez, E. W. Maier, K. N. Choi, and H. F. Taylor, J. Lightwave Technol. 23, 2081 (2005).\n[Crossref]\n3. K. Y. Song and K. Hotate, IEEE Photon. Technol. Lett. 19, 1928 (2007).\n[Crossref]\n4. A. Masoudi, M. Belal, and T. P. Newson, Meas. Sci. Technol. 24, 085204 (2013).\n[Crossref]\n5. A. Masoudi and T. P. Newson, Rev. Sci. Instrum. 87, 011501 (2016).\n[Crossref]\n6. C. Wang, C. Wang, Y. Shang, X. Liu, and G. Peng, Opt. Commun. 346, 172 (2015).\n[Crossref]\n7. K. De Souza and T. P. Newson, Opt. Express 12, 2656 (2004).\n[Crossref]\n8. G. B. Hocker, Appl. Opt. 18, 1445 (1979).\n[Crossref]\n9. http://doi.org/10.5258/SOTON/401728 .\n\n#### 2016 (1)\n\nA. Masoudi and T. P. Newson, Rev. Sci. Instrum. 87, 011501 (2016).\n[Crossref]\n\n#### 2015 (1)\n\nC. Wang, C. Wang, Y. Shang, X. Liu, and G. Peng, Opt. Commun. 346, 172 (2015).\n[Crossref]\n\n#### 2013 (1)\n\nA. Masoudi, M. Belal, and T. P. Newson, Meas. Sci. Technol. 24, 085204 (2013).\n[Crossref]\n\n#### 2007 (1)\n\nK. Y. Song and K. Hotate, IEEE Photon. Technol. Lett. 19, 1928 (2007).\n[Crossref]\n\n#### Belal, M.\n\nA. Masoudi, M. Belal, and T. P. Newson, Meas. Sci. Technol. 24, 085204 (2013).\n[Crossref]\n\n#### Hotate, K.\n\nK. Y. Song and K. Hotate, IEEE Photon. Technol. Lett. 19, 1928 (2007).\n[Crossref]\n\n#### Lee, C. E.\n\nH. F. Taylor and C. E. Lee, “Apparatus and method for fiber optic intrusion sensing,” U.S. patent5,194,847 (March16, 1993).\n\n#### Liu, X.\n\nC. Wang, C. Wang, Y. Shang, X. Liu, and G. Peng, Opt. Commun. 346, 172 (2015).\n[Crossref]\n\n#### Masoudi, A.\n\nA. Masoudi and T. P. Newson, Rev. Sci. Instrum. 87, 011501 (2016).\n[Crossref]\n\nA. Masoudi, M. Belal, and T. P. Newson, Meas. Sci. Technol. 24, 085204 (2013).\n[Crossref]\n\n#### Newson, T. P.\n\nA. Masoudi and T. P. Newson, Rev. Sci. Instrum. 87, 011501 (2016).\n[Crossref]\n\nA. Masoudi, M. Belal, and T. P. Newson, Meas. Sci. Technol. 24, 085204 (2013).\n[Crossref]\n\n#### Peng, G.\n\nC. Wang, C. Wang, Y. Shang, X. Liu, and G. Peng, Opt. Commun. 346, 172 (2015).\n[Crossref]\n\n#### Shang, Y.\n\nC. Wang, C. Wang, Y. Shang, X. Liu, and G. Peng, Opt. Commun. 346, 172 (2015).\n[Crossref]\n\n#### Song, K. Y.\n\nK. Y. Song and K. Hotate, IEEE Photon. Technol. Lett. 19, 1928 (2007).\n[Crossref]\n\n#### Taylor, H. F.\n\nH. F. Taylor and C. E. Lee, “Apparatus and method for fiber optic intrusion sensing,” U.S. patent5,194,847 (March16, 1993).\n\n#### Wang, C.\n\nC. Wang, C. Wang, Y. Shang, X. Liu, and G. Peng, Opt. Commun. 346, 172 (2015).\n[Crossref]\n\nC. Wang, C. Wang, Y. Shang, X. Liu, and G. Peng, Opt. Commun. 346, 172 (2015).\n[Crossref]\n\n#### IEEE Photon. Technol. Lett. (1)\n\nK. Y. Song and K. Hotate, IEEE Photon. Technol. Lett. 19, 1928 (2007).\n[Crossref]\n\n#### Meas. Sci. Technol. (1)\n\nA. Masoudi, M. Belal, and T. P. Newson, Meas. Sci. Technol. 24, 085204 (2013).\n[Crossref]\n\n#### Opt. Commun. (1)\n\nC. Wang, C. Wang, Y. Shang, X. Liu, and G. Peng, Opt. Commun. 346, 172 (2015).\n[Crossref]\n\n#### Rev. Sci. Instrum. (1)\n\nA. Masoudi and T. P. Newson, Rev. Sci. Instrum. 87, 011501 (2016).\n[Crossref]\n\n#### Other (2)\n\nH. F. Taylor and C. E. Lee, “Apparatus and method for fiber optic intrusion sensing,” U.S. patent5,194,847 (March16, 1993).\n\nhttp://doi.org/10.5258/SOTON/401728 .\n\n### Supplementary Material (1)\n\nNameDescription\nDataset 1 Datasets for all the figures.\n\n### Cited By\n\nOptica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.\n\n### Figures (5)\n\nFig. 1. Experimental setup. DFB, distributed feedback; PC, polarization controller; EOM, electro-optic modulator; EDFA, erbium-doped fiber amplifier; AOM, acousto-optic modulator; C, circulator; FBG, fiber Bragg grating; PD, photodetector; DAQ, data acquisition system.\nFig. 2. Frequency components present along a 300 m stretch of the sensing fiber between 4500 and 4800 m.\nFig. 3. PZT input voltage versus sensor output for 1 kHz sinusoidal signal.\nFig. 4. Frequency response of the large PZT measured by the distributed sensor (red data points) and MZI (solid line).\nFig. 5. 2D representation of the 3D diagram depicting the output of the sensor. (a) Spatial distribution of strain along the sensing fiber at 1 kHz. (b) Frequency spectrum of the dynamic fluctuations at 4578 m for 1500 Hz signal (blue trace) and 1505 Hz signal (red trace).\n\n### Equations (7)\n\n$Δ φ ( ℓ ) = ϵ ℓ [ β − 1 2 β n 2 [ ( 1 − μ ) p 12 − μ p 11 ] ] ,$\n$Δ φ ( ℓ ) = ϵ ℓ β × 0.78 .$\n$I 1 = I 0 [ M + N cos ( Δ φ ( ℓ ) ) ] , I 2 = I 0 [ M + N cos ( Δ φ ( ℓ ) + 2 π 3 ) ] , I 3 = I 0 [ M + N cos ( Δ φ ( ℓ ) − 2 π 3 ) ] ,$\n$S = I 1 + I 2 + I 3 3 = I 0 M .$\n$I ˙ 1 = I 0 N cos ( Δ φ ( ℓ ) ) , I ˙ 2 = I 0 N cos ( Δ φ ( ℓ ) + 2 π 3 ) , I ˙ 3 = I 0 N cos ( Δ φ ( ℓ ) − 2 π 3 ) .$\n$Δ φ = arctan ( I ˙ 2 − I ˙ 3 I ˙ 1 ) .$\n$ϵ = Δ φ 0.78 ℓ β = Δ φ 0.78 ℓ · λ 2 π n ,$" ]	[ null, "https://www.osapublishing.org/images/ajax-loader-big.gif", null, "https://www.osapublishing.org/images/ajax-loader-big.gif", null, "https://www.osapublishing.org/images/ajax-loader-big.gif", null, "https://www.osapublishing.org/images/ajax-loader-big.gif", null, "https://www.osapublishing.org/images/ajax-loader-big.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9282054,"math_prob":0.90719765,"size":15006,"snap":"2021-43-2021-49","text_gpt3_token_len":3381,"char_repetition_ratio":0.14984669,"word_repetition_ratio":0.015052889,"special_character_ratio":0.21824603,"punctuation_ratio":0.10953563,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9747051,"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\":\"2021-10-18T04:27:47Z\",\"WARC-Record-ID\":\"<urn:uuid:4828d1b0-3764-46e0-873c-235123c73c29>\",\"Content-Length\":\"160457\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7f32a8b5-9c9f-40fe-8780-d6ad05ea4535>\",\"WARC-Concurrent-To\":\"<urn:uuid:4c844b24-a1cc-45db-9960-776080641e73>\",\"WARC-IP-Address\":\"38.95.177.60\",\"WARC-Target-URI\":\"https://www.osapublishing.org/ol/fulltext.cfm?uri=ol-42-2-290&id=357170\",\"WARC-Payload-Digest\":\"sha1:GEZMR3O7ZUAAP7XXD34CY7NFKJW6JEWD\",\"WARC-Block-Digest\":\"sha1:277YSOP5XSQOPWHNKP4ODLIWNG5FDJIZ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585196.73_warc_CC-MAIN-20211018031901-20211018061901-00218.warc.gz\"}"}
http://qiancy.com/2016/11/	[ "# 终结工具 – Synergy", null, "## 1. SNE概要", null, "### 1.1 高维数据的相似度概率分布\n\nSNE将数据点之间高维的欧氏距离转换为表示相似度的条件概率，即用条件概率$$p(j\|i)$$表示点$$x_j$$到点$$x_i$$的相似度，这个含义可以理解为：若以$$x_i$$为中心的高斯分布来选取邻居，则$$x_i$$选择$$x_j$$作为自己邻居的概率是$$p(j\|i)$$。若数据点相距较近，则$$p(j\|i)$$较大，相反若数据点相距非常远，$$p(j\|i)$$则可以接近无穷小。条件概率$$p(j\|i)$$定义如下：" ]	[ null, "http://qiancy.com/wp-content/uploads/2016/11/14793612077732.jpg", null, "http://qiancy.com/wp-content/uploads/2016/11/14720411795869.jpg", null ]	{"ft_lang_label":"__label__zh","ft_lang_prob":0.9614572,"math_prob":1.0000085,"size":648,"snap":"2020-34-2020-40","text_gpt3_token_len":594,"char_repetition_ratio":0.10869565,"word_repetition_ratio":0.0,"special_character_ratio":0.18827161,"punctuation_ratio":0.026666667,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999956,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,2,null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-08-10T15:07:35Z\",\"WARC-Record-ID\":\"<urn:uuid:1306f55c-6594-4466-aa69-c73f27825fce>\",\"Content-Length\":\"27244\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c18ab974-b349-4274-a664-396c8046dcf1>\",\"WARC-Concurrent-To\":\"<urn:uuid:aaef52c9-c214-4cb0-9225-d635071e1067>\",\"WARC-IP-Address\":\"139.129.6.49\",\"WARC-Target-URI\":\"http://qiancy.com/2016/11/\",\"WARC-Payload-Digest\":\"sha1:YSP3YCOXD4DAMOP42V76NZSVMTMYIHLD\",\"WARC-Block-Digest\":\"sha1:V4SFO4KTPDK5DZMD2HWC2P7QVS3L26MJ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-34/CC-MAIN-2020-34_segments_1596439736057.87_warc_CC-MAIN-20200810145103-20200810175103-00163.warc.gz\"}"}
https://www.oscarsports.com/issue-001-confucius-lotto-forecast-award-52-single-inject-reference/	[ "Best seller\n\n# Issue 001 Confucius Lotto Forecast Award: 5+2 Single Inject Reference\n\nLottery retrospective: Sports Lottery Lotto No. 22150 issued the prize number: 06 07 26 29 30+03 12, the size ratio ratio of the front area is 3: 2,012 to 2: 1: 2, and the quality combination ratio is 2: 3.The ratio ratio of the strange coupling is 2: 3, the ratio ratio of the rear area is 1: 1,012 to 2: 0: 0, and the mass -to -match ratio is 1: 1, and the strange coupling ratio is 1: 1.\n\nAnalysis of Protocol Number of Phase 2023001 Phase 2023001:", null, "First place: The prize number 06 was opened in the previous period. After the first 10 times opened the number 06, the prize number was issued in the next issue: 06-02-15-01-07-07-02-05-04,Among them, the number ratio ratio is 5: 5, the size ratio is 0: 10,012 to 3: 4: 3, and the mass -to -quality ratio is 6: 4., Pay attention to the number of numbers in this issue, select one yard 08.\n\nSecond place: The prize number 07 was opened in the previous period. After the first 10 times opened the number 07, the prize number was issued in the next issue: 12-11-14-10-15-09-12-05-06,Among them, the size ratio ratio of the number is 0: 10,012 to 5: 1: 4, and the mass -ratio is 2: 8, and the strange coupling ratio is 4: 6.In this issue, the second place is optimistic about the 0 number, and the gallbladder is concerned 15.\n\nThird place: The prize number 26 in the previous period, after the number 26 of the first 10 times, the prize number of the next issue: 30-27-13-28-23-11-26-22-16-16,Among them, the number 012 is 2: 5: 3, the qualitative ratio is 3: 7, and the puppet ratio is 4: 6, and the size ratio is 6: 4., The third place in this issue pays attention to the odd number, selects one yard 27.\n\nFourth place: The prize number 29 was opened in the previous period, and the number 29 was opened in the first 10 times.Among them, the size ratio ratio of the number is 10: 0,012 to 5: 3: 2, and the mass -to -line ratio is 1: 9, and the strange coupling ratio is 3: 7.In this issue, the fourth reference of the odd number number, selected gallbladder 31.\n\nFifth place: 30 prize number 30 in the previous period, after the number 30 in the first 10 times, the prize number of the next period: 32-28-20-33-31-32-30-19-27, andAmong them, the number 012 is 3: 3: 4, the qualitative ratio is 2: 8, and the coupling ratio is 4: 6, and the size ratio is 10: 0., The fifth place in this issue, the bile code is optimistic about 33.", null, "Sixth place: The prize number 03 was opened in the previous period, and after the first 10 times opened the number 03, the next issue of the prize number: 09-09-04-03-06-07-01-07,Among them, the number combined ratio is 5: 5, the strange coupling ratio is 6: 4, and the size ratio is 4: 6,012 to 4: 5: 1. The sixth place in this issue is optimistic about the quality digits, and the bile code follows 01.\n\nSeventh place: The prize number 12 was opened in the previous period, and the number 12 was opened in the first 10 times.Among them, the number 012 is 3: 5: 2, the qualitative ratio is 4: 6, and the puppet ratio is 5: 5, and the size ratio is 7: 3. The seventh -digit digital number in this issue. The gallbladder code is optimistic about 03.", null, "Confucius Lotto No. 2023001 Number Recommendation:\n\nThe front area kill number: 01 03 09 10 19 22 29 34\n\nReference in the front area: 06 08 13 14 15 16 17 21 24 26 27 30 33 35\n\nFive yards reference in the back area: 01 03 09 10 11\n\n5+2 Single Note Reference: 08 15 27 31 33+01 03\n\n[Sweep the code download app, and more than 10 million experts are here!]", null, "We will be happy to hear your thoughts", null, "Enable registration in settings - general" ]	[ null, "data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI2MDUiIGhlaWdodD0iMjE3IiB2aWV3Qm94PSIwIDAgNjA1IDIxNyI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgc3R5bGU9ImZpbGw6I2NmZDRkYjtmaWxsLW9wYWNpdHk6IDAuMTsiLz48L3N2Zz4=", null, "data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI3MDQiIGhlaWdodD0iNDAxIiB2aWV3Qm94PSIwIDAgNzA0IDQwMSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgc3R5bGU9ImZpbGw6I2NmZDRkYjtmaWxsLW9wYWNpdHk6IDAuMTsiLz48L3N2Zz4=", null, "data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI1ODMiIGhlaWdodD0iNTY0IiB2aWV3Qm94PSIwIDAgNTgzIDU2NCI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgc3R5bGU9ImZpbGw6I2NmZDRkYjtmaWxsLW9wYWNpdHk6IDAuMTsiLz48L3N2Zz4=", null, "data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxODgiIGhlaWdodD0iMTg4IiB2aWV3Qm94PSIwIDAgMTg4IDE4OCI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgc3R5bGU9ImZpbGw6I2NmZDRkYjtmaWxsLW9wYWNpdHk6IDAuMTsiLz48L3N2Zz4=", null, "data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNjAiIGhlaWdodD0iNTAiIHZpZXdCb3g9IjAgMCAxNjAgNTAiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9213183,"math_prob":0.99051553,"size":3544,"snap":"2023-40-2023-50","text_gpt3_token_len":1120,"char_repetition_ratio":0.22881356,"word_repetition_ratio":0.1891892,"special_character_ratio":0.3535553,"punctuation_ratio":0.17253122,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97164863,"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\":\"2023-12-01T10:50:38Z\",\"WARC-Record-ID\":\"<urn:uuid:00001fa5-9022-4183-ae56-1a7de0a7c679>\",\"Content-Length\":\"148640\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7d639256-5686-43f9-b464-bf2c5970c523>\",\"WARC-Concurrent-To\":\"<urn:uuid:daaac7a6-dbe7-4061-9da0-1eb2d0537a19>\",\"WARC-IP-Address\":\"172.111.38.73\",\"WARC-Target-URI\":\"https://www.oscarsports.com/issue-001-confucius-lotto-forecast-award-52-single-inject-reference/\",\"WARC-Payload-Digest\":\"sha1:47ATCL7BT2DGU37O2TYSGZTRETKWHHWH\",\"WARC-Block-Digest\":\"sha1:ZF5YKEGVOSRHCEKWEGPRLF3FDA6ISGXE\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100286.10_warc_CC-MAIN-20231201084429-20231201114429-00610.warc.gz\"}"}
https://sciencekick.blogspot.com/2010/11/wrong-wrong-wrong.html	[ "## Wednesday, November 3, 2010\n\n### Wrong, Wrong, Wrong!\n\nA certain incident has been nibbling away at my consciousness for ages like a kind of psychic termite. It happened in my freshman physics class when the teacher was introducing Newton's law of gravitation. In order to protect the reputation of said teacher, I shall refer to him here as Professor X.\n\nProfessor X presented the equation for finding the force exerted on an object by the earth's gravity. It's a classic, lovely little equation...\nMe = the mass of the earth, about 1.3 x 1025 lbs or 5.97 x 1024 kilograms. (Do we remember our metric conversions and our scientific notation?)\nm = the mass, in kilograms, of some object.\nr = how far that object is from the center of the earth.\nG = the universal gravitation constant, a sort of cosmic fudge factor, equal to\n6.673 x 10-11 N•m2/kg2\n(the \"N\" stands for \"Newtons\", the unit of measure for force; read as \"Newton meters squared per kilograms squared\").\n\nA student asked a question: \"So, if r = 0, the force is infinite?\"\n\nKnowing that we had been taught in our math classes that anytime you divide a number by 0 the answer is , Professor X said, \"Yes, if r is 0, Fg is infinitely large.\"\n\nThere are a few reasons why the professor might have responded to the student's question with such an incredibly wrong answer:\n1) The student who asked the question was kind of annoying and may have been asking a smart-alecky question which, naturally, demanded a smart-alecky answer;\n2) The professor may have simply been joking, oblivious to the fact that some people in the class might have taken him seriously; or, what I think is the most likely answer,\n3) The professor wanted to get beyond gravitation and move on to the next subject without introducing additional, complicated material which would have blown the minds of unprepared freshman physics students.\nPhysicists are sometimes taken to task by other scientists for playing fast and loose with mathematics; assumptions and leaps of logic that -- even as they are proven true beyond a reasonable doubt by experiment after experiment -- leave mathematicians flabbergasted by their audacity and seem a lot like cheating. It happens often enough that, sometimes, physicists get caught with their panties down and major flaws are found in their reasoning and their lovely little equations. The Fg equation works. Mostly. For the purposes of an undergraduate physics student, at least, it works just fine. But then you start to look at the assumptions being made and you realize the equation comes with some strings attached.\n\nThe big assumption being made in regard to the gravity equation and the earth is that you are dealing with a point mass, a planet-sized spherical mass of uniform density where all its gravity is coming from a single point in the center, which is kind of a funny idea when you think about it. If you're talking about objects near or above the surface of the earth, you can get decent approximations of Fg from the equation. But when you start to dig down, things get complicated.\n\nEvery part of the earth, from the crust to the core, has mass. Start boring a hole toward the center of the earth and eventually, you'll find yourself with a mass of earth-stuff above you that's large enough to exert a significant gravitational force. (How far down do you have to go? That's a future post.) The gravity you felt would be the sum of countless sources of gravitational attraction acting on you.\n\nThe total force of gravity acting on you deep below the earth's surface would be the total Fg from below you and minus the total Fg from above you (from the sides, it would mostly cancel out... more on that later). This is because of something called the superposition of forces. The total force of gravity you would experience would still be drawing you toward the center of the earth, but Fg (from beneath you) would be less than if you were on the surface.\n\nSince you experience the sum of all the gravitational forces acting on you, two equal forces attracting you from opposite directions cancel each other out. Thanks to the principle of superposition, we know that if you were at the center of the earth, for all the mass in any direction exerting a gravitational force on you, there is an equal mass on the other side of you canceling it out. Newton's formula isn't wrong, it just needs to be applied in a slightly different way. Fg at the center of the earth, where r = 0, isn't ∞ like Professor X said, it's actually the sum total of all the \"gravities\" acting on you which add up to 0.\n\nBut it's not as if we could actually test this. It's about 6367 km (3956 miles) to the center of the earth and there's a helluva lot of stuff to dig through to get there, not to mention the fact that the middle of our planet is crazy hot (estimated at around 4000° C /7230° F). As any geophysicist will tell you, gravity gets complicated when you go underground.\n\nPicture yourself standing pretty much anywhere on the earth's surface. We can describe the force of gravity that keeps you from flying off into space with a vector, an arrow pointing downward that represents both the direction of the force and how strong it is.", null, "A force like the force due to gravity, is often represented in diagrams by a simple arrow which indicates the direction in which the force acts.\n\nWhen you're underground, the force of gravity you experience as a result of being surrounded by earth can be described by a three-dimensional vector field that represents the total gravitational attraction from all the sources of mass that surround you. Vector fields are something a freshman physics student would probably only know about if they had taken a multivariable calculus class (which would be one or two semesters in the future for most students). There is no such thing as a quick lesson on vector fields... re-read this paragraph and you'll realize I really haven't told you anything about them except that they exist, so I suppose Professor X can be forgiven for not wanting to bring nasty old vector fields into our pleasant discussion of gravity law. I guess I can let him off the hook... for this at least.\n\nSources\n• Geodynamics: Applications of Continuum Physics to Geological Problems by Donald L. Turcotte & Gerald Schubert. ©1982, John Wiley & Sons, Inc.\n• Geophysical Methods in Geology, 2nd Ed. by P.V. Sharma. ©1986, Elsevier Science Publishing Co., Inc." ]	[ null, "https://2.bp.blogspot.com/_CnSUJiRChqA/TNFsmZISx-I/AAAAAAAAAJE/C6ALl4-1F3s/s400/xavier_costume2.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9683348,"math_prob":0.8994379,"size":6917,"snap":"2022-40-2023-06","text_gpt3_token_len":1482,"char_repetition_ratio":0.11688124,"word_repetition_ratio":0.23738627,"special_character_ratio":0.21555588,"punctuation_ratio":0.10007413,"nsfw_num_words":1,"has_unicode_error":false,"math_prob_llama3":0.9671886,"pos_list":[0,1,2],"im_url_duplicate_count":[null,3,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-09-28T10:48:57Z\",\"WARC-Record-ID\":\"<urn:uuid:f57c908b-1e8e-411d-8663-0fa275b411fe>\",\"Content-Length\":\"68772\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f1cf1855-c346-44ba-869d-931a5a7369e7>\",\"WARC-Concurrent-To\":\"<urn:uuid:c49b2b6b-cf94-451d-8238-54c85902e524>\",\"WARC-IP-Address\":\"172.253.62.132\",\"WARC-Target-URI\":\"https://sciencekick.blogspot.com/2010/11/wrong-wrong-wrong.html\",\"WARC-Payload-Digest\":\"sha1:VGRN7SCBYW2DWD3NLVG6ZYXDZ4ZE6RMW\",\"WARC-Block-Digest\":\"sha1:M6WUI6D4GJNFCAGKCOZR355A7DQYBNBC\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030335190.45_warc_CC-MAIN-20220928082743-20220928112743-00205.warc.gz\"}"}
https://aspirantszone.com/reasoning-questions-based-on-data-sufficiency/	[ "# Reasoning: Data Sufficiency Set 29\n\n1. In each of the questions below consists of a question and two statements numbered a and b given below it. You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and give answer:\n\nSix friends Manish, Nitin, Ojasvee, Priyank, Queen and Risabh went to office in different days from Monday to Saturday of the same week but not necessarily in the same order. Who among the following went to office on Wednesday?\nSTATEMENTS:\na)Only one person went to office after Nitin. Two days gap between when Manish and Queen went to office.\nb)Priyank went to office after Nitin. Ojasvee went to office before the day Queen went to office. At least one day gap between the days when Manish and Risabh went to office.\n\nIf the data in statement a alone are sufficient to answer the question, while the data in statement b alone are not sufficient to answer the question\nIf the data in statement b alone are sufficient to answer the question, while the data in statement a alone are not sufficient to answer the question\nIf the data either in statement a alone or in statement b alone are sufficient to answer the question\nIf the data given in both statements a and b together are not sufficient to answer the question\nIf the data in both statements a and b together are necessary to answer the question.\nOption E\n\n2. Seven friends i.e. Akash, Bani, Ceom, Diksha, Elvish, Farah, and Garima are sitting in a row facing north but not necessarily in the same order. Who among the following sits 2nd to the right of Farah?\nSTATEMENTS:\na) Akash sits one of the extreme ends of the row. Three persons sit between Akash and Garima.\nb) Diksha sits immediate left of Garima. Two persons sit between Ceom and Farah. Elvish doesn’t sit at any of the extreme end.\nIf the data in statement a alone are sufficient to answer the question, while the data in statement b alone are not sufficient to answer the question\nIf the data in statement b alone are sufficient to answer the question, while the data in statement a alone are not sufficient to answer the question\nIf the data either in statement a alone or in statement b alone are sufficient to answer the question\nIf the data given in both statements a and b together are not sufficient to answer the question\nIf the data in both statements a and b together are necessary to answer the question.\nOption D\n\n3. Five persons Farooq, Garima, Heena , Kamini and Lokesh like different mobiles viz.Apple , Samsung , Lenovo , Redmi and Asus but not\nnecessarily in the same order. Find who among the following likes lenovo?\nSTATEMENTS:\na) Among Garima, Heena and Kamini no one like Samsung. Garima neither like Asus nor Redmi\nb) Heena is neither like Asus nor Lenovo. Kamini does not like lenovo.\nIf the data in statement a alone are sufficient to answer the question, while the data in statement b alone are not sufficient to answer the question\nIf the data in statement b alone are sufficient to answer the question, while the data in statement a alone are not sufficient to answer the question\nIf the data either in statement a alone or in statement b alone are sufficient to answer the question\nIf the data given in both statements a and b together are not sufficient to answer the question\nIf the data in both statements a and b together are necessary to answer the question.\nOption D\n\n4. In a certain code language How the word “Love” is coded?\nSTATEMENTS:\na) “money love fun common” is coded as “sd fr gr da” and “ransom jump love fun” is coded as “cg nb fr gr”\nb) “must ransom burden common” is coded as “tj pl da cg” and “wan fun house” is coded as “qs sw gr”\nIf the data in statement a alone are sufficient to answer the question, while the data in statement b alone are not sufficient to answer the question\nIf the data in statement b alone are sufficient to answer the question, while the data in statement a alone are not sufficient to answer the question\nIf the data either in statement a alone or in statement b alone are sufficient to answer the question\nIf the data given in both statements a and b together are not sufficient to answer the question\nIf the data in both statements a and b together are necessary to answer the question.\nOption E\n\n5. Six persons Priyal, Queen, Risabh, Srikant, Tanu, and Unnati are sitting in row according to their age in descending order from left to right but not necessarily in the same order. Find who among the following is 2nd youngest?\nSTATEMENTS:\na) At least two people are older than Tanu. Queen is older than Unnati but younger than Priyal.\nb) Srikant is older than Priyal but younger than Risabh. Priyal is younger than Tanu.\nIf the data in statement a alone are sufficient to answer the question, while the data in statement b alone are not sufficient to answer the question\nIf the data in statement b alone are sufficient to answer the question, while the data in statement a alone are not sufficient to answer the question\nIf the data either in statement a alone or in statement b alone are sufficient to answer the question\nIf the data given in both statements a and b together are not sufficient to answer the question\nIf the data in both statements a and b together are necessary to answer the question.\nOption E\n\n6. Each of the questions below consists of a question and two statements numbered a and b given below it. You have to decide whether the data provided in the statement are sufficient to answer the question. Read both the statements and give answer:\n\nWho sits immediate left of Manika, if all persons sit in a row are facing north?\na) Akansha sits third to the right of Nitin. There is only three persons sit between Nitin and Manika.\nb) Kamini sits second to the left of Manika. Not more than eight persons sit in a row.\n\nIf the data in statement a alone are sufficient to answer the question, while the data in statement b alone are not sufficient to answer the question.\nIf the data in statement b alone are sufficient to answer the question, while the data in statement a alone are not sufficient to answer the question.\nIf the data either in statement a alone or in statement b alone are sufficient to answer the question.\nIf the data even in both statements a and b together are not sufficient to answer the question.\nIf the data in both statement a and b together are necessary to answer the question.\nOption E\n\n7. Who amongst Priyank, Queen, Risabh, Srikant, Tanu and Unnati is the shortest person?\na)Risabh is taller than Srikant but shorter than only 2 persons. Queen is shorter than Priyank.\nb) Tanu is not as tall as Unnati who is not the tallest.\nIf the data in statement a alone are sufficient to answer the question, while the data in statement b alone are not sufficient to answer the question.\nIf the data in statement b alone are sufficient to answer the question, while the data in statement a alone are not sufficient to answer the question.\nIf the data either in statement a alone or in statement b alone are sufficient to answer the question.\nIf the data even in both statements a and b together are not sufficient to answer the question.\nIf the data in both statement a and b together are necessary to answer the question.\nOption D\n\n8. How is Av related to Bv?\na)Av has only two children. Cv is sister of Ev. Dv is father in law of Bv.\nb) Ev is husband of Bv. Av is married to Dv.\nIf the data in statement a alone are sufficient to answer the question, while the data in statement b alone are not sufficient to answer the question.\nIf the data in statement b alone are sufficient to answer the question, while the data in statement a alone are not sufficient to answer the question.\nIf the data either in statement a alone or in statement b alone are sufficient to answer the question.\nIf the data even in both statements a and b together are not sufficient to answer the question.\nIf the data in both statement a and b together are necessary to answer the question.\nOption E\n\n9. What is the code of ‘tommorrow’ in a certain code language?\na) ‘tommorrow burden summon’ is coded as ‘zq xr mz’ and ‘pistol tommorrow not’ is coded as ‘mz yq hn’\nb) ‘tommorrow fun sunk’ is coded as ‘xr mz zq’ and ‘ever sunk tommorrow’ is coded as ‘zq mz am’\nIf the data in statement a alone are sufficient to answer the question, while the data in statement b alone are not sufficient to answer the question.\nIf the data in statement b alone are sufficient to answer the question, while the data in statement a alone are not sufficient to answer the question.\nIf the data either in statement a alone or in statement b alone are sufficient to answer the question.\nIf the data even in both statements a and b together are not sufficient to answer the question.\nIf the data in both statement a and b together are necessary to answer the question.\nOption A\n\n10. Six students- Jc, Kc, Lc, Mc, Nc sitting in a row. Among them how many persons are facing to the north?\na)There are only two persons sit between Jc and Kc. Mc sits fourth to the left of Jc.\nb) Oc sits immediate left of Nc but not an immediate neighbour of Jc.\nIf the data in statement a alone are sufficient to answer the question, while the data in statement b alone are not sufficient to answer the question.\nIf the data in statement b alone are sufficient to answer the question, while the data in statement a alone are not sufficient to answer the question.\nIf the data either in statement a alone or in statement b alone are sufficient to answer the question.\nIf the data even in both statements a and b together are not sufficient to answer the question.\nIf the data in both statement a and b together are necessary to answer the question.\nOption D" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9194589,"math_prob":0.6651542,"size":9652,"snap":"2020-34-2020-40","text_gpt3_token_len":2113,"char_repetition_ratio":0.3553068,"word_repetition_ratio":0.63431543,"special_character_ratio":0.20171985,"punctuation_ratio":0.074384496,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9801454,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-01T08:37:54Z\",\"WARC-Record-ID\":\"<urn:uuid:58e50008-49c4-4e69-86e2-6d663f2eb804>\",\"Content-Length\":\"130399\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:dfa75b99-6d0e-44a1-907a-f0e14d1f7e7f>\",\"WARC-Concurrent-To\":\"<urn:uuid:aa5cab35-7050-410e-aa15-cbf6e1d00dc4>\",\"WARC-IP-Address\":\"104.31.91.195\",\"WARC-Target-URI\":\"https://aspirantszone.com/reasoning-questions-based-on-data-sufficiency/\",\"WARC-Payload-Digest\":\"sha1:TURJVZS7XXK4XPQSGLG2UUFPOIZB7WE3\",\"WARC-Block-Digest\":\"sha1:T4NA3TX5GLCXBL62GJTPELR7DGD732EB\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600402124756.81_warc_CC-MAIN-20201001062039-20201001092039-00174.warc.gz\"}"}
https://mathtuition88.com/2013/05/09/h2-maths-tuition-foot-of-perpendicular-from-point-to-line-part-ii/	[ "## H2 Maths Tuition: Foot of Perpendicular (from point to plane) (Part II)\n\nThis is a continuation from H2 Maths Tuition: Foot of Perpendicular (from point to line) (Part I).\n\n## Foot of Perpendicular (from point to plane)\n\nFrom point (B) to Plane (", null, "$p$)", null, "## Equation (I):\n\nWhere does F lie?\n\nF lies on the plane", null, "$p$.", null, "$\\overrightarrow{\\mathit{OF}}\\cdot \\mathbf{n}=d$\n\n## Equation (II):\n\nPerpendicular", null, "$\\overrightarrow{\\mathit{BF}}=k\\mathbf{n}$", null, "$\\overrightarrow{\\mathit{OF}}-\\overrightarrow{\\mathit{OB}}=k\\mathbf{n}$", null, "$\\overrightarrow{\\mathit{OF}}=k\\mathbf{n}+\\overrightarrow{OB}$\n\n## Final Step\n\nSubstitute Equation (II) into Equation (I) and solve for k.\n\n## Example\n\n[VJC 2010 P1Q8i]\n\nThe planes", null, "$\\Pi _{1}$ and", null, "$\\Pi _{2}$ have equations", null, "$\\mathbf{r\\cdot(i+j-k)}=6$ and", null, "$\\mathbf{r\\cdot(2i-4j+k)}=-12$ respectively. The point", null, "$A$ has position vector", null, "$\\mathbf{{9i-7j+5k}}$ .\n\n(i) Find the position vector of the foot of perpendicular from", null, "$A$ to", null, "$\\Pi _{2}$ .\n\n## Solution\n\nLet the foot of perpendicular be F.\n\n### Equation (I)", null, "$\\overrightarrow{\\mathit{OF}}\\cdot \\left(\\begin{matrix}2\\\\-4\\\\1\\end{matrix}\\right)=-12$\n\n### Equation (II)", null, "$\\overrightarrow{\\mathit{OF}}=k\\left(\\begin{matrix}2\\\\-4\\\\1\\end{matrix}\\right)+\\left(\\begin{matrix}9\\\\-7\\\\5\\end{matrix}\\right)=\\left(\\begin{matrix}2k+9\\\\-4k-7\\\\k+5\\end{matrix}\\right)$\n\nSubst. (II) into (I)", null, "$2(2k+9)-4(-4k-7)+(k+5)=-12$\n\nSolve for k,", null, "$k=-3$ .", null, "$\\overrightarrow{\\mathit{OF}}=\\left(\\begin{matrix}3\\\\5\\\\2\\end{matrix}\\right)$\n\n## H2 Maths Tuition\n\nIf you are looking for Maths Tuition, contact Mr Wu at:\n\nSMS: 98348087\n\nEmail: [email protected]", null, "" ]	[ null, "https://s0.wp.com/latex.php", null, "https://mathtuition88.files.wordpress.com/2013/05/vectorfoot-img2.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, "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://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://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://0.gravatar.com/avatar/f7133b9d7f511c9c866589ec1c51e0d4", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8229459,"math_prob":0.9998777,"size":801,"snap":"2019-43-2019-47","text_gpt3_token_len":215,"char_repetition_ratio":0.17565872,"word_repetition_ratio":0.029850746,"special_character_ratio":0.25842696,"punctuation_ratio":0.13071896,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000015,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],"im_url_duplicate_count":[null,null,null,8,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-17T21:20:06Z\",\"WARC-Record-ID\":\"<urn:uuid:57ee5609-d66c-49ba-ad28-4091dfd8f057>\",\"Content-Length\":\"98227\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ab58e009-83fe-4cdc-a8db-191191933036>\",\"WARC-Concurrent-To\":\"<urn:uuid:53febb11-f426-4832-b0d3-1f399eadb070>\",\"WARC-IP-Address\":\"192.0.78.25\",\"WARC-Target-URI\":\"https://mathtuition88.com/2013/05/09/h2-maths-tuition-foot-of-perpendicular-from-point-to-line-part-ii/\",\"WARC-Payload-Digest\":\"sha1:6WDEG6IRKUPSSSW3JTTWJH4U7VGO7QWG\",\"WARC-Block-Digest\":\"sha1:7NMAASEXXE4GG6JIPKCE5TDXCMCB4RHS\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496669276.41_warc_CC-MAIN-20191117192728-20191117220728-00159.warc.gz\"}"}
https://mmp.susu.ru/article/en/341	[ "Volume 8, no. 2Pages 127 - 132\n\n# Computational Experiment for One Mathematical Model of Ion-acoustic Waves\n\nA.A. Zamyshlyaeva, A.S. Muravyev\nIn the article the mathematical model of ion-acoustic waves in a plasma in an external magnetic field is studied. This model can be reduced to a Cauchy problem for a Sobolev type equation of the fourth order with polynomially (A,p)-bounded operator pencil. Therefore abstract results on solvability of the Cauchy problem for such equation can be used. In the article a theorem on the unique solvability of the Cauchy - Dirichlet problem is mentioned. Based on the theoretical results there was developed an algorithm for the numerical solution of the problem, using a modified Galerkin method. The algorithm is implemented in Maple. The article includes description of this algorithm. It is illustrated by model examples showing the work of the developed program.\nFull text\nKeywords\nmathematical model; ion-acoustic waves; Galerkin method.\nReferences\n1. Zamyshlyaeva A.A. Lineynye uravneniya sobolevskogo tipa vysokogo poryadka [Linear Sobolev Type Equations of High Order]. Chelyabinsk, Publ. Center of the South Ural State University, 2012.\n2. Sveshnikov A.G., Al'shin A.B., Korpusov M.O., Pletner Yu.D. Lineynye i nelineynye uravneniya sobolevskogo tipa [Linear and Non-Linear Equations of Sobolev Type]. Moscow, FIZMATLIT, 2007.\n3. Zamyshlyaeva A.A. [Stochastic Mathematical Model of Ion-Acoustic Waves in a Plasma]. Estestvennye i tekhnicheskie nauki [Natural and Technical Sciences], 2013, no. 4, pp. 284-292. (in Russian)\n4. Zamyshlyaeva A.A. [A Mathematical Model of Ion-Acoustic Waves in a Plasma in a Magnetic Field]. Materialy mezhdunarodnoy konferentsii 'Voronezhskaya zimnyaya matematicheskaya shkola S.G. Kreyna - 2014', 2014, pp. 142-144. (in Russian)" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.79277885,"math_prob":0.7548533,"size":1811,"snap":"2023-14-2023-23","text_gpt3_token_len":510,"char_repetition_ratio":0.10293304,"word_repetition_ratio":0.022988506,"special_character_ratio":0.2252899,"punctuation_ratio":0.1871345,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9891712,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-06-04T07:04:18Z\",\"WARC-Record-ID\":\"<urn:uuid:de3e4384-5805-4a5c-a2fe-9e0278e20d4f>\",\"Content-Length\":\"4198\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:bfce13fc-644d-479c-8763-4f233d4ccb49>\",\"WARC-Concurrent-To\":\"<urn:uuid:c38f9941-f18f-4ea1-be4e-d822a23a3025>\",\"WARC-IP-Address\":\"37.75.250.13\",\"WARC-Target-URI\":\"https://mmp.susu.ru/article/en/341\",\"WARC-Payload-Digest\":\"sha1:VQULHRNC2ENODHCDSL2AC67QN7ETLGLK\",\"WARC-Block-Digest\":\"sha1:IX3CDNSPFCQE4GV3EYX5CQ5ZOG24GT63\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-23/CC-MAIN-2023-23_segments_1685224649518.12_warc_CC-MAIN-20230604061300-20230604091300-00595.warc.gz\"}"}
https://www.codea.io/talk/discussion/3350/algorithm-for-coloring-a-line-based-on-its-length	[ "#### Howdy, Stranger!\n\nIt looks like you're new here. If you want to get involved, click one of these buttons!\n\n# Algorithm for coloring a line based on its length\n\nedited August 2013 Posts: 38\n\nI'm having trouble figuring out the math involved in changing a line's stroke color based on its variable length. The goal is that when the line gets shorter it will be gradually drawn red and when it gets longer it gradually draws blue. And there is a middle ground where in between shorter and longer it is just white. For example:\n\n``````-- line test\nsupportedOrientations(LANDSCAPE_ANY)\n\nfunction setup()\nparameter.integer( \"length\", 100, 600, 350 )\nparameter.watch( \"i\" )\n\nRED_MIN = 200\nRED_MAX = 100\nBLUE_MIN = 500\nBLUE_MAX = 600\nend\n\nfunction draw()\n\nif length <= RED_MIN then\n-- i = 255 - (255 * a value between 0 and 1, where 0 is f(length=RED_MIN) and 1 is f(length=RED_MAX)\nc = color(255, i, i, 255)\nelseif length >= BLUE_MIN then\n-- i = 255 - (255 * a value between 0 and 1, where 0 is f(length=BLUE_MIN) and 1 is f(length=BLUE_MAX)\nc = color(i, i, 255, 255)\nelse\nc = color(255,255,255,255)\nend\n\nbackground(40, 40, 50)\n\nstrokeWidth(5)\nstroke(c)\nline(70, HEIGHT/2, 70+length, HEIGHT/2)\nend\n\n``````\n\nThanks again guys!\n\nTagged:\n\n• Posts: 2,161\n``````function interp(l,m,n)\nreturn math.max(0,math.min(1,(l-m)/(n-m))\nend\n``````\n\ncall as `interp(length,RED_MIN,RED_MAX)` and you get out a number between 0 and 1 so that if `length` is in between then it is the correct proportion, but it is \"clamped\" to the end points in that it never goes below 0 or above 1.\n\n• Posts: 3,297\n\nCheck this:\n\n``````-- line test\nsupportedOrientations(LANDSCAPE_ANY)\n\nfunction setup()\nparameter.integer( \"length\", 100, 600, 350 )\nparameter.watch( \"i\" )\n\nRED_MIN = 200\nRED_MAX = 100\nBLUE_MIN = 500\nBLUE_MAX = 600\nend\n\nfunction draw()\nbackground(40, 40, 50)\nstrokeWidth(5)\nlocal r,g,b,i\nif length < RED_MAX then\nr,g,b = 255,0,0\nelseif length < RED_MIN then\ni = 255(length-RED_MAX)/(RED_MIN-RED_MAX)\nr,g,b = 255,i,i\nelseif length < BLUE_MIN then\nr,g,b = 255,255,255\nelseif length < BLUE_MAX then\ni = 255(length-BLUE_MAX)/(BLUE_MIN-BLUE_MAX)\nr,g,b = i,i,255\nelse\nr,g,b = 0,0,255\nend\nc = color(r,g,b,255)\nstroke(c)\nline(70, HEIGHT/2, 70+length, HEIGHT/2)\nend\n``````\n• Posts: 38\n\nExcellent solutions guys! Exactly what I was hoping for from this forum! Thanks so much!" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8522905,"math_prob":0.9967725,"size":1455,"snap":"2020-34-2020-40","text_gpt3_token_len":424,"char_repetition_ratio":0.10475534,"word_repetition_ratio":0.104417674,"special_character_ratio":0.31890035,"punctuation_ratio":0.12416107,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99668056,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-08-15T04:42:18Z\",\"WARC-Record-ID\":\"<urn:uuid:1b3006ac-cc2c-4b8e-9993-eaff57488386>\",\"Content-Length\":\"27092\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2e2bd3bf-6346-400e-821c-f638bdfe8942>\",\"WARC-Concurrent-To\":\"<urn:uuid:eae00d2a-22f2-448b-bb74-2241e220a765>\",\"WARC-IP-Address\":\"192.184.94.102\",\"WARC-Target-URI\":\"https://www.codea.io/talk/discussion/3350/algorithm-for-coloring-a-line-based-on-its-length\",\"WARC-Payload-Digest\":\"sha1:5I6OG3BCG2YTFJMTIIH3QEOWELPUHIP7\",\"WARC-Block-Digest\":\"sha1:VBUGFSMV5VI7RTP7RXRANMOMSAFSNSR3\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-34/CC-MAIN-2020-34_segments_1596439740679.96_warc_CC-MAIN-20200815035250-20200815065250-00494.warc.gz\"}"}
https://homework.cpm.org/category/CCI_CT/textbook/pc3/chapter/5/lesson/5.2.5/problem/5-124	[ "", null, "", null, "### Home > PC3 > Chapter 5 > Lesson 5.2.5 > Problem5-124\n\n5-124.\n\nSylvie needs to solve $\\cos(x)=−0.3$ on a math test. She knows that $\\cos^{−1}(x)$ is the inverse function for cosine, so she applies this function to both sides of the equation. This gives her $x=\\cos^{-1}(−0.3)\\approx1.875$. But when her test is returned, she learns she earned only partial credit on the problem.\n\n1. Why did Sylvie only get partial credit?\n\nThere are multiple solutions." ]	[ null, "https://homework.cpm.org/dist/7d633b3a30200de4995665c02bdda1b8.png", null, 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SMAK/8hRgLL8S6SwvMcWDQzvascJkuopwm+szYqyA2SH3kRum89v6EE33NrjKLdwLy0Ffh2G4qUg32uVon3YtWxXrWXUEd8FCqftTH765n3cuqEC7zXUczvGyW8W5TzFrwvFmda1k/5wn0wEqelQJ7qWX/XlHC9Jr6z9hLrr0LRKws9tPhJS4FKutaTFjbUcSQcIhO48vcP7F9sZHWJhA58zshvpW/D9SoNNFAIMkRXQ27yHInWkL+ADa2LqTyGCXv+6ciz9GLs7aWfxLT3s4GIAxq8x5n2oALpQCB38X7PeXlw5bNM/2mmfdY59jz/38HjPr7BfFwVk4ejeXxG4NhHeN2XJJr/AOWJlfWOK/IO7D0v8fbv4z0Xnvlv3vNAfsf07+exh6ic+cR5Ae9jPVbYvijwbhDvMZv32jMmz0fy/FsK1P+TmZ9rCjz7VF7nm72ou7vElAfK6RGWq0/4tzL9PwJ1Au/04zH3QnDrLyRaCvkVvtvZRd7tRL7/13gOzv2l9OwGRPndXCBfuO8nipSFfbffKpBmBtNMLXKtk5gOsUTDlKYU/WmhZ2MIvbNCefqQ00BmaG3tE9Nozab2HCLoNY5G7Fp3owNp0T0wpgzFoFLYjB6Mnfn/VeYRDc6lEi0aM9GxEDZhwybcZxeoBfHbYMVT2ABZLX8bCqam/WlMPr4i+eF7Q4rkGaMbtuS76QqUWcJpxOud/HY69cfm91iS6IWedY38xgUsDuXxVd7+/VlvhrNsXmR5oSG+nedMi7EyJ/P4ZCoSqx2PyFjHE5Ry6ppb31c639P2tIirPCX4VxKtBgjMo/W1PZ/9Uzy2wrnODvRWYA6HCQEr3JbDigIWHIJGtyWxX0GPgA+U89Ysq3JRRyXGWrJZx1BA3vYyciiVsLWO8rgd03YG6vBRVODvcu6D7+MevosMFTYowntQcPw7Xt6+4xDnElrmyOsJLG8onU85dXIrJ1+2TXHzdQzzNTNG0Z1MRWwyvYAhq34sy+Ub/BbfiCnT8/jemjYy40PxHrTQQ+iqoFtoNK2PI9kQ7BtDtLDkf+6QiA806D8q4X7PsdFMDED5X83GaIFEa7uPpxxPUsAwv9O9cgZ+xgZ/R/4iNuA2ktN0yc++57pZz2BjEfIQuKMFisUjWCI7xcmDK+PZ+LrXQgO8k5Nmd8fC/j6f3ffQxE3qkw4QKkj8Jv7+kff6MJXDHzLNZVSQfNgpi4VKneuheJjPY8t5MvfPoQJkn/dwrx52eN/Dt0jYq1incc4H+X6XkbAv9JTmDsfrcEGJ5eBiJz4b0OwoE6FvN84zVgz2/UKp2I1ltAOf78tU9A/y6rDN77leHd6dym09CXGYo1TdSDKczfLYieV3GdOc79WhfRwyv5RpbZ14gG3M9Z4HzObrvJh81Xn58pXJcY6XZq8i3w6I+rSYNJ93PAgdou52xQAQ+kBgKt1icV6GIbRKFhS5DhqDtwcg/2igPsftMyVa/jXDjxgW5ZU8dnbAbbmazzWPv3B7TqIS00wLxMeOtH58wHrbtBf5X+TkwZW5bMh90niNx+fTMsJ8BLMc5aAv+CS9Bkv4PHNYlktIpo+wrp8ZOHcij83l/0nOsTbut+X8hkN+9nlej7G0xCGkE7l9Cb0IHSyTu0ggQqKPc69+m5ZoOTiGHoV5zO+kfqzLackHvM7n9g2S78I4WnpOKLXUq8OoEyfxnYEcd2G63aiItbKePM93i/7w7xm5m+lOdK5tn/XPVBiX8ZyX6alq4/UPCTwL7v8vL1+TuB+KcqhLwN77Nf6eUEKZTQ54C1EPz1JaUgw0oW/oRUlg2V5cJE2t89HH4T5q300DUPZoHBpp3TweOD6dpPftwHtKxlhLL3M7zl39TU8Bgqvwq45VWA7K6a6B5VoT2P9bx5rsSx3awfG2LA0cn0Kiv9Xb30yLKMuyWUhLb8uY+6Sc56ktMW9Qlmx/+gOB4w+R3DeR9fvdq0g8C3jfH5dxT6Q71lEGXqVC8MF+qstx5fG04wWqLaH+LCVxAkMdi1eoWL0WOOde/m7r7NveO+biLXrAzohRxEL5Wu7UK1/p2oyKwTpes4WK+ogSPJH+PBoHSnwMgULRL4Qeck03SnhseiXRzgbxMDZSxQjIRr+jEX8wcBxW0jkFnqm/Yee1XynhaG7sn0Fr3Y+E7o7xSNh+8IXesQdo2XzMs0pgOW1HC/8fZea/EjETbzl5b+jDdWwjG+dpQUAUgsf+GmhA4SlBlwC6CeBih2v1iAq+5yaSWafk+9r9et1CIqnzvrMsLbZVtCi/U+I94fL9AOsBvAD3U2Hqr9EdWQlH2u/rELVfx0PR+weQjLO08oHhzjUk5juxdci2aU1F6sPdVJifCRwL5etAyceCvOwd+yy/ZVjyCGJDtwCi8A8t0Hb+kt/w1x3FxSrcwEyJjw1SKCpiZbkNUKjRapJ8UE9fAGviSoeQYXku4wf+ai8UljQVgNmelfgTiSJJB7rsu6T8/stNaNW6VuC32OgsCxAXgv4w8c+1THc3G3jr3kMU9GllNN7AFWwwk16D9b2YhlJilCrrceiLhZ4sUDcLwbpGf+80pCdy/3SpzOp5SckPLQzFBXQ7+xMBJe0JiVzXeEfnUvF4usg9j3eIK81fBGIhIvxyqVwAq1uXMT/FWueZP8P8WgLzyxJW7OZMm6FX5EQqP4gHedF7t+uKKJZJpwxD9WFXfjdZJ13I6j/Cy9dYenf8fPllfadThw5mHZoRk2d8n2OoKEyi9wWWOUZ9wN3/fxLFZWj/uaLfCT2k9Q7nR+AT+v5s4NNO5QSp3sCPI4TFrNCVBAgGQTBnOhbs1AEue7dhKddDcDLFByL7vyw9o5mHsnFBfy2Gtu1GBeyjtDhmUukpB3EL8/y0DEJ3yyJbobIsFWioD2KjbUdVII5hCZ9tl148R2/ec7H3D+/Xj0jGu7Px372AEjhC8gFwv+bvoxL1Ce9A6/3+CtdlfP+PxRybwW/Px3HSc8hZG7/9s5xyK/ZuE166uHNQhhO8c690lA6LYwKeDHjIEIB7tqeYjGd5tku+L38W0+9PBXtujBJyNQkdVvr/UuGCAYKA1/kyMF5DxSAk9BcC+6C9fs2z8rDvssBHBFxVwPqp7qdnRV6OYkOOhV2WD3DZ9+WDfZtKSZKNACwjuPxulsi1HipTuG2voyJzjuOt+G82pMky84358Z+UvFswUaB+FPKgDFRZHk6yhJvddjesIrmfxkb9mQrlLdGH57CW4mkkzY+TBBbFXOMztEThfXrEsW7RdQOX/cR+IPRuWq7dfKcZEtmdjlLhA11hiB9AVx2i4D9EMjy1l+82UeQcxGu8QuPCkm1XgXwlWc7IF0ZOTAmktYGHs0jCwJtMj2NHSj641QW6l+5gvUM3GQJz0RXWQkLfSqlJsaEI/a8kR/+jQXAV+o7gEkRf4BdjyBxE9KCEg6T6E8v4cR0vPYOjBgJtzsddI4XXhk94FsgvJN//Xw5gZaCf7mj+XyDR+OjeAIQxu49lYPu+OyTvUrWKRZzClw4oA+scS7FURcK6SuGh2JPfQkbyoyKg/F1c5L2Ugg5aZPUSjhOwM9+JxA/Vs+WNbo6LJBri9ouYdLYb4SXvuawCcBjLaWUF6/JKWqpryzgHwai3OSQICxf90RjG+ZyTrt3xMoUwxClnW286vPplFVeLmwsQ+h+db+JNtmeH0ZvldtHVOJb8K3z+JOuntcqhPP1Qes7SZ2daRJ5ukXyA73S2Ux9QalL0Br2xkBBA9ZeYY0fzY/lpDJkDP6FLKjUAz3ujQ2YDjVX8qEfHNFZoQOACnik9I2t7a9kulfUnl7mOjXBvrldXgTKw0elLnEbYTuoyJuacTZ3ycz0WwLiYc6ZQibya/3eSfDQxJtV5lMdhrf+A+xE1vW8FnnEFSQllHJo2eRRJqU16Dvfzgbw9zXNs95Gr6CHP+3H7C95zXeeU38H94G0q1zho8Ej0CSo2/ph7G/W+eUybMc6rD1lHWdk65t7betcOKQhW6XhM8rP8uXBHDZxHb8iD/D2f+6Gc7FqgDOyshlYpvVYpSbGhCd0O8elNANzj1EIH0ipevJGU/Rx6K+okP3TMfS/Q2g8gma8ONKC9xfW0gEAMN/XhOi1lpE1Lz0AsDEeyE7Xc5+x/mL8TAoQKIjuJ2+5qfU84SpAfXTyWFu2+TkNvXaVv0Br7jSP4/6pDin3FUsfiDAUens73PUcKj2e3jf43aFmGukg+T6JEEOTtged6vsBztffxOftSJ9P0PgBwU3/CMyDWkZxPCNSHL3h1QBzP0XHSc6w3vAC7sx17rEi+YO3b2QWP8IwU6+GZS0+DW9b4P9/zBMV5by6nV+g6Cfe3KxQlo7f91a+wgt9awCoKWfbHSt9dmO8VrGUjdj01fFikGGJUS9I6hA3Kd6Uy0dYWi9lgurOR9QYns4FLBOoUvAovelb1+ZJ3PW5FTwkaW7g1f+aR80zWL/R7wmWJvkaMrf86FYGF9LZYPMWG9Bg2pldTYRlH5RPW3WtsNF1X6eUSng4XZT+Lv2OkbxMPZfme9yPBQIGzUd/HOXkBcZQy2uFJWuoXBAh1IrevlfA0txNIdgfwHSxwjkHhCc15kKLy9Eg/fw/38N1/gs/2WYcwf05FBvVkRyp9GP+Ncd8Y5vaW5GeNBG6gVwZu9XtZHkizN89JUZl9roR8WSt9Ar/FQ6lkH+5Y578LnIeI/RlUsnBea8z1URf+UKaCrFBUlNCFHzg+kMvYKMW5YGHJ3yzR0JvVXgPUHEhf7rKmdpUjH0PLuEbcilH93c8PMkFUMmaz+hLFAtbk2bJ+P7V1B5Y6ZrsupkxDQ4CaS3hmt6xPLZBuCQndXmszkqePZ+ideMuziibz3EMCxPQyFZ63A+ckaeH5i6y8SOsObtmjqBRkJD9TnY+H+Qyb0AK8xiub5hiLtNqpey4xoovqFF7ncIcMrKcDBHaHsy/pvOOQJY5vDv26OzvvAwqDndp2ZsxzQcnBzHbbsq5d6NxnP8m7631MjyF06wIfVoa3z9az2oCVPo1K7aFU6OxznMO6jzI8V9aPTH+ZyqXr3XiLRHozy+hG716/ooLgoqlIvv7A+ngg68WmrE9xAYb30usxjnVyRoF7rIkp16GiY9EVG4jQhZYSgt8QbIbpRnciQWXo9kODfZ/0nOjEupum8eNIO/mZ1wt33Q9oSaWdRnCJlD4U6kESjjseGNd4dgO8g8tpBdg5vrtpOaCBn+OlvZ3l83AZStc0elSKWZFX0QouZLV08nqjC3gNkpJ3f2Jq3qmyflBQgiSGYw9IeEz0clpoIL6DmS8ohugT/rX07IKwjeJRJDpEem9BpegR75x2PkMhFze8J6eTIBd75DGNhNEZ4/24hPfw83gTlbOJJJkEy+D2wPtZRpJHw7405tuBBXi8971cwW8t7n2jfqPvfU/nPFiIr0p+oZQQad8Xc715VC7WluF5g7W8jazvIreAgnUWyTLlKaCnsqxQJ7Zk+T7EfS0xyuIEltFeJMc3SMx/jsnXdgXydSYV03rWtWl8f3HBhVA4v0KPwhpHMYIy9XiRMprH72ZlActeoehpcWWz5Q3/3WrX0wZ7kUmiKjjC62w25NdrtVIoFJXG/KemayEo+tVCH3x0noiN/XlaCg87UigUCoVi47HQFQqFQqFQbHzQgAuFQqFQKJTQFQqFQqFQKKErFAqFQqGoCP4jwADQNvw20jA5ogAAAABJRU5ErkJggg==", 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https://calcpercentage.com/what-is-449-percent-of-90	[ "# PercentageCalculator, What is 449% of 90?\n\n## What is 449 percent of 90? 449% of 90 is equal to 404.1\n\n%\n\n### How to Calculate 449 Percent of 90?\n\n• F\n\nFormula\n\n(449 ÷ 100) x 90 = 404.1\n\n• 1\n\nConvert percent to decimal\n\n449% to decimal is 449 ÷ 100 = 4.49\n\n• 2\n\nMultiply the decimal number with the second number\n\n4.49 x 90 = 404.1\n\n#### Example\n\nFor example, John needs 449 percent of the shares to be in power. The number of shares is 90. How many shares does John need to buy? What is 449 percent of 90? 449 percent to decimal is equal 449 / 100 = 4.49 4.49 x 90 = 404.1 so John need to buy 404.1 shares" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.89326805,"math_prob":0.99619347,"size":424,"snap":"2022-27-2022-33","text_gpt3_token_len":142,"char_repetition_ratio":0.15714286,"word_repetition_ratio":0.023255814,"special_character_ratio":0.4268868,"punctuation_ratio":0.1262136,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99951386,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-06-28T14:48:17Z\",\"WARC-Record-ID\":\"<urn:uuid:762e3dba-957b-4ec0-b52a-be7d5767a639>\",\"Content-Length\":\"11927\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:fe909998-5247-4026-8970-d079ee483290>\",\"WARC-Concurrent-To\":\"<urn:uuid:e0db7e92-2a98-43a4-a61d-ecfc10ce7f36>\",\"WARC-IP-Address\":\"76.76.21.9\",\"WARC-Target-URI\":\"https://calcpercentage.com/what-is-449-percent-of-90\",\"WARC-Payload-Digest\":\"sha1:UXZKP4F3XFCCSZBICV6PZQGHJY3PHCEE\",\"WARC-Block-Digest\":\"sha1:GCD76CVAG2YMU3XFOI7UJ5HK6TOQW2HK\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656103556871.29_warc_CC-MAIN-20220628142305-20220628172305-00058.warc.gz\"}"}
http://euler.stephan-brumme.com/279/	[ "<< problem 278 - Linear Combinations of Semiprimes", null, "Ant and seeds - problem 280 >>\n\n# Problem 279: Triangles with integral sides and an integral angle\n\nHow many triangles are there with integral sides, at least one integral angle (measured in degrees), and a perimeter that does not exceed 10^8 ?\n\n# My Algorithm\n\nNiven's theorem (see en.wikipedia.org/wiki/Niven's_theorem) states that for rational angles sin(alpha) the only rational values are \\{ 0, +1/2, -1/2, +1, -1 \\}.\nSince sin(90 + alpha) = cos(alpha) this theorem can be applied to cos(alpha), too.\n\nThus only 3 angles can be part of a valid triangle:\n(1) cos( 60) = +1/2\n(2) cos( 90) = 0\n(3) cos(120) = -1/2\n\nEach triangle satisfies (see en.wikipedia.org/wiki/Law_of_cosines):\n(4) c^2 = a^2 + b^2 - 2ab cos(gamma)\n\nIf gamma = 60 then (4) can be simplified to\n(5) c^2 = a^2 + b^2 - ab\n\nIf gamma = 90 then (4) can be simplified to\n(6) c^2 = a^2 + b^2\n\nIf gamma = 120 then (4) can be simplified to\n(7) c^2 = a^2 + b^2 + ab.\n\nThese equations have their own Wikipedia pages, too: en.wikipedia.org/wiki/Eisenstein_triple and en.wikipedia.org/wiki/Pythagorean_triple .\nBut more interesting is en.wikipedia.org/wiki/Integer_triangle which gives away nice formulas to generate all such triangles.\nThe formulas need two variables m and n where 0 < n < m.\n\nThe case cos(60) boils down to:\na = m^2 - mn + n^2\nb = 2mn - n^2\nc = m^2 - n^2\n\nI solved cos(90) in problem 75 (see en.wikipedia.org/wiki/Pythagorean_triple):\na = m^2 - n^2\nb = 2mn\nc = m^2 + n^2\n\nAnd finally cos(120):\na = m^2 + mn + n^2\nb = 2mn + n^2\nc = m^2 - n^2\n\nThe triples (a,b,c) are not necessarily basic triples - but their greatest common divisor is always either 1 or 3.\nNow I have all the tools to generate all basic triples. Those basic triples and all their multiples are solutions.\nThere are 10^8 / perimeter multiples of each basic triples which satisfy the condition that the perimeter must not exceed 10^8.\n\nThe functions search60(), search90() and search120() implement each set of equations for 60, 90 and 120.\n\n## Note\n\nFortunately I didn't find a triangle with angles 60 + 90 + 30 and integer sides.\nIf there were any, I would have needed to avoid double-counting (as they would be counted as 60 and 90 separately).\n\nSince today is my lucky day, I added a few OpenMP pragmas which reduce the execution time from 4 to less than 1 second on my computer (enable #define PARALLEL).\n\n# Interactive test\n\nYou can submit your own input to my program and it will be instantly processed at my server:\n\nInput data (separated by spaces or newlines):\n\nThis is equivalent to\necho 100 \| ./279\n\nOutput:\n\nNote: the original problem's input 100000000 cannot be entered\nbecause just copying results is a soft skill reserved for idiots.\n\n(this interactive test is still under development, computations will be aborted after one second)\n\n# My code\n\n… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too. Or just jump to my GitHub repository.\n\n #include <iostream>\n#include <cmath>\n\n#define PARALLEL\nconst auto numCores = 8; // 4 = four cores, 8 = eight cores, etc.\n\n// ---------- code copied from my toolbox ----------\n\n// find greatest common divisor\ntemplate <typename T>\nT gcd(T a, T b)\n{\nwhile (a != 0)\n{\nT c = a;\na = b % a;\nb = c;\n}\nreturn b;\n}\n\n// ---------- problem-specific code ----------\n\n// 90 degrees (based on code from problem 75)\nunsigned int search90(unsigned int limit)\n{\nunsigned int result = 0;\n\nunsigned int last = sqrt(limit / 2);\n#ifdef PARALLEL\n#pragma omp parallel for num_threads(numCores) reduction(+:result) schedule(dynamic)\n#endif\nfor (unsigned int m = 2; m < last; m++)\nfor (unsigned int n = (m & 1) + 1; n < m; n += 2) // m + n must be odd\n{\n// only valid m and n\nif (gcd(m, n) != 1)\ncontinue;\n\n// compute basic triplet\nauto a = mm - nn;\nauto b = 2mn;\nauto c = mm + nn;\n\n// too large ?\nauto sum = a + b + c;\nif (sum > limit)\nbreak;\n\n// all its multiples are valid, too\nauto numMultiples = limit / sum;\nresult += numMultiples;\n}\n\nreturn result;\n}\n\n// 60 degrees\nunsigned int search60(unsigned int limit)\n{\nunsigned int last = sqrt(limit3/2);\nunsigned int result = 0;\n#ifdef PARALLEL\n#pragma omp parallel for num_threads(numCores) reduction(+:result) schedule(dynamic)\n#endif\nfor (unsigned int m = 2; m < last; m++)\nfor (unsigned int n = 1; 2n <= m; n++)\n{\n// only valid m and n\nif (gcd(m, n) != 1)\ncontinue;\n\n// compute next triplet\nauto a = mm - mn + nn;\nauto b = 2mn - nn;\nauto c = mm - nn;\n\nauto sum = a + b + c;\n// divide by their greatest common divisor to get the basic triplet\n// gcd(a,b,c) is either 1 or 3\nif (a % 3 == 0 && b % 3 == 0 && c % 3 == 0)\nsum /= 3;\nif (sum > 3limit)\nbreak;\n\nif (sum <= limit)\n{\n// all its multiples are valid, too\nauto numMultiples = limit / sum;\nresult += numMultiples;\n}\n}\n\nreturn result;\n}\n\n// 120 degrees\nunsigned int search120(unsigned int limit)\n{\nunsigned int result = 0;\n\nunsigned int last = sqrt(limit3/2);\n#ifdef PARALLEL\n#pragma omp parallel for num_threads(numCores) reduction(+:result) schedule(dynamic)\n#endif\nfor (unsigned int m = 2; m < last; m++)\nfor (unsigned int n = 1; 2n <= m; n++)\n{\n// only valid m and n\nif (gcd(m, n) != 1)\ncontinue;\n\n// compute next triplet\nauto a = mm + mn + nn;\nauto b = 2mn + nn;\nauto c = mm - nn;\n\n// skip mirrored triangles\nif (b > c)\nbreak;\n// note: a is always the longest side\n\nauto sum = a + b + c;\n// divide by their greatest common divisor to get the basic triplet\n// gcd(a,b,c) is either 1 or 3\nif (a % 3 == 0 && b % 3 == 0 && c % 3 == 0)\nsum /= 3;\nif (sum > 3limit)\nbreak;\n\n// all its multiples are valid, too\nauto numMultiples = limit / sum;\nresult += numMultiples;\n}\n\nreturn result;\n}\n\nint main()\n{\nunsigned int limit = 100000000;\nstd::cin >> limit;\n\nauto num60 = search60 (limit);\nauto num90 = search90 (limit);\nauto num120 = search120(limit);\n\nauto result = num60 + num90 + num120;\nstd::cout << result << std::endl;\nreturn 0;\n}\n\n\nThis solution contains 25 empty lines, 22 comments and 12 preprocessor commands.\n\n# Benchmark\n\nThe correct solution to the original Project Euler problem was found in 3.9 seconds.\nThe code can be accelerated with OpenMP but the timings refer to the single-threaded version on an Intel® Core™ i7-2600K CPU @ 3.40GHz.\n(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=gnu++11 -DORIGINAL)\n\nSee here for a comparison of all solutions.\n\nNote: interactive tests run on a weaker (=slower) computer. Some interactive tests are compiled without -DORIGINAL.\n\n# Changelog\n\nDecember 27, 2017 submitted solution\n\n# Difficulty\n\nProject Euler ranks this problem at 60% (out of 100%).\n\n# Heatmap\n\nPlease click on a problem's number to open my solution to that problem:\n\n green solutions solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too yellow solutions score less than 100% at Hackerrank (but still solve the original problem easily) gray problems are already solved but I haven't published my solution yet blue solutions are relevant for Project Euler only: there wasn't a Hackerrank version of it (at the time I solved it) or it differed too much orange problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte red problems are not solved yet but I wrote a simulation to approximate the result or verified at least the given example - usually I sketched a few ideas, too black problems are solved but access to the solution is blocked for a few days until the next problem is published [new] the flashing problem is the one I solved most recently\n\nI stopped working on Project Euler problems around the time they released 617.\n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100\n 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200\n 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300\n 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400\n 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500\n 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600\n 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708\nThe 310 solved problems (that's level 12) had an average difficulty of 32.6% at Project Euler and\nI scored 13526 points (out of 15700 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.\n\nMy username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.\n\nLook at my progress and performance pages to get more details.\n\n << problem 278 - Linear Combinations of Semiprimes", null, "Ant and seeds - problem 280 >>\nmore about me can be found on my homepage, especially in my coding blog.\nsome names mentioned on this site may be trademarks of their respective owners.\nthanks to the KaTeX team for their great typesetting library !" ]	[ null, "https://projecteuler.net/profile/stephanbrumme.png", null, "https://projecteuler.net/profile/stephanbrumme.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6785877,"math_prob":0.9711925,"size":12437,"snap":"2020-34-2020-40","text_gpt3_token_len":3929,"char_repetition_ratio":0.09233492,"word_repetition_ratio":0.15391229,"special_character_ratio":0.5036584,"punctuation_ratio":0.07813765,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9728593,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-08-14T07:38:29Z\",\"WARC-Record-ID\":\"<urn:uuid:21937c7d-4cd0-4f3e-baed-718b50278369>\",\"Content-Length\":\"98500\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f765c85a-a127-4689-9a65-f5d17644e3bb>\",\"WARC-Concurrent-To\":\"<urn:uuid:7213f0b7-3a3a-4a5c-b9b8-6b61e0577733>\",\"WARC-IP-Address\":\"217.160.0.76\",\"WARC-Target-URI\":\"http://euler.stephan-brumme.com/279/\",\"WARC-Payload-Digest\":\"sha1:XFTJHKNIO5GYXRJUS3MA3BZHZIRVTJEL\",\"WARC-Block-Digest\":\"sha1:MPEG5S4JJEPPHQE2FPFBTTPFCQKVPRGI\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-34/CC-MAIN-2020-34_segments_1596439739182.35_warc_CC-MAIN-20200814070558-20200814100558-00069.warc.gz\"}"}
https://codeforces.com/topic/22406/diff/en3/en4	[ "Counting Divisors of a Number in $O(N^{\\frac{1}{3}})$ [tutorial]\nDifference between en3 and en4, changed 168 character(s)\n\nTopic : Counting Divisors of a Number↵\n\nPre Requisites : Basic Maths , $O(\\sqrt[]N)$ Factorisation , Primality testing↵\n\nMotivation Problem :↵\n\nThere are $T$ test cases. Each test case contains a number $N$. For each test case , output the number of factors of $N$.↵\n\n$1 <= T <= 10$↵\n\n$1 <= N <= 10^{18}$↵\n\n_Note_ : $1$ and $N$ are also treated as factors of the number $N$.↵\n\nSolution :↵\n\nNaive Solution — _$O(\\sqrt[]N)$ Factorisation_↵\n\nThe most naive solution here is to do the $O(\\sqrt[]N)$ factorisation. But as we can see, it will surely receive the Time Limit Exceeded verdict. You may try to compute the prime numbers till required range and loop over them , but that too exceeds the usual limit of $10^8$ operations. Some optimisations and heuristics may allow you to squeeze through your solution, but usually at the cost of a few TLE's.↵\n\nSupposing you fit in the above description and cannot think of anything better than the naive solution, I would like to present to you a very simple algorithm which helps you count the number of divisors in $O(N^{\\frac{1}{3}})$, which would be useful for this kind of questions.\nThis algorithm only gives us the count of factors, and not the factors itself.\n\nFirstly, let's do a quick recap of how we obtain the $O(\\sqrt[]N)$ algorithm for any number $N$ :↵\n\nWe write $N$ as product of two numbers $P$ and $Q$.↵\n\n$PQ = N , where \\hspace{1mm} P <= Q$↵\n\n$Maximum \\hspace{1mm} value \\hspace{1mm} of \\hspace{1mm} P = \\sqrt[]{N}$↵\n\nLooping over all values of $P$ gives us the count of factors of $N$.↵\n\nBefore you move forward to the next section, it would be useful if you try to come up with an algorithm that finds the\ncount of factors in $O(N^{\\frac{1}{3}})$. As a hint, the way of thinking is same as that of getting to the $O(\\sqrt[]N)$ algorithm.↵\n<br/><br/>↵\n\nCounting factors in $\\mathbf{O(N^{\\frac{1}{3}})}$ Factorisation\n-------------------------------------------↵\n\nLet's start off with some maths to reduce our $O(\\sqrt[]N)$ factorisation to $O(N^{\\frac{1}{3}})$\nfor counting factors :↵\n\nWe write $N$ as product of three numbers $P$, $Q$ and $R$.↵\n\n$PQR = N , where \\hspace{1mm} P <= Q <= R$↵\n\n$Maximum \\hspace{1mm} value \\hspace{1mm} of \\hspace{1mm} P = N^{\\frac{1}{3}}$↵\n\nWe can loop over all prime numbers in range $[2,N^{\\frac{1}{3}}]$ and try to reduce $N$ to it's prime factorisation, which would help us count the number of factors of $N$.↵\n\nTo know how to calculate divisors using prime factorisation, [click here](http://mathschallenge.net/library/number/number_of_divisors).↵\n\nWe will split our number $N$ into two numbers $X$ and $Y$ such that $XY=N$. Further, $X$ contains only prime factors in range $[2,N^{\\frac{1}{3}}]$ and $Y$ deals with higher prime factors ($> N^{\\frac{1}{3}}$). Thus, gcd($X$ , $Y$) = 1. Let the count of divisors of a number $N$ be denoted by the function $F(N)$. It is easy to prove that this function is multiplicative in nature, i.e., $F(mn) = F(m)F(n)$, if gcd($M$,$N$) = 1. So, if we can find $F(X)$ and $F(Y)$, we can also find $F(XY)$ or $F(N)$ which is the required quantity.↵\n\nFor finding $F(X)$, we use the naive trial division to prime factorise $X$ and calculate the number of factors. Once this is done, we have $Y=N/X$ remaining to be factorised. This may look tough, but we can see that there are only three cases which will cover all possibilities of $Y$ :↵\n\n1. $\\mathbf{Y}$ is a prime number* : $F(Y)$ = $2$.↵\n2. $\\mathbf{Y}$ is square of a prime number : $F(Y)$ = $3$.↵\n3. $\\mathbf{Y}$ is product of two distinct prime numbers : $F(Y)$ = $4$.↵\n\nWe have only these three cases since there can be at max two prime factors of $Y$. If it would have had more than two prime factors, one of them would surely have been $<= N^{\\frac{1}{3}}$, and hence it would be included in $X$ and not in $Y$.↵\n\nSo once we are done with finding $F(X)$ and $F(Y)$, we are also done with finding $F(XY)$ or $F(N)$.↵\n\nPseudo Code* :↵\n\n~~~~~↵\nN = input()↵\nprimes = array containing primes till 10^6↵\nans = 1↵\nfor all p in primes :↵\nif ppp > N:↵\nbreak↵\ncount = 1↵\nwhile N divisible by p:↵\nN = N/p↵\ncount = count + 1↵\nans = ans * count↵\nif N is prime:↵\nans = ans * 2↵\nelse if N is square of a prime:↵\nans = ans * 3↵\nelse if N != 1:↵\nans = ans * 4↵\n~~~~~↵\n\nChecking for primality can be done quickly using Miller Rabin. Thus, the time complexity is $O(N^{\\frac{1}{3}})$ for every test case and hence we can solve our problem efficiently.↵\n\nAt this point, you may think that in a similar way, we can reduce this to $O(N^{\\frac{1}{4}})$ by handling some cases. I have not thought much on it, but the number of cases to be handled are high since after trial division $N$ could be factorised into one, two or three primes. This is easy enough to code in contest environment, which is our prime objective.↵\n\nThis trick is not quite commonly known and people tend to make bugs in handling the three cases. A problem in regionals which uses this trick directly :↵\n\n[Problem F \| Codeforces Gym](http://codeforces.com/gym/100753)↵\n\nYou can try also this technique on problems requiring $\\sqrt[]{N}$ factorisation for practice purposes.↵\n\nHope you found this useful! Please suggest more problems to be added as well as any edits, if required.↵\n\nHappy Coding!\n\n#### History\n\nRevisions", null, "Rev. Lang. By When Δ Comment\nen5", null, "himanshujaju 2015-12-26 17:09:26 20\nen4", null, "himanshujaju 2015-12-26 17:01:34 168\nen3", null, "himanshujaju 2015-12-26 15:12:53 25 title\nen2", null, "himanshujaju 2015-12-26 15:08:47 60 first iteration (published)\nen1", null, "himanshujaju 2015-12-26 15:05:51 5527 Initial revision (saved to drafts)" ]	[ null, "https://sta.codeforces.com/s/63915/images/icons/control.png", null, "https://sta.codeforces.com/s/63915/images/flags-16/en.png", null, "https://sta.codeforces.com/s/63915/images/flags-16/en.png", null, "https://sta.codeforces.com/s/63915/images/flags-16/en.png", null, "https://sta.codeforces.com/s/63915/images/flags-16/en.png", null, "https://sta.codeforces.com/s/63915/images/flags-16/en.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.87707716,"math_prob":0.993127,"size":10756,"snap":"2020-10-2020-16","text_gpt3_token_len":3608,"char_repetition_ratio":0.12872024,"word_repetition_ratio":0.9905263,"special_character_ratio":0.31907773,"punctuation_ratio":0.10504396,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999697,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12],"im_url_duplicate_count":[null,null,null,5,null,5,null,5,null,5,null,5,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-04-05T13:50:35Z\",\"WARC-Record-ID\":\"<urn:uuid:eeae9ff6-6384-4e4e-88a1-6f2fef5ea7df>\",\"Content-Length\":\"85475\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:136619ff-523d-4a5f-94cc-b79b2f43c15b>\",\"WARC-Concurrent-To\":\"<urn:uuid:ddbedb64-7b64-4702-a4ae-23dfef884b2e>\",\"WARC-IP-Address\":\"81.27.240.126\",\"WARC-Target-URI\":\"https://codeforces.com/topic/22406/diff/en3/en4\",\"WARC-Payload-Digest\":\"sha1:QMJSXL7Y2ODZ2D344ENRRIKBGXJTUV54\",\"WARC-Block-Digest\":\"sha1:343LEHPXOGMAX2XAQ7IVIYBX3LAHQQ2E\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-16/CC-MAIN-2020-16_segments_1585371604800.52_warc_CC-MAIN-20200405115129-20200405145629-00360.warc.gz\"}"}
https://jupyter.brynmawr.edu/services/public/dblank/CS110%20Intro%20to%20Computing/2015-Spring/Lectures/Visualization2.ipynb	[ "# 1. Visualizations, Part 2¶\n\nIn this notebook, we continue to look at methods of displaying information visually. See Part 1\n\n## 1.1 Preliminaries¶\n\nTo explore these ideas, we need to introduce a new construct in Processing: the array.\n\n### 1.1.1 Arrays¶\n\nArrays are contiguous blocks of memory used to store similar things. Effectively this allows you to have variables that you can reference by a number.\n\nYou can define a variable to hold an unspecified number of integers with:\n\nint [] values = new int[] { 1, 2, 3 };\n\n\nYou can also assign the variable with an array of values at the same time. We use the new keyword to create the array that holds the elements 1, 2, and 3:\n\nint [] values = new int[] { 1, 2, 3 };\n\n\nTo refer to the first and second items, we use:\n\nvalues\nvalues[i]\n\n\nLet's see the array in use.\n\n## 1.2 Albers Conic Map Projection¶\n\nIn the last lecture, we used a SVG image of the united states.\n\nIn [ ]:\n%download http://upload.wikimedia.org/wikipedia/commons/archive/3/32/20091105194402%21Blank_US_Map.svg\n\n\nAnd a meta-command to rename the file:\n\nIn [ ]:\n! mv 20091105194402%21Blank_US_Map.svg usa-wikipedia.svg\n\n\nIn this example, we want to use the states as before, but also be able to plot longitude and latitude on the map.\n\nSince the map was just a picture (and not drawn) I wasn't sure what coordinate system it was in.\n\nAfter a bit of trial and error, I found that the Albers conic projection was a good fit.\n\nIn this example, I plot some longitude and latitude points, and kept adjusting the xoffset, xscale, yoffset, and yscale until the points lined up pretty well. Normally, we would know these values, but the wikipedia map did not provide those.\n\nIf you click, it will plot those points, and some cites.\n\nNotice that the albers function returns an array containing the x and the y values:\n\nIn :\nPShape usa;\nPShape michigan;\nPShape ohio;\n\nvoid setup() {\nsize(959, 593);\nbackground(255);\nmichigan = usa.getChild(\"MI\");\nohio = usa.getChild(\"OH\");\n}\n\nvoid draw() {\n// Draw the full map\nshape(usa, 0, 0);\nnoLoop();\n}\n\nfloat[] albers(lat, lng) {\nlat0 = 23.0 * (PI/180); // Latitude_Of_Origin\nlng0 = -96.0 * (PI/180); // Central_Meridian\nphi1 = 30.0 * (PI/180); // Standard_Parallel_1\nphi2 = 50.0 * (PI/180); // Standard_Parallel_2\n\nn = 0.5 * (sin(phi1) + sin(phi2));\nc = cos(phi1);\nC = c * c + 2 * n * sin(phi1);\np0 = sqrt(C - 2 * n * sin(lat0)) / n;\ntheta = n * (lng * PI/180 - lng0);\np = sqrt(C - 2 * n * sin(lat * PI/180)) / n;\nx = p * sin(theta);\ny = p0 - p * cos(theta);\nreturn new float { x, y };\n}\n\nvoid plot(lat, lon, c) {\n// Values to scale the lat, lon to fit on the USA map:\nxoffset = 485;\nxscale = 1245;\nyoffset = 630;\nyscale = 1250;\n\nfloat xy = albers(lat, lon);\nfill(c);\nellipse(xoffset + xy * xscale, yoffset - xy * yscale, 10, 10);\n}\n\nvoid mousePressed() {\nfor (lat = 30; lat <= 50; lat += 5) {\nfor (lon = 70; lon <= 130; lon += 5) {\nplot(lat, -lon, color(255));\n}\n}\n// Some Cites:\nplot(39.790942, -86.147685, color(255, 0, 128));\nplot(47.042418, -122.893077, color(255, 128, 0));\nplot(30.4518, -84.27277, color(255, 0, 0));\nplot(44.323535, -69.765261, color(255, 0, 255));\nplot(33.448457, -112.073844, color(255, 255, 0));\n}\n\nSketch #82:\n\nYou can use this to overlay data, such as population density, locations of events, etc. You might want to use the 4th element of color(), the alpha value, to make the colors transparent.\n\n## 1.3 Genomics¶\n\nHere is a diagram style that is used a lot in genomic and biological data:\n\nhttp://circos.ca/intro/genomic_data/", null, "How could we make such a chart?\n\nFirst, let's make some tests using different curves.\n\nI found that the bezier, with control points at the center looks pretty good:\n\nIn :\nnoFill();\nstroke(255, 0, 0);\nbezier(10, 10, 50, 50, 50, 50, 90, 10);\n\nSketch #92:\n\nFor this example, we'll reuse the rotateAround functions from the last assignment. This is used to find the starting and stopping points around a center.\n\nThen, we draw a bezier curve between them:\n\nIn :\nint rotateAroundX(int x1, int y1, int length, int angle) {\nreturn x1 + length * cos(angle);\n}\n\nint rotateAroundY(int x1, int y1, int length, int angle) {\nreturn y1 - length * sin(angle);\n}\n\nvoid setup() {\nsize(300, 300);\nbackground();\n}\n\nvoid draw() {\n// Center:\ncx = width/2;\ncy = height/2;\n\n// Random degree, converted to radians:\np1 = random(360) * PI/180;\n// Random degree, converted to radians:\np2 = random(360) * PI/180;\n\n// Find where p1 would be:\nx1 = rotateAroundX(cx, cy, width/2, p1);\ny1 = rotateAroundY(cx, cy, width/2, p1);\n\n// Find where p2 would be:\nx2 = rotateAroundX(cx, cy, width/2, p2);\ny2 = rotateAroundY(cx, cy, width/2, p2);\n\n// Connect p1 to p2:\nnoFill();\nstroke(random(255), random(255), random(255));\nbezier(x1, y1, cx, cy, cx, cy, x2, y2);\n}\n\nSketch #103:\n\nWhat could you use this to represent?\n\n## 1.4 Relationships¶\n\nIn this example, we construct three arrays:\n\n• name - to hold the name of a person\n• from - to hold the index of the to-person\n• to - to hold the index of the from-person\n\nThe idea is we are representing information about a relationship. For example, it might represent a tweet (from one person, sent to (or mentioning) another).\n\nIn :\nint rotateAroundX(int x1, int y1, int length, int angle) {\nreturn x1 + length * cos(angle);\n}\n\nint rotateAroundY(int x1, int y1, int length, int angle) {\nreturn y1 - length * sin(angle);\n}\n\nvoid setup() {\nsize(300, 300);\nbackground();\n}\n\nvoid draw() {\n// Center:\ncx = width/2;\ncy = height/2;\n\nint[] from = new int[] {0, 1, 2, 3, 4, 5, 6, 7, 8, 9 };\nint[] to = new int[] {1, 3, 3, 6, 3, 1, 2, 1, 2, 3 };\nstring[] name = new string[] {\"Helen\",\n\"Mary\",\n\"Teyvonia\",\n\"Glenda\",\n\"Bertrude\",\n\"Elvie\",\n\"Sarah\",\n\"Mary\",\n\"Trisha\",\n\"Karen\" };\n\n// First, let's draw the names around the circle:\nfor (int i = 0; i < 10; i++) {\np1 = from[i] * 36 * PI/180;\np2 = to[i] * 36 * PI/180;\n\nx1 = rotateAroundX(cx, cy, width/2, p1);\ny1 = rotateAroundY(cx, cy, width/2, p1);\nfill(0);\ntext(name[i], x1, y1);\n}\n// Next, we connect up the people:\nfor (int i = 0; i < 10; i++) {\np1 = from[i] * 36 * PI/180;\np2 = to[i] * 36 * PI/180;\n\n// Find where p1 would be:\nx1 = rotateAroundX(cx, cy, width/2, p1);\ny1 = rotateAroundY(cx, cy, width/2, p1);\n\n// Find where p2 would be:\nx2 = rotateAroundX(cx, cy, width/2, p2);\ny2 = rotateAroundY(cx, cy, width/2, p2);\n\n// Connect p1 to p2:\nnoFill();\nstrokeWeight(random(5));\nstroke(random(255), random(255), random(255));\nbezier(x1, y1, cx, cy, cx, cy, x2, y2);\n}\nnoLoop();\n}\n\nSketch #1:" ]	[ null, "http://circos.ca/intro/genomic_data/img/circos-conde-nast-large.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7170263,"math_prob":0.99396944,"size":6481,"snap":"2023-40-2023-50","text_gpt3_token_len":2063,"char_repetition_ratio":0.10745715,"word_repetition_ratio":0.23503521,"special_character_ratio":0.38034254,"punctuation_ratio":0.23444976,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9958497,"pos_list":[0,1,2],"im_url_duplicate_count":[null,3,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-08T09:51:07Z\",\"WARC-Record-ID\":\"<urn:uuid:f04fe62f-08fd-4df1-af25-7533afdb459b>\",\"Content-Length\":\"331906\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4b1b1c8e-d07e-403a-a2b2-a23fd5835f07>\",\"WARC-Concurrent-To\":\"<urn:uuid:eb910c00-f89b-4faf-8bce-d8f6d27d65cf>\",\"WARC-IP-Address\":\"165.106.90.202\",\"WARC-Target-URI\":\"https://jupyter.brynmawr.edu/services/public/dblank/CS110%20Intro%20to%20Computing/2015-Spring/Lectures/Visualization2.ipynb\",\"WARC-Payload-Digest\":\"sha1:76D367OBIIE6DDOTTRWXC4TVDTBMJV7W\",\"WARC-Block-Digest\":\"sha1:EWCW7FJHFLO7KSNY6HQADJSUN4YTTAP7\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100739.50_warc_CC-MAIN-20231208081124-20231208111124-00676.warc.gz\"}"}
https://spiderlabweb.com/python-program-display-print-prime-numbers-between-range-interval/	[ "# Python Program to Display or Print Prime Numbers Between a Range or an Interval\n\nIn this program, we will learn how to display or print the Prime Numbers between a given range using the Python programming language.\n\nI have already discussed how to check whether a number is prime or not in previous posts. So, check it out for better understanding.\n\nI will be discussing different programs\n\n• Between an Interval.\n• In Range 1 to N\n\n### Python Programming Code to Display or Print Prime Numbers Between a Range or an Interval\n\nIn both the programs below, I have used the different approaches in the inner for loop.\n\nIn one, I have traversed until half of the number (i/2). And in other, I have traversed till the square root of the number (sqrt(i)).\n\n### Between an Interval\n\nIn this program, user is asked to give the start and end of the range or interval.\n\nAnd then I apply the for loop in that range and check every number in between the range including the ends also whether they are a prime number or not.\n\n### Code:-\n\n```start = int(input(\"Enter starting number of the interval : \"))\nend = int(input(\"Enter ending number of the interval : \"))\n\nprint(\"Prime Numbers Between \", start, \" and \", end, \" : \")\nfor i in range(start, end+1):\nisPrime = True\n\nif(i <= 1):\nisPrime = False\nelse:\nfor j in range(2, int(i/2)):\nif(i % j == 0):\nisPrime = False\nbreak\n\nif (isPrime):\nprint(i, end=\" \")\n\nprint()```\nEnter starting number of the interval : 5\nEnter ending number of the interval : 60\nPrime Numbers Between 5 and 60 :\n5 7 11 13 17 19 23 29 31 37 41 43 47 53 59\n\n### In Range 1 to N\n\nIn this program, I have asked for only the end of the range.\n\nAnd then with the help of for loop, numbers are checked.\n\n### Code:-\n\n```import math\n\nend = int(input(\"Enter the end of the range : \"))\n\nprint(\"Prime Numbers Between 1\", \" and \", end, \" : \")\n\nfor i in range(2, end+1):\nisPrime = True\n\nfor j in range(2, int(math.sqrt(i)+1)):\nif(i % j == 0):\nisPrime = False\nbreak\n\nif (isPrime):\nprint(i, end=\" \")\n\nprint()```\n\n### Output:-\n\nEnter the end of the range : 25\nPrime Numbers Between 1 and 25 :\n2 3 5 7 11 13 17 19 23\n\n#### Best Books for learning Python with Data Structure, Algorithms, Machine learning and Data Science.\n\nThe following two tabs change content below.", null, "#### Amit Rawat\n\nFounder and Developer at SpiderLabWeb\nI love to work on new projects and also work on my ideas. My main field of interest is Web Development.", null, "" ]	[ null, "https://secure.gravatar.com/avatar/e0c5242b46deda0e13fcdd7a594be755", null, "https://secure.gravatar.com/avatar/e0c5242b46deda0e13fcdd7a594be755", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7788193,"math_prob":0.9732417,"size":2407,"snap":"2021-43-2021-49","text_gpt3_token_len":631,"char_repetition_ratio":0.147732,"word_repetition_ratio":0.18322296,"special_character_ratio":0.2978812,"punctuation_ratio":0.12127236,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9905188,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-16T12:20:53Z\",\"WARC-Record-ID\":\"<urn:uuid:94910a7c-41fb-4fb3-8079-295ad0a34484>\",\"Content-Length\":\"116407\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4778653b-74cb-4e39-8516-041f4748dc06>\",\"WARC-Concurrent-To\":\"<urn:uuid:6c862cd1-b703-4a7a-bab2-588073647bc7>\",\"WARC-IP-Address\":\"185.61.152.63\",\"WARC-Target-URI\":\"https://spiderlabweb.com/python-program-display-print-prime-numbers-between-range-interval/\",\"WARC-Payload-Digest\":\"sha1:O4TMJ7UMCS5ULTMNZ76HJGC72B6E4DAR\",\"WARC-Block-Digest\":\"sha1:E6KYFX6OENBWVQRT7SGPVNFOF632TGKO\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323584567.81_warc_CC-MAIN-20211016105157-20211016135157-00351.warc.gz\"}"}
https://vivian.worldvista.org/dox/Routine_FBAAUTL4_source.html	[ "Home Package List Routine Alphabetical List Global Alphabetical List FileMan Files List FileMan Sub-Files List Package Component Lists Package-Namespace Mapping\nRoutine: FBAAUTL4\n\n# FBAAUTL4.m\n\nGo to the documentation of this file.\n```FBAAUTL4 ;AISC/CMR,dmk,WCIOFO/SAB-UTILITY ROUTINE ; 8/21/12 3:39pm\n```\n``` ;;3.5;FEE BASIS;4,32,77,81,144;JAN 30, 1995;Build 8\n```\n``` ;;Per VHA Directive 10-93-142, this routine should not be modified.\n```\n``` ;\n```\n```CPT(X,Y,FBSRVDT) ;return external format of CPT code\n```\n``` ;INPUT X = ien of CPT\n```\n``` ;optional Y I Y return description, I 'Y return external format of CPT\n```\n``` ;optional FBSRVDT - date of service\n```\n``` ;OUTPUT external format of CPT code or description of CPT code\n```\n``` I '\\$G(X) Q \"\"\n```\n``` N Z\n```\n``` S Z=\\$\\$CPT^ICPTCOD(X,\\$S(\\$G(FBSRVDT)>0:+\\$G(FBSRVDT),1:\"\"),1)\n```\n``` Q \\$S('\\$G(Y):\\$P(Z,U,2),1:\\$P(Z,U,3))\n```\n``` ;\n```\n```MOD(X,Y,FBSRVDT) ;return external format of modifier\n```\n``` ;INPUT X = ien of modifier\n```\n``` ;optional Y I Y return description, I 'Y return external format of mod\n```\n``` ;optional FBSRVDT - date of service\n```\n``` ;OUTPUT external format of modifier or description of CPT code\n```\n``` I '\\$G(X) Q \"\"\n```\n``` N Z\n```\n``` S Z=\\$\\$MOD^ICPTMOD(X,\"I\",\\$S(\\$G(FBSRVDT)>0:+\\$G(FBSRVDT),1:\"\"),1)\n```\n``` Q \\$S('\\$G(Y):\\$P(Z,U,2),1:\\$P(Z,U,3))\n```\n``` ;\n```\n```CPTDATA(W,X,Y,Z) ;get internal value of CPT\n```\n``` ; input\n```\n``` ; W = IEN of PATIENT in file 162\n```\n``` ; X = IEN of VENDOR multiple in file 162\n```\n``` ; Y = IEN of INITIAL TREATMENT DATE multiple in file 162\n```\n``` ; Z = IEN of SERVICE PROVIDED multiple in file 162\n```\n``` ; returns\n```\n``` ; value of SERVICE PROVIDED (internal)\n```\n``` ;\n```\n``` I '\\$G(W)!('\\$G(X))!('\\$G(Y))!('\\$G(Z)) Q \"\"\n```\n``` Q \\$P(\\$G(^FBAAC(W,1,X,1,Y,1,Z,0)),U)\n```\n``` ;\n```\n```MODDATA(W,X,Y,Z) ;get internal values of CPT Modifier\n```\n``` ; input\n```\n``` ; W = IEN of PATIENT in file 162\n```\n``` ; X = IEN of VENDOR multiple in file 162\n```\n``` ; Y = IEN of INITIAL TREATMENT DATE multiple in file 162\n```\n``` ; Z = IEN of SERVICE PROVIDED multiple in file 162\n```\n``` ; output\n```\n``` ; FBMODA( array of CPT MODIFIERs\n```\n``` ; FBMODA(#)=CPT MODIFIER (internal value)\n```\n``` ; where # is the IEN for an entry in the CPT MODIFIER multiple\n```\n``` K FBMODA\n```\n``` I '\\$G(W)!('\\$G(X))!('\\$G(Y))!('\\$G(Z)) Q\n```\n``` N FBI,FBMOD\n```\n``` S FBI=0 F S FBI=\\$O(^FBAAC(W,1,X,1,Y,1,Z,\"M\",FBI)) Q:'FBI D\n```\n``` . S FBMOD=\\$P(\\$G(^FBAAC(W,1,X,1,Y,1,Z,\"M\",FBI,0)),U)\n```\n``` . Q:FBMOD=\"\"\n```\n``` . S FBMODA(FBI)=FBMOD\n```\n``` Q\n```\n``` ;\n```\n```APS(FBJ,FBK,FBL,FBM) ; amount paid symbol\n```\n``` ; input\n```\n``` ; FBJ = IEN of PATIENT in file 162\n```\n``` ; FBK = IEN of VENDOR multiple in file 162\n```\n``` ; FBL = IEN of INITIAL TREATMENT DATE multiple in file 162\n```\n``` ; FBM = IEN of SERVICE PROVIDED multiple in file 162\n```\n``` ; returns symbol\n```\n``` ; where value is M (Mill Bill emergency care - 38 U.S.C. 1725)\n```\n``` ; R (RBRVS fee schedule amount)\n```\n``` ; F (VA fee schedule amount)\n```\n``` ; C (contracted service amount)\n```\n``` ; U (usual & customary - claimed)\n```\n``` ; null if no amount paid\n```\n``` N FBAP,FBRET,FBY0,FBY2\n```\n``` S FBRET=\"\"\n```\n``` S FBY0=\\$G(^FBAAC(FBJ,1,FBK,1,FBL,1,FBM,0))\n```\n``` S FBY2=\\$G(^FBAAC(FBJ,1,FBK,1,FBL,1,FBM,2))\n```\n``` S FBAP=\\$P(FBY0,U,3)\n```\n``` I FBAP>0 D\n```\n``` . ; FB3.5144 Changed order of evaluation, setting Mill-Bill first as\n```\n``` . ; this coding takes precedence.\n```\n``` . ; Mill Bill payments\n```\n``` . I \"^39^52^\"[(U_\\$P(\\$G(^FBAA(161.82,+\\$P(FBY0,U,18),0)),U,3)_U) S FBRET=\"M\" Q\n```\n``` . ; use fee schedule info for payment (if any)\n```\n``` . I +FBAP=+\\$P(FBY2,U,12) S FBRET=\\$P(FBY2,U,13) Q:FBRET]\"\"\n```\n``` . ; if no fee schedule info then calc 75th percentile and check\n```\n``` . I \\$P(FBY2,U,12)=\"\" D Q:FBRET]\"\"\n```\n``` . . S FBCPT=\\$\\$CPT(\\$P(FBY0,U))\n```\n``` . . S FBMODL=\\$\\$MODL(\"^FBAAC(\"_FBJ_\",1,\"_FBK_\",1,\"_FBL_\",1,\"_FBM_\",\"\"M\"\")\",\"E\")\n```\n``` . . S FBDOS=\\$P(\\$G(^FBAAC(FBJ,1,FBK,1,FBL,0)),U)\n```\n``` . . I +FBAP=+\\$\\$PRCTL^FBAAFSF(FBCPT,FBMODL,FBDOS) S FBRET=\"F\"\n```\n``` . ; since not paid by a fee schedule, check prompt pay type\n```\n``` . I \\$P(FBY2,U,2) S FBRET=\"C\" Q\n```\n``` . ; all other payments considered u&c\n```\n``` . S FBRET=\"U\"\n```\n``` Q FBRET\n```\n``` ;\n```\n```CHKBI(X,Y) ;called to determine if batch number or invoice number\n```\n``` ;already exists\n```\n``` ;X= next batch/invoice number\n```\n``` ;Y=1 if Batch\n```\n``` ;Y undefined if invoice number passed\n```\n``` ;returns a truth if X is ok for next batch/invoice #\n```\n``` ;\n```\n``` I 'X Q \"\"\n```\n``` I \\$G(Y) Q \\$S(\\$D(^FBAA(161.7,\"B\",X)):\"\",1:1)\n```\n``` I '\\$G(Y) Q \\$S(\\$D(^FBAA(162.1,\"B\",X)):\"\",\\$D(^FBAAI(\"B\",X)):\"\",\\$D(^FBAAC(\"C\",X)):\"\",1:1)\n```\n``` ;\n```\n```MODL(FBAN,FBFLAG) ;return sorted list given array of modifiers\n```\n``` ; Input\n```\n``` ; FBAN - closed root of array containing modifiers\n```\n``` ; the data must be in nodes descendent from this root.\n```\n``` ; The subscripts of the nodes below FBAN must be\n```\n``` ; positive numbers. The CPT MODIFIER internal value\n```\n``` ; must be the first piece in these nodes or in the\n```\n``` ; 0-node descendent from these nodes.\n```\n``` ; i.e.\n```\n``` ; ARRAY(number)=CPT MODIFIER (internal value)\n```\n``` ; OR\n```\n``` ; ARRAY(number,0)=CPT MODIFIER (internal value)\n```\n``` ; FBFLAG - (optional) flag, E or I, default I\n```\n``` ; I to return internal values of modifiers\n```\n``` ; E to return external values of modifiers\n```\n``` ; Returns string of sorted modifiers (e.g. \"1,3,7\")\n```\n``` ;\n```\n``` N FBI,FBRET,FBSORT,FBX,FBZERO\n```\n``` S FBRET=\"\"\n```\n``` S FBFLAG=\\$G(FBFLAG,\"I\")\n```\n``` ;\n```\n``` ; if any descendent data then determine if it is 0-node descendent\n```\n``` S FBZERO=0 I \\$O(@FBAN@(0)),\\$D(@FBAN@(\\$O(@FBAN@(0)),0))#2 S FBZERO=1\n```\n``` ;\n```\n``` ; loop thru input array and place modifiers in a sort array\n```\n``` S FBI=0 F S FBI=\\$O(@FBAN@(FBI)) Q:'FBI D\n```\n``` . ; get the cpt modifier\n```\n``` . I FBZERO S FBX=\\$P(@FBAN@(FBI,0),U)\n```\n``` . E S FBX=\\$P(@FBAN@(FBI),U)\n```\n``` . I FBFLAG=\"E\" D\n```\n``` . . ; convert to external value\n```\n``` . . S FBX=\\$\\$MOD^ICPTMOD(FBX,\"I\")\n```\n``` . . I FBX>0 S FBX=\\$P(FBX,U,2)\n```\n``` . . E S FBX=\"\"\n```\n``` . I FBX]\"\" S FBSORT(FBX)=\"\"\n```\n``` ;\n```\n``` ; loop thru sorted array and add the modifiers to return value\n```\n``` S FBX=\"\" F S FBX=\\$O(FBSORT(FBX)) Q:FBX=\"\" S FBRET=FBRET_\",\"_FBX\n```\n``` ;\n```\n``` ; strip leading comma (if any)\n```\n``` I \\$E(FBRET)=\",\" S FBRET=\\$E(FBRET,2,999)\n```\n``` ;\n```\n``` ; return value\n```\n``` Q FBRET\n```\n``` ;\n```\n```REPMOD(FBJ,FBK,FBL,FBM) ; Replace CPT Modifier(s) in payment\n```\n``` ; input\n```\n``` ; FBJ = IEN of PATIENT in file 162\n```\n``` ; FBK = IEN of VENDOR multiple in file 162\n```\n``` ; FBL = IEN of INITIAL TREATMENT DATE multiple in file 162\n```\n``` ; FBM = IEN of SERVICE PROVIDED multiple in file 162\n```\n``` ; FBMODA( array of modifiers\n```\n``` ; where FBMODA(number)=CPT Modifier (internal)\n```\n``` ;\n```\n``` N FBI,FBIENS,FBFDA\n```\n``` S FBIENS=FBM_\",\"_FBL_\",\"_FBK_\",\"_FBJ_\",\"\n```\n``` ;\n```\n``` ; delete any existing CPT MODIFIER entries from global\n```\n``` I \\$O(^FBAAC(FBJ,1,FBK,1,FBL,1,FBM,\"M\",0)) D\n```\n``` . K FBFDA S FBI=0\n```\n``` . F S FBI=\\$O(^FBAAC(FBJ,1,FBK,1,FBL,1,FBM,\"M\",FBI)) Q:'FBI D\n```\n``` . . S FBFDA(162.06,FBI_\",\"_FBIENS,.01)=\"@\"\n```\n``` . D FILE^DIE(\"\",\"FBFDA\") D MSG^DIALOG()\n```\n``` ;\n```\n``` ; create CPT MODIFIER entries from data in array FBMODA\n```\n``` I \\$O(FBMODA(0)) D\n```\n``` . K FBFDA S FBI=0 F S FBI=\\$O(FBMODA(FBI)) Q:'FBI D\n```\n``` . . S FBFDA(162.06,\"+\"_FBI_\",\"_FBIENS,.01)=FBMODA(FBI)\n```\n``` . D UPDATE^DIE(\"\",\"FBFDA\") D MSG^DIALOG()\n```\n``` ;\n```\n``` Q\n```\n``` ;\n```\n``` ;FBAAUTL4\n```" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5061884,"math_prob":0.99065423,"size":13465,"snap":"2021-31-2021-39","text_gpt3_token_len":5280,"char_repetition_ratio":0.12963375,"word_repetition_ratio":0.43776825,"special_character_ratio":0.38277015,"punctuation_ratio":0.22439452,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99628294,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-09-26T09:51:03Z\",\"WARC-Record-ID\":\"<urn:uuid:92c77368-ee50-41e5-a213-a652c4967270>\",\"Content-Length\":\"70149\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:af712a9c-848a-4e6e-b7d5-a1bacd43a664>\",\"WARC-Concurrent-To\":\"<urn:uuid:bab8c7cc-15af-4602-9dfb-b43caa543546>\",\"WARC-IP-Address\":\"156.96.113.49\",\"WARC-Target-URI\":\"https://vivian.worldvista.org/dox/Routine_FBAAUTL4_source.html\",\"WARC-Payload-Digest\":\"sha1:5HEUIQOPJNZM3NXASGKZC3ONRTZ47PEU\",\"WARC-Block-Digest\":\"sha1:FTGVL7D45NFP2SHMQVYJRHQK36WENKJP\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780057857.27_warc_CC-MAIN-20210926083818-20210926113818-00018.warc.gz\"}"}
https://distill.pub/2017/momentum/?utm_campaign=Data%20Machina&utm_medium=email&utm_source=Revue%20newsletter	[ "# Why Momentum Really Works\n\nHereŌĆÖs a popular story about momentum [1, 2, 3]: gradient descent is a man walking down a hill. He follows the steepest path downwards; his progress is slow, but steady. Momentum is a heavy ball rolling down the same hill. The added inertia acts both as a smoother and an accelerator, dampening oscillations and causing us to barrel through narrow valleys, small humps and local minima.\n\nThis standard story isnŌĆÖt wrong, but it fails to explain many important behaviors of momentum. In fact, momentum can be understood far more precisely if we study it on the right model.\n\nOne nice model is the convex quadratic. This model is rich enough to reproduce momentumŌĆÖs local dynamics in real problems, and yet simple enough to be understood in closed form. This balance gives us powerful traction for understanding this algorithm.\n\nWe begin with gradient descent. The algorithm has many virtues, but speed is not one of them. It is simpleŌĆēŌĆöŌĆēwhen optimizing a smooth function $f$, we make a small step in the gradient $w^{k+1} = w^k-\\alpha\\nabla f(w^k).$ For a step-size small enough, gradient descent makes a monotonic improvement at every iteration. It always converges, albeit to a local minimum. And under a few weak curvature conditions it can even get there at an exponential rate.\n\nBut the exponential decrease, though appealing in theory, can often be infuriatingly small. Things often begin quite wellŌĆēŌĆöŌĆēwith an impressive, almost immediate decrease in the loss. But as the iterations progress, things start to slow down. You start to get a nagging feeling youŌĆÖre not making as much progress as you should be. What has gone wrong?\n\nThe problem could be the optimizerŌĆÖs old nemesis, pathological curvature. Pathological curvature is, simply put, regions of $f$ which arenŌĆÖt scaled properly. The landscapes are often described as valleys, trenches, canals and ravines. The iterates either jump between valleys, or approach the optimum in small, timid steps. Progress along certain directions grind to a halt. In these unfortunate regions, gradient descent fumbles.\n\nMomentum proposes the following tweak to gradient descent. We give gradient descent a short-term memory: \\begin{aligned} z^{k+1}&=\\beta z^{k}+\\nabla f(w^{k})\\\\[0.4em] w^{k+1}&=w^{k}-\\alpha z^{k+1} \\end{aligned} The change is innocent, and costs almost nothing. When $\\beta = 0$ , we recover gradient descent. But for $\\beta = 0.99$ (sometimes $0.999$, if things are really bad), this appears to be the boost we need. Our iterations regain that speed and boldness it lost, speeding to the optimum with a renewed energy.\n\nOptimizers call this minor miracle ŌĆ£accelerationŌĆØ.\n\nThe new algorithm may seem at first glance like a cheap hack. A simple trick to get around gradient descentŌĆÖs more aberrant behaviorŌĆēŌĆöŌĆēa smoother for oscillations between steep canyons. But the truth, if anything, is the other way round. It is gradient descent which is the hack. First, momentum gives up to a quadratic speedup on many functions. 1 This is no small matterŌĆēŌĆöŌĆēthis is similar to the speedup you get from the Fast Fourier Transform, Quicksort, and GroverŌĆÖs Algorithm. When the universe gives you quadratic speedups, you should start to pay attention.\n\nBut thereŌĆÖs more. A lower bound, courtesy of Nesterov , states that momentum is, in a certain very narrow and technical sense, optimal. Now, this doesnŌĆÖt mean it is the best algorithm for all functions in all circumstances. But it does satisfy some curiously beautiful mathematical properties which scratch a very human itch for perfection and closure. But more on that later. LetŌĆÖs say this for nowŌĆēŌĆöŌĆēmomentum is an algorithm for the book.\n\nWe begin by studying gradient descent on the simplest model possible which isnŌĆÖt trivialŌĆēŌĆöŌĆēthe convex quadratic, $f(w) = \\tfrac{1}{2}w^TAw - b^Tw, \\qquad w \\in \\mathbf{R}^n.$ Assume $A$ is symmetric and invertible, then the optimal solution $w^{\\star}$ occurs at $w^{\\star} = A^{-1}b.$ Simple as this model may be, it is rich enough to approximate many functions (think of $A$ as your favorite model of curvatureŌĆēŌĆöŌĆēthe Hessian, Fisher Information Matrix , etc) and captures all the key features of pathological curvature. And more importantly, we can write an exact closed formula for gradient descent on this function.\n\nThis is how it goes. Since $\\nabla f(w)=Aw - b$, the iterates are $w^{k+1}=w^{k}- \\alpha (Aw^{k} - b).$ HereŌĆÖs the trick. There is a very natural space to view gradient descent where all the dimensions act independentlyŌĆēŌĆöŌĆēthe eigenvectors of $A$.\n\nEvery symmetric matrix $A$ has an eigenvalue decomposition $A=Q\\ \\text{diag}(\\lambda_{1},\\ldots,\\lambda_{n})\\ Q^{T},\\qquad Q = [q_1,\\ldots,q_n],$ and, as per convention, we will assume that the $\\lambda_i$ŌĆÖs are sorted, from smallest $\\lambda_1$ to biggest $\\lambda_n$. If we perform a change of basis, $x^{k} = Q^T(w^{k} - w^\\star)$, the iterations break apart, becoming: \\begin{aligned} x_{i}^{k+1} & =x_{i}^{k}-\\alpha \\lambda_ix_{i}^{k} \\\\[0.4em] &= (1-\\alpha\\lambda_i)x^k_i=(1-\\alpha \\lambda_i)^{k+1}x^0_i \\end{aligned} Moving back to our original space $w$, we can see that $w^k - w^\\star = Qx^k=\\sum_i^n x^0_i(1-\\alpha\\lambda_i)^k q_i$ and there we have itŌĆēŌĆöŌĆēgradient descent in closed form.\n\n### Decomposing the Error\n\nThe above equation admits a simple interpretation. Each element of $x^0$ is the component of the error in the initial guess in the $Q$-basis. There are $n$ such errors, and each of these errors follows its own, solitary path to the minimum, decreasing exponentially with a compounding rate of $1-\\alpha\\lambda_i$. The closer that number is to $1$, the slower it converges.\n\nFor most step-sizes, the eigenvectors with largest eigenvalues converge the fastest. This triggers an explosion of progress in the first few iterations, before things slow down as the smaller eigenvectorsŌĆÖ struggles are revealed. By writing the contributions of each eigenspaceŌĆÖs error to the loss $f(w^{k})-f(w^{\\star})=\\sum(1-\\alpha\\lambda_{i})^{2k}\\lambda_{i}[x_{i}^{0}]^2$ we can visualize the contributions of each error component to the loss.\n\n### Choosing A Step-size\n\nThe above analysis gives us immediate guidance as to how to set a step-size $\\alpha$. In order to converge, each $\|1-\\alpha \\lambda_i\|$ must be strictly less than 1. All workable step-sizes, therefore, fall in the interval $0<\\alpha\\lambda_i<2.$ The overall convergence rate is determined by the slowest error component, which must be either $\\lambda_1$ or $\\lambda_n$: \\begin{aligned}\\text{rate}(\\alpha) & ~=~ \\max_{i}\\left\|1-\\alpha\\lambda_{i}\\right\|\\\\[0.9em] & ~=~ \\max\\left\\{\|1-\\alpha\\lambda_{1}\|,~ \|1-\\alpha\\lambda_{n}\|\\right\\} \\end{aligned}\n\nThis overall rate is minimized when the rates for $\\lambda_1$ and $\\lambda_n$ are the sameŌĆēŌĆöŌĆēthis mirrors our informal observation in the previous section that the optimal step-size causes the first and last eigenvectors to converge at the same rate. If we work this through we get: \\begin{aligned} \\text{optimal }\\alpha ~=~{\\mathop{\\text{argmin}}\\limits_\\alpha} ~\\text{rate}(\\alpha) & ~=~\\frac{2}{\\lambda_{1}+\\lambda_{n}}\\\\[1.4em] \\text{optimal rate} ~=~{\\min_\\alpha} ~\\text{rate}(\\alpha) & ~=~\\frac{\\lambda_{n}/\\lambda_{1}-1}{\\lambda_{n}/\\lambda_{1}+1} \\end{aligned}\n\nNotice the ratio $\\lambda_n/\\lambda_1$ determines the convergence rate of the problem. In fact, this ratio appears often enough that we give it a name, and a symbolŌĆēŌĆöŌĆēthe condition number. $\\text{condition number} := \\kappa :=\\frac{\\lambda_n}{\\lambda_1}$ The condition number means many things. It is a measure of how close to singular a matrix is. It is a measure of how robust $A^{-1}b$ is to perturbations in $b$. And, in this context, the condition number gives us a measure of how poorly gradient descent will perform. A ratio of $\\kappa = 1$ is ideal, giving convergence in one step (of course, the function is trivial). Unfortunately the larger the ratio, the slower gradient descent will be. The condition number is therefore a direct measure of pathological curvature.\n\n## Example: Polynomial Regression\n\nThe above analysis reveals an insight: all errors are not made equal. Indeed, there are different kinds of errors, $n$ to be exact, one for each of the eigenvectors of $A$. And gradient descent is better at correcting some kinds of errors than others. But what do the eigenvectors of $A$ mean? Surprisingly, in many applications they admit a very concrete interpretation.\n\nLets see how this plays out in polynomial regression. Given 1D data, $\\xi_i$, our problem is to fit the model $\\text{model}(\\xi)=w_{1}p_{1}(\\xi)+\\cdots+w_{n}p_{n}(\\xi)\\qquad p_{i}=\\xi\\mapsto\\xi^{i-1}$ to our observations, $d_i$. This model, though nonlinear in the input $\\xi$, is linear in the weights, and therefore we can write the model as a linear combination of monomials, like:\n\nBecause of the linearity, we can fit this model to our data $\\xi_i$ using linear regression on the model mismatch $\\text{minimize}_w \\qquad\\tfrac{1}{2}\\sum_i (\\text{model}(\\xi_{i})-d_{i})^{2} ~~=~~ \\tfrac{1}{2}\\\|Zw - d\\\|^2$ where $Z=\\left(\\begin{array}{ccccc} 1 & \\xi_{1} & \\xi_{1}^{2} & \\ldots & \\xi_{1}^{n-1}\\\\ 1 & \\xi_{2} & \\xi_{2}^{2} & \\ldots & \\xi_{2}^{n-1}\\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots\\\\ 1 & \\xi_{m} & \\xi_{m}^{2} & \\ldots & \\xi_{m}^{n-1} \\end{array}\\right).$\n\nThe path of convergence, as we know, is elucidated when we view the iterates in the space of $Q$ (the eigenvectors of $Z^T Z$). So letŌĆÖs recast our regression problem in the basis of $Q$. First, we do a change of basis, by rotating $w$ into $Qw$, and counter-rotating our feature maps $p$ into eigenspace, $\\bar{p}$. We can now conceptualize the same regression as one over a different polynomial basis, with the model $\\text{model}(\\xi)~=~x_{1}\\bar{p}_{1}(\\xi)~+~\\cdots~+~x_{n}\\bar{p}_{n}(\\xi)\\qquad \\bar{p}_{i}=\\sum q_{ij}p_j.$ This model is identical to the old one. But these new features $\\bar{p}$ (which I call ŌĆ£eigenfeaturesŌĆØ) and weights have the pleasing property that each coordinate acts independently of the others. Now our optimization problem breaks down, really, into $n$ small 1D optimization problems. And each coordinate can be optimized greedily and independently, one at a time in any order, to produce the final, global, optimum. The eigenfeatures are also much more informative:\n\nThe observations in the above diagram can be justified mathematically. From a statistical point of view, we would like a model which is, in some sense, robust to noise. Our model cannot possibly be meaningful if the slightest perturbation to the observations changes the entire model dramatically. And the eigenfeatures, the principal components of the data, give us exactly the decomposition we need to sort the features by its sensitivity to perturbations in $d_i$ŌĆÖs. The most robust components appear in the front (with the largest eigenvalues), and the most sensitive components in the back (with the smallest eigenvalues).\n\nThis measure of robustness, by a rather convenient coincidence, is also a measure of how easily an eigenspace converges. And thus, the ŌĆ£pathological directionsŌĆØŌĆēŌĆöŌĆēthe eigenspaces which converge the slowestŌĆēŌĆöŌĆēare also those which are most sensitive to noise! So starting at a simple initial point like $0$ (by a gross abuse of language, letŌĆÖs think of this as a prior), we track the iterates till a desired level of complexity is reached. LetŌĆÖs see how this plays out in gradient descent.\n\nThis effect is harnessed with the heuristic of early stopping : by stopping the optimization early, you can often get better generalizing results. Indeed, the effect of early stopping is very similar to that of more conventional methods of regularization, such as Tikhonov Regression. Both methods try to suppress the components of the smallest eigenvalues directly, though they employ different methods of spectral decay.2 But early stopping has a distinct advantage. Once the step-size is chosen, there are no regularization parameters to fiddle with. Indeed, in the course of a single optimization, we have the entire family of models, from underfitted to overfitted, at our disposal. This gift, it seems, doesnŌĆÖt come at a price. A beautiful free lunch indeed.\n\n## The Dynamics of Momentum\n\nLetŌĆÖs turn our attention back to momentum. Recall that the momentum update is \\begin{aligned} z^{k+1}&=\\beta z^{k}+\\nabla f(w^{k})\\\\[0.4em] w^{k+1}&=w^{k}-\\alpha z^{k+1}. \\end{aligned} Since $\\nabla f(w^k) = Aw^k - b$, the update on the quadratic is \\begin{aligned} z^{k+1}&=\\beta z^{k}+ (Aw^{k}-b)\\\\[0.4em] w^{k+1}&=w^{k}-\\alpha z^{k+1}. \\end{aligned} Following , we go through the same motions, with the change of basis $x^{k} = Q(w^{k} - w^\\star)$ and $y^{k} = Qz^{k}$, to yield the update rule \\begin{aligned} y_{i}^{k+1}&=\\beta y_{i}^{k}+\\lambda_{i}x_{i}^{k}\\\\[0.4em] x_{i}^{k+1}&=x_{i}^{k}-\\alpha y_{i}^{k+1}. \\end{aligned} in which each component acts independently of the other components (though $x^k_i$ and $y^k_i$ are coupled). This lets us rewrite our iterates as 3 $\\left(\\!\\!\\begin{array}{c} y_{i}^{k}\\\\ x_{i}^{k} \\end{array}\\!\\!\\right)=R^k\\left(\\!\\!\\begin{array}{c} y_{i}^{0}\\\\ x_{i}^{0} \\end{array}\\!\\!\\right) \\qquad R = \\left(\\!\\!\\begin{array}{cc} \\beta & \\lambda_{i}\\\\ -\\alpha\\beta & 1-\\alpha\\lambda_{i} \\end{array}\\!\\!\\right).$ There are many ways of taking a matrix to the $k^{th}$ power. But for the $2 \\times 2$ case there is an elegant and little known formula in terms of the eigenvalues of $R$, $\\sigma_1$ and $\\sigma_2$. $\\color{#AAA}{\\color{black}{R^{k}}=\\begin{cases} \\color{black}{\\sigma_{1}^{k}}R_{1}-\\color{black}{\\sigma_{2}^{k}}R_{2} & \\sigma_{1}\\neq\\sigma_{2}\\\\ \\sigma_{1}^{k}(kR/\\sigma_1-(k-1)I) & \\sigma_{1}=\\sigma_{2} \\end{cases},\\qquad R_{j}=\\frac{R-\\sigma_{j}I}{\\sigma_{1}-\\sigma_{2}}}$ This formula is rather complicated, but the takeaway here is that it plays the exact same role the individual convergence rates, $1-\\alpha\\lambda_i$ do in gradient descent. But instead of one geometric series, we have two coupled series, which may have real or complex values. The convergence rate is therefore the slowest of the two rates, $\\max\\{\|\\sigma_{1}\|,\|\\sigma_{2}\|\\}$ 4. By plotting this out, we see there are distinct regions of the parameter space which reveal a rich taxonomy of convergence behavior :\n\nFor what values of $\\alpha$ and $\\beta$ does momentum converge? Since we need both $\\sigma_1$ and $\\sigma_2$ to converge, our convergence criterion is now $\\max\\{\|\\sigma_{1}\|,\|\\sigma_{2}\|\\} < 1$. The range of available step-sizes work out 5 to be $0<\\alpha\\lambda_{i}<2+2\\beta \\qquad \\text{for} \\qquad 0 \\leq \\beta < 1$ We recover the previous result for gradient descent when $\\beta = 0$. But notice an immediate boon we get. Momentum allows us to crank up the step-size up by a factor of 2 before diverging.\n\n### The Critical Damping Coefficient\n\nThe true magic happens, however, when we find the sweet spot of $\\alpha$ and $\\beta$. Let us try to first optimize over $\\beta$.\n\nMomentum admits an interesting physical interpretation when $\\alpha$ is small: it is a discretization of a damped harmonic oscillator. Consider a physical simulation operating in discrete time (like a video game).\n\nWe can break this equation apart to see how each component affects the dynamics of the system. Here we plot, for $150$ iterates, the particleŌĆÖs velocity (the horizontal axis) against its position (the vertical axis), in a phase diagram.\n\nThis system is best imagined as a weight suspended on a spring. We pull the weight down by one unit, and we study the path it follows as it returns to equilibrium. In the analogy, the spring is the source of our external force $\\lambda_ix^k_i$, and equilibrium is the state when both the position $x^k_i$ and the speed $y^k_i$ are 0. The choice of $\\beta$ crucially affects the rate of return to equilibrium.\n\nThe critical value of $\\beta = (1 - \\sqrt{\\alpha \\lambda_i})^2$ gives us a convergence rate (in eigenspace $i$) of $1 - \\sqrt{\\alpha\\lambda_i}.$ A square root improvement over gradient descent, $1-\\alpha\\lambda_i$! Alas, this only applies to the error in the $i^{th}$ eigenspace, with $\\alpha$ fixed.\n\n### Optimal parameters\n\nTo get a global convergence rate, we must optimize over both $\\alpha$ and $\\beta$. This is a more complicated affair,6 but they work out to be $\\alpha = \\left(\\frac{2}{\\sqrt{\\lambda_{1}}+\\sqrt{\\lambda_{n}}}\\right)^{2} \\quad \\beta = \\left(\\frac{\\sqrt{\\lambda_{n}}-\\sqrt{\\lambda_{1}}}{\\sqrt{\\lambda_{n}}+\\sqrt{\\lambda_{1}}}\\right)^{2}$ Plug this into the convergence rate, and you get\n\nWith barely a modicum of extra effort, we have essentially square rooted the condition number! These gains, in principle, require explicit knowledge of $\\lambda_1$ and $\\lambda_n$. But the formulas reveal a simple guideline. When the problemŌĆÖs conditioning is poor, the optimal $\\alpha$ is approximately twice that of gradient descent, and the momentum term is close to $1$. So set $\\beta$ as close to $1$ as you can, and then find the highest $\\alpha$ which still converges. Being at the knifeŌĆÖs edge of divergence, like in gradient descent, is a good place to be.\n\nWhile the loss function of gradient descent had a graceful, monotonic curve, optimization with momentum displays clear oscillations. These ripples are not restricted to quadratics, and occur in all kinds of functions in practice. They are not cause for alarm, but are an indication that extra tuning of the hyperparameters is required.\n\n## Example: The Colorization Problem\n\nLetŌĆÖs look at how momentum accelerates convergence with a concrete example. On a grid of pixels let $G$ be the graph with vertices as pixels, $E$ be the set of edges connecting each pixel to its four neighboring pixels, and $D$ be a small set of a few distinguished vertices. Consider the problem of minimizing\n\nThe optimal solution to this problem is a vector of all $1$ŌĆÖs 7. An inspection of the gradient iteration reveals why we take a long time to get there. The gradient step, for each component, is some form of weighted average of the current value and its neighbors: $w_{i}^{k+1}=w_{i}^{k}-\\alpha\\sum_{j\\in N}(w_{i}^{k}-w_{j}^{k})-\\begin{cases} \\alpha(w_{i}^{k}-1) & i\\in D\\\\ 0 & i\\notin D \\end{cases}$ This kind of local averaging is effective at smoothing out local variations in the pixels, but poor at taking advantage of global structure. The updates are akin to a drop of ink, diffusing through water. Movement towards equilibrium is made only through local corrections and so, left undisturbed, its march towards the solution is slow and laborious. Fortunately, momentum speeds things up significantly.\n\nIn vectorized form, the colorization problem is\n\nThe Laplacian matrix, $L_G$ 8, which dominates the behavior of the optimization problem, is a valuable bridge between linear algebra and graph theory. This is a rich field of study, but one fact is pertinent to our discussion here. The conditioning of $L_G$, here defined as the ratio of the second eigenvector to the last (the first eigenvalue is always 0, with eigenvector equal to the matrix of all 1ŌĆ▓s), is directly connected to the connectivity of the graph.\n\nThese observations carry through to the colorization problem, and the intuition behind it should be clear. Well connected graphs allow rapid diffusion of information through the edges, while graphs with poor connectivity do not. And this principle, taken to the extreme, furnishes a class of functions so hard to optimize they reveal the limits of first order optimization.\n\n## The Limits of Descent\n\nLetŌĆÖs take a step back. We have, with a clever trick, improved the convergence of gradient descent by a quadratic factor with the introduction of a single auxiliary sequence. But is this the best we can do? Could we improve convergence even more with two sequences? Could one perhaps choose the $\\alpha$ŌĆÖs and $\\beta$ŌĆÖs intelligently and adaptively? It is tempting to ride this wave of optimism - to the cube root and beyond!\n\nUnfortunately, while improvements to the momentum algorithm do exist, they all run into a certain, critical, almost inescapable lower bound.\n\nTo understand the limits of what we can do, we must first formally define the algorithmic space in which we are searching. HereŌĆÖs one possible definition. The observation we will make is that both gradient descent and momentum can be ŌĆ£unrolledŌĆØ. Indeed, since $\\begin{array}{lll} w^{1} & \\!= & \\!w^{0} ~-~ \\alpha\\nabla f(w^{0})\\\\[0.35em] w^{2} & \\!= & \\!w^{1} ~-~ \\alpha\\nabla f(w^{1})\\\\[0.35em] & \\!= & \\!w^{0} ~-~ \\alpha\\nabla f(w^{0}) ~-~ \\alpha\\nabla f(w^{1})\\\\[0.35em] & ~ \\!\\vdots \\\\ w^{k+1} & \\!= & \\!w^{0} ~-~ \\alpha\\nabla f(w^{0}) ~-~~~~ \\cdots\\cdots ~~~~-~ \\alpha\\nabla f(w^{k}) \\end{array}$ we can write gradient descent as $w^{k+1} ~~=~~ w^{0} ~-~ \\alpha\\sum_i^k\\nabla f(w^{i}).$ A similar trick can be done with momentum: $w^{k+1} ~~=~~ w^{0} ~+~ \\alpha\\sum_i^k\\frac{(1-\\beta^{k+1-i})}{1-\\beta}\\nabla f(w^i).$ In fact, all manner of first order algorithms, including the Conjugate Gradient algorithm, AdaMax, Averaged Gradient and more, can be written (though not quite so neatly) in this unrolled form. Therefore the class of algorithms for which $w^{k+1} ~~=~~ w^{0} ~+~ \\sum_{i}^{k}\\gamma_{i}^{k}\\nabla f(w^{i}) \\qquad \\text{ for some } \\gamma_{i}^{k}$ contains momentum, gradient descent and a whole bunch of other algorithms you might dream up. This is what is assumed in Assumption 2.1.4 of Nesterov. But letŌĆÖs push this even further, and expand this class to allow different step-sizes for different directions. $w^{k+1} ~~=~~ w^{0} ~+~ \\sum_{i}^{k}\\Gamma_{i}^{k}\\nabla f(w^{i}) \\quad \\text{ for some diagonal matrix } \\Gamma_{i}^{k} .$ This class of methods covers most of the popular algorithms for training neural networks, including ADAM and AdaGrad. We shall refer to this class of methods as ŌĆ£Linear First Order MethodsŌĆØ, and we will show a single function all these methods ultimately fail on.\n\n### The Resisting Oracle\n\nEarlier, when we talked about the colorizer problem, we observed that wiry graphs cause bad conditioning in our optimization problem. Taking this to its extreme, we can look at a graph consisting of a single pathŌĆēŌĆöŌĆēa function so badly conditioned that Nesterov called a variant of it ŌĆ£the worst function in the worldŌĆØ. The function follows the same structure as the colorizer problem, and we shall call this the Convex Rosenbrock,\n\nThe optimal solution of this problem is $w_{i}^{\\star}=\\left(\\frac{\\sqrt{\\kappa}-1}{\\sqrt{\\kappa}+1}\\right)^{i}$ and the condition number of the problem $f^n$ approaches $\\kappa$ as $n$ goes to infinity. Now observe the behavior of the momentum algorithm on this function, starting from $w^0 = 0$.\n\nThe observations made in the above diagram are true for any Linear First Order algorithm. Let us prove this. First observe that each component of the gradient depends only on the values directly before and after it: $\\nabla f(x)_{i}=2w_{i}-w_{i-1}-w_{i+1} +\\frac{4}{\\kappa-1} w_{i}, \\qquad i \\neq 1.$ Therefore the fact we start at 0 guarantees that that component must remain stoically there till an element either before or after it turns nonzero. And therefore, by induction, for any linear first order algorithm,\n\n$\\begin{array}{lllllllll} w^{0} & = & [~~0, & 0, & 0, & \\ldots & 0, & 0, & \\ldots & 0~]\\\\[0.35em] w^{1} & = & [~w_{1}^{1}, & 0, & 0, & \\ldots & 0, & 0, & \\ldots & 0~]\\\\[0.35em] w^{2} & = & [~w_{1}^{2}, & w_{2}^{2}, & 0, & \\ldots & 0, & 0, & \\ldots & 0~]\\\\[0.35em] & ~ \\vdots \\\\ w^{k} & = & [~w_{1}^{k}, & w_{2}^{k}, & w_{3}^{k}, & \\ldots & w_{k}^{k}, & 0, & \\ldots & 0~].\\\\ \\end{array}$\n\nThink of this restriction as a ŌĆ£speed of lightŌĆØ of information transfer. Error signals will take at least $k$ steps to move from $w_0$ to $w_k$. We can therefore sum up the errors which cannot have changed yet9: \\begin{aligned} \\\|w^{k}-w^{\\star}\\\|_{\\infty}&\\geq\\max_{i\\geq k+1}\\{\|w_{i}^{\\star}\|\\}\\\\[0.9em]&=\\left(\\frac{\\sqrt{\\kappa}-1}{\\sqrt{\\kappa}+1}\\right)^{k+1}\\\\[0.9em]&=\\left(\\frac{\\sqrt{\\kappa}-1}{\\sqrt{\\kappa}+1}\\right)^{k}\\\|w^{0}-w^{\\star}\\\|_{\\infty}. \\end{aligned} As $n$ gets large, the condition number of $f^n$ approaches $\\kappa$. And the gap therefore closes; the convergence rate that momentum promises matches the best any linear first order algorithm can do. And we arrive at the disappointing conclusion that on this problem, we cannot do better.\n\nLike many such lower bounds, this result must not be taken literally, but spiritually. It, perhaps, gives a sense of closure and finality to our investigation. But this is not the final word on first order optimization. This lower bound does not preclude the possibility, for example, of reformulating the problem to change the condition number itself! There is still much room for speedups, if you understand the right places to look.\n\nThere is a final point worth addressing. All the discussion above assumes access to the true gradientŌĆēŌĆöŌĆēa luxury seldom afforded in modern machine learning. Computing the exact gradient requires a full pass over all the data, the cost of which can be prohibitively expensive. Instead, randomized approximations of the gradient, like minibatch sampling, are often used as a plug-in replacement of $\\nabla f(w)$. We can write the approximation in two parts,\n\nIt is helpful to think of our approximate gradient as the injection of a special kind of noise into our iteration. And using the machinery developed in the previous sections, we can deal with this extra term directly. On a quadratic, the error term cleaves cleanly into a separate term, where 10\n\nThe error term, $\\epsilon^k$, with its dependence on the $w^k$, is a fairly hairy object. Following , we model this as independent 0-mean Gaussian noise. In this simplified model, the objective also breaks into two separable components, a sum of a deterministic error and a stochastic error 11, visualized here.\n\nNote that there are a set of unfortunate tradeoffs which seem to pit the two components of error against each other. Lowering the step-size, for example, decreases the stochastic error, but also slows down the rate of convergence. And increasing momentum, contrary to popular belief, causes the errors to compound. Despite these undesirable properties, stochastic gradient descent with momentum has still been shown to have competitive performance on neural networks. As has observed, the transient phase seems to matter more than the fine-tuning phase in machine learning. And in fact, it has been recently suggested that this noise is a good thingŌĆēŌĆöŌĆēit acts as a implicit regularizer, which, like early stopping, prevents overfitting in the fine-tuning phase of optimization.\n\n## Onwards and Downwards\n\nThe study of acceleration is seeing a small revival within the optimization community. If the ideas in this article excite you, you may wish to read , which fully explores the idea of momentum as the discretization of a certain differential equation. But other, less physical, interpretations exist. There is an algebraic interpretation of momentum in terms of approximating polynomials [3, 14]. Geometric interpretations are emerging [15, 16], connecting momentum to older methods, like the Ellipsoid method. And finally, there are interpretations relating momentum to duality , perhaps providing a clue as how to accelerate second order methods and Quasi Newton (for a first step, see ). But like the proverbial blind men feeling an elephant, momentum seems like something bigger than the sum of its parts. One day, hopefully soon, the many perspectives will converge into a satisfying whole.\n\n### Acknowledgments\n\nI am deeply indebted to the editorial contributions of Shan Carter and Chris Olah, without which this article would be greatly impoverished. Shan Carter provided complete redesigns of many of my original interactive widgets, a visual coherence for all the figures, and valuable optimizations to the pageŌĆÖs performance. Chris Olah provided impeccable editorial feedback at all levels of detail and abstraction - from the structure of the content, to the alignment of equations.\n\nI am also grateful to Michael Nielsen for providing the title of this article, which really tied the article together. Marcos Ginestra provided editorial input for the earliest drafts of this article, and spiritual encouragement when I needed it the most. And my gratitude extends to my reviewers, Matt Hoffman and Anonymous Reviewer B for their astute observations and criticism. I would like to thank Reviewer B, in particular, for pointing out two non-trivial errors in the original manuscript (discussion here). The contour plotting library for the hero visualization is the joint work of Ben Frederickson, Jeff Heer and Mike Bostock.\n\nMany thanks to the numerous pull requests and issues filed on github. Thanks in particular, to Osemwaro Pedro for spotting an off by one error in one of the equations. And also to Dan Schmidt who did an editing pass over the whole project, correcting numerous typographical and grammatical errors.\n\n### Footnotes\n\n1. It is possible, however, to construct very specific counterexamples where momentum does not converge, even on convex functions. See for a counterexample.\n2. In Tikhonov Regression we add a quadratic penalty to the regression, minimizing $\\text{minimize}\\qquad\\tfrac{1}{2}\\\|Zw-d\\\|^{2}+\\frac{\\eta}{2}\\\|w\\\|^{2}=\\tfrac{1}{2}w^{T}(Z^{T}Z+\\eta I)w-(Zd)^{T}w$ Recall that $Z^{T}Z=Q\\ \\text{diag}(\\Lambda_{1},\\ldots,\\Lambda_{n})\\ Q^T$. The solution to Tikhonov Regression is therefore $(Z^{T}Z+\\eta I)^{-1}(Zd)=Q\\ \\text{diag}\\left(\\frac{1}{\\lambda_{1}+\\eta},\\cdots,\\frac{1}{\\lambda_{n}+\\eta}\\right)Q^T(Zd)$ We can think of regularization as a function which decays the largest eigenvalues, as follows: $\\text{Tikhonov Regularized } \\lambda_i = \\frac{1}{\\lambda_{i}+\\eta}=\\frac{1}{\\lambda_{i}}\\left(1-\\left(1+\\lambda_{i}/\\eta\\right)^{-1}\\right).$ Gradient descent can be seen as employing a similar decay, but with the decay rate $\\text{ Gradient Descent Regularized } \\lambda_i = \\frac{1}{\\lambda_i} \\left( 1-\\left(1-\\alpha\\lambda_{i}\\right)^{k} \\right)$ instead. Note that this decay is dependent on the step-size.\n3. This is true as we can write updates in matrix form as $\\left(\\!\\!\\begin{array}{cc} 1 & 0\\\\ \\alpha & 1 \\end{array}\\!\\!\\right)\\Bigg(\\!\\!\\begin{array}{c} y_{i}^{k+1}\\\\ x_{i}^{k+1} \\end{array}\\!\\!\\Bigg)=\\left(\\!\\!\\begin{array}{cc} \\beta & \\lambda_{i}\\\\ 0 & 1 \\end{array}\\!\\!\\right)\\left(\\!\\!\\begin{array}{c} y_{i}^{k}\\\\ x_{i}^{k} \\end{array}\\!\\!\\right)$ which implies, by inverting the matrix on the left, $\\Bigg(\\!\\!\\begin{array}{c} y_{i}^{k+1}\\\\ x_{i}^{k+1} \\end{array}\\!\\!\\Bigg)=\\left(\\!\\!\\begin{array}{cc} \\beta & \\lambda_{i}\\\\ -\\alpha\\beta & 1-\\alpha\\lambda_{i} \\end{array}\\!\\!\\right)\\left(\\!\\!\\begin{array}{c} y_{i}^{k}\\\\ x_{i}^{k} \\end{array}\\!\\!\\right)=R^{k+1}\\left(\\!\\!\\begin{array}{c} x_{i}^{0}\\\\ y_{i}^{0} \\end{array}\\!\\!\\right)$\n4. We can write out the convergence rates explicitly. The eigenvalues are \\begin{aligned} \\sigma_{1} & =\\frac{1}{2}\\left(1-\\alpha\\lambda+\\beta+\\sqrt{(-\\alpha\\lambda+\\beta+1)^{2}-4\\beta}\\right)\\\\[0.6em] \\sigma_{2} & =\\frac{1}{2}\\left(1-\\alpha\\lambda+\\beta-\\sqrt{(-\\alpha\\lambda+\\beta+1)^{2}-4\\beta}\\right) \\end{aligned} When the $(-\\alpha\\lambda+\\beta+1)^{2}-4\\beta<0$ is less than zero, then the roots are complex and the convergence rate is \\begin{aligned} \|\\sigma_{1}\|=\|\\sigma_{2}\| & =\\sqrt{(1-\\alpha\\lambda+\\beta)^{2}+\|(-\\alpha\\lambda+\\beta+1)^{2}-4\\beta\|}=2\\sqrt{\\beta} \\end{aligned} Which is, surprisingly, independent of the step-size or the eigenvalue $\\alpha\\lambda$. When the roots are real, the convergence rate is $\\max\\{\|\\sigma_{1}\|,\|\\sigma_{2}\|\\}=\\tfrac{1}{2}\\max\\left\\{ \|1-\\alpha\\lambda_{i}+\\beta\\pm\\sqrt{(1-\\alpha\\lambda_{i}+\\beta)^{2}-4\\beta}\|\\right\\}$\n5. This can be derived by reducing the inequalities for all 4 + 1 cases in the explicit form of the convergence rate above.\n6. We must optimize over $\\min_{\\alpha,\\beta}\\max\\left\\{ \\bigg\\\| \\! \\left(\\begin{array}{cc} \\beta & \\lambda_{i}\\\\ -\\alpha\\beta & 1-\\alpha\\lambda_{i} \\end{array}\\right) \\! \\bigg\\\|,\\ldots,\\bigg\\\| \\! \\left(\\begin{array}{cc} \\beta & \\lambda_{n}\\\\ -\\alpha\\beta & 1-\\alpha\\lambda_{n} \\end{array}\\right)\\! \\bigg\\\|\\right\\}.$ ( $\\\|\\cdot \\\|$ here denotes the magnitude of the maximum eigenvalue), and occurs when the roots of the characteristic polynomial are repeated for the matrices corresponding to the extremal eigenvalues.\n7. The above optimization problem is bounded from below by $0$, and vector of all $1$ŌĆÖs achieve this.\n8. This can be written explicitly as $[L_{G}]_{ij}=\\begin{cases} \\text{degree of vertex }i & i=j\\\\ -1 & i\\neq j,(i,j)\\text{ or }(j,i)\\in E\\\\ 0 & \\text{otherwise} \\end{cases}$\n9. We use the infinity norm to measure our error, similar results can be derived for the 1 and 2 norms.\n10. The momentum iterations are \\begin{aligned} z^{k+1}&=\\beta z^{k}+ A w^{k} + \\text{error}(w^k) \\\\[0.4em] w^{k+1}&=w^{k}-\\alpha z^{k+1}. \\end{aligned} which, after a change of variables, become $\\left(\\!\\!\\begin{array}{cc} 1 & 0\\\\ \\alpha & 1 \\end{array}\\!\\!\\right)\\Bigg(\\!\\!\\begin{array}{c} y_{i}^{k+1}\\\\ x_{i}^{k+1} \\end{array}\\!\\!\\Bigg)=\\left(\\!\\!\\begin{array}{cc} \\beta & \\lambda_{i}\\\\ 0 & 1 \\end{array}\\!\\!\\right)\\left(\\!\\!\\begin{array}{c} y_{i}^{k}\\\\ x_{i}^{k} \\end{array}\\!\\!\\right)+\\left(\\!\\!\\begin{array}{c} \\epsilon_{i}^{k}\\\\ 0 \\end{array}\\!\\!\\right)$ Inverting the $2 \\times 2$ matrix on the left, and applying the formula recursively yields the final solution.\n11. On the 1D function $f(x)=\\frac{\\lambda}{2}x^{2}$, the objective value is \\begin{aligned} \\mathbf{E}f(x^{k})&=\\frac{\\lambda}{2}\\mathbf{E}[(x^{k})^{2}]\\\\&=\\frac{\\lambda}{2}\\mathbf{E}\\left(e_{2}^{T}R^{k}\\left(\\begin{array}{c} y^{0}\\\\ x^{0} \\end{array}\\right)+\\epsilon^{k}e_{2}^{T}\\sum_{i=1}^{k}R^{k-i}\\left(\\begin{array}{c} 1\\\\ -\\alpha \\end{array}\\right)\\right)^{2}\\\\&=\\frac{\\lambda}{2}e_{2}^{T}R^{k}\\left(\\begin{array}{c} y^{0}\\\\ x^{0} \\end{array}\\right)+\\frac{\\lambda}{2}\\mathbf{E}\\left(\\epsilon^{k}e_{2}^{T}\\sum_{i=1}^{k}R^{k-i}\\left(\\begin{array}{c} 1\\\\ -\\alpha \\end{array}\\right)\\right)^{2}\\\\&=\\frac{\\lambda}{2}e_{2}^{T}R^{k}\\left(\\begin{array}{c} y^{0}\\\\ x^{0} \\end{array}\\right)+\\frac{\\lambda}{2}\\mathbf{E}[\\epsilon^{k}]\\,\\cdot\\,\\sum_{i=1}^{k}\\left(e_{2}^{T}R^{k-i}\\left(\\begin{array}{c} 1\\\\ -\\alpha \\end{array}\\right)\\right)^{2}\\\\&=\\frac{\\lambda}{2}e_{2}^{T}R^{k}\\left(\\begin{array}{c} y^{0}\\\\ x^{0} \\end{array}\\right)+\\frac{\\lambda\\mathbf{E}[\\epsilon^{k}}{2}\\cdot\\sum_{i=1}^{k}\\gamma_{i}^{2}, \\qquad \\gamma_i = e_{2}^{T}R^{k-i}\\left(\\begin{array}{c} 1\\\\ -\\alpha \\end{array}\\right) \\end{aligned} The third inequality uses the fact that $\\mathbf{E} \\epsilon^k = 0$ and the fourth uses the fact they are uncorrelated.\n\n### References\n\n1. On the importance of initialization and momentum in deep learning. ŌĆé[PDF]\nSutskever, I., Martens, J., Dahl, G.E. and Hinton, G.E., 2013. ICML (3), Vol 28, pp. 1139ŌĆö1147.\n2. Some methods of speeding up the convergence of iteration methods ŌĆé[PDF]\nPolyak, B.T., 1964. USSR Computational Mathematics and Mathematical Physics, Vol 4(5), pp. 1ŌĆö17. Elsevier. DOI: 10.1016/0041-5553(64)90137-5\nRutishauser, H., 1959. Refined iterative methods for computation of the solution and the eigenvalues of self-adjoint boundary value problems, pp. 24ŌĆö49. Springer. DOI: 10.1007/978-3-0348-7224-9_2\n4. Analysis and design of optimization algorithms via integral quadratic constraints ŌĆé[PDF]\nLessard, L., Recht, B. and Packard, A., 2016. SIAM Journal on Optimization, Vol 26(1), pp. 57ŌĆö95. SIAM.\n5. Introductory lectures on convex optimization: A basic course\nNesterov, Y., 2013. , Vol 87. Springer Science \\& Business Media. DOI: 10.1007/978-1-4419-8853-9\nAmari, S., 1998. Neural computation, Vol 10(2), pp. 251ŌĆö276. MIT Press. DOI: 10.1162/089976698300017746\n7. Deep Learning, NIPSŌĆ▓2015 Tutorial ŌĆé[PDF]\nHinton, G., Bengio, Y. and LeCun, Y., 2015.\nOŌĆÖDonoghue, B. and Candes, E., 2015. Foundations of computational mathematics, Vol 15(3), pp. 715ŌĆö732. Springer. DOI: 10.1007/s10208-013-9150-3\n9. The Nth Power of a 2x2 Matrix. ŌĆé[PDF]\nWilliams, K., 1992. Mathematics Magazine, Vol 65(5), pp. 336. MAA. DOI: 10.2307/2691246\n10. From Averaging to Acceleration, There is Only a Step-size. ŌĆé[PDF]\nFlammarion, N. and Bach, F.R., 2015. COLT, pp. 658ŌĆö695.\n11. On the momentum term in gradient descent learning algorithms ŌĆé[PDF]\nQian, N., 1999. Neural networks, Vol 12(1), pp. 145ŌĆö151. Elsevier. DOI: 10.1016/s0893-6080(98)00116-6\n12. Understanding deep learning requires rethinking generalization ŌĆé[PDF]\nZhang, C., Bengio, S., Hardt, M., Recht, B. and Vinyals, O., 2016. arXiv preprint arXiv:1611.03530.\n13. A differential equation for modeling NesterovŌĆÖs accelerated gradient method: Theory and insights ŌĆé[PDF]\nSu, W., Boyd, S. and Candes, E., 2014. Advances in Neural Information Processing Systems, pp. 2510ŌĆö2518.\n14. The Zen of Gradient Descent ŌĆé[HTML]\nHardt, M., 2013.\n15. A geometric alternative to NesterovŌĆÖs accelerated gradient descent ŌĆé[PDF]\nBubeck, S., Lee, Y.T. and Singh, M., 2015. arXiv preprint arXiv:1506.08187.\n16. An optimal first order method based on optimal quadratic averaging ŌĆé[PDF]\nDrusvyatskiy, D., Fazel, M. and Roy, S., 2016. arXiv preprint arXiv:1604.06543.\n17. Linear coupling: An ultimate unification of gradient and mirror descent ŌĆé[PDF]\nAllen-Zhu, Z. and Orecchia, L., 2014. arXiv preprint arXiv:1407.1537.\n18. Accelerating the cubic regularization of NewtonŌĆÖs method on convex problems ŌĆé[PDF]\nNesterov, Y., 2008. Mathematical Programming, Vol 112(1), pp. 159ŌĆö181. Springer. DOI: 10.1007/s10107-006-0089-x\n\nView all changes to this article since it was first published. If you see a mistake or want to suggest a change, please create an issue on GitHub.\n\n### Citations and Reuse\n\nDiagrams and text are licensed under Creative Commons Attribution CC-BY 2.0, unless noted otherwise, with the source available on GitHub. The figures that have been reused from other sources don't fall under this license and can be recognized by a note in their caption: ŌĆ£Figure from ŌĆ”ŌĆØ.\n\nGoh, \"Why Momentum Really Works\", Distill, 2017. http://doi.org/10.23915/distill.00006\n@article{goh2017why,\n}" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.84139323,"math_prob":0.9979371,"size":35188,"snap":"2022-27-2022-33","text_gpt3_token_len":9759,"char_repetition_ratio":0.12738745,"word_repetition_ratio":0.1747937,"special_character_ratio":0.24033761,"punctuation_ratio":0.13688639,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99951935,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-08-20T05:55:25Z\",\"WARC-Record-ID\":\"<urn:uuid:410e02a6-22d5-409c-9c84-d7f39e041e78>\",\"Content-Length\":\"1049262\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b1837123-fa9a-4e86-80df-7646a6539f5a>\",\"WARC-Concurrent-To\":\"<urn:uuid:cc292608-f8b6-4ad5-b3dc-054966129dce>\",\"WARC-IP-Address\":\"151.101.65.195\",\"WARC-Target-URI\":\"https://distill.pub/2017/momentum/?utm_campaign=Data%20Machina&utm_medium=email&utm_source=Revue%20newsletter\",\"WARC-Payload-Digest\":\"sha1:APXKUKLEJHYRNW5DJP6IYXXX362XVZSM\",\"WARC-Block-Digest\":\"sha1:43E5ABXVULTLFQJPQEOJQHPDUIKSZP3B\",\"WARC-Truncated\":\"length\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-33/CC-MAIN-2022-33_segments_1659882573908.30_warc_CC-MAIN-20220820043108-20220820073108-00447.warc.gz\"}"}
https://tipcalc.net/how-much-is-a-10-percent-tip-on-162.9	[ "# Tip Calculator\n\nHow much is a 10 percent tip on \\$162.9?\n\nTIP:\n\\$ 0\nTOTAL:\n\\$ 0\n\nTIP PER PERSON:\n\\$ 0\nTOTAL PER PERSON:\n\\$ 0\n\n## How much is a 10 percent tip on \\$162.9? How to calculate this tip?\n\nAre you looking for the answer to this question: How much is a 10 percent tip on \\$162.9? Here is the answer.\n\nLet's see how to calculate a 10 percent tip when the amount to be paid is 162.9. Tip is a percentage, and a percentage is a number or ratio expressed as a fraction of 100. This means that a 10 percent tip can also be expressed as follows: 10/100 = 0.1 . To get the tip value for a \\$162.9 bill, the amount of the bill must be multiplied by 0.1, so the calculation is as follows:\n\n1. TIP = 162.910% = 162.90.1 = 16.29\n\n2. TOTAL = 162.9+16.29 = 179.19\n\n3. Rounded to the nearest whole number: 179\n\nIf you want to know how to calculate the tip in your head in a few seconds, visit the Tip Calculator Home.\n\n## So what is a 10 percent tip on a \\$162.9? The answer is 16.29!\n\nOf course, it may happen that you do not pay the bill or the tip alone. A typical case is when you order a pizza with your friends and you want to split the amount of the order. For example, if you are three, you simply need to split the tip and the amount into three. In this example it means:\n\n1. Total amount rounded to the nearest whole number: 179\n\n2. Split into 3: 59.67\n\nSo in the example above, if the pizza order is to be split into 3, you’ll have to pay \\$59.67 . Of course, you can do these settings in Tip Calculator. You can split the tip and the total amount payable among the members of the company as you wish. So the TipCalc.net page basically serves as a Pizza Tip Calculator, as well.\n\n## Tip Calculator Examples (BILL: \\$162.9)\n\nHow much is a 5% tip on \\$162.9?\nHow much is a 10% tip on \\$162.9?\nHow much is a 15% tip on \\$162.9?\nHow much is a 20% tip on \\$162.9?\nHow much is a 25% tip on \\$162.9?\nHow much is a 30% tip on \\$162.9?" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9674502,"math_prob":0.99343973,"size":2607,"snap":"2022-27-2022-33","text_gpt3_token_len":856,"char_repetition_ratio":0.32308874,"word_repetition_ratio":0.24598931,"special_character_ratio":0.4050633,"punctuation_ratio":0.16332377,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99777305,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-08-16T16:40:24Z\",\"WARC-Record-ID\":\"<urn:uuid:72cd8c58-d1bd-431a-98b9-0df2ef524073>\",\"Content-Length\":\"10973\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4c627ba3-3bef-4b8b-a86f-1d259e170428>\",\"WARC-Concurrent-To\":\"<urn:uuid:269b2ef9-f150-4ccb-b490-aab3357b71e0>\",\"WARC-IP-Address\":\"161.35.97.186\",\"WARC-Target-URI\":\"https://tipcalc.net/how-much-is-a-10-percent-tip-on-162.9\",\"WARC-Payload-Digest\":\"sha1:FBXHS3XLIY7LJMMLCDI2TFYYQQX7RSRB\",\"WARC-Block-Digest\":\"sha1:2UEXFZ4KV3LDZRVHBOOHETLNAQIYDPYK\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-33/CC-MAIN-2022-33_segments_1659882572408.31_warc_CC-MAIN-20220816151008-20220816181008-00550.warc.gz\"}"}
https://spec.polkadot.network/chap-host-api	[ "# Appendix B: Host API\n\nDescription of the expected environment available for import by the Polkadot Runtime\n\n## B.1. Preliminaries\n\nThe Polkadot Host API is a set of functions that the Polkadot Host exposes to Runtime to access external functions needed for various reasons, such as the Storage of the content, access and manipulation, memory allocation, and also efficiency. The encoding of each data type is specified or referenced in this section. If the encoding is not mentioned, then the default Wasm encoding is used, such as little-endian byte ordering for integers.\n\n###### Definition 214. Exposed Host API\n\nBy $\\text{RE}_{{B}}$ we refer to the API exposed by the Polkadot Host, which interacts, manipulates, and responds based on the state storage whose state is set at the end of the execution of block ${B}$.\n\n###### Definition 215. Runtime Pointer\n\nThe Runtime pointer type is an unsigned 32-bit integer representing a pointer to data in memory. This pointer is the primary way to exchange data of fixed/known size between the Runtime and Polkadot Host.\n\n###### Definition 216. Runtime Pointer Size\n\nThe Runtime pointer-size type is an unsigned 64-bit integer representing two consecutive integers. The least significant is Runtime pointer (Definition 215). The most significant provides the size of the data in bytes. This representation is the primary way to exchange data of arbitrary/dynamic sizes between the Runtime and the Polkadot Host.\n\n###### Definition 217. Lexicographic ordering\n\nLexicographic ordering refers to the ascending ordering of bytes or byte arrays, such as:\n\n${\\left[{0},{0},{2}\\right]}<{\\left[{0},{1},{1}\\right]}<{\\left[{1}\\right]}<{\\left[{1},{1},{0}\\right]}<{\\left[{2}\\right]}<{\\left[\\ldots\\right]}$\n\nThe functions are specified in each subsequent subsection for each category of those functions.\n\n## B.2. Storage\n\nInterface for accessing the storage from within the runtime.\n\ndanger\n\nAs of now, the storage API should silently ignore any keys that start with the :child_storage:default: prefix. This applies to reading and writing. If the function expects a return value, then None (Definition 200) should be returned. See substrate issue #12461.\n\n###### Definition 218. State Version\n\nThe state version, ${v}$, dictates how a Merkle root should be constructed. The data structure is a varying type of the following format:\n\n${v}={\\left\\lbrace\\begin{matrix}{0}&\\text{full values}\\\\{1}&\\text{node hashes}\\end{matrix}\\right.}$\n\nwhere ${0}$ indicates that the values of the keys should be inserted into the trie directly, and ${1}$ makes use of \"node hashes\" when calculating the Merkle proof (Definition 28).\n\n### B.2.1. ext_storage_set\n\nSets the value under a given key into storage.\n\n(func $ext_storage_set_version_1 (param$key i64) (param $value i64)) Arguments • key: a pointer-size (Definition 216) containing the key. • value: a pointer-size (Definition 216) containing the value. ### B.2.2. ext_storage_get Retrieves the value associated with the given key from storage. #### B.2.2.1. Version 1 - Prototype (func$ext_storage_get_version_1 (param $key i64) (result i64)) Arguments ### B.2.3. ext_storage_read Gets the given key from storage, placing the value into a buffer and returning the number of bytes that the entry in storage has beyond the offset. #### B.2.3.1. Version 1 - Prototype (func$ext_storage_read_version_1 (param $key i64) (param$value_out i64) (param $offset i32) (result i64)) Arguments • key: a pointer-size (Definition 216) containing the key. • value_out: a pointer-size (Definition 216) containing the buffer to which the value will be written to. This function will never write more then the length of the buffer, even if the value’s length is bigger. • offset: an u32 integer (typed as i32 due to wasm types) containing the offset beyond the value should be read from. • result: a pointer-size (Definition 216) pointing to a SCALE encoded Option value (Definition 200) containing an unsigned 32-bit integer representing the number of bytes left at supplied offset. Returns None if the entry does not exist. ### B.2.4. ext_storage_clear Clears the storage of the given key and its value. Non-existent entries are silently ignored. #### B.2.4.1. Version 1 - Prototype (func$ext_storage_clear_version_1 (param $key_data i64)) Arguments • key: a pointer-size (Definition 216) containing the key. ### B.2.5. ext_storage_exists Checks whether the given key exists in storage. #### B.2.5.1. Version 1 - Prototype (func$ext_storage_exists_version_1 (param $key_data i64) (return i32)) Arguments • key: a pointer-size (Definition 216) containing the key. • return: an i32 integer value equal to 1 if the key exists or a value equal to 0 if otherwise. ### B.2.6. ext_storage_clear_prefix Clear the storage of each key/value pair where the key starts with the given prefix. #### B.2.6.1. Version 1 - Prototype (func$ext_storage_clear_prefix_version_1 (param $prefix i64)) Arguments • prefix: a pointer-size (Definition 216) containing the prefix. #### B.2.6.2. Version 2 - Prototype (func$ext_storage_clear_prefix_version_2 (param $prefix i64) (param$limit i64) (return i64))\n\nArguments\n\n• prefix: a pointer-size (Definition 216) containing the prefix.\n\n• limit: a pointer-size (Definition 216) to an Option type (Definition 200) containing an unsigned 32-bit integer indicating the limit on how many keys should be deleted. No limit is applied if this is None. Any keys created during the current block execution do not count toward the limit.\n\n• return: a pointer-size (Definition 216) to the following variant, ${k}$:\n\n${k}={\\left\\lbrace\\begin{matrix}{0}&\\rightarrow{c}\\\\{1}&\\rightarrow{c}\\end{matrix}\\right.}$\n\nwhere 0 indicates that all keys of the child storage have been removed, followed by the number of removed keys, ${c}$. The variant 1 indicates that there are remaining keys, followed by the number of removed keys.\n\n### B.2.7. ext_storage_append\n\nAppend the SCALE encoded value to a SCALE encoded sequence (Definition 202) at the given key. This function assumes that the existing storage item is either empty or a SCALE-encoded sequence and that the value to append is also SCALE encoded and of the same type as the items in the existing sequence.\n\nTo improve performance, this function is allowed to skip decoding the entire SCALE encoded sequence and instead can just append the new item to the end of the existing data and increment the length prefix ${\\text{Enc}_{{\\text{SC}}}^{{\\text{Len}}}}$.\n\ncaution\n\nIf the storage item does not exist or is not SCALE encoded, the storage item will be set to the specified value, represented as a SCALE-encoded byte array.\n\n(func $ext_storage_append_version_1 (param$key i64) (param $value i64)) Arguments • key: a pointer-size (Definition 216) containing the key. • value: a pointer-size (Definition 216) containing the value to be appended. ### B.2.8. ext_storage_root Compute the storage root. #### B.2.8.1. Version 1 - Prototype (func$ext_storage_root_version_1 (return i64))\n\nArguments\n\n• return: a pointer-size (Definition 216) to a buffer containing the 256-bit Blake2 storage root.\n\n#### B.2.8.2. Version 2 - Prototype\n\n(func $ext_storage_root_version_2 (param$version i32) (return i64))\n\nArguments\n\n• version: the state version (Definition 218).\n\n• return: a pointer-size (Definition 216) to the buffer containing the 256-bit Blake2 storage root.\n\n### B.2.9. ext_storage_changes_root\n\ninfo\n\nThis function is not longer used and only exists for compatibility reasons.\n\n#### B.2.9.1. Version 1 - Prototype\n\n(func $ext_storage_changes_root_version_1 (param$parent_hash i64) (return i64))\n\nArguments\n\n### B.2.10. ext_storage_next_key\n\nGet the next key in storage after the given one in lexicographic order (Definition 217). The key provided to this function may or may not exist in storage.\n\n#### B.2.10.1. Version 1 - Prototype\n\n(func $ext_storage_next_key_version_1 (param$key i64) (return i64))\n\nArguments\n\n• key: a pointer-size (Definition 216) to the key.\n\n• return: a pointer-size (Definition 216) to the SCALE encoded Option value (Definition 200) containing the next key in lexicographic order.\n\n### B.2.11. ext_storage_start_transaction\n\nStart a new nested transaction. This allows to either commit or roll back all changes that are made after this call. For every transaction, there must be a matching call to either ext_storage_rollback_transaction (Section B.2.12.) or ext_storage_commit_transaction (Section B.2.13.). This is also effective for all values manipulated using the child storage API (Section B.3.). It’s legal to call this function multiple times in a row.\n\ncaution\n\nThis is a low-level API that is potentially dangerous as it can easily result in unbalanced transactions. Runtimes should use high-level storage abstractions.\n\nArguments\n\n• None.\n\n### B.2.13. ext_storage_commit_transaction\n\nCommit the last transaction started by ext_storage_start_transaction (Section B.2.11.). Any changes made during that transaction are committed to the main state. It’s legal to call this function multiple times in a row.\n\ncaution\n\nPanics if ext_storage_start_transaction (Section B.2.11.) was not called.\n\nArguments\n\n### B.3.3. ext_default_child_storage_read\n\nGets the given key from storage, placing the value into a buffer and returning the number of bytes that the entry in storage has beyond the offset.\n\nArguments\n\n### B.3.5. ext_default_child_storage_storage_kill\n\nClears an entire child storage.\n\n#### B.3.5.1. Version 1 - Prototype\n\n(func $ext_default_child_storage_storage_kill_version_1 (param$child_storage_key i64))\n\nArguments\n\n(func $ext_default_child_storage_storage_kill_version_2 (param$child_storage_key i64) (param $limit i64) (return i32)) Arguments • child_storage_key: a pointer-size (Definition 216) to the child storage key (Definition 219). • limit: a pointer-size (Definition 216) to an Option type (Definition 200) containing an unsigned 32-bit integer indicating the limit on how many keys should be deleted. No limit is applied if this is None. Any keys created during the current block execution do not count toward the limit. • return: a value equal to 1 if all the keys of the child storage have been deleted or a value equal to 0 if there are remaining keys. #### B.3.5.3. Version 3 - Prototype (func$ext_default_child_storage_storage_kill_version_3 (param $child_storage_key i64) (param$limit i64) (return i64))\n\nArguments\n\n• child_storage_key: a pointer-size (Definition 216) to the child storage key (Definition 219).\n\n• limit: a pointer-size (Definition 216) to an Option type (Definition 200) containing an unsigned 32-bit integer indicating the limit on how many keys should be deleted. No limit is applied if this is None. Any keys created during the current block execution do not count toward the limit.\n\n• return: a pointer-size (Definition 216) to the following variant, ${k}$:\n\n${k}={\\left\\lbrace\\begin{matrix}{0}&\\rightarrow{c}\\\\{1}&\\rightarrow{c}\\end{matrix}\\right.}$\n\nwhere 0 indicates that all keys of the child storage have been removed, followed by the number of removed keys, ${c}$. The variant 1 indicates that there are remaining keys, followed by the number of removed keys.\n\n### B.3.6. ext_default_child_storage_exists\n\nChecks whether the given key exists in the child storage.\n\nArguments\n\n#### B.3.7.2. Version 2 - Prototype\n\n(func $ext_default_child_storage_clear_prefix_version_2 (param$child_storage_key i64) (param $prefix i64) (param$limit i64) (return i64))\n\nArguments\n\n• child_storage_key: a pointer-size (Definition 216) to the child storage key (Definition 219).\n\n• prefix: a pointer-size (Definition 216) to the prefix.\n\n• limit: a pointer-size (Definition 216) to an Option type (Definition 200) containing an unsigned 32-bit integer indicating the limit on how many keys should be deleted. No limit is applied if this is None. Any keys created during the current block execution do not count towards the limit.\n\n• return: a pointer-size (Definition 216) to the following variant, ${k}$:\n\n${k}={\\left\\lbrace\\begin{matrix}{0}&\\rightarrow{c}\\\\{1}&\\rightarrow{c}\\end{matrix}\\right.}$\n\nwhere 0 indicates that all keys of the child storage have been removed, followed by the number of removed keys, ${c}$. The variant 1 indicates that there are remaining keys, followed by the number of removed keys.\n\n### B.3.8. ext_default_child_storage_root\n\nCommits all existing operations and computes the resulting child storage root.\n\n#### B.3.8.1. Version 1 - Prototype\n\n(func $ext_default_child_storage_root_version_1 (param$child_storage_key i64) (return i64))\n\nArguments\n\n(func $ext_default_child_storage_root_version_2 (param$child_storage_key i64) (param $version i32) (return i64)) Arguments ### B.3.9. ext_default_child_storage_next_key Gets the next key in storage after the given one in lexicographic order (Definition 217). The key provided to this function may or may not exist in storage. #### B.3.9.1. Version 1 - Prototype (func$ext_default_child_storage_next_key_version_1 (param $child_storage_key i64) (param$key i64) (return i64))\n\nArguments\n\n• child_storage_key: a pointer-size (Definition 216) to the child storage key (Definition 219).\n\n• key: a pointer-size (Definition 216) to the key.\n\n• return: a pointer-size (Definition 216) to the SCALE encoded as defined in Definition 200 containing the next key in lexicographic order. Returns if the entry cannot be found.\n\n## B.4. Crypto\n\nInterfaces for working with crypto related types from within the runtime.\n\n###### Definition 220. Key Type Identifier\n\nCryptographic keys are stored in separate key stores based on their intended use case. The separate key stores are identified by a 4-byte ASCII key type identifier. The following known types are available:\n\n###### Table 5. Table of known key type identifiers\nIdDescription\naccoKey type for the controlling accounts\nbabeKey type for the Babe module\ngranKey type for the Grandpa module\nimonKey type for the ImOnline module\naudiKey type for the AuthorityDiscovery module\nparaKey type for the Parachain Validator Key\nasgnKey type for the Parachain Assignment Key\n###### Definition 221. ECDSA Verify Error\n\nEcdsaVerifyError is a varying data type (Definition 198) that specifies the error type when using ECDSA recovery functionality. The following values are possible:\n\n###### Table 6. Table of error types in ECDSA recovery\nIdDescription\n0Incorrect value of R or S\n1Incorrect value of V\n2Invalid signature\n\n### B.4.1. ext_crypto_ed25519_public_keys\n\nReturns all ed25519 public keys for the given key identifier from the keystore.\n\n#### B.4.1.1. Version 1 - Prototype\n\n(func $ext_crypto_ed25519_public_keys_version_1 (param$key_type_id i32) (return i64))\n\nArguments\n\n### B.4.2. ext_crypto_ed25519_generate\n\nGenerates an ed25519 key for the given key type using an optional BIP-39 seed and stores it in the keystore.\n\ncaution\n\nPanics if the key cannot be generated, such as when an invalid key type or invalid seed was provided.\n\nArguments\n\n### B.4.8. ext_crypto_sr25519_sign\n\nSigns the given message with the sr25519 key that corresponds to the given public key and key type in the keystore.\n\n#### B.4.8.1. Version 1 - Prototype\n\n(func $ext_crypto_sr25519_sign_version_1 (param$key_type_id i32) (param $key i32) (param$msg i64) (return i64))\n\nArguments\n\n• key_type_id: a pointer (Definition 215) to the key identifier (Definition 220).\n\n• key: a pointer to the buffer containing the 256-bit public key.\n\n• msg: a pointer-size (Definition 216) to the message that is to be signed.\n\n• return: a pointer-size (Definition 216) to the SCALE encoded Option value (Definition 200) containing the 64-byte signature. This function returns None if the public key cannot be found in the key store.\n\n### B.4.9. ext_crypto_sr25519_verify\n\nVerifies an sr25519 signature.\n\n#### B.4.9.1. Version 1 - Prototype\n\n(func $ext_crypto_sr25519_verify_version_1 (param$sig i32) (param $msg i64) (param$key i32) (return i32))\n\nArguments\n\n• sig: a pointer (Definition 215) to the buffer containing the 64-byte signature.\n\n• msg: a pointer-size (Definition 216) to the message that is to be verified.\n\n• key: a pointer to the buffer containing the 256-bit public key.\n\n• return: a i32 integer value equal to 1 if the signature is valid or a value equal to 0 if otherwise.\n\n#### B.4.9.2. Version 2 - Prototype\n\n(func $ext_crypto_sr25519_verify_version_2 (param$sig i32) (param $msg i64) (param$key i32) (return i32))\n\nArguments\n\n• sig: a pointer (Definition 215) to the buffer containing the 64-byte signature.\n\n• msg: a pointer-size (Definition 216) to the message that is to be verified.\n\n• key: a pointer to the buffer containing the 256-bit public key.\n\n• return: a i32 integer value equal to 1 if the signature is valid or a value equal to 0 if otherwise.\n\n### B.4.10. ext_crypto_sr25519_batch_verify\n\nRegisters a sr25519 signature for batch verification. Batch verification is enabled by calling ext_crypto_start_batch_verify (Section B.4.20.). The result of the verification is returned by ext_crypto_finish_batch_verify (Section B.4.21.). If batch verification is not enabled, the signature is verified immediately.\n\n#### B.4.10.1. Version 1\n\n(func $ext_crypto_sr25519_batch_verify_version_1 (param$sig i32) (param $msg i64) (param$key i32) (return i32))\n\nArguments\n\n• sig: a pointer (Definition 215) to the buffer containing the 64-byte signature.\n\n• msg: a pointer-size (Definition 216) to the message that is to be verified.\n\n• key: a pointer to the buffer containing the 256-bit public key.\n\n• return: an i32 integer value equal to 1 if the signature is valid or batched or a value equal 0 to if otherwise.\n\n### B.4.11. ext_crypto_ecdsa_public_keys\n\nReturns all ecdsa public keys for the given key id from the keystore.\n\n#### B.4.11.1. Version 1 - Prototype\n\n(func $ext_crypto_ecdsa_public_key_version_1 (param$key_type_id i64) (return i64))\n\nArguments\n\n### B.4.12. ext_crypto_ecdsa_generate\n\nGenerates an ecdsa key for the given key type using an optional BIP-39 seed and stores it in the keystore.\n\ncaution\n\nPanics if the key cannot be generated, such as when an invalid key type or invalid seed was provided.\n\n(func $ext_crypto_ecdsa_generate_version_1 (param$key_type_id i32) (param $seed i64) (return i32)) Arguments • key_type_id: a pointer (Definition 215) to the key identifier (Definition 220). • seed: a pointer-size (Definition 216) to the SCALE encoded Option value (Definition 200) containing the BIP-39 seed which must be valid UTF8. • return: a pointer (Definition 215) to the buffer containing the 33-byte compressed public key. ### B.4.13. ext_crypto_ecdsa_sign Signs the hash of the given message with the ecdsa key that corresponds to the given public key and key type in the keystore. #### B.4.13.1. Version 1 - Prototype (func$ext_crypto_ecdsa_sign_version_1 (param $key_type_id i32) (param$key i32) (param $msg i64) (return i64)) Arguments • key_type_id: a pointer (Definition 215) to the key identifier (Definition 220). • key: a pointer to the buffer containing the 33-byte compressed public key. • msg: a pointer-size (Definition 216) to the message that is to be signed. • return: a pointer-size (Definition 216) to the SCALE encoded Option value (Definition 200) containing the signature. The signature is 65-bytes in size, where the first 512-bits represent the signature and the other 8 bits represent the recovery ID. This function returns if the public key cannot be found in the key store. ### B.4.14. ext_crypto_ecdsa_sign_prehashed Signs the prehashed message with the ecdsa key that corresponds to the given public key and key type in the keystore. #### B.4.14.1. Version 1 - Prototype (func$ext_crypto_ecdsa_sign_prehashed_version_1 (param $key_type_id i32) (param$key i32) (param $msg i64) (return i64)) Arguments • key_type_id: a pointer-size (Definition 215) to the key identifier (Definition 220). • key: a pointer to the buffer containing the 33-byte compressed public key. • msg: a pointer-size (Definition 216) to the message that is to be signed. • return: a pointer-size (Definition 216) to the SCALE encoded Option value (Definition 200) containing the signature. The signature is 65-bytes in size, where the first 512-bits represent the signature and the other 8 bits represent the recovery ID. This function returns if the public key cannot be found in the key store. ### B.4.15. ext_crypto_ecdsa_verify Verifies the hash of the given message against an ECDSA signature. #### B.4.15.1. Version 1 - Prototype This function allows the verification of non-standard, overflowing ECDSA signatures, an implementation specific mechanism of the Rust libsecp256k1 library, specifically the parse_overflowing function. (func$ext_crypto_ecdsa_verify_version_1 (param $sig i32) (param$msg i64) (param $key i32) (return i32)) Arguments • sig: a pointer (Definition 215) to the buffer containing the 65-byte signature. The signature is 65-bytes in size, where the first 512-bits represent the signature and the other 8 bits represent the recovery ID. • msg: a pointer-size (Definition 216) to the message that is to be verified. • key: a pointer to the buffer containing the 33-byte compressed public key. • return: a i32 integer value equal 1 to if the signature is valid or a value equal to 0 if otherwise. #### B.4.15.2. Version 2 - Prototype Does not allow the verification of non-standard, overflowing ECDSA signatures. (func$ext_crypto_ecdsa_verify_version_2 (param $sig i32) (param$msg i64) (param $key i32) (return i32)) Arguments • sig: a pointer (Definition 215) to the buffer containing the 65-byte signature. The signature is 65-bytes in size, where the first 512-bits represent the signature and the other 8 bits represent the recovery ID. • msg: a pointer-size (Definition 216) to the message that is to be verified. • key: a pointer to the buffer containing the 33-byte compressed public key. • return: a i32 integer value equal 1 to if the signature is valid or a value equal to 0 if otherwise. ### B.4.16. ext_crypto_ecdsa_verify_prehashed Verifies the prehashed message against a ECDSA signature. #### B.4.16.1. Version 1 - Prototype (func$ext_crypto_ecdsa_verify_prehashed_version_1 (param $sig i32) (param$msg i32) (param $key i32) (return i32)) Arguments • sig: a pointer (Definition 215) to the buffer containing the 65-byte signature. The signature is 65-bytes in size, where the first 512-bits represent the signature and the other 8 bits represent the recovery ID. • msg: a pointer to the 32-bit prehashed message to be verified. • key: a pointer to the 33-byte compressed public key. • return: a i32 integer value equal 1 to if the signature is valid or a value equal to 0 if otherwise. ### B.4.17. ext_crypto_ecdsa_batch_verify Registers a ECDSA signature for batch verification. Batch verification is enabled by calling ext_crypto_start_batch_verify (Section B.4.20.). The result of the verification is returned by ext_crypto_finish_batch_verify (Section B.4.21.). If batch verification is not enabled, the signature is verified immediately. #### B.4.17.1. Version 1 (func$ext_crypto_ecdsa_batch_verify_version_1 (param $sig i32) (param$msg i64) (param $key i32) (return i32)) Arguments • sig: a pointer (Definition 215) to the buffer containing the 64-byte signature. • msg: a pointer-size (Definition 216) to the message that is to be verified. • key: a pointer to the buffer containing the 256-bit public key. • return: a i32 integer value equal to 1 if the signature is valid or batched or a value equal 0 to if otherwise. ### B.4.18. ext_crypto_secp256k1_ecdsa_recover Verify and recover a secp256k1 ECDSA signature. #### B.4.18.1. Version 1 - Prototype This function can handle non-standard, overflowing ECDSA signatures, an implemenation specific mechanism of the Rust libsecp256k1 library, specifically the parse_overflowing function. (func$ext_crypto_secp256k1_ecdsa_recover_version_1 (param $sig i32) (param$msg i32) (return i64))\n\nArguments\n\n• sig: a pointer (Definition 215) to the buffer containing the 65-byte signature in RSV format. V should be either 0/1 or 27/28.\n\n• msg: a pointer (Definition 215) to the buffer containing the 256-bit Blake2 hash of the message.\n\n• return: a pointer-size (Definition 216) to the SCALE encoded Result (Definition 201). On success it contains the 64-byte recovered public key or an error type (Definition 221) on failure.\n\n#### B.4.18.2. Version 2 - Prototype\n\nDoes not handle non-standard, overflowing ECDSA signatures.\n\n(func $ext_crypto_secp256k1_ecdsa_recover_version_2 (param$sig i32) (param $msg i32) (return i64)) Arguments • sig: a pointer (Definition 215) to the buffer containing the 65-byte signature in RSV format. V should be either or . • msg: a pointer (Definition 215) to the buffer containing the 256-bit Blake2 hash of the message. • return: a pointer-size (Definition 216) to the SCALE encoded Result (Definition 201). On success it contains the 64-byte recovered public key or an error type (Definition 221) on failure. ### B.4.19. ext_crypto_secp256k1_ecdsa_recover_compressed Verify and recover a secp256k1 ECDSA signature. #### B.4.19.1. Version 1 - Prototype This function can handle non-standard, overflowing ECDSA signatures, an implemenation specific mechanism of the Rust libsecp256k1 library, specifically the parse_overflowing function. (func$ext_crypto_secp256k1_ecdsa_recover_compressed_version_1 (param $sig i32) (param$msg i32) (return i64))\n\nArguments\n\n• sig: a pointer (Definition 215) to the buffer containing the 65-byte signature in RSV format. V should be either 0/1 or 27/28.\n\n• msg: a pointer (Definition 215) to the buffer containing the 256-bit Blake2 hash of the message.\n\n• return: a pointer-size (Definition 216) to the SCALE encoded Result value (Definition 201). On success it contains the 33-byte recovered public key in compressed form on success or an error type (Definition 221) on failure.\n\n#### B.4.19.2. Version 2 - Prototype\n\nDoes not handle non-standard, overflowing ECDSA signatures.\n\nArguments\n\n• None.\n\n### B.4.21. ext_crypto_finish_batch_verify\n\nFinish verifying the batch of signatures since the last call to this function. Blocks until all the signatures are verified.\n\ncaution\n\nPanics if ext_crypto_start_batch_verify (Section B.4.20.) was not called.\n\n#### B.4.21.1. Version 1 - Prototype\n\n(func $ext_crypto_finish_batch_verify_version_1 (return i32)) Arguments • return: an i32 integer value equal to 1 if all the signatures are valid or a value equal to 0 if one or more of the signatures are invalid. ## B.5. Hashing Interface that provides functions for hashing with different algorithms. ### B.5.1. ext_hashing_keccak_256 Conducts a 256-bit Keccak hash. #### B.5.1.1. Version 1 - Prototype (func$ext_hashing_keccak_256_version_1 (param $data i64) (return i32)) Arguments • data: a pointer-size (Definition 216) to the data to be hashed. • return: a pointer (Definition 215) to the buffer containing the 256-bit hash result. ### B.5.2. ext_hashing_keccak_512 Conducts a 512-bit Keccak hash. #### B.5.2.1. Version 1 - Prototype (func$ext_hashing_keccak_512_version_1 (param $data i64) (return i32)) Arguments • data: a pointer-size (Definition 216) to the data to be hashed. • return: a pointer (Definition 215) to the buffer containing the 512-bit hash result. ### B.5.3. ext_hashing_sha2_256 Conducts a 256-bit Sha2 hash. #### B.5.3.1. Version 1 - Prototype (func$ext_hashing_sha2_256_version_1 (param $data i64) (return i32)) Arguments • data: a pointer-size (Definition 216) to the data to be hashed. • return: a pointer (Definition 215) to the buffer containing the 256-bit hash result. ### B.5.4. ext_hashing_blake2_128 Conducts a 128-bit Blake2 hash. #### B.5.4.1. Version 1 - Prototype (func$ext_hashing_blake2_128_version_1 (param $data i64) (return i32)) Arguments • data: a pointer-size (Definition 216) to the data to be hashed. • return: a pointer (Definition 215) to the buffer containing the 128-bit hash result. ### B.5.5. ext_hashing_blake2_256 Conducts a 256-bit Blake2 hash. #### B.5.5.1. Version 1 - Prototype (func$ext_hashing_blake2_256_version_1 (param $data i64) (return i32)) Arguments • data: a pointer-size (Definition 216) to the data to be hashed. • return: a pointer (Definition 215) to the buffer containing the 256-bit hash result. ### B.5.6. ext_hashing_twox_64 Conducts a 64-bit xxHash hash. #### B.5.6.1. Version 1 - Prototype (func$ext_hashing_twox_64_version_1 (param $data i64) (return i32)) Arguments • data: a pointer-size (Definition 216) to the data to be hashed. • return: a pointer (Definition 215) to the buffer containing the 64-bit hash result. ### B.5.7. ext_hashing_twox_128 Conducts a 128-bit xxHash hash. #### B.5.7.1. Version 1 - Prototype (func$ext_hashing_twox_128 (param $data i64) (return i32)) Arguments • data: a pointer-size (Definition 216) to the data to be hashed. • return: a pointer (Definition 215) to the buffer containing the 128-bit hash result. ### B.5.8. ext_hashing_twox_256 Conducts a 256-bit xxHash hash. #### B.5.8.1. Version 1 - Prototype (func$ext_hashing_twox_256 (param $data i64) (return i32)) Arguments • data: a pointer-size (Definition 216) to the data to be hashed. • return: a pointer (Definition 215) to the buffer containing the 256-bit hash result. ## B.6. Offchain The Offchain Workers allow the execution of long-running and possibly non-deterministic tasks (e.g. web requests, encryption/decryption and signing of data, random number generation, CPU-intensive computations, enumeration/aggregation of on-chain data, etc.) which could otherwise require longer than the block execution time. Offchain Workers have their own execution environment. This separation of concerns is to make sure that the block production is not impacted by the long-running tasks. All data and results generated by Offchain workers are unique per node and nondeterministic. Information can be propagated to other nodes by submitting a transaction that should be included in the next block. As Offchain workers runs on their own execution environment they have access to their own separate storage. There are two different types of storage available which are defined in Definition 222 and Definition 223. ###### Definition 222. Persisted Storage Persistent storage is non-revertible and not fork-aware. It means that any value set by the offchain worker is persisted even if that block (at which the worker is called) is reverted as non-canonical (meaning that the block was surpassed by a longer chain). The value is available for the worker that is re-run at the new (different block with the same block number) and future blocks. This storage can be used by offchain workers to handle forks and coordinate offchain workers running on different forks. ###### Definition 223. Local Storage Local storage is revertible and fork-aware. It means that any value set by the offchain worker triggered at a certain block is reverted if that block is reverted as non-canonical. The value is NOT available for the worker that is re-run at the next or any future blocks. ###### Definition 224. HTTP Status Code An enumerated data type that holds a finite set of distinct variants that gets SCALE-encoded as described in (Definition 199) and returned by offchain http functions. The set of variants is defined as follows. IdNameDescription 0DeadlineReachedthe deadline for the started request was reached. 1IoErroran error has occurred during the request. 2Invalidthe specified request identifier is invalid. 3Finished(http_status)the request has finished with the given HTTP status code. where • http_status: a 16-bit unsigned integer type representing the HTTP status code to be returned. ###### Definition 225. HTTP Error HTTP error, ${E}$, is a varying data type (Definition 198) and specifies the error types of certain HTTP functions. Following values are possible: ${E}={\\left\\lbrace\\begin{matrix}{0}&\\text{The deadile was reached}\\\\{1}&\\text{There was an IO error while processing the request}\\\\{2}&\\text{The Id of the request is invalid}\\end{matrix}\\right.}$ ### B.6.1. ext_offchain_is_validator Check whether the local node is a potential validator. Even if this function returns 1, it does not mean that any keys are configured or that the validator is registered in the chain. #### B.6.1.1. Version 1 - Prototype (func$ext_offchain_is_validator_version_1 (return i32))\n\nArguments\n\n• return: a i32 integer which is equal to 1 if the local node is a potential validator or a integer equal to 0 if it is not.\n\n### B.6.2. ext_offchain_submit_transaction\n\nGiven a SCALE encoded extrinsic, this function submits the extrinsic to the Host’s transaction pool, ready to be propagated to remote peers.\n\n#### B.6.2.1. Version 1 - Prototype\n\n(func $ext_offchain_submit_transaction_version_1 (param$data i64) (return i64))\n\nArguments\n\n• data: a pointer-size (Definition 216) to the byte array storing the encoded extrinsic.\n\n• return: a pointer-size (Definition 216) to the SCALE encoded Result value (Definition 201). Neither on success or failure is there any additional data provided. The cause of a failure is implementation specific.\n\n### B.6.3. ext_offchain_network_state\n\nReturns the SCALE encoded, opaque information about the local node’s network state.\n\n###### Definition 226. Opaque Network State\n\nThe Opaque network state structure, ${S}$, is a SCALE encoded blob holding information about the the libp2p PeerId, ${P}_{{\\text{id}}}$, of the local node and a list of libp2p Multiaddresses, ${\\left({M}_{{0}},\\ldots{M}_{{n}}\\right)}$, the node knows it can be reached at:\n\n${S}={\\left({P}_{{\\text{id}}},{\\left({M}_{{0}},\\ldots{M}_{{n}}\\right)}\\right)}$\n\nwhere\n\n${P}_{{\\text{id}}}={\\left({b}_{{0}},\\ldots{b}_{{n}}\\right)}$\n${M}={\\left({b}_{{0}},\\ldots{b}_{{n}}\\right)}$\n\nThe information contained in this structure is naturally opaque to the caller of this function.\n\n(func $ext_offchain_network_state_version_1 (result i64)) Arguments • result: a pointer-size (Definition 216) to the SCALE encoded Result value (Definition 201). On success it contains the Opaque network state structure (Definition 226). On failure, an empty value is yielded where its cause is implementation specific. ### B.6.4. ext_offchain_timestamp Returns the current timestamp. #### B.6.4.1. Version 1 - Prototype (func$ext_offchain_timestamp_version_1 (result i64))\n\nArguments\n\n• result: an u64 integer (typed as i64 due to wasm types) indicating the current UNIX timestamp (Definition 191).\n\n### B.6.5. ext_offchain_sleep_until\n\nPause the execution until the deadline is reached.\n\n#### B.6.5.1. Version 1 - Prototype\n\n(func $ext_offchain_sleep_until_version_1 (param$deadline i64))\n\nArguments\n\n• deadline: an u64 integer (typed as i64 due to wasm types) specifying the UNIX timestamp (Definition 191).\n\n### B.6.6. ext_offchain_random_seed\n\nGenerates a random seed. This is a truly random non deterministic seed generated by the host environment.\n\n(func $ext_offchain_random_seed_version_1 (result i32)) Arguments • result: a pointer (Definition 215) to the buffer containing the 256-bit seed. ### B.6.7. ext_offchain_local_storage_set Sets a value in the local storage. This storage is not part of the consensus, it’s only accessible by the offchain worker tasks running on the same machine and is persisted between runs. #### B.6.7.1. Version 1 - Prototype (func$ext_offchain_local_storage_set_version_1 (param $kind i32) (param$key i64) (param $value i64)) Arguments • kind: an i32 integer indicating the storage kind. A value equal to 1 is used for a persistent storage (Definition 222) and a value equal to 2 for local storage (Definition 223). • key: a pointer-size (Definition 216) to the key. • value: a pointer-size (Definition 216) to the value. ### B.6.8. ext_offchain_local_storage_clear Remove a value from the local storage. #### B.6.8.1. Version 1 - Prototype (func$ext_offchain_local_storage_clear_version_1 (param $kind i32) (param$key i64))\n\nArguments\n\n• kind: an i32 integer indicating the storage kind. A value equal to 1 is used for a persistent storage (Definition 222) and a value equal to 2 for local storage (Definition 223).\n\n• key: a pointer-size (Definition 216) to the key.\n\n### B.6.9. ext_offchain_local_storage_compare_and_set\n\nSets a new value in the local storage if the condition matches the current value.\n\n(fund $ext_offchain_local_storage_compare_and_set_version_1 (param$kind i32) (param $key i64) (param$old_value i64) (param $new_value i64) (result i32)) Arguments • kind: an i32 integer indicating the storage kind. A value equal to 1 is used for a persistent storage (Definition 222) and a value equal to 2 for local storage (Definition 223). • key: a pointer-size (Definition 216) to the key. • old_value: a pointer-size (Definition 216) to the SCALE encoded Option value (Definition 200) containing the old key. • new_value: a pointer-size (Definition 216) to the new value. • result: an i32 integer equal to 1 if the new value has been set or a value equal to 0 if otherwise. ### B.6.10. ext_offchain_local_storage_get Gets a value from the local storage. #### B.6.10.1. Version 1 - Prototype (func$ext_offchain_local_storage_get_version_1 (param $kind i32) (param$key i64) (result i64))\n\nArguments\n\n• kind: an i32 integer indicating the storage kind. A value equal to 1 is used for a persistent storage (Definition 222) and a value equal to 2 for local storage (Definition 223).\n\n• key: a pointer-size (Definition 216) to the key.\n\n• result: a pointer-size (Definition 216) to the SCALE encoded Option value (Definition 200) containing the value or the corresponding key.\n\n### B.6.11. ext_offchain_http_request_start\n\nInitiates a HTTP request given by the HTTP method and the URL. Returns the Id of a newly started request.\n\n#### B.6.11.1. Version 1 - Prototype\n\n(func $ext_offchain_http_request_start_version_1 (param$method i64) (param $uri i64) (param$meta i64) (result i64))\n\nArguments\n\n• method: a pointer-size (Definition 216) to the HTTP method. Possible values are “GET” and “POST”.\n\n• uri: a pointer-size (Definition 216) to the URI.\n\n• meta: a future-reserved field containing additional, SCALE encoded parameters. Currently, an empty array should be passed.\n\n• result: a pointer-size (Definition 216) to the SCALE encoded Result value (Definition 201) containing the i16 ID of the newly started request. On failure no additionally data is provided. The cause of failure is implementation specific.\n\n### B.6.12. ext_offchain_http_request_add_header\n\nAppend header to the request. Returns an error if the request identifier is invalid, http_response_wait has already been called on the specified request identifier, the deadline is reached or an I/O error has happened (e.g. the remote has closed the connection).\n\n#### B.6.12.1. Version 1 - Prototype\n\n(func $ext_offchain_http_request_add_header_version_1 (param$request_id i32) (param $name i64) (param$value i64) (result i64))\n\nArguments\n\n• request_id: an i32 integer indicating the ID of the started request.\n\n• name: a pointer-size (Definition 216) to the HTTP header name.\n\n• value: a pointer-size (Definition 216) to the HTTP header value.\n\n• result: a pointer-size (Definition 216) to the SCALE encoded Result value (Definition 201). Neither on success or failure is there any additional data provided. The cause of failure is implementation specific.\n\n### B.6.13. ext_offchain_http_request_write_body\n\nWrites a chunk of the request body. Returns a non-zero value in case the deadline is reached or the chunk could not be written.\n\n#### B.6.13.1. Version 1 - Prototype\n\n(func $ext_offchain_http_request_write_body_version_1 (param$request_id i32) (param $chunk i64) (param$deadline i64) (result i64))\n\nArguments\n\n• request_id: an i32 integer indicating the ID of the started request.\n\n• chunk: a pointer-size (Definition 216) to the chunk of bytes. Writing an empty chunk finalizes the request.\n\n• deadline: a pointer-size (Definition 216) to the SCALE encoded Option value (Definition 200) containing the UNIX timestamp (Definition 191). Passing None blocks indefinitely.\n\n• result: a pointer-size (Definition 216) to the SCALE encoded Result value (Definition 201). On success, no additional data is provided. On error it contains the HTTP error type (Definition 225).\n\n### B.6.14. ext_offchain_http_response_wait\n\nReturns an array of request statuses (the length is the same as IDs). Note that if deadline is not provided the method will block indefinitely, otherwise unready responses will produce DeadlineReached status.\n\n(func $ext_offchain_http_response_wait_version_1 (param$ids i64) (param $deadline i64) (result i64)) Arguments ### B.6.15. ext_offchain_http_response_headers Read all HTTP response headers. Returns an array of key/value pairs. Response headers must be read before the response body. #### B.6.15.1. Version 1 - Prototype (func$ext_offchain_http_response_headers_version_1 (param $request_id i32) (result i64)) Arguments • request_id: an i32 integer indicating the ID of the started request. • result: a pointer-size (Definition 216) to a SCALE encoded array of key/value pairs. ### B.6.16. ext_offchain_http_response_read_body Reads a chunk of body response to the given buffer. Returns the number of bytes written or an error in case a deadline is reached or the server closed the connection. If 0 is returned it means that the response has been fully consumed and the request_id is now invalid. This implies that response headers must be read before draining the body. #### B.6.16.1. Version 1 - Prototype (func$ext_offchain_http_response_read_body_version_1 (param $request_id i32) (param$buffer i64) (param $deadline i64) (result i64)) Arguments • request_id: an i32 integer indicating the ID of the started request. • buffer: a pointer-size (Definition 216) to the buffer where the body gets written to. • deadline: a pointer-size (Definition 216) to the SCALE encoded Option value (Definition 200) containing the UNIX timestamp (Definition 191). Passing None will block indefinitely. • result: a pointer-size (Definition 216) to the SCALE encoded Result value (Definition 201). On success it contains an i32 integer specifying the number of bytes written or a HTTP error type (Definition 225) on failure. ## B.7. Offchain Index Interface that provides functions to access the Offchain DB through offchain indexing. ### B.7.1. Offchain_index_set Write a key-value pair to the Offchain DB in a buffered fashion. #### B.7.1.1. Version 1 - Prototype (func$ext_offchain_index_set_version_1 (param $key i64) (param$value i64))\n\nArguments\n\n• key: a pointer-size (Definition 216) containing the key.\n\n• value: a pointer-size (Definition 216) containing the value.\n\n### B.7.2. Offchain_index_clear\n\nRemove a key and its associated value from the Offchain DB.\n\n#### B.7.2.1. Version 1 - Prototype\n\n(func $ext_offchain_index_clear_version_1 (param$key i64))\n\nArguments\n\n• key: a pointer-size (Definition 216) containing the key.\n\n## B.8. Trie\n\nInterface that provides trie related functionality.\n\n### B.8.1. ext_trie_blake2_256_root\n\nCompute a 256-bit Blake2 trie root formed from the iterated items.\n\n#### B.8.1.1. Version 1 - Prototype\n\n(func $ext_trie_blake2_256_root_version_1 (param$data i64) (result i32))\n\nArguments\n\n• data: a pointer-size (Definition 216) to the iterated items from which the trie root gets formed. The items consist of a SCALE encoded array containing arbitrary key/value pairs (tuples).\n\n• result: a pointer (Definition 215) to the buffer containing the 256-bit trie root.\n\n(func $ext_trie_blake2_256_root_version_2 (param$data i64) (param $version i32) (result i32)) Arguments • data: a pointer-size (Definition 216) to the iterated items from which the trie root gets formed. The items consist of a SCALE encoded array containing arbitrary key/value pairs (tuples). • version: the state version (Definition 218). • result: a pointer (Definition 215) to the buffer containing the 256-bit trie root. ### B.8.2. ext_trie_blake2_256_ordered_root Compute a 256-bit Blake2 trie root formed from the enumerated items. #### B.8.2.1. Version 1 - Prototype (func$ext_trie_blake2_256_ordered_root_version_1 (param $data i64) (result i32)) Arguments • data: a pointer-size (Definition 216) to the enumerated items from which the trie root gets formed. The items consist of a SCALE encoded array containing only values, where the corresponding key of each value is the index of the item in the array, starting at 0. The keys are compact encoded integers (Definition 208). • result: a pointer (Definition 215) to the buffer containing the 256-bit trie root result. #### B.8.2.2. Version 2 - Prototype (func$ext_trie_blake2_256_ordered_root_version_2 (param $data i64) (param$version i32) (result i32))\n\nArguments\n\n• data: a pointer-size (Definition 216) to the enumerated items from which the trie root gets formed. The items consist of a SCALE encoded array containing only values, where the corresponding key of each value is the index of the item in the array, starting at 0. The keys are compact encoded integers (Definition 208).\n\n• version: the state version (Definition 218).\n\n• result: a pointer (Definition 215) to the buffer containing the 256-bit trie root result.\n\n### B.8.3. ext_trie_keccak_256_root\n\nCompute a 256-bit Keccak trie root formed from the iterated items.\n\n#### B.8.3.1. Version 1 - Prototype\n\n(func $ext_trie_keccak_256_root_version_1 (param$data i64) (result i32))\n\nArguments\n\n• data: a pointer-size (Definition 216) to the iterated items from which the trie root gets formed. The items consist of a SCALE encoded array containing arbitrary key/value pairs.\n\n• result: a pointer (Definition 215) to the buffer containing the 256-bit trie root.\n\n(func $ext_trie_keccak_256_root_version_2 (param$data i64) (param $version i32) (result i32)) Arguments • data: a pointer-size (Definition 216) to the iterated items from which the trie root gets formed. The items consist of a SCALE encoded array containing arbitrary key/value pairs. • version: the state version (Definition 218). • result: a pointer (Definition 215) to the buffer containing the 256-bit trie root. ### B.8.4. ext_trie_keccak_256_ordered_root Compute a 256-bit Keccak trie root formed from the enumerated items. #### B.8.4.1. Version 1 - Prototype (func$ext_trie_keccak_256_ordered_root_version_1 (param $data i64) (result i32)) Arguments • data: a pointer-size (Definition 216) to the enumerated items from which the trie root gets formed. The items consist of a SCALE encoded array containing only values, where the corresponding key of each value is the index of the item in the array, starting at 0. The keys are compact encoded integers (Definition 208). • result: a pointer (Definition 215) to the buffer containing the 256-bit trie root result. #### B.8.4.2. Version 2 - Prototype (func$ext_trie_keccak_256_ordered_root_version_2 (param $data i64) (param$version i32) (result i32))\n\nArguments\n\n• data: a pointer-size (Definition 216) to the enumerated items from which the trie root gets formed. The items consist of a SCALE encoded array containing only values, where the corresponding key of each value is the index of the item in the array, starting at 0. The keys are compact encoded integers (Definition 208).\n\n• version: the state version (Definition 218).\n\n• result: a pointer (Definition 215) to the buffer containing the 256-bit trie root result.\n\n### B.8.5. ext_trie_blake2_256_verify_proof\n\nVerifies a key/value pair against a Blake2 256-bit merkle root.\n\n(func $ext_trie_blake2_256_verify_proof_version_1 (param$root i32) (param $proof i64) (param$key i64) (param $value i64) (result i32)) Arguments • root: a pointer to the 256-bit merkle root. • proof: a pointer-size (Definition 216) to an array containing the node proofs. • key: a pointer-size (Definition 216) to the key. • value: a pointer-size (Definition 216) to the value. • return: a value equal to 1 if the proof could be successfully verified or a value equal to 0 if otherwise. #### B.8.5.2. Version 2 - Prototype (func$ext_trie_blake2_256_verify_proof_version_2 (param $root i32) (param$proof i64) (param $key i64) (param$value i64) (param $version i32) (result i32)) Arguments • root: a pointer to the 256-bit merkle root. • proof: a pointer-size (Definition 216) to an array containing the node proofs. • key: a pointer-size (Definition 216) to the key. • value: a pointer-size (Definition 216) to the value. • version: the state version (Definition 218). • return: a value equal to 1 if the proof could be successfully verified or a value equal to 0 if otherwise. ### B.8.6. ext_trie_keccak_256_verify_proof Verifies a key/value pair against a Keccak 256-bit merkle root. #### B.8.6.1. Version 1 - Prototype (func$ext_trie_keccak_256_verify_proof_version_1 (param $root i32) (param$proof i64) (param $key i64) (param$value i64) (result i32))\n\nArguments\n\n• root: a pointer to the 256-bit merkle root.\n\n• proof: a pointer-size (Definition 216) to an array containing the node proofs.\n\n• key: a pointer-size (Definition 216) to the key.\n\n• value: a pointer-size (Definition 216) to the value.\n\n• return: a value equal to 1 if the proof could be successfully verified or a value equal to 0 if otherwise.\n\n#### B.8.6.2. Version 2 - Prototype\n\n(func $ext_trie_keccak_256_verify_proof_version_2 (param$root i32) (param $proof i64) (param$key i64) (param $value i64) (param$version i32) (result i32))\n\nArguments\n\n• root: a pointer to the 256-bit merkle root.\n\n• proof: a pointer-size (Definition 216) to an array containing the node proofs.\n\n• key: a pointer-size (Definition 216) to the key.\n\n• value: a pointer-size (Definition 216) to the value.\n\n• version: the state version (Definition 218).\n\n• return: a value equal to 1 if the proof could be successfully verified or a value equal to 0 if otherwise.\n\n## B.9. Miscellaneous\n\nInterface that provides miscellaneous functions for communicating between the runtime and the node.\n\n### B.9.1. ext_misc_print_num\n\nPrint a number.\n\n#### B.9.1.1. Version 1 - Prototype\n\n(func $ext_misc_print_num_version_1 (param$value i64))\n\nArguments\n\n• value: the number to be printed.\n\n### B.9.2. ext_misc_print_utf8\n\nPrint a valid UTF8 encoded buffer.\n\n#### B.9.2.1. Version 1 - Prototype\n\n(func $ext_misc_print_utf8_version_1 (param$data i64))\n\nArguments:\n\n### B.9.3. ext_misc_print_hex\n\nPrint any buffer in hexadecimal representation.\n\n#### B.9.3.1. Version 1 - Prototype\n\n(func $ext_misc_print_hex_version_1 (param$data i64))\n\nArguments:\n\n• data: a pointer-size (Definition 216) to the buffer to be printed.\n\n### B.9.4. ext_misc_runtime_version\n\nExtract the Runtime version of the given Wasm blob by calling Core_version (Section C.4.1.). Returns the SCALE encoded runtime version or None (Definition 200) if the call fails. This function gets primarily used when upgrading Runtimes.\n\ncaution\n\nCalling this function is very expensive and should only be done very occasionally. For getting the runtime version, it requires instantiating the Wasm blob (Section 2.6.2.) and calling the Core_version function (Section C.4.1.) in this blob.\n\n#### B.9.4.1. Version 1 - Prototype\n\n(func $ext_misc_runtime_version_version_1 (param$data i64) (result i64))\n\nArguments\n\n• data: a pointer-size (Definition 216) to the Wasm blob.\n\n• result: a pointer-size (Definition 216) to the SCALE encoded Option value (Definition 200) containing the Runtime version of the given Wasm blob which is encoded as a byte array.\n\n## B.10. Allocator\n\nThe Polkadot Runtime does not include a memory allocator and relies on the Host API for all heap allocations. The beginning of this heap is marked by the __heap_base symbol exported by the Polkadot Runtime. No memory should be allocated below that address, to avoid clashes with the stack and data section. The same allocator made accessible by this Host API should be used for any other WASM memory allocations and deallocations outside the runtime e.g. when passing the SCALE-encoded parameters to Runtime API calls.\n\n### B.10.1. ext_allocator_malloc\n\nAllocates the given number of bytes and returns the pointer to that memory location.\n\n#### B.10.1.1. Version 1 - Prototype\n\n(func $ext_allocator_malloc_version_1 (param$size i32) (result i32))\n\nArguments\n\n• size: the size of the buffer to be allocated.\n\n• result: a pointer (Definition 215) to the allocated buffer.\n\n### B.10.2. ext_allocator_free\n\nFree the given pointer.\n\n#### B.10.2.1. Version 1 - Prototype\n\n(func $ext_allocator_free_version_1 (param$ptr i32))\n\nArguments\n\n• ptr: a pointer (Definition 215) to the memory buffer to be freed.\n\n## B.11. Logging\n\nInterface that provides functions for logging from within the runtime.\n\n###### Definition 227. Log Level\n\nThe Log Level, ${L}$, is a varying data type (Definition 198) and implies the emergency of the log. Possible log levels and the corresponding identifier is as follows:\n\n${L}={\\left\\lbrace\\begin{matrix}{0}&\\text{Error = 1}\\\\{1}&\\text{Warn = 2}\\\\{2}&\\text{Info = 3}\\\\{3}&\\text{Debug = 4}\\\\{4}&\\text{Trace = 5}\\end{matrix}\\right.}$\n\n### B.11.1. ext_logging_log\n\nRequest to print a log message on the host. Note that this will be only displayed if the host is enabled to display log messages with given level and target.\n\n#### B.11.1.1. Version 1 - Prototype\n\n(func $ext_logging_log_version_1 (param$level i32) (param $target i64) (param$message i64))\n\nArguments\n\n• level: the log level (Definition 227).\n\n• target: a pointer-size (Definition 216) to the string which contains the path, module or location from where the log was executed.\n\n• message: a pointer-size (Definition 216) to the UTF-8 encoded log message.\n\n### B.11.2. ext_logging_max_level\n\nReturns the max logging level used by the host.\n\n(func $ext_logging_max_level_version_1 (result i32)) Arguments • None Returns • result: the max log level (Definition 227) used by the host. ## B.12. Abort Handler Interface for aborting the execution of the runtime. ### B.12.1. ext_panic_handler_abort_on_panic Aborts the execution of the runtime with a given message. Note that the message will be only displayed if the host is enabled to display those types of messages, which is implementation specific. #### B.12.1.1. Version 1 - Prototype (func$ext_panic_handler_abort_on_panic_version_1 (param \\$message i64))\n\nArguments\n\n• message: a pointer-size (Definition 216) to the UTF-8 encoded message." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6075522,"math_prob":0.87096107,"size":62213,"snap":"2023-40-2023-50","text_gpt3_token_len":15825,"char_repetition_ratio":0.2552846,"word_repetition_ratio":0.52895236,"special_character_ratio":0.27000788,"punctuation_ratio":0.14618029,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9583082,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-04T12:18:33Z\",\"WARC-Record-ID\":\"<urn:uuid:063fb128-6d6d-4dac-9437-6a09aec29361>\",\"Content-Length\":\"345051\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0b118557-2ce1-4293-8eb6-97e50c91a2ee>\",\"WARC-Concurrent-To\":\"<urn:uuid:3a8b7f25-ad27-4646-b108-bfa52385035e>\",\"WARC-IP-Address\":\"172.67.70.100\",\"WARC-Target-URI\":\"https://spec.polkadot.network/chap-host-api\",\"WARC-Payload-Digest\":\"sha1:7LTGAX3OMASOGLB4JBBFYEAEAE2UPRND\",\"WARC-Block-Digest\":\"sha1:QKRPOIJLR2A6BXJKGOAZMQ5GRDASZRRT\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100529.8_warc_CC-MAIN-20231204115419-20231204145419-00748.warc.gz\"}"}
https://books.google.com.gi/books?qtid=4f7e78de&id=C0gAAAAAYAAJ&lr=&sa=N&start=90	[ "Books Books", null, "Every circumference of a. circle, whether the circle be large or small, is supposed to be divided into 360 equal parts called degrees. Each degree is divided into 60 equal parts called minutes, and each minute into 60 equal parts called seconds.", null, "Easy Lessons in Geography and History: Designed for the Use of the Younger ... - Page 39\nby Joseph Allen - 1832 - 116 pages", null, "## Elementary Geometry: With Applications in Mensuration\n\nCharles Davies - Geometry - 1850 - 236 pages\n...measurement of angles. For this purpose it is divided into 360 equal parts called degrees, each degree into 60 equal parts called minutes, and each minute into 60 equal parts called seconds. The degrees, minutes, and seconds are marked thus ° ' \" ; and 9° 18' 16\", are read, 9 degrees 18...", null, "## Elementary Geometry: With Applications in Mensuration\n\nCharles Davies - Geometry - 1850 - 218 pages\n...measurement of angles. For this purpose it is divided into 360 equal parts called degrees, each degree into 60 equal parts called minutes, and each minute into 60 equal parts called seconds. The degrees, minutes, and seconds are marked thus ° ' \" ; and 9° 18' 16\", are read, 9 degrees 18...", null, "## The University Arithmetic: Embracing the Science of Numbers, and Their ...\n\nCharles Davies - Arithmetic - 1850 - 412 pages\n...OF TIME. 40. Every circle is supposed to be divided into 360 equal parts called degrees, each degree into 60 equal parts called minutes, and each minute into .60 equal parts called seconds. For astronomical purposes, the circumference of the circle is also supposed to be divided into 12 equal...", null, "## Mensuration, Mechanical Powers, and Machinery: The Principles of Mensuration ...\n\nDaniel Adams - Measurement - 1850 - 144 pages\n...10. The circumference of every circle is divided into 360 equal parts, called Degrees ; each degree into 60 equal parts, called Minutes ; and each minute into 60 equal parts, called Seconds. 11. Degrees, minutes, and seconds, are marked respectively °, ', \" ; they are used in mensuration...", null, "## Arithmetic on the Productive System: Accompanied by a Key and ..., Volume 1\n\nRoswell Chamberlain Smith - Arithmetic - 1850 - 314 pages\n...through the centre of the earth. 15 CORRESPOND, [L. corrcspondeo.] To answer to; communicate with. parts, called minutes, and each minute into 60 equal parts, called seconds. 76. The circumference1 of the earth is a great circle of 360 de grees. On this circle every minute...", null, "## A Manual of Surveying for India, Detailing the Mode of Operations on the ...\n\nSir Henry Edward Landor Thuillier - Surveying - 1851 - 828 pages\n...circle is supposed to be divided off into 360 equal parts, called degrees, each degree is subdivided into 60 equal parts, called minutes, and each minute into 60 equal parts, called seconds. Degrees are expressed thus:0 minutes, thus/ seconds, thus:\" A Quadrant of a circle will therefore contain...", null, "## The Decimal System of Numbers: Illustrated and Practically Applied, by a ...\n\nDana Pond Colburn - Arithmetic - 1852 - 228 pages\n...whether the circle be large or small, is supposed to be divided into 360 equal parts called degrees. Each degree is divided into 60 equal parts called...and each minute into 60 equal parts called seconds. A degree is to be regarded simply as the 360th part of the circumference of the circle considered....", null, "## A complete treatise on practical land-surveying\n\nAnthony Nesbit - 1870 - 578 pages\n...circumference of every circle is supposed to be divided into 360 equal parts, called degrees; each degree into 60 equal parts, called minutes ; and each minute into 60 equal parts, called seconds. 40. The diameter of a circle is a right line drawn through the centre, A and terminating in the circumference...", null, "## Arithmetic and Its Applications ...\n\nDana Pond Colburn - 1871 - 392 pages\n...whether the circle be 1vge or small, is supposed to be divided into 360 equal pwrtsf called degrees. Each degree is divided into 60 equal parts, called...minutes, and each minute into 60 equal parts, called èeconds. (g.) A degree is to be regarded simply as the 860th part of the circumference of the circle...", null, "" ]	[ null, "https://books.google.com.gi/googlebooks/quote_l.gif", null, "https://books.google.com.gi/googlebooks/quote_r.gif", null, "https://books.google.com.gi/books/content", null, "https://books.google.com.gi/books/content", null, "https://books.google.com.gi/books/content", null, "https://books.google.com.gi/books/content", null, "https://books.google.com.gi/books/content", null, "https://books.google.com.gi/books/content", null, "https://books.google.com.gi/books/content", null, "https://books.google.com.gi/books/content", null, "https://books.google.com.gi/books/content", null, "https://books.google.com.gi/books/content", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.92034096,"math_prob":0.9822907,"size":3462,"snap":"2023-14-2023-23","text_gpt3_token_len":845,"char_repetition_ratio":0.23597455,"word_repetition_ratio":0.3979933,"special_character_ratio":0.28798383,"punctuation_ratio":0.22130014,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.986683,"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],"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],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-06-06T17:24:25Z\",\"WARC-Record-ID\":\"<urn:uuid:1cf395a0-f555-4110-9096-8d8e2f0666ef>\",\"Content-Length\":\"33464\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:8351662f-2733-441b-9d9b-346445ee4c9f>\",\"WARC-Concurrent-To\":\"<urn:uuid:0b790d8d-9231-4726-9346-c858d8a0dfb0>\",\"WARC-IP-Address\":\"142.251.167.100\",\"WARC-Target-URI\":\"https://books.google.com.gi/books?qtid=4f7e78de&id=C0gAAAAAYAAJ&lr=&sa=N&start=90\",\"WARC-Payload-Digest\":\"sha1:2UZ33DT45SG3ISWZIXKVJLMRZGD73YBB\",\"WARC-Block-Digest\":\"sha1:SFW65NGVUFBRNKPHBOYGZ7ATRTNZFQMV\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-23/CC-MAIN-2023-23_segments_1685224652959.43_warc_CC-MAIN-20230606150510-20230606180510-00229.warc.gz\"}"}
https://classes.golem.ph.utexas.edu/category/2009/07/generalized_homotopy_theory.html	[ "## July 6, 2009\n\n### Generalized Homotopy Theory\n\n#### Posted by David Corfield", null, "Over at $n$Lab we’re itching for some discussion as to whether there can be something which is to homotopy as Nonabelian (unstable) cohomology is to cohomology. Can we free things up so we don’t just map spheres into spaces? At the entry homotopy (as an operation) you can read the suggestion of a ‘homotopy with co-coefficients in $B$’, rather than a sphere.\n\nI suppose Moore spaces as domain would be a start as suggested here for spaces of type $(A, 2)$ and suspensions.\n\nA couple of things to check out perhaps:\n\nI have the feeling they’ll be a huge amount out there to learn, e.g., about cofibrant co-grouplike objects.\n\nPosted at July 6, 2009 5:01 PM UTC\n\nTrackBack URL for this Entry: https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2009\n\n### Re: Generalized Homotopy Theory\n\nIn a 1996 discussion on the cat-list André Joyal writes\n\nPeter Hilton and Beno Eckmann…proved that any cogroup in the category of groups is free.\n\nI would like to say that the younger generation is not always aware that Eckmann and Hilton have fundamental contributions to category theory. They have provided basic examples of objects equipped with algebraic structures in categories. Consequently opening the road to further abstractions, like the concept of algebraic theory in the sense of Lawvere.\n\nAmong other things, Eckmann and Hilton were interested in identifying all cogroups in the homotopy category $hTop_$ of pointed topological spaces. A basic example of such cogroup is the circle $S^1$. It explains why the functor $\\pi_1(-)=[S^1,-]: hTop_ \\to Sets$ has a natural group structure. If $G$ is a cogroup in $hTop_$ then so is the smash product $X \\wedge G$ for any pointed topological space $X$. This is because $X \\wedge (-)$ preserves coproduct since it has a right adjoint (here we are supposing that Top is a convenient category of topological space). In particular, the spheres $S^{n+1} = S^n \\wedge S^1$ have a cogroup structure. Any wedge (topologists call the coproduct the wedge) of cogroups is obviously a cogroup. In particular, any wedge of spheres of dimension $n \\geq 1$ has a co-group structure.\n\nAll the known examples of cogroups in $hTop_$ are obtained by taking a smash $X \\wedge S^1$. It was conjectured by Eckmann and Hilton that all cogroups in $hTop^$ are of the form $X \\wedge S^1$. They observe that $\\pi_1(G)$ is a cogroup in $Groups$ when $G$ is a cogroup in $hTop_$. This is because the functor $\\pi_1: hTop_* \\to Groups$ preserves coproducts by Van Kampen theorem. In support to their conjecture they proved that any cogroup in $Groups$ is free. It follows that all cogroups in $Groups$ are of the form $\\pi_1(X \\wedge S^1)$ where $X$ is a pointed set.\n\nPosted by: David Corfield on July 6, 2009 9:45 PM \| Permalink \| Reply to this\n\n### Re: Generalized Homotopy Theory\n\nAs an example of the sort of theory that is available for this, Baues in his book Algebraic Homotopy’ (section II.6, starting on page 115), develops the notion of homotopy groups in a cofibration category, using suspensions $\\pi_n^A(U) = [\\Sigma^n A,U].$ These have all the usual elementary properties with relative forms, long exact sequences etc. $A$ has to be a based object for it to work.\n\nThe development is quite detailed and lengthy.\n\nPosted by: Tim Porter on July 7, 2009 12:26 PM \| Permalink \| Reply to this\n\n### Re: Generalized Homotopy Theory\n\nSo these are abelian for $n \\gt 1$. Presumably stablization of homotopy groups with coefficients goes through.\n\nPosted by: David Corfield on July 7, 2009 2:22 PM \| Permalink \| Reply to this\n\n### Re: Generalized Homotopy Theory\n\nDavid you say: “Presumably stablization of homotopy groups with coefficients goes through.” I could not find a mention of stabilisation with in Baues’ book. (Perhaps he treats that in another text.)\n\nI do not quite see what you mean by “homotopy groups with coefficients”. Are the $A$ somehow the coefficient? (I suppose the duality could lead either to (co)$^2$efficients or worse efficients’ in that case … I know I should have resisted the obvious pun.)\n\nPosted by: Tim Porter on July 7, 2009 2:39 PM \| Permalink \| Reply to this\n\n### Re: Generalized Homotopy Theory\n\nUrs is using the phrase ‘homotopy of $X$ with co-coefficients in $B$’ at the speculative page homotopy (as an operation), and is also tempted by that bad pun.\n\nHatcher in section 4.H talks about ‘Homotopy Groups with Coefficients’, using Moore spaces as domain. Presumably in this case $[\\Sigma^n (M(k, G)), \\Sigma^n (X)]$ stabilizes as $n \\to \\infty$.\n\nPosted by: David Corfield on July 7, 2009 3:06 PM \| Permalink \| Reply to this\n\n### Re: Generalized Homotopy Theory\n\nthe notion of homotopy groups [with co-co-efficients] in a cofibration category, using suspensions\n\n$\\pi_n^A(U) = [\\Sigma^n A, U]$\n\nHey, that’s great. That’s pretty much what David was looking for!\n\nWe are gonna work this into [[homotopy]] to make it become more completely the abstract dual of [[cohomology]].\n\nWhile I am looking at the book:\n\n“cofibration category” is that [[cat of fib objects]] opposed or is it [[Waldhausen category]]?\n\nPosted by: Urs Schreiber on July 7, 2009 7:12 PM \| Permalink \| Reply to this\n\n### Re: Generalized Homotopy Theory\n\nFrom memory it’s a category of cofibrant objects, or the same with minor variations. More suited to $sSet$ than $Top$\n\nPosted by: David Roberts on July 8, 2009 5:27 AM \| Permalink \| Reply to this\n\n### Re: Generalized Homotopy Theory\n\nYes, it is categories of cofibrant objects (in the Baues definition in which the axioms are the duals of Ken Brown’s axioms with some slight additions). Many of the arguments used are really just the Dold-Puppe fibration / cofibration long exact sequences suitably abstracted.\n\n(I tried to post this yesterday but the system went AWOL!)\n\nPosted by: Tim Porter on July 8, 2009 10:52 AM \| Permalink \| Reply to this\n\n### Re: Generalized Homotopy Theory\n\nYes, these should be sorted out. It all looks almost trivially equivalent, but it seems one has to exercis a bit of care here and there for precisely relating these axiom systems.\n\nI see that Baues in his remark (1a.6) comments briefly and roughly on the slight variations in the axioms of\n\n-Brown cat of fibs / dually of cofibs\n\n- Anderson cat of cofibs\n\n- Heller and Shitanda who do something else\n\n- Waldhausen of cofibs.\n\nOne easy thing is how the factorization lemma assumed by Baues implies the existence of (co)path objects assumed by Brown (as in his remark on p. 422), while Brown of course proves the converse. So that bit is clear.\n\nI have yet to think about the other differences.\n\nIt would be good to eventually list the relations somewhere, it’s slightly annyong that with each author one needs to start with slightly new axioms.\n\nPosted by: Urs Schreiber on July 8, 2009 1:06 PM \| Permalink \| Reply to this\n\nPost a New Comment" ]	[ null, "https://golem.ph.utexas.edu/~distler/blog/images/MathML.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.91864425,"math_prob":0.86455,"size":855,"snap":"2022-40-2023-06","text_gpt3_token_len":222,"char_repetition_ratio":0.10810811,"word_repetition_ratio":0.0,"special_character_ratio":0.21169591,"punctuation_ratio":0.10119048,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97617346,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-02-04T02:50:27Z\",\"WARC-Record-ID\":\"<urn:uuid:778f99a9-4b87-4654-ab13-bdf60952ac0b>\",\"Content-Length\":\"36498\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:016010f0-49f7-4e43-9936-9de9caf65e2e>\",\"WARC-Concurrent-To\":\"<urn:uuid:1ec560d3-91c4-4009-92f2-2c50046e7900>\",\"WARC-IP-Address\":\"128.83.16.204\",\"WARC-Target-URI\":\"https://classes.golem.ph.utexas.edu/category/2009/07/generalized_homotopy_theory.html\",\"WARC-Payload-Digest\":\"sha1:UHSCR3JPNHHDWBHUH3INRGWWIBI4DHEM\",\"WARC-Block-Digest\":\"sha1:CRVJJNSYUHKGAZJ5MYJMAS5KATILQ7XF\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764500080.82_warc_CC-MAIN-20230204012622-20230204042622-00259.warc.gz\"}"}
https://profdoc.um.ac.ir/paper-abstract-1016553.html	[ "ROMAI Journal, Volume (4), No (2), Year (2008-6) , Pages (191-203)\n\n#### Title : ( A Numerical methods for obtaining the Generalized derivatives of non differentiable function )\n\nCitation: BibTeX \| EndNote\n\n#### Abstract\n\nA numerical method to achieve generalized derivative for continuous functions which are nondi®erentiable, is introduced. At ¯rst, it is proved that the relation between a continuously di®erentiable function and its derivative can be shown with an in¯nite moment problem (IMP). Then, by approximating the IMP to a ¯nite moment problem (FMP), we obtain the relation between continuous and nondi®erentiable function and its generalized derivative. Thus, by solving the FMP with numerical method, the generalized derivative is achieved. Finally, e±ciency of our approach is con¯rmed by some numerical examples.\n\n#### Keywords\n\n, generalized derivative, non-di®erentiable functions, infinite moment problem, finite moment problem.\nبرای دانلود از شناسه و رمز عبور پرتال پویا استفاده کنید.", null, "@article{paperid:1016553,\ntitle = {A Numerical methods for obtaining the Generalized derivatives of non differentiable function},\njournal = {ROMAI Journal},\nyear = {2008},\nvolume = {4},\nnumber = {2},\nmonth = {June},\nissn = {1841-5512},\npages = {191--203},\nnumpages = {12},\nkeywords = {generalized derivative; non-di®erentiable functions; infinite moment problem;finite moment problem.},\n}" ]	[ null, "https://profdoc.um.ac.ir/images/ajax-loader.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8561688,"math_prob":0.9242213,"size":1631,"snap":"2020-24-2020-29","text_gpt3_token_len":409,"char_repetition_ratio":0.15119852,"word_repetition_ratio":0.09777778,"special_character_ratio":0.23543838,"punctuation_ratio":0.16788322,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98821414,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-10T12:12:04Z\",\"WARC-Record-ID\":\"<urn:uuid:5d1c26bb-9bf4-4c5e-bb77-1801103bd78c>\",\"Content-Length\":\"25060\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:dea5a739-04a1-4e01-ad7e-e5b4bf6b32fd>\",\"WARC-Concurrent-To\":\"<urn:uuid:c0f13364-afe4-4729-8922-ef1f4559201b>\",\"WARC-IP-Address\":\"109.122.252.57\",\"WARC-Target-URI\":\"https://profdoc.um.ac.ir/paper-abstract-1016553.html\",\"WARC-Payload-Digest\":\"sha1:RUYP2GOK3XUUCSRWRGOZ35VEW52A2B7Y\",\"WARC-Block-Digest\":\"sha1:PGIF2PZUSNHNHM2BGWW5IC62DJ6R37I2\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593655908294.32_warc_CC-MAIN-20200710113143-20200710143143-00243.warc.gz\"}"}
https://flashgene.com/archives/161714.html	[ "## 一、基于ID3算法手动构建决策树，并通过Matplotlib进行可视化", null, "\\begin{aligned} & Gain(D, 年龄) = 0.083 \\\\ & Gain(D, 工作)=0.324 \\\\ & Gain(D, 房子)=0.420 \\\\ & Gain(D, 信用)=0.363 \\end{aligned}\n\n\\begin{aligned} & Gain(D_1, 年龄) = 0.251 \\\\ & Gain(D_1, 工作)=0.918 \\\\ & Gain(D_1, 信用)=0.474 \\end{aligned}", null, "{\"房子\": {\n\"1\": \"Y\",\n\"0\"：{\"工作\": {\n\"1\": \"Y\",\n\"0\": \"N\"\n}}\n}}\n\n\"\"\"\nAuthor: Taoye\n微信公众号: 玩世不恭的Coder\nExplain：创建训数据集\n\"\"\"\ndef establish_data():\ndata = [[0, 0, 0, 0, 'N'], # 样本数据集相关信息，前面几项代表属性特征，最后一项表示是否放款\n[0, 0, 0, 1, 'N'],\n[0, 1, 0, 1, 'Y'],\n[0, 1, 1, 0, 'Y'],\n[0, 0, 0, 0, 'N'],\n[1, 0, 0, 0, 'N'],\n[1, 0, 0, 1, 'N'],\n[1, 1, 1, 1, 'Y'],\n[1, 0, 1, 2, 'Y'],\n[1, 0, 1, 2, 'Y'],\n[2, 0, 1, 2, 'Y'],\n[2, 0, 1, 1, 'Y'],\n[2, 1, 0, 1, 'Y'],\n[2, 1, 0, 2, 'Y'],\n[2, 0, 0, 0, 'N']]\nlabels = [\"年纪\", \"工作\", \"房子\", \"信用\"]\nreturn np.array(data), labels\n\n\"\"\"\nAuthor: Taoye\n微信公众号: 玩世不恭的Coder\nExplain：找出对应属性特征值的样本，比如找出所有年纪为青年的样本数据集\n\"\"\"\ndef handle_data(data, axis, value):\nresult_data = list()\nfor item in data:\nif item[axis] == value:\nreduced_data = item[: axis].tolist()\nreduced_data.extend(item[axis + 1:])\nresult_data.append(reduced_data)\nreturn result_data\n\n\"\"\"\nAuthor: Taoye\n微信公众号: 玩世不恭的Coder\nExplain：创建决策树\n\"\"\"\ndef establish_decision_tree(data, labels, feat_labels):\ncat_list = [item[-1] for item in data]\nif (cat_list.count(cat_list) == len(cat_list)): return cat_list # 数据集中的类别只有一种\nbest_feature_index = calc_information_gain(data) # 通过信息增益优先选取最好的属性特征\nbest_label = labels[best_feature_index] # 属性特征对应的标签内容\n# feat_labels表示已选取的属性；新建一个决策树节点；将属性标签列表中删除已选取的属性\nfeat_labels.append(best_label); decision_tree = {best_label: dict()}; del(labels[best_feature_index])\nfeature_values = [item[best_feature_index] for item in data]\nunique_values = set(feature_values) # 获取最优属性对应值的set集合\nfor value in unique_values:\nsub_label = labels[:]\ndecision_tree[best_label][value] = establish_decision_tree(np.array(handle_data(data, best_feature_index, value)), sub_label, feat_labels)\nreturn decision_tree\n\nimport numpy as np\nimport pandas as pd\nnp.__version__\npd.__version__\n\"\"\"\nAuthor: Taoye\n微信公众号: 玩世不恭的Coder\nExplain：创建训数据集\n\"\"\"\ndef establish_data():\ndata = [[0, 0, 0, 0, 'N'], # 样本数据集相关信息，前面几项代表属性特征，最后一项表示是否放款\n[0, 0, 0, 1, 'N'],\n[0, 1, 0, 1, 'Y'],\n[0, 1, 1, 0, 'Y'],\n[0, 0, 0, 0, 'N'],\n[1, 0, 0, 0, 'N'],\n[1, 0, 0, 1, 'N'],\n[1, 1, 1, 1, 'Y'],\n[1, 0, 1, 2, 'Y'],\n[1, 0, 1, 2, 'Y'],\n[2, 0, 1, 2, 'Y'],\n[2, 0, 1, 1, 'Y'],\n[2, 1, 0, 1, 'Y'],\n[2, 1, 0, 2, 'Y'],\n[2, 0, 0, 0, 'N']]\nlabels = [\"年纪\", \"工作\", \"房子\", \"信用\"]\nreturn np.array(data), labels\n\"\"\"\nAuthor: Taoye\n微信公众号: 玩世不恭的Coder\nExplain：计算信息熵\n\"\"\"\ndef calc_information_entropy(data):\ndata_number, _ = data.shape\ninformation_entropy = 0\nfor item in pd.DataFrame(data).groupby(_ - 1):\nproportion = item.shape / data_number\ninformation_entropy += - proportion * np.log2(proportion)\nreturn information_entropy\n\"\"\"\nAuthor: Taoye\n微信公众号: 玩世不恭的Coder\nExplain：找出对应属性特征值的样本，比如找出所有年纪为青年的样本数据集\n\"\"\"\ndef handle_data(data, axis, value):\nresult_data = list()\nfor item in data:\nif item[axis] == value:\nreduced_data = item[: axis].tolist()\nreduced_data.extend(item[axis + 1:])\nresult_data.append(reduced_data)\nreturn result_data\n\"\"\"\nAuthor: Taoye\n微信公众号: 玩世不恭的Coder\nExplain：计算最大的信息增益，并输出其所对应的特征索引\n\"\"\"\ndef calc_information_gain(data):\nfeature_number = data.shape - 1 # 属性特征的数量\nbase_entropy = calc_information_entropy(data) # 计算总体数据集的信息熵\nmax_information_gain, best_feature = 0.0, -1 # 初始化最大信息增益和对应的特征索引\nfor index in range(feature_number):\nfeat_list = [item[index] for item in data]\nfeat_set = set(feat_list)\nnew_entropy = 0.0\nfor set_item in feat_set: # 计算属性特征划分后的信息增益\nsub_data = handle_data(data, index, set_item)\nproportion = len(sub_data) / float(data.shape) # 计算子集的比例\nnew_entropy += proportion * calc_information_entropy(np.array(sub_data))\ntemp_information_gain = base_entropy - new_entropy # 计算信息增益\nprint(\"第%d个属性特征所对应的的增益为%.3f\" % (index + 1, temp_information_gain)) # 输出每个特征的信息增益\nif (temp_information_gain > max_information_gain):\nmax_information_gain, best_feature = temp_information_gain, index # 更新信息增益，确定的最大的信息增益对应的索引\nreturn best_feature\n\"\"\"\nAuthor: Taoye\n微信公众号: 玩世不恭的Coder\nExplain：创建决策树\n\"\"\"\ndef establish_decision_tree(data, labels, feat_labels):\ncat_list = [item[-1] for item in data]\nif (cat_list.count(cat_list) == len(cat_list)): return cat_list # 数据集中的类别只有一种\nbest_feature_index = calc_information_gain(data) # 通过信息增益优先选取最好的属性特征\nbest_label = labels[best_feature_index] # 属性特征对应的标签内容\n# feat_labels表示已选取的属性；新建一个决策树节点；将属性标签列表中删除已选取的属性\nfeat_labels.append(best_label); decision_tree = {best_label: dict()}; del(labels[best_feature_index])\nfeature_values = [item[best_feature_index] for item in data]\nunique_values = set(feature_values) # 获取最优属性对应值的set集合\nfor value in unique_values:\nsub_label = labels[:]\ndecision_tree[best_label][value] = establish_decision_tree(np.array(handle_data(data, best_feature_index, value)), sub_label, feat_labels)\nreturn decision_tree\nif __name__ == \"__main__\":\ndata, labels = establish_data()\nprint(establish_decision_tree(data, labels, list()))\n\n{'房子': {'1': 'Y', '0': {'工作': {'1': 'Y', '0': 'N'}}}}\n\nimport numpy as np\nimport pandas as pd\n\"\"\"\nAuthor: Taoye\n微信公众号: 玩世不恭的Coder\nExplain：创建训数据集\n\"\"\"\ndef establish_data():\ndata = [[0, 0, 0, 0, 'N'], # 样本数据集相关信息，前面几项代表属性特征，最后一项表示是否放款\n[0, 0, 0, 1, 'N'],\n[0, 1, 0, 1, 'Y'],\n[0, 1, 1, 0, 'Y'],\n[0, 0, 0, 0, 'N'],\n[1, 0, 0, 0, 'N'],\n[1, 0, 0, 1, 'N'],\n[1, 1, 1, 1, 'Y'],\n[1, 0, 1, 2, 'Y'],\n[1, 0, 1, 2, 'Y'],\n[2, 0, 1, 2, 'Y'],\n[2, 0, 1, 1, 'Y'],\n[2, 1, 0, 1, 'Y'],\n[2, 1, 0, 2, 'Y'],\n[2, 0, 0, 0, 'N']]\nlabels = [\"年纪\", \"工作\", \"房子\", \"信用\"]\nreturn np.array(data), labels\n\"\"\"\nAuthor: Taoye\n微信公众号: 玩世不恭的Coder\nExplain：计算信息熵\n\"\"\"\ndef calc_information_entropy(data):\ndata_number, _ = data.shape\ninformation_entropy = 0\nfor item in pd.DataFrame(data).groupby(_ - 1):\nproportion = item.shape / data_number\ninformation_entropy += - proportion * np.log2(proportion)\nreturn information_entropy\n\"\"\"\nAuthor: Taoye\n微信公众号: 玩世不恭的Coder\nExplain：找出对应属性特征值的样本，比如找出所有年纪为青年的样本数据集\n\"\"\"\ndef handle_data(data, axis, value):\nresult_data = list()\nfor item in data:\nif item[axis] == value:\nreduced_data = item[: axis].tolist()\nreduced_data.extend(item[axis + 1:])\nresult_data.append(reduced_data)\nreturn result_data\n\"\"\"\nAuthor: Taoye\n微信公众号: 玩世不恭的Coder\nExplain：计算最大的信息增益，并输出其所对应的特征索引\n\"\"\"\ndef calc_information_gain(data):\nfeature_number = data.shape - 1 # 属性特征的数量\nbase_entropy = calc_information_entropy(data) # 计算总体数据集的信息熵\nmax_information_gain, best_feature = 0.0, -1 # 初始化最大信息增益和对应的特征索引\nfor index in range(feature_number):\nfeat_list = [item[index] for item in data]\nfeat_set = set(feat_list)\nnew_entropy = 0.0\nfor set_item in feat_set: # 计算属性特征划分后的信息增益\nsub_data = handle_data(data, index, set_item)\nproportion = len(sub_data) / float(data.shape) # 计算子集的比例\nnew_entropy += proportion * calc_information_entropy(np.array(sub_data))\ntemp_information_gain = base_entropy - new_entropy # 计算信息增益\nprint(\"第%d个属性特征所对应的的增益为%.3f\" % (index + 1, temp_information_gain)) # 输出每个特征的信息增益\nif (temp_information_gain > max_information_gain):\nmax_information_gain, best_feature = temp_information_gain, index # 更新信息增益，确定的最大的信息增益对应的索引\nreturn best_feature\n\"\"\"\nAuthor: Taoye\n微信公众号: 玩世不恭的Coder\nExplain：创建决策树\n\"\"\"\ndef establish_decision_tree(data, labels, feat_labels):\ncat_list = [item[-1] for item in data]\nif (cat_list.count(cat_list) == len(cat_list)): return cat_list # 数据集中的类别只有一种\nbest_feature_index = calc_information_gain(data) # 通过信息增益优先选取最好的属性特征\nbest_label = labels[best_feature_index] # 属性特征对应的标签内容\n# feat_labels表示已选取的属性；新建一个决策树节点；将属性标签列表中删除已选取的属性\nfeat_labels.append(best_label); decision_tree = {best_label: dict()}; del(labels[best_feature_index])\nfeature_values = [item[best_feature_index] for item in data]\nunique_values = set(feature_values) # 获取最优属性对应值的set集合\nfor value in unique_values:\nsub_label = labels[:]\ndecision_tree[best_label][value] = establish_decision_tree(np.array(handle_data(data, best_feature_index, value)), sub_label, feat_labels)\nreturn decision_tree\n\"\"\"\nAuthor: Taoye\n微信公众号: 玩世不恭的Coder\nExplain：统计决策树当中的叶子节点数目，以及决策树的深度\n\"\"\"\ndef get_leaf_number_and_tree_depth(decision_tree):\nleaf_number, first_key, tree_depth = 0, next(iter(decision_tree)), 0; second_dict = decision_tree[first_key]\nfor key in second_dict.keys():\nif type(second_dict.get(key)).__name__ == \"dict\":\ntemp_number, temp_depth = get_leaf_number_and_tree_depth(second_dict[key])\nleaf_number, curr_depth = leaf_number + temp_number, 1 + temp_depth\nelse: leaf_number += 1; curr_depth = 1\nif curr_depth > tree_depth: tree_depth = curr_depth\nreturn leaf_number, tree_depth\nfrom matplotlib.font_manager import FontProperties\n\"\"\"\nAuthor: Taoye\n微信公众号: 玩世不恭的Coder\nExplain：绘制节点\n\"\"\"\ndef plot_node(node_text, center_pt, parent_pt, node_type):\narrow_args = dict(arrowstyle = \"<-\")\nfont = FontProperties(fname=r\"c:\\windows\\fonts\\simsun.ttc\", size=14) # 设置字体\ncreate_plot.ax1.annotate(node_text, xy=parent_pt, xycoords='axes fraction',\nxytext=center_pt, textcoords='axes fraction',\nva=\"center\", ha=\"center\", bbox=node_type, arrowprops=arrow_args, FontProperties=font)\n\"\"\"\nAuthor: Taoye\n微信公众号: 玩世不恭的Coder\nExplain：标注有向边的值\n\"\"\"\ndef tag_text(cntr_pt, parent_pt, node_text):\nx_mid = (parent_pt - cntr_pt) / 2.0 + cntr_pt\ny_mid = (parent_pt - cntr_pt) / 2.0 + cntr_pt\ncreate_plot.ax1.text(x_mid, y_mid, node_text, va=\"center\", ha=\"center\", rotation=30)\n\"\"\"\nAuthor: Taoye\n微信公众号: 玩世不恭的Coder\nExplain：绘制决策树\n\"\"\"\ndef plot_tree(decision_tree, parent_pt, node_text):\ndecision_node = dict(boxstyle=\"sawtooth\", fc=\"0.8\")\nleaf_node = dict(boxstyle = \"round4\", fc = \"0.8\")\nleaf_number, tree_depth = get_leaf_number_and_tree_depth(decision_tree)\nfirst_key = next(iter(decision_tree))\ncntr_pt = (plot_tree.xOff + (1.0 + float(leaf_number)) / 2.0 / plot_tree.totalW, plot_tree.yOff)\ntag_text(cntr_pt, parent_pt, node_text); plot_node(first_key, cntr_pt, parent_pt, decision_node)\nsecond_dict = decision_tree[first_key]\nplot_tree.yOff = plot_tree.yOff - 1.0 / plot_tree.totalD\nfor key in second_dict.keys():\nif type(second_dict[key]).__name__ == 'dict': plot_tree(second_dict[key], cntr_pt, str(key))\nelse:\nplot_tree.xOff = plot_tree.xOff + 1.0 / plot_tree.totalW\nplot_node(second_dict[key], (plot_tree.xOff, plot_tree.yOff), cntr_pt, leaf_node)\ntag_text((plot_tree.xOff, plot_tree.yOff), cntr_pt, str(key))\nplot_tree.yOff = plot_tree.yOff + 1.0 / plot_tree.totalD\nfrom matplotlib import pyplot as plt\n\"\"\"\nAuthor: Taoye\n微信公众号: 玩世不恭的Coder\nExplain：创建决策树\n\"\"\"\ndef create_plot(in_tree):\nfig = plt.figure(1, facecolor = \"white\")\nfig.clf()\naxprops = dict(xticks = [], yticks = [])\ncreate_plot.ax1 = plt.subplot(111, frameon = False, **axprops)\nleaf_number, tree_depth = get_leaf_number_and_tree_depth(in_tree)\nplot_tree.totalW, plot_tree.totalD = float(leaf_number), float(tree_depth)\nplot_tree.xOff = -0.5 / plot_tree.totalW; plot_tree.yOff = 1.0\nplot_tree(in_tree, (0.5,1.0), '')\nplt.show()\n\nif __name__ == \"__main__\":\ndata, labels = establish_data()\ndecision_tree = establish_decision_tree(data, labels, list())\nprint(decision_tree)\nprint(\"决策树的叶子节点数和深度：\", get_leaf_number_and_tree_depth(decision_tree))\ncreate_plot(decision_tree)", null, "get_leaf_number_and_tree_depth 主要用于统计决策树当中的叶子节点数目，以及决策树的深度。选取 key 对应的 value ，判断 value 是否为一个字典类型，否的话说明是一个叶子节点，是的话说明非叶子节点，不同情况做不同处理\nplot_node 方法用于绘制节点，这里设置了font类型是在windows下的，如果是linux则需要额外设置\ntag_text 用于标注有向边的属性值，在这里主要是用1和0来进行标注，1代表对属性的肯定，0表示否定\nplot_tree 遍历绘制决策树，在这里需要调用前面所定义的几个方法\n\n## 二、基于已经构建好的决策树进行分类预测\n\n\"\"\"\nAuthor: Taoye\n微信公众号: 玩世不恭的Coder\nExplain：通过决策树模型对测试数据进行分类\n\"\"\"\ndef classify(decision_tree, best_feature_labels, test_data):\nfirst_node = next(iter(decision_tree))\nsecond_dict = decision_tree[first_node]\nfeat_index = best_feature_labels.index(first_node)\nfor key in second_dict.keys():\nif int(test_data[feat_index]) == int(key):\nif type(second_dict[key]).__name__ == \"dict\": # 为字典说明还没到叶子节点\nresult_label = classify(second_dict[key], best_feature_labels, test_data)\nelse: result_label = second_dict[key]\nreturn result_label", null, "## 三、构建好的决策树模型应当如何保存和读取？\n\nimport pickle\nwith open(\"DecisionTreeModel.txt\", \"wb\") as f:\npickle.dump(decision_tree, f) # 保存决策树模型\nf = open(\"DecisionTreeModel.txt\", \"rb\")\ndecision_tree = pickle.load(f) # 加载决策树模型\n\n## 四、通过鸢尾花（Iris）数据集，使用Sklearn构建决策树", null, "criterion ：属性选取的标准，默认采用的是 gini ，也可以自行选择 entropygini 是基尼值， entropy 是信息熵，这两个我们在上篇文章中已经讲到过了\nsplitter ：特征划分节点的选择标准，默认是best，可以设置为random。默认的”best”适合样本量不大的时候，而如果样本数据量非常大，此时决策树构建推荐”random”。\nmax_depth ：决策树最大深度，默认是None。需要注意一点的是，该深度是不包含根节点的。一般来说，数据少或者特征少的时候可以不管这个值。如果模型样本量多，特征也多的情况下，推荐限制这个最大深度，具体的取值取决于数据的分布。\nmax_features ：划分时考虑的最大特征数，默认是None。一般来说，如果样本特征数不多，比如小于50，我们用默认的”None”就可以了，如果特征数非常多，我们可以灵活使用其他取值来控制划分时考虑的最大特征数，以控制决策树的生成时间。用到的时候查看下文档即可。\nmin_samples_split ：内部节点再划分所需最小样本数，默认为2。意思就是说，比如我们某个属性对应样本数目小于 min_samples_split ，即使它满足优先选取条件，依然会被剔除掉。\nmin_samples_leaf ：叶子节点最少样本数，默认是1。这个值限制了叶子节点最少的样本数，如果某叶子节点数目小于样本数，则会和兄弟节点一起被剪枝。叶结点需要最少的样本数，也就是最后到叶结点，需要多少个样本才能算一个叶结点。如果设置为1，哪怕这个类别只有1个样本，决策树也会构建出来。\nmax_leaf_nodes ：最大叶子节点数，默认是None。通过限制最大叶子节点数，可以防止过拟合。如果加了限制，算法会建立在最大叶子节点数内最优的决策树。如果特征不多，可以不考虑这个值，但是如果特征分成多的话，可以加以限制，具体的值可以通过交叉验证得到。\nrandom_state ：随机数种子，默认是None。如果没有设置随机数，随机出来的数与当前系统时间有关，每个时刻都是不同的。如果设置了随机数种子，那幺相同随机数种子，不同时刻产生的随机数也是相同的。", null, "import numpy as np\nimport matplotlib.pyplot as plt\nfrom sklearn.tree import DecisionTreeClassifier, plot_tree\nclass IrisDecisionTree:\n\"\"\"\nExplain：属性的初始化\nParameters:\nn_classes：鸢尾花的类别数\nplot_colors: 不同类别花的颜色\nplot_step： meshgrid网格的步长\n\"\"\"\ndef __init__(self, n_classes, plot_colors, plot_step):\nself.n_classes = n_classes\nself.plot_colors = plot_colors\nself.plot_step = plot_step\n\n\"\"\"\n\"\"\"\ndef establish_data(self):\nreturn iris_info.data, iris_info.target, iris_info.feature_names, iris_info.target_names\n\n\"\"\"\nExplain：分类的可视化\n\"\"\"\ndef show_result(self, x_data, y_label, feature_names, target_names):\n# 选取两个属性来构建决策树，以方便可视化，其中列表内部元素代表属性对应的索引\nfor index, pair in enumerate([[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]]):\nsub_x_data, sub_y_label = x_data[:, pair], y_label\nclf = DecisionTreeClassifier().fit(sub_x_data, sub_y_label) # 选取两个属性构建决策树\nplt.subplot(2, 3, index + 1)\nx_min, x_max = sub_x_data[:, 0].min() - 1, sub_x_data[:, 0].max() + 1 # 第一个属性\ny_min, y_max = sub_x_data[:, 1].min() - 1, sub_x_data[:, 1].max() + 1 # 第二个属性\nxx, yy = np.meshgrid(np.arange(x_min, x_max, self.plot_step), np.arange(y_min, y_max, self.plot_step))\nZ = clf.predict(np.c_[xx.ravel(), yy.ravel()]).reshape(xx.shape) # 预测meshgrid内部每个元素的分类\ncs = plt.contourf(xx, yy, Z, cmap = plt.cm.RdYlBu) # 绘制带有颜色的网格图\nplt.xlabel(feature_names[pair]); plt.ylabel(feature_names[pair]) # 标注坐标轴标签\nfor i, color in zip(range(self.n_classes), self.plot_colors):\nidx = np.where(sub_y_label == i)\nplt.scatter(sub_x_data[idx, 0], sub_x_data[idx, 1], c=color, label=target_names[i],\ncmap=plt.cm.RdYlBu, edgecolor='black', s=15) # 绘制数据样本集的散点图\n\nfrom matplotlib.font_manager import FontProperties\nfont = FontProperties(fname=r\"c:\\windows\\fonts\\simsun.ttc\", size=14) # 定义中文字体\nplt.suptitle(\"通过决策树对鸢尾花数据进行可视化\", fontproperties=font)\nplt.axis(\"tight\")\nplt.figure()\nclf = DecisionTreeClassifier().fit(x_data, y_label) # 针对鸢尾花数据集的多重属性来构建决策树\nplot_tree(clf, filled=True)\nplt.show()\n\nif __name__ == \"__main__\":\niris_decision_tree = IrisDecisionTree(3, \"ryb\", 0.02)\nx_data, y_label, feature_names, target_names = iris_decision_tree.establish_data()\niris_decision_tree.show_result(x_data, y_label, feature_names, target_names)", null, "graphviz 不能采用pip进行安装，采用anaconda安装的时候也会很慢，甚至多次尝试都可能安装失败，前几天帮同学安装就出现这种情况（windows下是这样的，linux环境下会很方便），所以这里我们采用直接通过 whl 文件来安装。", null, "pip install graphviz‑0.15‑py3‑none‑any.whl\n\n$sudo apt install graphviz # Ubuntu$ sudo apt install graphviz # Debian\n\n\"\"\"\nExplain：通过graphviz实现决策树的可视化\n\"\"\"\ndef show_result_by_graphviz(self, x_data, y_label):\nclf = DecisionTreeClassifier().fit(x_data, y_label)\niris_dot_data = tree.export_graphviz(clf, out_file=None,\nfeature_names=iris.feature_names,\nclass_names=iris.target_names,\nfilled=True, rounded=True,\nspecial_characters=True)\nimport graphviz\ngraph = graphviz.Source(iris_dot_data); graph.render(\"iris\")\n\n#### 参考资料：\n\n 《机器学习实战》：Peter Harrington 人民邮电出版社《统计学习方法》：李航第二版清华大学出版社《机器学习》：周志华清华大学出版社\n\n Python Extension Packages： https://www.lfd.uci.edu/~gohlke/pythonlibs/#wordcloud\n\n sklearn.tree.DecisionTreeClassifier： https://scikit-learn.org/stable/modules/generated/sklearn.tree.DecisionTreeClassifier.html\n\n Graphviz官网： https://graphviz.org/" ]	[ null, "data:image/svg+xml;base64,PHN2ZyBoZWlnaHQ9JzQwMScgd2lkdGg9JzY0OScgeG1sbnM9J2h0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnJyB2ZXJzaW9uPScxLjEnLz4=", null, "data:image/svg+xml;base64,PHN2ZyBoZWlnaHQ9JzYyMicgd2lkdGg9JzgwNicgeG1sbnM9J2h0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnJyB2ZXJzaW9uPScxLjEnLz4=", null, "data:image/svg+xml;base64,PHN2ZyBoZWlnaHQ9JzU1Nicgd2lkdGg9JzYxMicgeG1sbnM9J2h0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnJyB2ZXJzaW9uPScxLjEnLz4=", null, "data:image/svg+xml;base64,PHN2ZyBoZWlnaHQ9JzMwNicgd2lkdGg9JzU0MicgeG1sbnM9J2h0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnJyB2ZXJzaW9uPScxLjEnLz4=", null, "data:image/svg+xml;base64,PHN2ZyBoZWlnaHQ9JzgzNScgd2lkdGg9Jzk1OCcgeG1sbnM9J2h0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnJyB2ZXJzaW9uPScxLjEnLz4=", null, "data:image/svg+xml;base64,PHN2ZyBoZWlnaHQ9JzMzNycgd2lkdGg9Jzk0NCcgeG1sbnM9J2h0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnJyB2ZXJzaW9uPScxLjEnLz4=", null, "data:image/svg+xml;base64,PHN2ZyBoZWlnaHQ9JzQ0OCcgd2lkdGg9JzExMjEnIHhtbG5zPSdodHRwOi8vd3d3LnczLm9yZy8yMDAwL3N2ZycgdmVyc2lvbj0nMS4xJy8+", null, "data:image/svg+xml;base64,PHN2ZyBoZWlnaHQ9JzExMicgd2lkdGg9JzM5OCcgeG1sbnM9J2h0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnJyB2ZXJzaW9uPScxLjEnLz4=", null ]	{"ft_lang_label":"__label__zh","ft_lang_prob":0.50746655,"math_prob":0.97312,"size":21184,"snap":"2020-45-2020-50","text_gpt3_token_len":10794,"char_repetition_ratio":0.1401322,"word_repetition_ratio":0.4846154,"special_character_ratio":0.2735083,"punctuation_ratio":0.25108492,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9879364,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-12-05T08:16:38Z\",\"WARC-Record-ID\":\"<urn:uuid:111477a6-d2d7-4576-adb3-6d5464a3c460>\",\"Content-Length\":\"112908\",\"Content-Type\":\"application/http; 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https://hungary.pure.elsevier.com/en/publications/thermodynamic-significance-of-boiling-point-correlations-for-alky	[ "# Thermodynamic significance of boiling point correlations for alkylbenzenes in gas chromatography. Extension of Trouton's rule\n\nKároly Héberger, Teresa Kowalska\n\nResearch output: Contribution to journalArticle\n\n21 Citations (Scopus)\n\n### Abstract\n\nA more exact thermodynamic interpretation of the empirical correlations between retention parameters of solutes (i.e. the relative retention times: τ (s)) and their boiling points (T(B)) was established using Trouton-Hildebrandt-Everett's rule (the extension of Trouton's rule). These empirical correlations are known for C6-C12 alkylbenzenes for low and medium polar stationary phases. A statistical analysis has been made to compare the description by the old and newly developed models. The exponential relation suggested earlier [Chromatographia, 44 (1997) 179-186] can be extended into a form consisting of two variables T(M), T(B):τ((i)corr.)=AT(M(i)) exp(BT(B(i)))where: τ((i)corr.)=(t(R(i))-t0)/t(R(st)), A=(t0Φ/t(R(st))) exp(-4.0), T(M(i))=T(B(i))((T(B(i))/T-1)), B=4.0/T, t0 is the column dead time, Φ is the phase ratio, t(R(st)) is the retention time of the standard compound, R is the gas constant, T is the column temperature, and subscript '(i)' refers to the ith alkylbenzene. High correlation coefficients and small residual error indicate the superiority of the developed equation. Moreover, the residua show normal behavior, whereas curvature can be seen in the residua for earlier models. Validity of the approach proposed was confirmed through a comparison of the numerical values obtained from our computation (fitted values) with those stemming from theory. Copyright (C) 1999 Elsevier Science B.V.\n\nOriginal language English 13-20 8 Journal of Chromatography A 845 1-2 https://doi.org/10.1016/S0021-9673(99)00289-7 Published - Jun 11 1999\n\n### Keywords\n\n• Alkylbenzenes\n• Boiling points\n• Quantitative structure-retention relationships\n• Thermodynamic parameters\n• Trouton's rule\n\n### ASJC Scopus subject areas\n\n• Analytical Chemistry\n• Biochemistry\n• Organic Chemistry" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8170621,"math_prob":0.7761512,"size":1854,"snap":"2020-10-2020-16","text_gpt3_token_len":471,"char_repetition_ratio":0.0945946,"word_repetition_ratio":0.0,"special_character_ratio":0.24649407,"punctuation_ratio":0.093939394,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9838796,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-04-08T05:59:55Z\",\"WARC-Record-ID\":\"<urn:uuid:002a7cd8-87e7-425b-b47e-93ac7da53289>\",\"Content-Length\":\"41996\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f7497808-0cc6-4f5f-8b78-7870c6021b27>\",\"WARC-Concurrent-To\":\"<urn:uuid:e3816415-0a6a-476c-923d-71347423eca9>\",\"WARC-IP-Address\":\"52.51.22.49\",\"WARC-Target-URI\":\"https://hungary.pure.elsevier.com/en/publications/thermodynamic-significance-of-boiling-point-correlations-for-alky\",\"WARC-Payload-Digest\":\"sha1:7BLWU6H6YB2LH2LOI2SEEQUQ26RYK55U\",\"WARC-Block-Digest\":\"sha1:LUK2HXWYL5NITWH7SWA2YTU23Z2MYKUB\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-16/CC-MAIN-2020-16_segments_1585371810617.95_warc_CC-MAIN-20200408041431-20200408071931-00233.warc.gz\"}"}
https://de.maplesoft.com/support/help/Maple/view.aspx?path=plottools%2Ftransform	[ "plottools - Maple Programming Help\n\nHome : Support : Online Help : Graphics : Packages : Plot Tools : plottools/transform\n\nplottools\n\n transform\n generate procedure to transform plots\n\n Calling Sequence transform(f)\n\nParameters\n\n f - procedure\n\nDescription\n\n • The transform command generates a procedure that can be used to change plot data structures by applying the procedure f to all the points in the plot structure.\n • The procedure f represents a mapping\n\n$f:{R}^{m}\\to {R}^{n}$\n\n where m and n can take values 2 or 3. The procedure must take as input m arguments and output a list of n components.\n • Examples of the use of this command include embedding 2-D plots into 3-D space, changing coordinate systems or transforming curves in the complex plane into the Riemann Sphere.\n • The result of a call to transform is a procedure that takes a 2-D or 3-D plot data structure as parameter. You can assign the procedure to a variable, and apply the procedure to various plots. For more information about plot data structures, see plot/structure.\n • The transform command is usually used with [m, n] equal to [2, 2], [2, 3], or [3, 3]. It returns a transformation procedure in the case where m is 3 and n is 2. However, this procedure works only on those 3-D plot objects that have 2-D counterparts.\n • Several commands in the plottools package can transform plots. For a list, see the plottools help page. The plots[changecoords] and plots[display] commands can also be used to transform plots.\n • If the transform command is applied to a filled shape, such as an arrow or rectangle, the polygon that is generated might not resemble the result one would expect from a transformation of the entire shape. That is because the transform command works only on the points of the plot data structure, which are typically the vertices and not interior points.\n\nExamples\n\n > $\\mathrm{with}\\left(\\mathrm{plottools}\\right):$\n > $\\mathrm{with}\\left(\\mathrm{plots}\\right):$\n\nDraw contour plots onto axes of 3-D plots.\n\n > $p≔\\mathrm{plot3d}\\left(-\\frac{5x}{{x}^{2}+{y}^{2}+1},x=-3..3,y=-3..3,\\mathrm{style}=\\mathrm{contour},\\mathrm{contours}=8\\right):$\n > $q≔\\mathrm{contourplot}\\left(-\\frac{5x}{{x}^{2}+{y}^{2}+1},x=-3..3,y=-3..3,\\mathrm{filled}=\\mathrm{true}\\right):$\n > $f≔\\mathrm{transform}\\left(\\left(x,y\\right)→\\left[x,y,-3\\right]\\right):$\n > $\\mathrm{display}\\left(\\left\\{p,f\\left(q\\right)\\right\\},\\mathrm{orientation}=\\left[50,40\\right]\\right)$", null, "Change coordinate systems.\n\n > $p≔\\mathrm{plot3d}\\left(\\left[{1.3}^{x}\\mathrm{sin}\\left(y\\right),x,y\\right],x=-1..2\\mathrm{Pi},y=0..\\mathrm{Pi}\\right):$\n > $f≔\\mathrm{transform}\\left(\\left(r,\\mathrm{th},\\mathrm{ph}\\right)→\\left[r\\mathrm{sin}\\left(\\mathrm{ph}\\right)\\mathrm{cos}\\left(\\mathrm{th}\\right),r\\mathrm{sin}\\left(\\mathrm{ph}\\right)\\mathrm{sin}\\left(\\mathrm{th}\\right),r\\mathrm{cos}\\left(\\mathrm{ph}\\right)\\right]\\right):$\n > $\\mathrm{display}\\left(\\left\\{p,f\\left(p\\right)\\right\\},\\mathrm{orientation}=\\left[145,20\\right],\\mathrm{scaling}=\\mathrm{constrained}\\right)$", null, "" ]	[ null, "https://de.maplesoft.com/support/help/content/9054/plot166.png", null, "https://de.maplesoft.com/support/help/content/9054/plot185.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.77475977,"math_prob":0.9992706,"size":2423,"snap":"2019-35-2019-39","text_gpt3_token_len":676,"char_repetition_ratio":0.15254237,"word_repetition_ratio":0.0,"special_character_ratio":0.22657862,"punctuation_ratio":0.15238096,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99899286,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-09-20T12:32:00Z\",\"WARC-Record-ID\":\"<urn:uuid:c0989a7e-e121-4da8-b45a-f029105610a2>\",\"Content-Length\":\"189301\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a0023ace-6a3d-487f-9867-f8d935bf8730>\",\"WARC-Concurrent-To\":\"<urn:uuid:42eb186e-298e-4873-b458-bb5fbe08f931>\",\"WARC-IP-Address\":\"199.71.183.28\",\"WARC-Target-URI\":\"https://de.maplesoft.com/support/help/Maple/view.aspx?path=plottools%2Ftransform\",\"WARC-Payload-Digest\":\"sha1:U3JBFL7WGARAKZZDYLEPXJS7SLGPHEAZ\",\"WARC-Block-Digest\":\"sha1:HGMTVKB6PGCRIY4GKH43REO2A6VITOPZ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-39/CC-MAIN-2019-39_segments_1568514574018.53_warc_CC-MAIN-20190920113425-20190920135425-00047.warc.gz\"}"}
https://cbsemathssolutions.in/area-of-circle-formula-with-examples-explained/	[ "# Area of circle formula with examples explained\n\nLearn and know what is the area of circle formula. All of us know what is a circle and the important terms related to circle i.e. diameter, radius, chord, perimeter, sector, and secant and so on. In this one more important term is area.", null, "Now we will learn how to find the area of circle. For this we have a formula. So now we will learn the area of circle formula. Don’t worry the formula is very simple and easy to remember. Just read and write the formula 4 or 5 times, automatically you can remember it.\n\n## Area of circle formula as follows:\n\nThe formula for finding the area of a circle is given by\n\nA = π r2\n\nπ We have a standard value i.e. 22/7 or 3.14 and “r” means radius of circle.\n\nNote:\n\nArea of circle formula can also be written in terms of diameter also. For this we need to replace “r” with d/2 in the above formula.\n\nSuppose if radius is 1 unit then the circle becomes unit circle.\n\n### Example Problems:\n\n♦ Calculate the area of a circle when radius is 49 cm.\n\nSolution:\n\nGiven\n\nWe know that,\n\nA = π r2\n\nA = 22/7 × 492\n\nA = 22/7 × 49 × 49\n\nA = 22 × 7 × 49\n\nA = 7546 cm2\n\n♦ If the diameter of a circle is given as 28 cm then find the area of a circle.\n\nSolution:\n\nGiven\n\nDiameter = 28 cm.\n\nRadius = diameter/2 = 28/2 = 14 cm.\n\nWe know that,\n\nA = π r2\n\nA = 22/7 × 142\n\nA = 22/7 × 14 × 14\n\nA = 22 × 2 × 14\n\nA = 616 cm2\n\n♦ The area of a circle is 1386 m2. Find the radius of the circle.\n\nSolution:\n\nGiven\n\nArea = 1386 m2\n\nWe know that,\n\nA = π r2\n\n1386 = 22/7 r2\n\n1386 × 7/22 = r2\n\n9702/22 = r2\n\n441 = r2\n\n212 = r2\n\n21 = r\n\nTherefore, radius of a circle is 21 m.\n\nI hope you understood area of circle formula and the example problems based on it.", null, "" ]	[ null, "https://cbsemathssolutions.in/wp-content/uploads/2019/05/Area-of-circle-formula-with-examples-explained.png", null, "https://secure.gravatar.com/avatar/", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.86830425,"math_prob":0.99736047,"size":1639,"snap":"2019-26-2019-30","text_gpt3_token_len":514,"char_repetition_ratio":0.17981651,"word_repetition_ratio":0.075418994,"special_character_ratio":0.35204393,"punctuation_ratio":0.11023622,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999765,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,1,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-07-23T05:07:34Z\",\"WARC-Record-ID\":\"<urn:uuid:ff4b8ae4-02a0-44d3-8d03-367b287908e4>\",\"Content-Length\":\"87053\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a255cff4-4b69-4497-9aca-d2f28571c07b>\",\"WARC-Concurrent-To\":\"<urn:uuid:43da7102-0033-4773-97ea-50d14e95530c>\",\"WARC-IP-Address\":\"160.153.138.53\",\"WARC-Target-URI\":\"https://cbsemathssolutions.in/area-of-circle-formula-with-examples-explained/\",\"WARC-Payload-Digest\":\"sha1:DVH2J5AOJHIYLG3CRQ7ZA42N2NDB5I5C\",\"WARC-Block-Digest\":\"sha1:MYR2VJD2TXUK3Q6U3TIG7OPT62HJUQYX\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-30/CC-MAIN-2019-30_segments_1563195528869.90_warc_CC-MAIN-20190723043719-20190723065719-00152.warc.gz\"}"}
https://www.causeweb.org/cause/statistical-topic/categorical-methods	[ "# Categorical Methods\n\n• ### Statistics: Basic Concepts for Astronomers\n\nThis presentation was given by Aneta Siemiginowska at the 4th International X-ray Astronomy School (2005), held at the Harvard-Smithsonian Center for Astrophysics in Cambridge, MA.\n\n• ### Introduction to Correlated Binary Data\n\nThis presentation is a part of a series of lessons on the Analysis of Categorical Data. This lecture overs the following: covariance patterns and generalized estimating equations (GEE).\n\n• ### Conditional Logistic Models for Matched Pairs\n\nThis presentation is a part of a series of lessons on the Analysis of Categorical Data. This lecture overs the following: conditional logistic regression, conditional likelihood for matched pairs, the non-central hypergeometric, the conditional maximum likelihood estimator (CMLE), conditional confidence interval for odds ratios, and McNemar's statistic.\n\n• ### Models for Matched Pairs\n\nThis presentation is a part of a series of lessons on the Analysis of Categorical Data. This lecture overs the following: odds ratio, dependent proportion, marginal homogeneity, McNemar's Test, marginal homogeneity for greater than 2 levels, measures of agreement, and the kappa coefficient (weighted vs. unweighted).\n\n• ### Log-linear Models - Sparse Tables\n\nThis presentation is a part of a series of lessons on the Analysis of Categorical Data. This lecture covers the following: sparse tables, sampling zeros, structural zeros, and log-linear model (and limitations).\n\n• ### Log-linear Models for Multidimensional Contingency Tables\n\nThis presentation is a part of a series of lessons on the Analysis of Categorical Data. This lecture covers the following: partial/conditional tables, confounding, types of independence (mutual, joint, marginal, and conditional), identifiability constraints, partial odds ratios, hierarchical log-linear model, pairwise interaction log-linear model, conditional independence log-linear model, goodness of fit, and model building.\n\n• ### Introduction to Log-linear Models\n\nThis presentation is a part of a series of lessons on the Analysis of Categorical Data. This lecture covers the following: conditional independence, log-linear models for 2x2 tables, expected counts, logistic regression, odds ratio, parameters of interest for different designs and the MLEs, poisson log-linear model, double dichotomy, the multinomial, and the multinomial log-linear model.\n\n• ### Logit Models for Multinomial Responses Part II\n\nThis presentation is a part of a series of lessons on the Analysis of Categorical Data. This lecture covers the following: ordinal regression models, cumulative probabilities, non-proportional odds, score stat for proportionl odds, MLEs, the adjacent categories logit, and proportional odds model.\n\n• ### Logit Models for Multinomial Responses Part I\n\nThis presentation is a part of a series of lessons on the Analysis of Categorical Data. This lecture covers the following: generalized odds ratio, collapsed categories, polytomous (or multinomial) logistic regression, and maximum likelihood using the multinomial.\n\n• ### Conditional Logistic Regression\n\nThis presentation is a part of a series of lessons on the Analysis of Categorical Data. This lecture covers the following: unconditional likelihood, elimination of nuisance parameters, and Mantel-Haenzsel estimate." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8076332,"math_prob":0.831871,"size":3097,"snap":"2022-40-2023-06","text_gpt3_token_len":617,"char_repetition_ratio":0.13967022,"word_repetition_ratio":0.45813954,"special_character_ratio":0.18243462,"punctuation_ratio":0.15296367,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9973371,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-02-02T10:30:11Z\",\"WARC-Record-ID\":\"<urn:uuid:a85a5df8-933b-4fa1-ba09-470f793273ad>\",\"Content-Length\":\"60332\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:57743ae9-4551-465e-beed-0c85bd2f5f33>\",\"WARC-Concurrent-To\":\"<urn:uuid:285f902f-f6f0-4867-aaa5-544b8bea1f06>\",\"WARC-IP-Address\":\"128.118.15.93\",\"WARC-Target-URI\":\"https://www.causeweb.org/cause/statistical-topic/categorical-methods\",\"WARC-Payload-Digest\":\"sha1:DTHHMT3DRH6IZ6P3HZHCUZAA227CCPVK\",\"WARC-Block-Digest\":\"sha1:UXZII7IFV332SSED4O7EIAQTWKICSLRC\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764500017.27_warc_CC-MAIN-20230202101933-20230202131933-00358.warc.gz\"}"}
https://askworksheet.com/fun-multiplication-worksheets/	[ "", null, "# Fun Multiplication Worksheets\n\nPrintable math worksheets for fun multiplication 1. The twos fives and tens are easy.", null, "Multiplication Spin And Multiply 2 8 So Much Fun When You Can Make A Game Out Of Learni Multiplication Math Activities Elementary Common Core Math Fractions\n\n### 2 to 6.", null, "Fun multiplication worksheets. Multiplication tables and charts given here help children to solve these problems quickly. If you re looking for more practice on a particular set of multiplication facts you can find a variety of printable worksheets from k 5 learning. Worksheet number range online.\n\nFree holiday seasonal and themed multiplication worksheets to help teach the times tables. Giant collection of fun free printable multiplication worksheets in a typical school setting 3rd grade and 4th grade is the time to focus on multiplication drills and basic multiplication lessons. Practice row and column multiplication of simple and large digit numbers.\n\nUsing skip counting worksheets is a simple way to introduce the basic multiplication facts and to review them well. After all just because learning multiplication is difficult doesn t mean it can t also be fun. Up to ten.\n\n6 to 20. So they make a great first assignment. 3 to 15.\n\nThey use the same fact layouts as the spaceship math sheets above so try the first two sets of worksheets if you are looking for all of the multiplication facts or for practice without the easier problems or try the. 2 to 10. These do tend to get a little boring and tedious though so i d encourage you to mix it up with some of the games and interactive worksheets above.\n\nMultiplication worksheets worksheets multiplication mixed tables worksheets. Each subsequent number is a multiple of the original number. 8 to 30.\n\nFree printable multiplication worksheets provided here has numerous exercises to sharpen your child s multiplication skills. More printable multiplication facts worksheets. These multiplication worksheets present the facts in a spiral layout that provides fun a twist on memorizing the times tables.\n\nIndividual table worksheets. Some kids learn their multiplication skills a bit earlier and others a bit later but 3rd grade and 4th grade is the general time frame. 1 to 4.\n\nMultiplication worksheets for parents and teachers that you will want to print. 2 to 12. For additional multiplication help check out our age appropriate multiplication worksheets and find the activities that your child will find stimulating and engaging.\n\nMultiplication mastery is close at hand with these thorough and fun worksheets that cover multiplication facts whole numbers fractions decimals and word problems. 12 to 100.", null, "", null, "Multiplication Worksheets Coloring Rocks Times Tables Worksheets Maths Times Tables Math Worksheets", null, "Silly Turtle Multiplication Puzzle Multiplication Facts Worksheets Math Coloring Worksheets Math Coloring", null, "Math Worksheet Fun Printable K5 Worksheets Fun Math Worksheets Math Coloring Addition Coloring Worksheet", null, "Multiplication Coloring Pages Google Search Math Coloring Worksheets Fun Math Worksheets Math Coloring", null, "Autumn Fall Color By Multiplication Worksheets Multiplication Worksheets Math Coloring Worksheets Math Worksheets", null, "Free Worksheet From Worksheets For Math Com By Rock N Learn Free Addition Worksheets Free Subtraction Worksheets F Fun Math Worksheets Math Maze Fun Math", null, "Halloween Multiplication Worksheets Math Multiplication Worksheets Multiplication Worksheets Math Worksheets", null, "Fun Math Worksheets For 4th Grade In 2020 Math Coloring Worksheets Fun Math Worksheets Math Coloring", null, "Fun Easy Thanksgiving Coloring And Activities Pages For Kids Thanksgiving Math Worksheets Thanksgiving Math Thanksgiving Worksheets", null, "Fun Multiplication Worksheets To 10×10 Multiplication Worksheets Fun Math Worksheets Math Sheets", null, "Summer End Of The Year Multiplication Worksheets By Teaching Naturally Multiplication Worksheets Multiplication Learning Colors", null, "This Growing Bundle Of Multiplication Tables From 2 To 12 Is Designed To Help Students P Multiplication Facts Multiplication Worksheets Times Tables Worksheets", null, "Worksheetfun Free Printable Worksheets Multiplication Activities Multiplication Worksheets Fun Math Worksheets", null, "Multiplication Spin And Multiply Such A Fun Multiplication Math Game Found In The November No P 3rd Grade Math Math Multiplication Worksheets Multiplication", null, "Multiplication Coloring Printable Middle School Math Worksheets Fun Math Worksheets Middle School Math Worksheets Fun Math", null, "Multiplication Coloring Worksheets 4th Grade Mosaic Coloring Pages For In 2020 Math Coloring Groundhog Day Activities Fun Math", null, "Coloring Multiplication Worksheets Free Multiplication Coloring Excel Math Worksheets Pictures Addi Math Coloring Worksheets Fun Math Worksheets 3rd Grade Math", null, "Math Worksheets Printable Fun Multiplication To 10×10 2 Fun Math Worksheets 1st Grade Math Worksheets Math Coloring Worksheets", null, "Previous post English Worksheet For Class 2nd", null, "Next post Free Printable English Worksheets For 5th Grade" ]	[ null, "https://askworksheet.com/wp-content/uploads/2022/07/034d40fcdf0eeebe5909887375ea0327.png", null, 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https://developer.apple.com/documentation/metalperformanceshaders/mpscnnpooling?changes=_3	[ "Class\n\n# MPSCNNPooling\n\nA pooling kernel.\n\n## Overview\n\nPooling is a form of non-linear sub-sampling. Pooling partitions the input image into a set of rectangles (overlapping or non-overlapping) and, for each such sub-region, outputs a value. The pooling operation is used in computer vision to reduce the dimensionality of intermediate representations.\n\nThe encode methods in the `MPSCNNKernel` class can be used to encode an `MPSCNNPooling` object to a `MTLCommandBuffer` object. The exact location of the pooling window for each output value is determined as follows:\n\n• The pooling window center for the first (top left) output pixel of the clip rectangle is at spatial coordinates `(offset.x, offset.y)` in the input image.\n\n• From this, the top left corner of the pooling window is at `(offset.x - floor(kernelWidth/2)`, `offset.y - floor(kernelHeight/2))` and extends `(kernelWidth, kernelHeight) `pixels to the right and down direction, which means that the last pixel to be included into the pooling window is at `(offset.x + floor((kernelWidth-1)/2)`, `offset.y + floor((kernelHeight-1)/2))`, so that for even kernel sizes the pooling window extends one pixel more into the left and up direction.\n\n• The following pooling windows can be then easily deduced from the first one by simple shifting the source coordinates according to the values of the strideInPixelsX and strideInPixelsY properties.\n\nFor example, the pooling window center `w(x,y)` for the output value at coordinate `(x,y)` of the destination clip rectangle (`(x,y)` computed with regard to clipping rectangle origin) is at `w(x,y) = (offset.x + strideInPixelsX * x , offset.y + strideInPixelsY * y)`.\n\nQuite often it is desirable to distribute the pooling windows as evenly as possible in the input image. As explained above, if the `offset` is zero, then the center of the first pooling window is at the top left corner of the input image, which means that the left and top stripes of the pooling window are read from outside the input image boundaries (when filter size is larger than unity). Also it may mean that some values from the bottom and right stripes are not included at all in the pooling, resulting in loss of valuable information.\n\nA scheme used in some common libraries is to shift the source `offset` according to the following formula:\n\n• `offset.xy += {(int)ceil(((L.xy - 1) % s.xy) / 2)}`, for odd `f.xy`\n\n• `offset.xy += {(int)floor(((L.xy - 1) % s.xy) / 2) + 1},` for even `f.xy`\n\nWhere `L` is the size of the input image (or more accurately the size corresponding to the scaled `clipRect` value in source coordinates, which commonly coincides with the source image itself), `s.xy` is `(``strideInPixelsX`, `strideInPixelsY``)` and `f.xy` is `(kernelWidth, kernelHeight)`.\n\nThis offset distributes the pooling window centers evenly in the effective source `clipRect`, when the output size is rounded up with regards to stride (`output size = ceil(input size / stride)`) and is commonly used in CNN libraries (for example TensorFlow uses this offset scheme in its maximum pooling implementation `tf.nn.max_pool` with `'S``AME``'` - padding, for `'VALID' `padding one can simply set `offset.xy += floor(f.xy/2)` to get the first pooling window inside the source image completely).\n\nFor an `MPSCNNPoolingMax` object, the way the input image borders are handled can become important: if there are negative values in the source image near the borders of the image and the pooling window crosses the borders, then using a `MPSImageEdgeMode.zero` edge modemay cause the maximum pooling operation to override the negative input data values with zeros coming from outside the source image borders, resulting in large boundary effects. A simple way to avoid this is to use a `MPSImageEdgeMode.clamp` edge mode, which for an `MPSCNNPoolingMax` object effectively causes all pooling windows to remain within the source image.\n\n## Relationships\n\n### Pooling Layers\n\n`class MPSCNNPoolingAverageGradient`\n\n`class MPSCNNPoolingL2Norm`\n\nAn L2-norm pooling filter.\n\n`class MPSCNNDilatedPoolingMax`\n\nA dilated max pooling filter.\n\n`class MPSCNNDilatedPoolingMaxGradient`\n\nA gradient dilated max pooling filter.\n\n`class MPSCNNPoolingL2NormGradient`\n\n`class MPSCNNPoolingMaxGradient`" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.82769656,"math_prob":0.92319185,"size":3740,"snap":"2020-10-2020-16","text_gpt3_token_len":851,"char_repetition_ratio":0.15604925,"word_repetition_ratio":0.015873017,"special_character_ratio":0.22272727,"punctuation_ratio":0.09388336,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9633788,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-04-07T01:42:04Z\",\"WARC-Record-ID\":\"<urn:uuid:7319d588-31f5-481c-9251-977efab78a16>\",\"Content-Length\":\"57674\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:82c1947a-247e-4696-bfd0-04cc809b32c3>\",\"WARC-Concurrent-To\":\"<urn:uuid:45c71004-4ae9-4944-9361-9aedc3a3d07c>\",\"WARC-IP-Address\":\"17.253.21.201\",\"WARC-Target-URI\":\"https://developer.apple.com/documentation/metalperformanceshaders/mpscnnpooling?changes=_3\",\"WARC-Payload-Digest\":\"sha1:FKGI643YW4FASASWP7RAUTDSWV4E2CQL\",\"WARC-Block-Digest\":\"sha1:CUDN7GW4GIPITWVPBNHJ2CAEXOBA4ICT\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-16/CC-MAIN-2020-16_segments_1585371662966.69_warc_CC-MAIN-20200406231617-20200407022117-00431.warc.gz\"}"}
https://tonybreyal.wordpress.com/2011/11/02/code-optimization-one-r-problem-ten-solutions-%E2%80%93-now-eleven-2/	[ "# Consistently Infrequent\n\n## November 2, 2011\n\n### Code Optimization: One R Problem, Ten Solutions – Now Eleven!\n\nFiled under: R — Tags: , , , , , , , , , , — BD @ 11:47 pm\n\nEarlier this year I came across a rather interesting page about optimisation in R from rwiki. The goal was to find the most efficient code to produce strings which follow the pattern below given a single integer input `n`:\n\n```# n = 4\n \"i001.002\" \"i001.003\" \"i001.004\" \"i002.003\" \"i002.004\" \"i003.004\"\n# n = 5\n \"i001.002\" \"i001.003\" \"i001.004\" \"i001.005\" \"i002.003\" \"i002.004\" \"i002.005\" \"i003.004\"\n \"i003.005\" \"i004.005\"\n# n = 6\n \"i001.002\" \"i001.003\" \"i001.004\" \"i001.005\" \"i001.006\" \"i002.003\" \"i002.004\" \"i002.005\"\n \"i002.006\" \"i003.004\" \"i003.005\" \"i003.006\" \"i004.005\" \"i004.006\" \"i005.006\"\n```\n\nFrom this we can see that the general pattern for `n` is:\n\n```# pseudo code\nFOR j = 1 to n-1\nFOR k = j+1 to n\nPRINT \"i\" + format(j, digits = 3) + \".\" + format(k, digits = 3)\nk = k + 1\nEND FOR\nj = j + 1\nEND FOR\n# this produces (n-1)n/2 strings.\n```\n\nIt is rather heart warming to go though that rwiki page and see how we can sequentially optimise the algorithm in R to more efficiently produce the desired string sequence. I learned quite a lot from this page about R and how fun these types of challenges can be!\n\nLooking at the tenth solution, it achieves its speed by recognising that there are a total of n unique strings (e.g. “001”, “002”) to the pattern. These can be assigned to a character vector which we subsequently call by working out the sequence of indices we require and passing them to the character vector:\n\n```generateIndex10 <- function(n) {\ns <- sprintf(\"%03d\", seq_len(n));\npaste(\n\"i\",\nrep(s[1:(n-1)], (n-1):1),\n\".\",\nunlist(lapply(2:n, function(k) s[k:n]), use.names=FALSE),\nsep=\"\"\n)\n}\ngenerateIndex10(7)\n \"i001.002\" \"i001.003\" \"i001.004\" \"i001.005\" \"i001.006\" \"i001.007\" \"i002.003\" \"i002.004\"\n \"i002.005\" \"i002.006\" \"i002.007\" \"i003.004\" \"i003.005\" \"i003.006\" \"i003.007\" \"i004.005\"\n \"i004.006\" \"i004.007\" \"i005.006\" \"i005.007\" \"i006.007\"\n```\n\nPlaying around with the solution above, I noticed that a speed up would be possible if the following were implemented:\n\n1. `rep` on a string vector is slower than simply using `rep.int` to work out the indices first and then passing those into the character vector.\n2. Initialise the vectors to the correct size. This seems to produce a speed up though I’m not sure why it would in a vectorised solution. I could see the benefit if I were creating the vectors using an explicit R loop and concatenating the results but that is not the case here.\n3. The functions could be compiled into bytecodes using the new compiler package (NB given that tenth solution is from 2007, this option would not have been available back then).\n4. This problem could be set up to run in parallel. I didn’t do this because I’m still within running distance of being an R noob 😀\n\nGiven the first three points above, this is the solution I came up with:\n\n```generateIndex11 <- function(n) {\n# initialise vectors\nlen <- (n-1)n/2\ns <- vector(mode = \"character\", length = n)\nind.1 <- vector(mode = \"integer\", length = len)\nind.2 <- vector(mode = \"integer\", length = len)\n\n# set up strings\ns <- sprintf(\"%03d\", seq_len(n))\n\n# calculate indices\nind.1 <- rep.int(1:(n-1), (n-1):1)\nind.2 <- unlist(lapply(2:n, \":\", n), use.names = FALSE)\n\n# paste strings together\nreturn(paste(\"i\", s[ind.1], \".\", s[ind.2], sep = \"\"))\n}\n```\n\nBut is it faster? We can benchmark it and see!\n\n```\nlibrary(compiler)\nlibrary(rbenchmark)\n\n# compile functions\ncompiled_generateIndex10 <- cmpfun(generateIndex10)\ncompiled_generateIndex11 <- cmpfun(generateIndex11)\n\n# set size\nn = 5000\n\n# run benchmark\nbenchmark(generateIndex10(n), compiled_generateIndex10(n), generateIndex11(n), compiled_generateIndex11(n),\ncolumns = c(\"test\", \"replications\", \"elapsed\", \"relative\"),\norder = \"relative\",\nreplications = 10)\n\n###--- TIMINGS ---###\ntest replications elapsed relative\n4 compiled_generateIndex11(n) 10 51.46 1.000000\n3 generateIndex11(n) 10 51.67 1.004081\n2 compiled_generateIndex10(n) 10 54.35 1.056160\n1 generateIndex10(n) 10 61.20 1.189273\n\n```\n\nSo it seems that there is a speed up between `generateIndex10` and `generateIndex11`. If we call the benchmark for different values of n and plot using the ggplot2 pack we get the following:\n\n```\nlibrary(ggplot2)\nget_bm <- function(n) {\ngc()\ndf = benchmark(generateIndex10(n), compiled_generateIndex10(n), generateIndex11(n), compiled_generateIndex11(n),\ncolumns = c(\"test\", \"elapsed\"),\norder = \"elapsed\",\nreplications = 10)\ndf\\$n = n\nreturn(df)\n\n}\n\ndf.list <- lapply(seq(1000, 9000, by = 250), get_bm)\ndf <- do.call(\"rbind\", df.list)\nggplot(df, aes(x=n, y=elapsed, colour=test, shape=test)) + geom_point() + stat_smooth() + xlab(\"N\") + ylab(\"Time (seconds)\")\n```\n\nWhich produces the graph below. I don’t know what’s happening at the extreme values but I suspect that it has something to do with memory because on my 8GB machine, the RAM was mostly at 99% for the largest values of n. I also like how stable the compiled versions of the functions are:", null, "In conclusion, whilst the compiled eleventh solution is faster for larger values of `n`, for values below `1000` the tenth solution is sometimes faster.\n\nWriting this post, I noticed something and actually came up with an even faster twelfth solution. When I get time in the next couple of days I’ll make a post about it and also submit it to rwiki.\n\nThanks for reading, this was my first proper R blog post, constructive comments are most welcome! Also, please kindly bare in mind that I am not an R expert but rather just trying to improve my knowledge of the language 🙂\n\n1.", null, "Very shortly this web page will be famous among all blogging and\nsite-building visitors, due to it’s fastidious articles\n\nComment by heart disease fish oil — May 11, 2013 @ 11:24 am\n\n2.", null, "Good article. I absolutely love this website.\n\nStick with it!\n\nComment by keygen for dark souls — June 21, 2013 @ 10:11 pm" ]	[ null, "http://rwiki.sciviews.org/lib/exe/fetch.php", null, "https://1.gravatar.com/avatar/10297879e08a48e0b92e3aa5e964d490", null, "https://0.gravatar.com/avatar/c1a3dcf4aa208e83d4be51b1a9c19135", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.81126845,"math_prob":0.93572146,"size":5735,"snap":"2023-14-2023-23","text_gpt3_token_len":1659,"char_repetition_ratio":0.15058453,"word_repetition_ratio":0.018847007,"special_character_ratio":0.35850042,"punctuation_ratio":0.15490533,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9857472,"pos_list":[0,1,2,3,4,5,6],"im_url_duplicate_count":[null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-05-30T06:42:31Z\",\"WARC-Record-ID\":\"<urn:uuid:80cb2825-be1d-47d7-b424-710508daf90e>\",\"Content-Length\":\"92350\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a3353904-9659-4130-a7df-5e6033ddb049>\",\"WARC-Concurrent-To\":\"<urn:uuid:ea8fa4a9-2572-49fc-9254-c769ab50307a>\",\"WARC-IP-Address\":\"192.0.78.12\",\"WARC-Target-URI\":\"https://tonybreyal.wordpress.com/2011/11/02/code-optimization-one-r-problem-ten-solutions-%E2%80%93-now-eleven-2/\",\"WARC-Payload-Digest\":\"sha1:WN5H2QLEGIRY2GUKTHSN3PJEO76HL2HO\",\"WARC-Block-Digest\":\"sha1:ZYE6XXXSO2HNZZEAPFUOYVJQOB2H6DTF\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-23/CC-MAIN-2023-23_segments_1685224645417.33_warc_CC-MAIN-20230530063958-20230530093958-00387.warc.gz\"}"}