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https://www.gradesaver.com/textbooks/math/algebra/algebra-1/chapter-4-an-introduction-to-functions-4-4-graphing-a-function-rule-practice-and-problem-solving-exercises-page-257/14	[ "## Algebra 1", null, "Equation: y=-5x+12 The general form of a linear equation is y=mx+b where m is the slope and b is the y-intercept. In the equation above the y-intercept, b, is +12 so start by plotting the point (0,12). In the equation above the slope, m, is -5 and since slope= $\\frac{rise}{run}$ you go down 5 units and to the right one unit from each point starting from the y-intercept to get the final graph." ]	[ null, "https://gradesaver.s3.amazonaws.com/uploads/solution/0974fc0c-00be-4074-a38f-ed83f1b0b1fa/result_image/1514840405.PNG", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.90079826,"math_prob":0.99987185,"size":446,"snap":"2022-05-2022-21","text_gpt3_token_len":122,"char_repetition_ratio":0.14479639,"word_repetition_ratio":0.025316456,"special_character_ratio":0.2735426,"punctuation_ratio":0.097087376,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000013,"pos_list":[0,1,2],"im_url_duplicate_count":[null,3,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-05-25T03:17:15Z\",\"WARC-Record-ID\":\"<urn:uuid:622d588f-31d8-4d97-a79e-22da50bcf8ef>\",\"Content-Length\":\"81975\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:88deadc0-428a-432f-8f33-3ab91ed51efe>\",\"WARC-Concurrent-To\":\"<urn:uuid:bd2358a7-ac76-4417-89ee-77c55fb7a07e>\",\"WARC-IP-Address\":\"44.198.198.134\",\"WARC-Target-URI\":\"https://www.gradesaver.com/textbooks/math/algebra/algebra-1/chapter-4-an-introduction-to-functions-4-4-graphing-a-function-rule-practice-and-problem-solving-exercises-page-257/14\",\"WARC-Payload-Digest\":\"sha1:JWR3T3LGZ63BOYBSZB5E7KB5237TRRGR\",\"WARC-Block-Digest\":\"sha1:4TPNNMWLO7VMUXAZPQTYSYXYNL7Z5XJ5\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-21/CC-MAIN-2022-21_segments_1652662578939.73_warc_CC-MAIN-20220525023952-20220525053952-00226.warc.gz\"}"}
https://www.programming-helper.com/snippet/5aKLkGLPfQvN6rfFEa3g	[ "Generation\n\ngenerate functionSat, 26 Nov 2022\n\n# the user also specifies the number of coin tosses at the beginning of the program. tosses a coin and then determines that did it fall on the side of the head or the writing. After the coin is tossed, write the result on the screen, that is, how many times there were heads and how many times there were writing.\n\n``````// User enters the number of coin tosses\n\n// random generator of numbers\nRandom random = new Random();\n\n// variables for counting\nint tailsCount = 0;\n\n// loop for the number of coin tosses\nfor (int i = 0; i < numberOfTosses; i++)\n{\n// random generation of numbers\nint toss = random.Next(0, 2);\n\n// increase the number of heads\nif (toss == 0)\n{" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7754364,"math_prob":0.99561167,"size":902,"snap":"2022-40-2023-06","text_gpt3_token_len":216,"char_repetition_ratio":0.1714922,"word_repetition_ratio":0.045454547,"special_character_ratio":0.27161863,"punctuation_ratio":0.1497006,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9846149,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-01-30T04:52:25Z\",\"WARC-Record-ID\":\"<urn:uuid:97912ff7-b49f-4462-a708-b5ee3c8f1838>\",\"Content-Length\":\"27600\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:517042f3-9b80-4c94-bf18-e8510b40412c>\",\"WARC-Concurrent-To\":\"<urn:uuid:4991ba5b-e797-4589-8a6d-a800bb817316>\",\"WARC-IP-Address\":\"76.76.21.123\",\"WARC-Target-URI\":\"https://www.programming-helper.com/snippet/5aKLkGLPfQvN6rfFEa3g\",\"WARC-Payload-Digest\":\"sha1:2LLKPWRXNSEIFZAENLRPE5RDXQRJP6PA\",\"WARC-Block-Digest\":\"sha1:RJBEZT3CRKRU5KOG73BLI32Z3BEMB7ID\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764499801.40_warc_CC-MAIN-20230130034805-20230130064805-00337.warc.gz\"}"}
https://math.answers.com/questions/How_many_inches_are_there_in_58_centimetres	[ "", null, "", null, "", null, "", null, "0\n\n# How many inches are there in 58 centimetres?\n\nUpdated: 9/14/2023", null, "Wiki User\n\n6y ago\n\nThere are 2.54 centimetres in one inch. Therefore, rounded to two decimal places, 58 centimetres is equal to 58 / 2.54 = 22.83 inches.\n58 cm = 22.835 inches\n58 cm = 22.8 inches.\n58 cm is equal to 22.8346 inches.", null, "Wiki User\n\n6y ago", null, "", null, "", null, "Wiki User\n\n13y ago\n\n58 cm = 22.8 inches.", null, "", null, "", null, "Earn +20 pts\nQ: How many inches are there in 58 centimetres?\nSubmit\nStill have questions?", null, "", null, "Related questions\n\n### How many centimeters are there in 58 inches?\n\nThere are 2.54 centimetres in one inch. Therefore, 58 inches is equal to 58 x 2.54 = 147.32 centimetres.\n\n### What equals 58 inches?\n\n58 inches is 147.32 centimeters.\n\n### How many centimetres are in 20.5 inches?\n\nThere are 52.07 centimetres in 20.5 inches.\n\n### How many centimetres are in 6.7 inches?\n\nThere are 17.02 centimetres in 6.7 inches.\n\n### How many centimetres is 84 inches?\n\n84 inches = 213.36 centimetres\n\n### How many inches is 10 centimetres?\n\n10 centimetres = 3.9 inches\n\n58\n\n### How many inches in 58cm?\n\nBy unit of length and distance and conversion ,we can say that 1 in =2.54 cm 58 cm=22.83 in\n\n### 5 inches is how many centimetres?\n\n12.7 centimetres.\n\n### How many centimetres in 66 inches?\n\n66 inches = 167.6 centimetres\n\n### How many inches make up 10 centimetres?\n\nThere are 3.94 inches in 10 centimetres.\n\n### How many inches are in 24 inches?\n\nThere are 24 inches in 24 inches. Perhaps you are asking how many centimetres there are in 24 inches. In which case, there are 2.54 centimetres in one inch. 24 inches is equal to 24 x 2.54 = 60.96 centimetres." ]	[ null, "https://math.answers.com/icons/searchIcon.svg", null, "https://math.answers.com/icons/searchGlassWhiteIcon.svg", null, "https://math.answers.com/icons/notificationBellIcon.svg", null, "https://math.answers.com/icons/coinIcon.svg", null, "https://math.answers.com/images/avatars/default.png", null, "https://math.answers.com/images/avatars/default.png", null, "https://math.answers.com/images/avatars/default.png", null, "https://math.answers.com/icons/sendIcon.svg", null, "https://math.answers.com/images/avatars/default.png", null, "https://math.answers.com/images/avatars/default.png", null, "https://math.answers.com/icons/sendIcon.svg", null, "https://math.answers.com/icons/coinIcon.svg", null, "https://math.answers.com/icons/searchIcon.svg", null, "https://st.answers.com/html_test_assets/imp_-_pixel.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8761844,"math_prob":0.99196076,"size":1217,"snap":"2023-40-2023-50","text_gpt3_token_len":408,"char_repetition_ratio":0.2893652,"word_repetition_ratio":0.41284403,"special_character_ratio":0.3812654,"punctuation_ratio":0.1891892,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99914485,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-10-02T11:18:35Z\",\"WARC-Record-ID\":\"<urn:uuid:9e0f4744-3274-4ef6-a7bc-77783a5ee929>\",\"Content-Length\":\"169903\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f2638255-8374-46d6-9aa2-6a09c5972526>\",\"WARC-Concurrent-To\":\"<urn:uuid:22329727-38b4-4ce4-b64d-d40401c8f578>\",\"WARC-IP-Address\":\"146.75.32.203\",\"WARC-Target-URI\":\"https://math.answers.com/questions/How_many_inches_are_there_in_58_centimetres\",\"WARC-Payload-Digest\":\"sha1:2DV6FC5RNQUMVHOSLZ5H5JMPJ4FUQSLI\",\"WARC-Block-Digest\":\"sha1:W5RCIO2QQR75J2PDLYLZF7QGW2HFF2TC\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233510994.61_warc_CC-MAIN-20231002100910-20231002130910-00100.warc.gz\"}"}
https://uk.mathworks.com/help/coder/ref/coder.wref.html	[ "# coder.wref\n\nIndicate write-only data to pass by reference\n\n## Syntax\n\n``coder.wref(arg)``\n\n## Description\n\nexample\n\n````coder.wref(arg)` indicates that `arg` is a write-only expression or variable to pass by reference to an external C/C++ function. Use `coder.wref` only inside a `coder.ceval` call. This function enables the code generator to optimize the generated code by ignoring prior assignments to `arg` in your MATLAB® code, because the external function is assumed to not read from the data. Write to all the elements of `arg` in your external code to fully initialize the memory. NoteThe C/C++ function must fully initialize the memory referenced by `coder.wref(arg)`. Initialize the memory by assigning values to every element of `arg` in your C/C++ code. If the generated code tries to read from uninitialized memory, it can cause undefined run-time behavior. See also `coder.ref` and `coder.rref`.```\n\n## Examples\n\n### Pass Array by Reference as Write-Only\n\nSuppose that you have a C function `init_array`.\n\n```void init_array(double* array, int numel) { for(int i = 0; i < numel; i++) { array[i] = 42; } }```\n\nThe C function defines the input variable `array` as a pointer to a double.\n\nCall the C function `init_array` to initialize all elements of `y` to 42:\n\n```... Y = zeros(5, 10); coder.ceval('init_array', coder.wref(Y), int32(numel(Y))); ...```\n\n### Pass Multiple Arguments as a Write-Only Reference\n\n```... U = zeros(5, 10); V = zeros(5, 10); coder.ceval('my_fcn', coder.wref(U), int32(numel(U)), coder.wref(V), int32(numel(V))); ...```\n\n### Pass Class Property as a Write-Only Reference\n\n```... x = myClass; x.prop = 1; coder.ceval('foo', coder.wref(x.prop)); ...```\n\n### Pass Structure as a Write-Only Reference\n\nTo indicate that the structure type is defined in a C header file, use `coder.cstructname`.\n\nSuppose that you have the C function `init_struct`. This function writes to the input argument but does not read from it.\n\n```#include \"MyStruct.h\" void init_struct(struct MyStruct my_struct) { my_struct->f1 = 1; my_struct->f2 = 2; } ```\n\nThe C header file, `MyStruct.h`, defines a structure type named `MyStruct`:\n\n```#ifndef MYSTRUCT #define MYSTRUCT typedef struct MyStruct { double f1; double f2; } MyStruct; void init_struct(struct MyStruct my_struct); #endif ```\n\nIn your MATLAB function, pass a structure as a write-only reference to `init_struct`. Use `coder.cstructname` to indicate that the structure type for `s` has the name `MyStruct` that is defined in the C header file `MyStruct.h`.\n\n```function y = foo %#codegen y = 0; coder.updateBuildInfo('addSourceFiles','init_struct.c'); s = struct('f1',1,'f2',2); coder.cstructname(s,'MyStruct','extern','HeaderFile','MyStruct.h'); coder.ceval('init_struct', coder.wref(s)); ```\n\nTo generate standalone library code, enter:\n\n```codegen -config:lib foo -report ```\n\n### Pass Structure Field as a Write-Only Reference\n\n```... s = struct('s1', struct('a', [0 1])); coder.ceval('foo', coder.wref(s.s1.a)); ...```\n\nYou can also pass an element of an array of structures:\n\n```... c = repmat(struct('u',magic(2)),1,10); b = repmat(struct('c',c),3,6); a = struct('b',b); coder.ceval('foo', coder.wref(a.b(3,4).c(2).u)); ... ```\n\n## Input Arguments\n\ncollapse all\n\nArgument to pass by reference to an external C/C++ function. The argument cannot be a class, a System object™, a cell array, or an index into a cell array.\n\nData Types: `single` \| `double` \| `int8` \| `int16` \| `int32` \| `int64` \| `uint8` \| `uint16` \| `uint32` \| `uint64` \| `logical` \| `char` \| `struct`\nComplex Number Support: Yes\n\n## Limitations\n\n• You cannot pass these data types by reference:\n\n• Class or System object\n\n• Cell array or index into a cell array\n\n• If a property has a get method, a set method, or validators, or is a System object property with certain attributes, then you cannot pass the property by reference to an external function. See Passing By Reference Not Supported for Some Properties.\n\n## Tips\n\n• If `arg` is an array, then `coder.wref(arg)` provides the address of the first element of the array. The `coder.wref(arg)` function does not contain information about the size of the array. If the C function must know the number of elements of your data, pass that information as a separate argument. For example:\n\n```coder.ceval('myFun',coder.wref(arg),int32(numel(arg)); ```\n• When you pass a structure by reference to an external C/C++ function, use `coder.cstructname` to provide the name of a C structure type that is defined in a C header file.\n\n• In MATLAB, `coder.wref` results in an error. To parametrize your MATLAB code so that it can run in MATLAB and in generated code, use `coder.target`.\n\n• You can use `coder.opaque` to declare variables that you pass to and from an external C/C++ function.\n\n## Version History\n\nIntroduced in R2011a" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5762214,"math_prob":0.8184748,"size":4602,"snap":"2022-05-2022-21","text_gpt3_token_len":1179,"char_repetition_ratio":0.15593736,"word_repetition_ratio":0.034383953,"special_character_ratio":0.259887,"punctuation_ratio":0.19711043,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9825759,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-05-26T20:21:20Z\",\"WARC-Record-ID\":\"<urn:uuid:d4b1dba4-e3bf-48b4-8007-47fc798c5dd9>\",\"Content-Length\":\"85543\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:fa05a15c-06d9-4fe4-86cf-474828107151>\",\"WARC-Concurrent-To\":\"<urn:uuid:10029846-ea35-4428-8c1d-c227d7e1bded>\",\"WARC-IP-Address\":\"104.69.217.80\",\"WARC-Target-URI\":\"https://uk.mathworks.com/help/coder/ref/coder.wref.html\",\"WARC-Payload-Digest\":\"sha1:YWIERYGWVFDIMTO3TNDIDRC6GJ7OOKAN\",\"WARC-Block-Digest\":\"sha1:MBHHYXQB2KUPU2AYT22VAH23D3FSZ43U\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-21/CC-MAIN-2022-21_segments_1652662625600.87_warc_CC-MAIN-20220526193923-20220526223923-00746.warc.gz\"}"}
https://www.scientistsolutions.com/forum/anatomy-and-physiology-electrophysiology/what-vm-ion-permeability-test	[ "# What is the \"Vm\" in ion permeability test?\n\n7 posts / 0 new\npangeng", null, "What is the \"Vm\" in ion permeability test?\n\nI am still new in learning this method at present.\n\nIf what I understand is right, to test the relative ion permeability of one type of ion channels, we initially use the Goldman equation. We set the permeability of one ion (like K+) as 1 normalizing other ions permeability to it.\n\nSay like we are tesitng relative permeability of [Na+], technically we need values of [Na+]i, [Na+]o, [K+]o, [K+]i and Vm to apply to the equation and calculate the value of Pna/Pk.\n\nQuesiton is: should the Vm here resting membrane potential (current=0)? In my opinion the Vm should be the balanced membrane potential when there are only two ions Na+ and K+ in the solution of both side. I am not sure which is right.\n\nlazy", null, "If I understand correctly,\n\nIf I understand correctly, you should measure the membrane potential (Vm), and then you should apply the value of membrane potential (Vm), [K]o, [K]i, {Na}o, [Na]i to the Goldman equation.\n\nFrom that, you can calculate the permeability (Pna/Pk) which is multiplied to [Na]i and [Na]o.\n\nThus, Vm is the measured membrane potential.\n\nHope this makes sense.\n\npangeng", null, "one thing I still dont\n\none thing I still dont understand.\nIf what I know is right, the membrane potential (Vm) is a combination of reversal potential of all ions (while Pk+ is dominant). So if Vm means the cellular membane potential, all the ion cencentration should be included in the GHK equation, like Cl-, Ca2+, H+, etc., but not only Na+ and K+.\n\nlazy", null, "Pcl, Pca, etc. are ignorable\n\nPcl, Pca, etc. are ignorable compare with Pna in the resting potential.\n\nYou can see the GHK equation on lemi-log graph. In the range of lower [K]o concentration, the curve gets deviation from the Nernst equation for K ion ( a straight line) because of the Pna.\n\nHope this makes sense.\n\npangeng", null, "so does it mean when we\n\nso does it mean when we detect the permeability of Cl-, we should apply [Na]i and [Na]o in the equation?\n\nlazy", null, "If you detect the Cl\n\nIf you detect the Cl permeability in resting membrane, then you apply Pcl/Pk to GHK equation.\n\nlazy", null, "This website is explaining\n\nThis website is explaining all that you should calculate.\nFYI\n\nhttp://www.physiologyweb.com/calculators/ghk_equation_calculator.html" ]	[ null, "https://www.scientistsolutions.com/sites/default/files/pictures/no-picture.jpg", null, "https://www.scientistsolutions.com/sites/default/files/pictures/no-picture.jpg", null, "https://www.scientistsolutions.com/sites/default/files/pictures/no-picture.jpg", null, "https://www.scientistsolutions.com/sites/default/files/pictures/no-picture.jpg", null, "https://www.scientistsolutions.com/sites/default/files/pictures/no-picture.jpg", null, "https://www.scientistsolutions.com/sites/default/files/pictures/no-picture.jpg", null, "https://www.scientistsolutions.com/sites/default/files/pictures/no-picture.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8847313,"math_prob":0.97004294,"size":1992,"snap":"2021-43-2021-49","text_gpt3_token_len":510,"char_repetition_ratio":0.1498994,"word_repetition_ratio":0.0,"special_character_ratio":0.23293173,"punctuation_ratio":0.13012049,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9973632,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-26T08:46:39Z\",\"WARC-Record-ID\":\"<urn:uuid:370e11b2-e5e5-4c19-9bef-347021f08393>\",\"Content-Length\":\"34526\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:5faa2154-b938-4e2d-8651-4432903f2470>\",\"WARC-Concurrent-To\":\"<urn:uuid:848b438d-c76d-4d9c-bb24-e6789078a47f>\",\"WARC-IP-Address\":\"143.198.1.144\",\"WARC-Target-URI\":\"https://www.scientistsolutions.com/forum/anatomy-and-physiology-electrophysiology/what-vm-ion-permeability-test\",\"WARC-Payload-Digest\":\"sha1:5ZQDH623QZZZ6L23Q7HVHXYE6D25LZXJ\",\"WARC-Block-Digest\":\"sha1:HPWJCAYQB3AWZ3OZKNT2WX7NIZWO5PSI\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323587854.13_warc_CC-MAIN-20211026072759-20211026102759-00139.warc.gz\"}"}
https://patents.justia.com/patent/4104729	[ "# Digital multiplier\n\nThe digital multiplier is of the add and shift type with a matrix type input in which a number of serial data words can be applied simultaneously to the multiplier enabling the multiplier to compute simultaneously the sum of a number of products. This multiplier is intended for use in digital filters and single channel coders.\n\n## Latest International Standard Electric Corporation Patents:\n\nDescription\nBACKGROUND OF THE INVENTION\n\nThe present invention relates to digital multipliers of the \"add and shift\" type such as may be used in digital filters for telecommunications systems.\n\nIn such multipliers multiplication is achieved by performing a number of addition operations using an adder with an accumulator in which the adder output is stored and shifted one bit position for each addition operation. An adder comprises a series of cells, one for each bit in a digital word, each cell having inputs for the addend and the augend and outputs for the sum and carry. A so-called \"full adder cell\" has in addition a third input to which the carry from the cell of next lesser significance may be applied.\n\nIn a known form of multiplier where, say, a parallel coefficient B is to be multiplied by an incoming serial data word C to form the product A=B.times.C in the accumulator, the multiplicand B is applied in parallel to a number of AND gates for the duration of the word C. Each bit of the word C, starting with the least significant bit, is applied to all the AND gates. If a bit of word C is a logic \"1\" then the multiplicand B is added to the accumulator (if it is a logic \"0\" nothing is added) and the contents of the accumulator are right shifted before the next bit of word C is applied to the AND gates. This continues until all the bits of word C have been applied to the gates, then the multiplication is complete. The answer A is now in the accumulator and may be extracted as required by various known methods.\n\nNegative numbers can be handled by various known simple modifications. For example, if 2's complement data is used then the most significant bit of the data has a negative weight and for this bit only the multiplicand B is subtracted from instead of added to the accumulator.\n\nIn digital filters a digitally encoded sampled signal is filtered by combining together various delayed copies of the signal through suitable weighting coefficients. In general in a digital filter it is thus necessary to form an accumulated product such as:\n\nP=(D.sub.1 .times.E.sub.1) + (D.sub.2 .times.E.sub.2) + (D.sub.3 .times.E.sub.3) . . . (D.sub.n .times.E.sub.n),\n\nwhere D.sub.1, D.sub.2 etc. are filter signal serial data words and E.sub.1, E.sub.2 etc. are the filter tap coefficients. In the simple type of multiplier described above this type of operation can only be accomplished by performing first the multiplication D.sub.1 .times.E.sub.1, storing the result, changing the coefficient E, applied in parallel to the AND gates, performing the next multiplication D.sub.2 .times.E.sub.2, adding the result to the previously stored product D.sub.1 .times.E.sub.1, storing the sum, and repeating this process as many times as required. It is obvious that it would be advantageous to be able to perform all n multiplications simultaneously and produce the product P at the end of one serial data word period rather than to have to wait for n word periods.\n\nSUMMARY OF THE INVENTION\n\nAn object of the present invention is to provide an improved digital multiplier capable of being employed in digital filters and single channel coders.\n\nA feature of the present invention is the provision of a digital multiplier providing a calculated output signal comprising: a first adder having a plurality of first adder cells; an accumulator coupled to each of the first adder cells to store the output of each of the first adder cells and to shift the stored output one bit for each addition operation; and input means coupled to the first adder cells for applying simultaneously a plurality of serial data words to the first adder cells, the input means being arranged so that each of the serial words is applied to predetermined ones of the first adder cells but no two serial words are applied to the same one of the first adder cells.\n\nBRIEF DESCRIPTION OF THE DRAWING\n\nAbove-mentioned and other features and objects of this invention will become more apparent by reference to the following description taken in conjunction with the accompanying drawing, in which:\n\nFIG. 1 is a block schematic diagram of a simple digital multiplier according to the principles of the present invention using a single full adder to perform addition operations, a single accumulator in which the sum outputs of the full-adder are simultaneously accumulated and right shifted, and a basic form of wired input matrix;\n\nFIG. 2 is a block schematic diagram of a modification of the multiplier of FIG. 1 in which the basic input matrix is modified to improve flexibility;\n\nFIG. 3 is a block schematic diagram of a fourth order recursive digital filter section which utilizes a digital multiplier having a separate accumulator and transfer register with provision for overflow detection in accordance with the principles of the present invention;\n\nFIG. 4 is a block schematic diagram of a further modification of the basic multiplier of FIG. 1 using a second full adder with a modified input matrix to enable the number of inputs to the multiplier to be increased still further; and\n\nFIG. 5 is a block schematic diagram of a complete digital multiplier for handling digital data using 2's complement and carry-save addition with separate sum and carry accumulators and separate sum and carry transfer registers in accordance with the principles of the present invention.\n\nDESCRIPTION OF THE PREFERRED EMBODIMENTS\n\nIn the simple multiplier arrangement shown in FIG. 1 a 6-stage adder ADD is constructed with \"full-adder\" cells the carry from each cell being applied to the cell of next greater significance.\n\nA 7-stage accumulator ACC receives, in stages A0 to A5, the sum outputs .SIGMA..sub.0 . . . .SIGMA..sub.5 of the adder cells. The most significant stage A6 of the accumulator receives the carry output C.sub.6 of the most significant adder cell. The contents of the accumulator stages A6 . . . A1 are fed back at the data bit rate as inputs to the adder cells of next lesser significance while the multiplier output is taken serially from stage A0 of the accumulator ACC. Each adder cell has, in addition to the carry and feedback (augend) inputs a data (addend) input which is fed via an input matrix M with serial data words D.sub.1 and D.sub.2 simultaneously. Suppose that it is desired to perform the calculation: P = 5/8D.sub.1 + 1/4D.sub.2, where 5/8 and 1/4 are fixed coefficients.\n\nThe first part of the calculation, i.e., multiplying D.sub.1 by 5/8 can be broken down into two separate operations as follows:\n\n5/8D.sub.1 = 1/2D.sub.1 + 1/8D.sub.1\n\nmultiplying D.sub.1 by 1/2 is simply effected by right-shifting D.sub.1 by one bit position while multiplying D.sub.1 by 1/8 is effected by right-shifting D.sub.1 by three bit positions. If the input I.sub.5 to the most significant cell of the adder ADD is regarded as having a weighting value of 1, i.e., it has a value of 2.degree., then to multiply D.sub.1 by 1/2 it must be fed to input I.sub.4 which has a weight of 1/2 or 2.sup.-1. To multiply D.sub.1 by 1/8 it must be fed to input I.sub.1, which has a weight of 1/8 or 2.sup.-3. When D.sub.1 is fed simultaneously to these inputs I.sub.4 and I.sub.2 the result in the accumulator ACC will be 1/2D.sub.1 + 1/8D.sub.1 = 5/8D.\n\nSimilarly D.sub.2 is fed simultaneously to input I.sub.3 and the accumulator ACC will then contain the result of P of the full calculation. It will be noted that each word is fed to one or more inputs to the adder, but that no two words are fed to the same input.\n\nIt will be appreciated that such a simple arrangement as that shown in FIG. 1 has severe limitations in practice. Greater flexibility can be obtained by using ternary coding of the coefficients. Suppose that it is desired to perform the calculation:\n\nP = 7/8D.sub.1 + 9/16D.sub.2 - 1/4D.sub.3.\n\nthis can be achieved by expressing the coefficients in a ternary code as follows:\n\n7/8 = 100100 = 1 - 1/8\n\n9/16 = 010010 = 1/2 + 1/16\n\n-1/4 = 001000 = -1/4\n\nIn this table the use of the bar, i.e., 1, indicates that the relevant bit has a negative weight. This calculation can then be performed using the arrangement shown in FIG. 2. The adder ADD and accumulator ACC are the same as in FIG. 1, but the input matrix M' is more complex, each data word being fed either direct and/or inverted (complemented) to the appropriate adder inputs.\n\nWord D.sub.1 is fed directly to input I.sub.5 and in complemented form to input I.sub.2 by means of the inverter INV. In practice what this means is that although the coefficients are expressed in a ternary code the negative weights are realized by reversing the sign of the data. Thus, in effect 7/8D.sub.1 = (11/8)D.sub.1 = 1.times.D.sub.1 + 1/8.times.(-D.sub.1). D.sub.2 + D.sub.3 are similarly fed to the input matrix M' with inversion where appropriate. Once again, it is important to note that any one adder input receives not more than one word, either direct or inverted.\n\nFurther flexibility can be achieved by using serial adders for cases where the coefficients (which can be regarded as data words) have bits which overlap, even after re-expressing them in ternary form. In other words two incoming data words are required to be fed to the same adder cell. Consider as an example the calculation:\n\nP = (D.sub.1 .times.E.sub.1) + (D.sub.2 .times.E.sub.2) + (D.sub.3 .times.E.sub.3) + (D.sub.4 .times.E.sub.4) + (D.sub.5 .times.E.sub.5)\n\nwhere the coefficients E.sub.1 -E.sub.5 have the values given below and are re-expressed in the ternary forms shown in the following table.\n\n______________________________________ VALUE TERNARY FORM ______________________________________ E.sub.1 136/256 0 0 0 1 0 0 0 1 0 0 0 E.sub.2 686/256 0 1 1 0 -1 0 -1 0 0 -1 0 E.sub.3 -796/256 -1 0 1 0 0 -1 0 0 1 0 0 E.sub.4 446/256 0 1 0 0 -1 0 0 0 0 -1 0 E.sub.5 -103/256 0 0 0 -1 0 1 0 -1 0 0 1 ##STR1## ##STR2## ______________________________________\n\nIt can be seen that in nearly every column there are two overlapping ternary bits but nowhere are there more than two. This calculation can be realized by pre-adding together two data words where they are each to be multiplied by the same weight bit. In this case the result will be formed by performing the calculation as follows:\n\nP=2.sup.2 (-D.sub.3) + 2.sup.1 (D.sub.2 +D.sub.4) + 2.degree. (D.sub.2 +D.sub.3) + 2.sup.-1 (-D.sub.5 +D.sub.1) +2.sup.-2 (-D.sub.4 -D.sub.2) + 2.sup.-3 (D.sub.5 -D.sub.3) + 2.sup.-4 (-D.sub.2) +2.sup.-5 (-D.sub.5 +D.sub.1) + 2.sup.-6 (D.sub.3) + 2.sup.-7 (-D.sub.2 -D.sub.4) + 2.sup.-8 (D.sub.5)\n\nan arrangement for performing the calculation in this manner is shown in FIG. 3. The arrangement of FIG. 3 is designed as a fourth order recursive digital filter section in which in fact D.sub.1 is the only external input and D.sub.2 - D.sub.5 are derived from the multiplier output. D.sub.2 is the multiplier output proper, and D.sub.3 - D.sub.5 are successively delayed versions of D.sub.2, in each case delayed by one word period. To perform the calculation the input I.sub.10, having the weight of 2.sup.2, receives word D.sub.3 via an inverter, input I.sub.9 receives the serial sum of words D.sub.2 and D.sub.4 produced by the serial adder cell 30, input I.sub.8 receives the serial sum of words D.sub.2 and D.sub.3 produced by serial adder cell 31 and so on. To prevent overflow in the serial adder cells the data must be pre-limited to .+-.1/2-full scale range so that on addition of any two words the sum is still within the full range.\n\nThe filter section shown in FIG. 3 is designed to work with offset binary data, that is for an N-bit word the value is: ##EQU1## where the rth bit B.sub.r is worth \"0\" or \"1\". Thus, the weighted value of that bit is -2.sup.-N-1 or +2.sup.-M+1, respectively. Sign reversal (that is multiplication by -1) is achieved by complementary (inverting) the data bits.\n\nIn a practical digital filter the output under some circumstances may exceed the allowable data range and for that case overflow protection must be provided. This can be accomplished in the following manner. When multiplication starts the least significant bits of the answer start coming from the least significant adder output. At the completion of the multiplication period the remaining most significant bits of the answer are in the accumulator (A0 - A10) and must be transferred over to a transfer register (T.sub.1 - T.sub.11) so that the accumulator can be cleared to start a new multiplication. While the multiplier is working on this new multiplication the bits in the transfer register are shifted out to complete the previous answer. In general, however, when working with fixed point arithmetic, it is necessary to limit the maximum answer to the available range expressible by the data format. In digital filters, particularly of the recursive types the answer may exceed the available data range so that if the most significant few bits of the accumulated result are simply ignored an undesirable overflow characteristic may occur leading to instability. However, by storing the bits to be dropped and checking the answer to see if the allowable range has been exceeded the multiplier can be made to saturate giving a maximum positive or negative result. This is done in the arrangement of FIG. 3 by checking the states of bits T.sub.8 to T.sub.11 in overflow detector OD to detect whether overflow will occur if T.sub.9 to T.sub.11 are dropped and using the result to control the multiplier output in an overflow correction circuit OC. If no overflow occurs then the normal output is allowed to flow out but if overflow has occurred then a maximum positive or negative data signal is substituted for the completed answer according to the sign.\n\nAnother way of achieving flexibility while adhering to the general rule stated earlier is to provide multiple inputs per stage of the multiplier using a second rank of parallel adder cells. Where there are overlapping bits in the ternary expressed coefficients the data can be fed into individual inputs without the need for serial pre-addition and, hence, pre-limiting of the data input magnitude. Such an arrangement is shown in FIG. 4.\n\nThe basic multiplier design of FIG. 4 is similar to that of the preceding figures but with the additional rank of adder cells ADD 2 preceding the final rank of adder cells ADD 1. Note that the final rank of adder cells ADD 1 has to have one more cell than the additional rank of adder cells ADD 2. The inputs to the multipliers are by way of the input matrix M\" feeding the cells of the addition rank of adder cells ADD 2.\n\nConsider the following calculation:\n\nP = 7/8D.sub.1 - 3/8D.sub.2 + 3/4D.sub.3\n\nby constructing a general purpose X-Y interconnection matrix as shown in FIG. 4 and making connections only at the appropriate crossings a very flexible arrangement results. Each input data line drives one input line directly and a second input line through a sign reverser (inverter) so that either data or minus data can be connected to give coefficients expressed in ternary. This scheme will work with any form of binary coded data, such as ordinary binary, 2's complement binary, offset-binary, negabinary (radix-2) etc., provided the logic in the sign reversers and adders is appropriate to the type of coding. Conversion between different codes can be simultaneously accomplished by feeding in appropriate conversion constants to unused inputs and inverting the polarity of data bits where necessary.\n\nIn the arrangement illustrated the least significant bit carry inputs of the adders are shown grounded. They may in fact be used (a) as extra data inputs, (b) for feeding in a rounding signal, (c) for providing an automatic clear of the accumulator ACC.\n\nThis last item can considerably simplify the hardware and timing of multipliers since it eliminates the need for individual clear elements in each accumulator cell.\n\nThe method will be described using a modified design of FIG. 4 with carry-save adders instead of fully parallel ripple or look-ahead carry types. The method can equally well be used with ripple through carry addition, but it is useful to consider a carry-save addition type which minimizes problems of logic propagation delay.\n\nFIG. 5 shows a complete multiplier design using 2's complement arithmetic and carry-save addition. This design allows two inputs per bit but this can be increased by additional ranks of ordinary or carry-save adders. Input data is routed to the adder inputs via the coefficient connection matrix M'\" previously described. Multiplication of data by -1 is achieved by inverting (complementing) the data bits. This is not strictly accurate with 2's complement data since it introduces an error of one least significant bit. This will cause a constant small error in the result. In cases where this is important the inverters can be replaced by true subtract from zero circuits or a compensation signal can be fed into an unused data input. Each carry-save adder cell is a normal full adder with three inputs each worth one unit and pseudo sum and carry outputs worth one and two units, respectively. The first rank takes two data inputs plus the fed back sum accumulator and feeds the second rank. The second rank takes the sum from the first rank plus the left shifted carry (shifted left since the carry is worth 2 units) plus the fed back carry accumulator. The outputs are loaded into the sum and carry accumulators A0-A6 and B0-B7, respectively. On feedback the sum is shifted one place right as in previous examples, but the carry shift right is neutralized by the fact that being worth twice as much as the sum it needs shifting one place left.\n\nOn completion of the multiplication the answer is the sum of the contents of the sum and carry accumulators plus the least significant bits which have flowed out the least-significant-bit-sum output through the selector and into the output delay. The contents of the sum and carry accumulators are loaded into sum and carry transfer registers SCTR and shifting begins, adding the two together in serial adder 50 and routing the result through the output delay 51. At the same time a new multiplication can start. However, since at this point the sum contained in the two transfer registers is the same as the sum contained in the two accumulators it is possible by feeding back minus the sum of the two transfer registers into the accumulators to remove the old result from the calculation without the physical necessity for clearing the accumulator cells A.sub.0 to A.sub.6 and B.sub.0 to B.sub.7. Additional refinements can include overflow detection and correction and also rounding the answer to a given number of bits.\n\nWhile I have described above the principles of my invention in connection with specific apparatus it is to be clearly understood that this description is made only by way of example and not as a limitation to the scope of my invention as set forth in the objects thereof and in the accompanying claims.\n\n## Claims\n\n1. A digital multiplier providing a calculated output signal comprising:\n\nan accumulator coupled to each of said first adder cells to store the output of each of said first adder cells and to shift said stored output one bit for each addition operation said calculated output signal being provided by said accumulator;\ninput means coupled to said first adder cells for applying simultaneously a plurality of serial data words to said first adder cells, said input means being arranged so that each of said serial words is applied to predetermined ones of said first adder cells but no two serial words are applied to the same one of said first adder cells;\nfirst means coupled to said accumulator for examining a given number of the most significant bits of said calculated output signal to determine if the complete calculated output signal value will exceed a predetermined range of values; and\nsecond means coupled to said accumulator and a given one of said first adder cells for substituting for said calculated output signal a selected one of a maximum positive and negative data signal according to the sign of said calculated output signal when the value of the latter exceeds said predetermined range of values.\n\n2. A multiplier according to claim 1, wherein said input means includes\n\nan input matrix having a plurality of column conductors and a plurality of row conductors each of said column conductors being coupled to a different one of said first adder cells and each of said row conductors being coupled to a different one of said serial words, the interconnection of each of said row conductors with said column conductors being arranged in a pattern corresponding to a predetermined digital word, said interconnection pattern for each of said row conductors being different from said interconnection pattern of every other one of said row conductors with no more than one of said row conductors having an interconnection with any one of said column conductors.\n\n3. A multiplier according to claim 2, further including:\n\nfirst means coupled to said accumulator for examining a given number of the most significant bits of said calculated output signal to determine if the complete calculated output signal value will exceed a predetermined range of values; and\nsecond means coupled to said accumulator and a given one of said first adder cells for substituting for said calculated output signal a selected one of a maximum positive and negative data signal according to the sign of said calculated output signal when the value of the latter exceeds said predetermined range of values.\n\n4. A multiplier according to claim 2, further including\n\na second adder having a plurality of second adder cells, said second adder cells being equal in number to one less than the number of said first adder cells, the output of each of said second adder cells being coupled to the input of a different one of less significant ones of said first adder cells, and wherein\nsaid input matrix couples said serial words to said second adder cells, said input matrix including\na plurality of two column conductors, each of said two column conductors being coupled to a different one of said second adder cells to enable two of said serial words that are simultaneously present on said row conductors to be serially added together to provide sum outputs at the output of certain of said second adder cells, said input matrix having not more than one of said row conductors with an interconnection with any one of said plurality of two column conductors.\n\n5. A multiplier according to claim 4 wherein\n\nfirst means coupled to said accumulator for examining a given number of the most significant bits of said calculated output signal to determine if the complete calculated output signal value will exceed a predetermined range of values; and\nsecond means coupled to said accumulator and a given one of said first adder cells for substituting for said calculated output signal a selected one of a maximum positive and negative data signal according to the sign of said calculated output signal when the value of the latter exceeds said predetermined range of values.\n\n6. A multiplier according to claim 2, further including\n\nat least one additional row conductor for each of said plurality of row conductors, each of said additional row conductors having a different one of said serial words coupled thereto, and\na plurality of complementing means each coupled to a different one of said additional row conductors for complementing the digits of an associated one of said serial words\nsaid interconnection pattern for each of said additional row conductors with said column conductors being different than said interconnection pattern of every other one of said additional row conductors with no more than one of said plurality of row conductors and said additional row conductors having an interconnection with any one of said column conductors.\n\n7. A multiplier according to claim 6, wherein\n\nfirst means coupled to said accumulator for examining a given number of the most significant bits of said calculated output signal to determine if the complete calculated output signal value will exceed a predetermined range of values; and\nsecond means coupled to said accumulator and a given one of said first adder cells for substituting for said calculated output signal a selected one of a maximum positive and negative data signal according to the sign of said calculated output signal when the value of the latter exceeds said predetermined range of values.\n\n8. A multiplier according to claim 6, further including\n\na second adder having a plurality of second adder cells, said second adder cells being equal in number to one less than the number of said first adder cells, the output of each of said second adder cells being coupled to the input of a different one of less significant ones of said first adder cells, and wherein\nsaid input matrix couples said serial words to said second adder cells, said input matrix including\na plurality of two column conductors each of said two column conductors being coupled to a different one of said second adder cells to enable two of said serial words that are simultaneously present on said row conductors to be serially added together to provide sum outputs at the output of certain of said second adder cells, said input matrix having not more than one of said row conductors with an interconnection with any one of said plurality of two column conductors.\n\n9. A multiplier according to claim 8, wherein\n\nfirst means coupled to said accumulator for examining a given number of the most significant bits of said calculated output signal to determine if the complete calculated output signal value will exceed a predetermined range of values; and\nsecond means coupled to said accumulator and a given one of said first adder cells for substituting for said calculated output signal a selected one of a maximum positive and negative data signal according to the sign of said calculated output signal when the value of the latter exceeds said predetermined range of values.\n\n10. A multiplier according to claim 8, further including\n\na sum accumulator;\na carry accumulator;\na sum transfer register;\na carry transfer register; and\na carry output from each of said second adder cells are each coupled as an input to one of said first adder cells of next greater significance than the significance of the associated one of said second adder cells;\na sum output from each of said first adder cells are each fed back as a carry input of one of said second adder cells of next lesser significance than the significance of the associated one of said first adder cells through said sum accumulator,\na carry output from each of said first adder cell is fed back to a carry input of the same one of said first adder cell through said carry accumulator,\na sum output of each of said first adder cells are coupled to said sum transfer register,\na carry output of each of said first adder cells are coupled to said carry transfer register, and\nsaid additional serial adder cell is coupled to said sum and carry transfer registers to add the contents of said sum and carry transfer registers as the contents thereof are transferred serially to an output of said multiplier.\n\n11. A digital multiplier according to claim 10, wherein\n\nfirst means coupled to said accumulator for examining a given number of the most significant bits of said calculated output signal to determine if the complete calculated output signal value will exceed a predetermined range of values; and\nsecond means coupled to said accumulator and a given one of said first adder cells for substituting for said calculated output signal a selected one of a maximum positive and negative data signal according to the sign of said calculated output signal when the value of the latter exceeds said predetermined range of values.\n\n12. A digital multiplier according to claim 10, wherein\n\nthe output of said additional serial adder cell is complemented and fed back as an input to the least significant one of said first adder cell.\n\n13. A digital multiplier according to claim 12, wherein\n\nfirst means coupled to said accumulator for examining a given number of the most significant bits of said calculated output signal to determine if the complete calculated output signal value will exceed a predetermined range of values; and\nsecond means coupled to said accumulator and a given one of said first adder cells for substituting for said calculated output signal a selected one of a maximum positive and negative data signal according to the sign of said calculated output signal when the value of the latter exceeds said predetermined range of values.\n\n14. A digital multiplier comprising:\n\ninput means for serially receiving a plurality of serial n-bit digital inputs at a predetermined clock rate for deriving a plurality of parallel n-bit serial digital inputs therefrom;\nweighting means for weighting each of said n-bit serial digital inputs with a plurality of constant coefficients, such that each of said n-bit serial digital inputs is weighted with one of said plurality of coefficients;\naccumulator means having said plurality of weighted n-bit serial digital inputs coupled thereto in parallel for adding said parallel n-bit serial inputs in parallel to obtain a calculated digital output; and\noutput means for serially coupling said calculated digital output from said accumulator at said predetermined clock rate.\n\n15. A digital multiplier in accordance with claim 14 wherein said input means includes a plurality of shift register delays each of which shift register delays being adapted to derive one of said parallel n-bit serial digital inputs.\n\n16. A digital multiplier in accordance with claim 15 wherein said weighting means includes a weighting matrix for deriving said constant coefficients, one of said constant coefficients being derived for each of said parallel n-bit serial inputs.\n\n17. A digital multiplier in accordance with claim 15 wherein the addition of said plurality of weighted parallel n-bit serial digital inputs by said accumulator to derive said calculated digital output is performed in an n-bit time duration.\n\nPatent History\nPatent number: 4104729\nType: Grant\nFiled: May 16, 1977\nDate of Patent: Aug 1, 1978\nAssignee: International Standard Electric Corporation (New York, NY)\nInventor: Michael J. Gingell (Sawbridgeworth)\nPrimary Examiner: Jerry Smith\nAttorneys: John T. O'Halloran, Jeffrey P. Morris\nApplication Number: 5/796,920\nClassifications\nCurrent U.S. Class: 364/757; 364/724\nInternational Classification: G06F 752;" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9213235,"math_prob":0.93486154,"size":31463,"snap":"2021-43-2021-49","text_gpt3_token_len":6851,"char_repetition_ratio":0.18287295,"word_repetition_ratio":0.26853782,"special_character_ratio":0.21955948,"punctuation_ratio":0.10447521,"nsfw_num_words":1,"has_unicode_error":false,"math_prob_llama3":0.98302346,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-24T23:25:52Z\",\"WARC-Record-ID\":\"<urn:uuid:dc657dc9-0fd3-4e55-a9c2-cd73a92e24a0>\",\"Content-Length\":\"87951\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:61871436-07ed-4a4d-9f9f-cd4d20b4a2a9>\",\"WARC-Concurrent-To\":\"<urn:uuid:e3ee930e-12a9-4f74-9afd-754e5995581d>\",\"WARC-IP-Address\":\"52.202.107.78\",\"WARC-Target-URI\":\"https://patents.justia.com/patent/4104729\",\"WARC-Payload-Digest\":\"sha1:KV642UV6GZ4GAFNHL4LY3BFLG44J4OIZ\",\"WARC-Block-Digest\":\"sha1:JO64W7JG4ZB2D23M3OFIF3RKIOL4BKUH\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323587606.8_warc_CC-MAIN-20211024204628-20211024234628-00107.warc.gz\"}"}
https://www.asknumbers.com/inch-to-cm/186-inches-to-cm.aspx	[ "# How Many Centimeters in 186 Inches?\n\n186 Inches to cm converter. How many centimeters in 186 inches?\n\n186 Inches equal to 472.44 cm or there are 472.44 centimeters in 186 inches.\n\n←→\nstep\nRound:\nEnter Inch\nEnter Centimeter\n\n## How to convert 186 inches to cm?\n\nThe conversion factor from inches to cm is 2.54. To convert any value of inches to cm, multiply the inch value by the conversion factor.\n\nTo convert 186 inches to cm, multiply 186 by 2.54, that makes 186 inches equal to 472.44 cm.\n\n186 inches to cm formula\n\ncm = inch value * 2.54\n\ncm = 186 * 2.54\n\ncm = 472.44\n\nCommon conversions from 186.x inches to cm:\n(rounded to 3 decimals)\n\n• 186 inches = 472.44 cm\n• 186.1 inches = 472.694 cm\n• 186.2 inches = 472.948 cm\n• 186.3 inches = 473.202 cm\n• 186.4 inches = 473.456 cm\n• 186.5 inches = 473.71 cm\n• 186.6 inches = 473.964 cm\n• 186.7 inches = 474.218 cm\n• 186.8 inches = 474.472 cm\n• 186.9 inches = 474.726 cm\n\nWhat is a Centimeter?\n\nCentimeter (centimetre) is a metric system unit of length. The symbol is \"cm\".\n\nWhat is a Inch?\n\nInch is an imperial and United States Customary systems unit of length, equal to 1/12 of a foot. 1 inch = 2.54 cm. The symbol is \"in\".\n\nCreate Conversion Table\nClick \"Create Table\". Enter a \"Start\" value (5, 100 etc). Select an \"Increment\" value (0.01, 5 etc) and select \"Accuracy\" to round the result." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7601809,"math_prob":0.99547863,"size":934,"snap":"2022-27-2022-33","text_gpt3_token_len":321,"char_repetition_ratio":0.24086021,"word_repetition_ratio":0.0,"special_character_ratio":0.44218415,"punctuation_ratio":0.18454936,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99649525,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-07-05T19:27:12Z\",\"WARC-Record-ID\":\"<urn:uuid:fcc1884f-5d2e-4014-a30e-f45e337844ba>\",\"Content-Length\":\"46155\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:01ee07c1-d451-4b3c-9281-20799d7f172a>\",\"WARC-Concurrent-To\":\"<urn:uuid:6acef65c-041e-4e8b-be12-3e49d3ce7efc>\",\"WARC-IP-Address\":\"104.21.33.54\",\"WARC-Target-URI\":\"https://www.asknumbers.com/inch-to-cm/186-inches-to-cm.aspx\",\"WARC-Payload-Digest\":\"sha1:NYBAV4NGX4ZHU2B2INJPQUOM56K4YUB5\",\"WARC-Block-Digest\":\"sha1:7AMR56CGO4MOF753P7TW4VLAP5K7VZRO\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656104597905.85_warc_CC-MAIN-20220705174927-20220705204927-00497.warc.gz\"}"}
https://help.matheass.eu/en/E201-Right-angled_Triangles.html	[ "## Right-angled Triangles\n\nA right-angled triangle is usually well defined by two properties and the right angle. The program will ask you are asked to choose two of the following elements and then to enter their values:\n\n• Catheti a and b\n• Hypotenuse c\n• Angles α and β\n• Hypotenuse segments p and q\n• Altitude h\n• Area A", null, "If you enter the values of two out of these nine properties the program computes the others.\n\n### Example 1:\n\n```Given:\n¯¯¯¯¯¯\nCathete a = 3\nHypotenuse c = 5\n\nResults :\n¯¯¯¯¯¯¯\nCathete b = 4\nAngle α = 36,869898°\nAngle β = 53,130102°\nHypot. segment p = 1,8\nHypot. segment q = 3,2\nAltitude h = 2,4\nArea A = 6```\n\n### Example 2:\n\n```Given:\n¯¯¯¯¯¯\nHypot. segment p = 1,8\nArea A = 6\n\nResults :\n¯¯¯¯¯¯¯\nCathete a = 3\nCathete b = 4\nHypotenuse c = 5\nAngle α = 36,869898°\nAngle β = 53,130102°\nHypot. segment q = 3,2\nAltitude h = 2,4\n```" ]	[ null, "https://help.matheass.eu/images/G2D-Dreieck1.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7343149,"math_prob":0.9953604,"size":819,"snap":"2023-40-2023-50","text_gpt3_token_len":273,"char_repetition_ratio":0.1411043,"word_repetition_ratio":0.14634146,"special_character_ratio":0.36630037,"punctuation_ratio":0.13294798,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99933255,"pos_list":[0,1,2],"im_url_duplicate_count":[null,3,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-26T21:34:35Z\",\"WARC-Record-ID\":\"<urn:uuid:1c152664-ec2a-47f0-b33d-ef1d9cbff846>\",\"Content-Length\":\"3356\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2cf34f1e-e38b-42f4-ac63-893d876eeb78>\",\"WARC-Concurrent-To\":\"<urn:uuid:d44070aa-2904-4ce7-8379-f6fdd0f0da45>\",\"WARC-IP-Address\":\"80.150.6.143\",\"WARC-Target-URI\":\"https://help.matheass.eu/en/E201-Right-angled_Triangles.html\",\"WARC-Payload-Digest\":\"sha1:GSDVB5KIUWMIKVZNGUU2C6IUBKB2747U\",\"WARC-Block-Digest\":\"sha1:VV6D2WDNDSOKHUYNFYMMLWF6PJEDSQMM\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233510225.44_warc_CC-MAIN-20230926211344-20230927001344-00472.warc.gz\"}"}
https://www.colorhexa.com/00c830	[ "# #00c830 Color Information\n\nIn a RGB color space, hex #00c830 is composed of 0% red, 78.4% green and 18.8% blue. Whereas in a CMYK color space, it is composed of 100% cyan, 0% magenta, 76% yellow and 21.6% black. It has a hue angle of 134.4 degrees, a saturation of 100% and a lightness of 39.2%. #00c830 color hex could be obtained by blending #00ff60 with #009100. Closest websafe color is: #00cc33.\n\n• R 0\n• G 78\n• B 19\nRGB color chart\n• C 100\n• M 0\n• Y 76\n• K 22\nCMYK color chart\n\n#00c830 color description : Strong lime green.\n\n# #00c830 Color Conversion\n\nThe hexadecimal color #00c830 has RGB values of R:0, G:200, B:48 and CMYK values of C:1, M:0, Y:0.76, K:0.22. Its decimal value is 51248.\n\nHex triplet RGB Decimal 00c830 `#00c830` 0, 200, 48 `rgb(0,200,48)` 0, 78.4, 18.8 `rgb(0%,78.4%,18.8%)` 100, 0, 76, 22 134.4°, 100, 39.2 `hsl(134.4,100%,39.2%)` 134.4°, 100, 78.4 00cc33 `#00cc33`\nCIE-LAB 70.538, -69.847, 59.9 21.186, 41.519, 9.694 0.293, 0.573, 41.519 70.538, 92.014, 139.384 70.538, -65.958, 79.644 64.436, -54.071, 36.185 00000000, 11001000, 00110000\n\n# Color Schemes with #00c830\n\n• #00c830\n``#00c830` `rgb(0,200,48)``\n• #c80098\n``#c80098` `rgb(200,0,152)``\nComplementary Color\n• #34c800\n``#34c800` `rgb(52,200,0)``\n• #00c830\n``#00c830` `rgb(0,200,48)``\n• #00c894\n``#00c894` `rgb(0,200,148)``\nAnalogous Color\n• #c80034\n``#c80034` `rgb(200,0,52)``\n• #00c830\n``#00c830` `rgb(0,200,48)``\n• #9400c8\n``#9400c8` `rgb(148,0,200)``\nSplit Complementary Color\n• #c83000\n``#c83000` `rgb(200,48,0)``\n• #00c830\n``#00c830` `rgb(0,200,48)``\n• #3000c8\n``#3000c8` `rgb(48,0,200)``\n• #98c800\n``#98c800` `rgb(152,200,0)``\n• #00c830\n``#00c830` `rgb(0,200,48)``\n• #3000c8\n``#3000c8` `rgb(48,0,200)``\n• #c80098\n``#c80098` `rgb(200,0,152)``\n• #007c1e\n``#007c1e` `rgb(0,124,30)``\n• #009524\n``#009524` `rgb(0,149,36)``\n• #00af2a\n``#00af2a` `rgb(0,175,42)``\n• #00c830\n``#00c830` `rgb(0,200,48)``\n• #00e236\n``#00e236` `rgb(0,226,54)``\n• #00fb3c\n``#00fb3c` `rgb(0,251,60)``\n• #16ff4e\n``#16ff4e` `rgb(22,255,78)``\nMonochromatic Color\n\n# Alternatives to #00c830\n\nBelow, you can see some colors close to #00c830. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #02c800\n``#02c800` `rgb(2,200,0)``\n• #00c80f\n``#00c80f` `rgb(0,200,15)``\n• #00c81f\n``#00c81f` `rgb(0,200,31)``\n• #00c830\n``#00c830` `rgb(0,200,48)``\n• #00c841\n``#00c841` `rgb(0,200,65)``\n• #00c851\n``#00c851` `rgb(0,200,81)``\n• #00c862\n``#00c862` `rgb(0,200,98)``\nSimilar Colors\n\n# #00c830 Preview\n\nThis text has a font color of #00c830.\n\n``<span style=\"color:#00c830;\">Text here</span>``\n#00c830 background color\n\nThis paragraph has a background color of #00c830.\n\n``<p style=\"background-color:#00c830;\">Content here</p>``\n#00c830 border color\n\nThis element has a border color of #00c830.\n\n``<div style=\"border:1px solid #00c830;\">Content here</div>``\nCSS codes\n``.text {color:#00c830;}``\n``.background {background-color:#00c830;}``\n``.border {border:1px solid #00c830;}``\n\n# Shades and Tints of #00c830\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, #000401 is the darkest color, while #effff3 is the lightest one.\n\n• #000401\n``#000401` `rgb(0,4,1)``\n• #001706\n``#001706` `rgb(0,23,6)``\n• #002b0a\n``#002b0a` `rgb(0,43,10)``\n• #003f0f\n``#003f0f` `rgb(0,63,15)``\n• #005214\n``#005214` `rgb(0,82,20)``\n• #006618\n``#006618` `rgb(0,102,24)``\n• #007a1d\n``#007a1d` `rgb(0,122,29)``\n• #008d22\n``#008d22` `rgb(0,141,34)``\n• #00a127\n``#00a127` `rgb(0,161,39)``\n• #00b42b\n``#00b42b` `rgb(0,180,43)``\n• #00c830\n``#00c830` `rgb(0,200,48)``\n• #00dc35\n``#00dc35` `rgb(0,220,53)``\n• #00ef39\n``#00ef39` `rgb(0,239,57)``\n• #04ff40\n``#04ff40` `rgb(4,255,64)``\n• #17ff4f\n``#17ff4f` `rgb(23,255,79)``\n• #2bff5e\n``#2bff5e` `rgb(43,255,94)``\n• #3fff6d\n``#3fff6d` `rgb(63,255,109)``\n• #52ff7c\n``#52ff7c` `rgb(82,255,124)``\n• #66ff8b\n``#66ff8b` `rgb(102,255,139)``\n• #7aff9a\n``#7aff9a` `rgb(122,255,154)``\n• #8dffa8\n``#8dffa8` `rgb(141,255,168)``\n• #a1ffb7\n``#a1ffb7` `rgb(161,255,183)``\n• #b4ffc6\n``#b4ffc6` `rgb(180,255,198)``\n• #c8ffd5\n``#c8ffd5` `rgb(200,255,213)``\n• #dcffe4\n``#dcffe4` `rgb(220,255,228)``\n• #effff3\n``#effff3` `rgb(239,255,243)``\nTint Color Variation\n\n# Tones of #00c830\n\nA tone is produced by adding gray to any pure hue. In this case, #5c6c60 is the less saturated color, while #00c830 is the most saturated one.\n\n• #5c6c60\n``#5c6c60` `rgb(92,108,96)``\n• #55735c\n``#55735c` `rgb(85,115,92)``\n• #4d7b58\n``#4d7b58` `rgb(77,123,88)``\n• #458354\n``#458354` `rgb(69,131,84)``\n• #3e8a50\n``#3e8a50` `rgb(62,138,80)``\n• #36924c\n``#36924c` `rgb(54,146,76)``\n• #2e9a48\n``#2e9a48` `rgb(46,154,72)``\n• #26a244\n``#26a244` `rgb(38,162,68)``\n• #1fa940\n``#1fa940` `rgb(31,169,64)``\n• #17b13c\n``#17b13c` `rgb(23,177,60)``\n• #0fb938\n``#0fb938` `rgb(15,185,56)``\n• #08c034\n``#08c034` `rgb(8,192,52)``\n• #00c830\n``#00c830` `rgb(0,200,48)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #00c830 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":0.5338449,"math_prob":0.85960203,"size":3655,"snap":"2023-40-2023-50","text_gpt3_token_len":1635,"char_repetition_ratio":0.13777047,"word_repetition_ratio":0.0073937154,"special_character_ratio":0.5592339,"punctuation_ratio":0.23190045,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98379457,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-10T01:35:49Z\",\"WARC-Record-ID\":\"<urn:uuid:099ddf51-f0b3-47ea-95d4-44ef38fd8217>\",\"Content-Length\":\"36151\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:05954e16-ff5d-43a4-a150-8c7d90d98c57>\",\"WARC-Concurrent-To\":\"<urn:uuid:aab437cb-eb34-4b0d-8d73-1615d729e125>\",\"WARC-IP-Address\":\"178.32.117.56\",\"WARC-Target-URI\":\"https://www.colorhexa.com/00c830\",\"WARC-Payload-Digest\":\"sha1:SX7PYKJO54FWES3FOPY6CJN67ZHMQJ7Y\",\"WARC-Block-Digest\":\"sha1:QNCX7FAR3CAFVLPTXUSAHYWNGTQK4662\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100989.75_warc_CC-MAIN-20231209233632-20231210023632-00344.warc.gz\"}"}
https://wordpress.stackexchange.com/questions/18725/custom-pagination-for-custom-post-types-by-names	[ "# Custom pagination for custom post types (by names)\n\nI have two custom post types that deal with people's names. Right now, in browsing views, it just lists them all alphabetically and the pagination breaks them down by numbers, which isn't terribly helpful when you're trying to find a specific person.\n\nSpecifically, I've been asked to create pagination links for people that look like this:\n\n• A-G\n• H-M\n• N-Q\n• R-Q\n\nMy problem - I can't figure out how I can query the custom post types by the first letter of one field. Then, I'm not sure how I can go about creating the pagination in this way. Does anybody have any suggestions? Thank you!\n\n• Interesting.. I will give it a shot but not very soon. I have a free slot after couple of days. Do share your solution if you find anything before that. Just to start look at posts_where filter to modify lookup and to do the pagination you need to play with rewrite rules, have a look at query_vars, query_posts and WP_Rewrite class. I am sure you will nail it with these things. – Hameedullah Khan May 30 '11 at 16:34\n• @mckeodm3 So what?? – kaiser Jan 12 '12 at 1:47\n\nInteresting question! I solved it by expanding the WHERE query with a bunch of post_title LIKE 'A%' OR post_title LIKE 'B%' ... clauses. You can also use a regular expression to do a range search, but I believe the database won't be able to use an index then.\n\nThis is the core of the solution: a filter on the WHERE clause:\n\nadd_filter( 'posts_where', 'wpse18703_posts_where', 10, 2 );\nfunction wpse18703_posts_where( $where, &$wp_query )\n{\nif ( $letter_range =$wp_query->get( 'wpse18703_range' ) ) {\nglobal $wpdb;$letter_clauses = array();\nforeach ( $letter_range as$letter ) {\n$letter_clauses[] =$wpdb->posts. '.post_title LIKE \\'' . $letter . '%\\''; }$where .= ' AND (' . implode( ' OR ', $letter_clauses ) . ') '; } return$where;\n}\n\n\nOf course you don't want to allow random external input in your query. That is why I have an input sanitization step on pre_get_posts, which converts two query variables into a valid range. (If you find a way to break this please leave a comment so I can correct it)\n\nadd_action( 'pre_get_posts', 'wpse18703_pre_get_posts' );\nfunction wpse18703_pre_get_posts( &$wp_query ) { // Sanitize input$first_letter = $wp_query->get( 'wpse18725_first_letter' );$last_letter = $wp_query->get( 'wpse18725_last_letter' ); if ($first_letter \|\| $last_letter ) {$first_letter = substr( strtoupper( $first_letter ), 0, 1 );$last_letter = substr( strtoupper( $last_letter ), 0, 1 ); // Make sure the letters are valid // If only one letter is valid use only that letter, not a range if ( ! ( 'A' <=$first_letter && $first_letter <= 'Z' ) ) {$first_letter = $last_letter; } if ( ! ( 'A' <=$last_letter && $last_letter <= 'Z' ) ) { if ($first_letter == $last_letter ) { // None of the letters are valid, don't do a range query return; }$last_letter = $first_letter; }$wp_query->set( 'posts_per_page', -1 );\n$wp_query->set( 'wpse18703_range', range($first_letter, $last_letter ) ); } } The final step is to create a pretty rewrite rule so you can go to example.com/posts/a-g/ or example.com/posts/a to see all posts beginning with this (range of) letter(s). add_action( 'init', 'wpse18725_init' ); function wpse18725_init() { add_rewrite_rule( 'posts/(\\w)(-(\\w))?/?', 'index.php?wpse18725_first_letter=$matches&wpse18725_last_letter=$matches', 'top' ); } add_filter( 'query_vars', 'wpse18725_query_vars' ); function wpse18725_query_vars($query_vars )\n{\n$query_vars[] = 'wpse18725_first_letter';$query_vars[] = 'wpse18725_last_letter';\nreturn $query_vars; } You can change the rewrite rule pattern to start with something else. If this is for a custom post type, be sure to add &post_type=your_custom_post_type to the substitution (the second string, which starts with index.php). Adding pagination links is left as an exercise for the reader :-) • Just a hint: like_escape() :) – kaiser Mar 1 '12 at 22:43 This will help you getting started. I don't know how you would break the query at specific letter and then tell WP that there's another page with more letters, but the following takes 99% of the rest. Don't forget to post your solution! query_posts( array( 'orderby' => 'title' ) ); // Build an alphabet array foreach( range( 'A', 'G' ) as$letter )\n$alphabet[] =$letter;\n\nforeach( range( 'H', 'M' ) as $letter )$alphabet[] = $letter; foreach( range( 'N', 'Q' ) as$letter )\n$alphabet[] =$letter;\n\nforeach( range( 'R', 'Z' ) as $letter )$alphabet[] = $letter; if ( have_posts() ) { while ( have_posts() ) { global$wp_query, $post;$max_paged = $wp_query->query_vars['max_num_pages'];$paged = $wp_query->query_vars['paged']; if ( !$paged )\n$paged = (int) 1; the_post();$first_title_letter = (string) substr( $post->post_title, 1 ); if ( in_array($first_title_letter, $alphabet ) ) { // DO STUFF } // Pagination if ($paged !== (int) 1 )\n{\necho 'Prev: '._wp_link_page( $paged - 1 ); } while ($i = 1; count($alphabet) <$max_paged; i++; )\n{\necho $i._wp_link_page($i );\n}\nif ( $paged !==$max_paged )\n{\necho 'Next: '._wp_link_page( $paged + 1 ); echo 'Last: '._wp_link_page($max_paged );\n}\n} // endwhile;\n} // endif;\n\n• It's not tested. – kaiser May 30 '11 at 18:52\n\nAn answer using @kaiser's example, with a custom post type as a function accepting alpha start and end params. This example is obviously for a short list of items, as it does not include secondary paging. I'm posting it so you can incorporate the concept into your functions.php if you like.\n\n// Dr Alpha Paging\n// Tyrus Christiana, Senior Developer, BFGInteractive.com\n// Call like alphaPageDr( \"A\",\"d\" );\nfunction alphaPageDr( $start,$end ) {\necho \"Alpha Start\";\n$loop = new WP_Query( 'post_type=physician&orderby=title&order=asc' ); // Build an alphabet array of capitalized letters from params foreach ( range($start, $end ) as$letter )\n$alphabet[] = strtoupper($letter );\nif ( $loop->have_posts() ) { echo \"Has Posts\"; while ($loop->have_posts() ) : $loop->the_post(); // Filter by the first letter of the last name$first_last_name_letter = ( string ) substr( get_field( \"last_name\" ), 0, 1 );\nif ( in_array( $first_last_name_letter,$alphabet ) ) {\n//Show things\necho \"<img class='sidebar_main_thumb' src= '\" .\nget_field( \"thumbnail\" ) . \"' />\";\necho \"<div class='sidesbar_dr_name'>\" .\nget_field( \"salutation\" ) . \" \" .\nget_field( 'first_name' ) . \" \" .\nget_field( 'last_name' ) . \"</div>\";\necho \"<div class='sidesbar_primary_specialty ' > Primary Specialty : \" .\nget_field( \"primary_specialty\" ) . \"</div>\";\n}\nendwhile;\n}\n}\n\n\nHere is a way to do this by using the query_vars and posts_where filters:\n\npublic function range_add($aVars) {$aVars[] = \"range\";\nreturn $aVars; } public function range_where($where, $args ) { if( !is_admin() ) {$range = ( isset($args->query_vars['range']) ?$args->query_vars['range'] : false );\nif( $range ) {$range = split(',',$range);$where .= \"AND LEFT(wp_posts.post_title,1) BETWEEN '$range' AND '$range'\";\n}\n}\nreturn $where; } add_filter( 'query_vars', array('atk','range_add') ); add_filter( 'posts_where' , array('atk','range_where') ); This is not so much an answer, but more a pointer to a direction to take. This will probably have to be 100% custom - and will be very involved. You'll need to create a custom sql query (using the wpdb classs) and then for pagination you'll pass these parameters to your custom query. You'll probably need to also create new rewrite rules for this as well. Some functions to look into: add_rewrite_tag( '%byletter%', '([^/]+)'); add_permastruct( 'byletter', 'byletter' . '/%byletter%' );$wp_rewrite->flush_rules();" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6093026,"math_prob":0.78201425,"size":2786,"snap":"2019-43-2019-47","text_gpt3_token_len":772,"char_repetition_ratio":0.18511862,"word_repetition_ratio":0.014423077,"special_character_ratio":0.34350324,"punctuation_ratio":0.17162472,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.964321,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-21T23:53:12Z\",\"WARC-Record-ID\":\"<urn:uuid:9ce5847e-0f05-47c4-9beb-adf1e09859e4>\",\"Content-Length\":\"169263\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f82a6bbb-382e-4a52-9e3d-050cbfd2ff07>\",\"WARC-Concurrent-To\":\"<urn:uuid:1826a945-7ba4-4c24-97d7-8f93df7cf0d0>\",\"WARC-IP-Address\":\"151.101.129.69\",\"WARC-Target-URI\":\"https://wordpress.stackexchange.com/questions/18725/custom-pagination-for-custom-post-types-by-names\",\"WARC-Payload-Digest\":\"sha1:IA3V45W62QV6J65H3U4B63XJYWHAIW3G\",\"WARC-Block-Digest\":\"sha1:Y74YRIBGLNF3IQECUXZU6WNXZBQA3CIO\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496671053.31_warc_CC-MAIN-20191121231600-20191122015600-00534.warc.gz\"}"}
https://foreach.id/EN/fluids/flowratemass/ton_metric%7Cminute-to-megagram%7Csecond.html	[ "# Convert ton (metric)/minute to megagram/second (t/min to Mg/s)\n\nBatch Convert\n• megagram/second [Mg/s]\n• ton (metric)/minute [t/min]\nCopy\n_\nCopy\n• megagram/second [Mg/s]\n• ton (metric)/minute [t/min]\n\n## Ton (metric)/minute to Megagram/second (t/min to Mg/s)\n\n### Ton (metric)/minute (Symbol or Abbreviation: t/min)\n\nTon (metric)/minute is one of mass flow rate units. Ton (metric)/minute abbreviated or symbolized by t/min. The value of 1 ton (metric)/minute is equal to 16.667 kilogram/second. In its relation with megagram/second, 1 ton (metric)/minute is equal to 0.016667 megagram/second.\n\n#### Relation with other units\n\n1 ton (metric)/minute equals to 16.667 kilogram/second\n\n1 ton (metric)/minute equals to 16,667 gram/second\n\n1 ton (metric)/minute equals to 1,000,000 gram/minute\n\n1 ton (metric)/minute equals to 60,000,000 gram/hour\n\n1 ton (metric)/minute equals to 1,440,000,000 gram/day\n\n1 ton (metric)/minute equals to 1,000,000,000 milligram/minute\n\n1 ton (metric)/minute equals to 60,000,000,000 milligram/hour\n\n1 ton (metric)/minute equals to 1,440,000,000,000 milligram/day\n\n1 ton (metric)/minute equals to 1,000 kilogram/minute\n\n1 ton (metric)/minute equals to 60,000 kilogram/hour\n\n1 ton (metric)/minute equals to 1,440,000 kilogram/day\n\n1 ton (metric)/minute equals to 1.6667e-14 exagram/second\n\n1 ton (metric)/minute equals to 1.6667e-11 petagram/second\n\n1 ton (metric)/minute equals to 1.6667e-8 teragram/second\n\n1 ton (metric)/minute equals to 0.000016667 gigagram/second\n\n1 ton (metric)/minute equals to 0.016667 megagram/second\n\n1 ton (metric)/minute equals to 166.67 hectogram/second\n\n1 ton (metric)/minute equals to 1,666.7 dekagram/second\n\n1 ton (metric)/minute equals to 166,670 decigram/second\n\n1 ton (metric)/minute equals to 1,666,700 centigram/second\n\n1 ton (metric)/minute equals to 16,667,000 milligram/second\n\n1 ton (metric)/minute equals to 16,667,000,000 microgram/second\n\n1 ton (metric)/minute equals to 0.016667 ton (metric)/second\n\n1 ton (metric)/minute equals to 60 ton (metric)/hour\n\n1 ton (metric)/minute equals to 1,440 ton (metric)/day\n\n1 ton (metric)/minute equals to 66.139 ton (short)/hour\n\n1 ton (metric)/minute equals to 36.744 pound/second\n\n1 ton (metric)/minute equals to 2,204.6 pound/minute\n\n1 ton (metric)/minute equals to 132,280 pound/hour\n\n1 ton (metric)/minute equals to 3,174,700 pound/day\n\n### Megagram/second (Symbol or Abbreviation: Mg/s)\n\nMegagram/second is one of mass flow rate units. Megagram/second abbreviated or symbolized by Mg/s. The value of 1 megagram/second is equal to 1000 kilogram/second. In its relation with ton (metric)/minute, 1 megagram/second is equal to 60 ton (metric)/minute.\n\n#### Relation with other units\n\n1 megagram/second equals to 1,000 kilogram/second\n\n1 megagram/second equals to 1,000,000 gram/second\n\n1 megagram/second equals to 60,000,000 gram/minute\n\n1 megagram/second equals to 3,600,000,000 gram/hour\n\n1 megagram/second equals to 86,400,000,000 gram/day\n\n1 megagram/second equals to 60,000,000,000 milligram/minute\n\n1 megagram/second equals to 3,600,000,000,000 milligram/hour\n\n1 megagram/second equals to 86,400,000,000,000 milligram/day\n\n1 megagram/second equals to 60,000 kilogram/minute\n\n1 megagram/second equals to 3,600,000 kilogram/hour\n\n1 megagram/second equals to 86,400,000 kilogram/day\n\n1 megagram/second equals to 1e-12 exagram/second\n\n1 megagram/second equals to 1e-9 petagram/second\n\n1 megagram/second equals to 0.000001 teragram/second\n\n1 megagram/second equals to 0.001 gigagram/second\n\n1 megagram/second equals to 10,000 hectogram/second\n\n1 megagram/second equals to 100,000 dekagram/second\n\n1 megagram/second equals to 10,000,000 decigram/second\n\n1 megagram/second equals to 100,000,000 centigram/second\n\n1 megagram/second equals to 1,000,000,000 milligram/second\n\n1 megagram/second equals to 1,000,000,000,000 microgram/second\n\n1 megagram/second equals to 1 ton (metric)/second\n\n1 megagram/second equals to 60 ton (metric)/minute\n\n1 megagram/second equals to 3,600 ton (metric)/hour\n\n1 megagram/second equals to 86,400 ton (metric)/day\n\n1 megagram/second equals to 3,968.3 ton (short)/hour\n\n1 megagram/second equals to 2,204.6 pound/second\n\n1 megagram/second equals to 132,280 pound/minute\n\n1 megagram/second equals to 7,936,600 pound/hour\n\n1 megagram/second equals to 190,480,000 pound/day\n\n### How to convert Ton (metric)/minute to Megagram/second (t/min to Mg/s):\n\n#### Conversion Table for Ton (metric)/minute to Megagram/second (t/min to Mg/s)\n\nton (metric)/minute (t/min) megagram/second (Mg/s)\n0.01 t/min 0.00016667 Mg/s\n0.1 t/min 0.0016667 Mg/s\n1 t/min 0.016667 Mg/s\n2 t/min 0.033333 Mg/s\n3 t/min 0.05 Mg/s\n4 t/min 0.066667 Mg/s\n5 t/min 0.083333 Mg/s\n6 t/min 0.1 Mg/s\n7 t/min 0.11667 Mg/s\n8 t/min 0.13333 Mg/s\n9 t/min 0.15 Mg/s\n10 t/min 0.16667 Mg/s\n20 t/min 0.33333 Mg/s\n25 t/min 0.41667 Mg/s\n50 t/min 0.83333 Mg/s\n75 t/min 1.25 Mg/s\n100 t/min 1.6667 Mg/s\n250 t/min 4.1667 Mg/s\n500 t/min 8.3333 Mg/s\n750 t/min 12.5 Mg/s\n1,000 t/min 16.667 Mg/s\n100,000 t/min 1,666.7 Mg/s\n1,000,000,000 t/min 16,667,000 Mg/s\n1,000,000,000,000 t/min 16,667,000,000 Mg/s\n\n#### Conversion Table for Megagram/second to Ton (metric)/minute (Mg/s to t/min)\n\nmegagram/second (Mg/s) ton (metric)/minute (t/min)\n0.01 Mg/s 0.6 t/min\n0.1 Mg/s 6 t/min\n1 Mg/s 60 t/min\n2 Mg/s 120 t/min\n3 Mg/s 180 t/min\n4 Mg/s 240 t/min\n5 Mg/s 300 t/min\n6 Mg/s 360 t/min\n7 Mg/s 420 t/min\n8 Mg/s 480 t/min\n9 Mg/s 540 t/min\n10 Mg/s 600 t/min\n20 Mg/s 1,200 t/min\n25 Mg/s 1,500 t/min\n50 Mg/s 3,000 t/min\n75 Mg/s 4,500 t/min\n100 Mg/s 6,000 t/min\n250 Mg/s 15,000 t/min\n500 Mg/s 30,000 t/min\n750 Mg/s 45,000 t/min\n1,000 Mg/s 60,000 t/min\n100,000 Mg/s 6,000,000 t/min\n1,000,000,000 Mg/s 60,000,000,000 t/min\n1,000,000,000,000 Mg/s 60,000,000,000,000 t/min\n\n#### Steps to Convert Ton (metric)/minute to Megagram/second (t/min to Mg/s)\n\n1. Example: Convert 828 ton (metric)/minute to megagram/second (828 t/min to Mg/s).\n2. 1 ton (metric)/minute is equivalent to 0.016667 megagram/second (1 t/min is equivalent to 0.016667 Mg/s).\n3. 828 ton (metric)/minute (t/min) is equivalent to 828 times 0.016667 megagram/second (Mg/s).\n4. Retrieved 828 ton (metric)/minute is equivalent to 13.8 megagram/second (828 t/min is equivalent to 13.8 Mg/s).\n\n▸▸" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6891006,"math_prob":0.9992331,"size":5993,"snap":"2021-21-2021-25","text_gpt3_token_len":2070,"char_repetition_ratio":0.35064286,"word_repetition_ratio":0.07634543,"special_character_ratio":0.40981144,"punctuation_ratio":0.13392857,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96554625,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-24T00:17:23Z\",\"WARC-Record-ID\":\"<urn:uuid:873156ff-142d-4cf3-869c-5a5ecfd04c90>\",\"Content-Length\":\"58698\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d8078c4b-ce4c-4a28-a584-7829ac594279>\",\"WARC-Concurrent-To\":\"<urn:uuid:f21131fc-a59e-4320-b8d1-8f8560478040>\",\"WARC-IP-Address\":\"172.67.130.66\",\"WARC-Target-URI\":\"https://foreach.id/EN/fluids/flowratemass/ton_metric%7Cminute-to-megagram%7Csecond.html\",\"WARC-Payload-Digest\":\"sha1:2OUBZKA6D6JBWNFLQ2OBEML2HUFF55SQ\",\"WARC-Block-Digest\":\"sha1:6RQEOMPNSUVTVMYCBRSJPVF5SV734IUX\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623488544264.91_warc_CC-MAIN-20210623225535-20210624015535-00510.warc.gz\"}"}
https://kr.mathworks.com/matlabcentral/cody/problems/155-all-your-base-are-belong-to-us/solutions/2271078	[ "Cody\n\n# Problem 155. All your base are belong to us\n\nSolution 2271078\n\nSubmitted on 11 May 2020 by Aron Frei\nThis solution is locked. To view this solution, you need to provide a solution of the same size or smaller.\n\n### Test Suite\n\nTest Status Code Input and Output\n1 Pass\nx = 1024; b = 2; y_correct = 10; assert(isequal(logb(x,b),y_correct))\n\nans = 10\n\n2 Pass\nx = 25; b = 5; y_correct = 2; assert(isequal(logb(x,b),y_correct))\n\nans = 2\n\n3 Pass\nx = 10000; b = 10; y_correct = 4; assert(isequal(logb(x,b),y_correct))\n\nans = 4" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.51025003,"math_prob":0.99705297,"size":456,"snap":"2020-34-2020-40","text_gpt3_token_len":165,"char_repetition_ratio":0.16814159,"word_repetition_ratio":0.0,"special_character_ratio":0.41885966,"punctuation_ratio":0.16666667,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9964381,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-08-06T22:32:08Z\",\"WARC-Record-ID\":\"<urn:uuid:2e43d40b-5693-4fcd-90dc-c129f6a505c4>\",\"Content-Length\":\"73287\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:6feeae5f-6124-4921-a752-98cbf4f8071c>\",\"WARC-Concurrent-To\":\"<urn:uuid:1b7bb3d9-bac7-48b3-a38f-11cc60dd4ec4>\",\"WARC-IP-Address\":\"23.223.252.57\",\"WARC-Target-URI\":\"https://kr.mathworks.com/matlabcentral/cody/problems/155-all-your-base-are-belong-to-us/solutions/2271078\",\"WARC-Payload-Digest\":\"sha1:7EJKV2EJ4O3FL3Y7KAVGFZYSNIP4DO3G\",\"WARC-Block-Digest\":\"sha1:D2VFXIKLN3KTGWICYYF6JUHMLO43R6RX\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-34/CC-MAIN-2020-34_segments_1596439737039.58_warc_CC-MAIN-20200806210649-20200807000649-00357.warc.gz\"}"}
https://domainedevilotte.com/how-many-hundreds-are-in-50000-update-new/	[ "Home » How Many Hundreds Are In 50000? Update New\n\n# How Many Hundreds Are In 50000? Update New\n\nLet’s discuss the question: how many hundreds are in 50000. We summarize all relevant answers in section Q&A of website Domainedevilotte.com in category: Blog Technology. See more related questions in the comments below.\n\n## How many thousands is 50000?\n\nAnswer: 50 thousands are there in 50000.\n\n## How many hundreds are there in 52000?\n\nThere are 520 hundreds in 52,000.\n\n### How many hundreds in this number\n\nHow many hundreds in this number\nHow many hundreds in this number\n\n## How much is 50000?\n\n50,000 (fifty thousand) is the natural number that comes after 49,999 and before 50,001.\n\n50,000.\n← 49999 50000 50001 →\nCardinal fifty thousand\nOrdinal 50000th (fifty thousandth)\nFactorization 24 × 55\nGreek numeral\n\n## How many hundreds are there in 2000?\n\nThere are 20 hundreds in 2000.\n\n## How is 100 thousand written?\n\nOne Hundred Thousand in numerals is written as 100000.\n\n## How many hundreds are there in 54000?\n\n90 × 600 (54,000; 540 hundreds are in 54,000)\n\n## How many hundreds are there in 42000?\n\nHence there are 420 hundreds in 42000.\n\n## How do you read 52000?\n\n52000 in words is written as Fifty-Two Thousand.\n\n## How many hundreds are there in 423?\n\nAnswer: There are 4 hundreds in the 423.\n\n## How many hundreds are in a million?\n\n» Hundred Conversions:\n\nHundred↔Million 1 Million = 10000 Hundred.\n\n## How many hundreds are there in 7000?\n\n7000 is the same as 70 hundreds\n\nWhich pair of numbers have the same digit at the hundreds place?\n\n### Famous Dex – Japan (Prod. JGramm) [Official Lyric Video]\n\nFamous Dex – Japan (Prod. JGramm) [Official Lyric Video]\nFamous Dex – Japan (Prod. JGramm) [Official Lyric Video]\n\n## What is the English meaning of 50000?\n\n50000 in words is Fifty Thousand.\n\n## How much is 50000 in millions?\n\nSo you want to convert 50 thousands into millions? If you’re in a rush and just need the answer, the calculator below is all you need. The answer is 0.05 millions.\n\n## What is a third of 50000?\n\nPercentage Calculator: What is 3. percent of 50000? = 1500.\n\n## How many hundreds are there in 28000?\n\nNumber of hundreds in 28,000 are 280.\n\nHope it helps you.\n\n## How many hundreds are there in 6000?\n\nthere are 60 hundreds in 6000. so correct answer is ’60’ hundreds.\n\n## How many hundreds are there in 890?\n\nHence, the number of hundreds in 890 is 8.\n\n## What number is after 999999?\n\nOne million (1,000,000), or one thousand thousand, is the natural number after 999,999 and before 1,000,001.\n\n## How much is \\$100000.00 in words?\n\nTherefore, the number 100000 in words is One Hundred Thousand.\n\n## How do you write 12000000 in words?\n\n12000000 in words = twelve million\n\nAnd, you may also be interested in how to spell 15000000; check it out now!\n\n## How many hundreds are there in 56000?\n\n1.2560 hundreds are there in 2,56,000.\n\n### Rich Homie Quan – Flex (Ooh, Ooh, Ooh)\n\nRich Homie Quan – Flex (Ooh, Ooh, Ooh)\nRich Homie Quan – Flex (Ooh, Ooh, Ooh)\n\n## How many hundreds are there in 6729?\n\nAnswer. As 7 is there in the hundreds place.. Hope it helps.\n\n## How many hundreds are there in 4500?\n\n45 hundreds\n\n45*100=4,500.\n\nRelated searches\n\n• how many 100 dollar bills make 5000\n• what is 100 of 50000\n• how many hundreds are there in 60000\n• how many hundreds are there in 50000\n• 50000 divided by 100\n• how many 100 in 50000\n\n## Information related to the topic how many hundreds are in 50000\n\nHere are the search results of the thread how many hundreds are in 50000 from Bing. You can read more if you want.\n\nYou have just come across an article on the topic how many hundreds are in 50000. If you found this article useful, please share it. Thank you very much." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.89421076,"math_prob":0.8551793,"size":3416,"snap":"2022-27-2022-33","text_gpt3_token_len":957,"char_repetition_ratio":0.2587925,"word_repetition_ratio":0.09539474,"special_character_ratio":0.33811477,"punctuation_ratio":0.14325452,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9823691,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-08-10T14:00:58Z\",\"WARC-Record-ID\":\"<urn:uuid:6387be4a-96ac-4055-9f7f-ab4d205b6dfd>\",\"Content-Length\":\"109529\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:6fcc9420-bed6-4dd1-93d7-33afb7f1a695>\",\"WARC-Concurrent-To\":\"<urn:uuid:9c5ce222-bcc7-4899-b11d-6f485e7b9730>\",\"WARC-IP-Address\":\"104.21.75.236\",\"WARC-Target-URI\":\"https://domainedevilotte.com/how-many-hundreds-are-in-50000-update-new/\",\"WARC-Payload-Digest\":\"sha1:3VUHQTZIG3KJJPVWJQK2D5OZYZYAML5S\",\"WARC-Block-Digest\":\"sha1:VDQQGT5C6K2XUBJLT3AWZHVEF7LUGUVP\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-33/CC-MAIN-2022-33_segments_1659882571190.0_warc_CC-MAIN-20220810131127-20220810161127-00098.warc.gz\"}"}
https://theuvocorp.com/a-using-the-two-upper-bounds-for-the-complementary-error-4/	[ "# A Using The Two Upper Bounds For The Complementary Error\n\nIn this problem we explore the approximations to the probability of an error, Pe, for the pair of antipodal signals shown in Figure in the presence of additive white Gaussian noise of power spectral density N0/2. The exact formula for Pe is Pe = ½ erfc (√Eb/N0). (This formula is derived in Section 6.3)\n\n(a) Using the two upper bounds for the complementary error function given in Note 3, derive the corresponding approximations to P.\n\n(b) Compare the approximations derived in part (a) for Pe to the exact formula for Eb/N0 = 9. For the exact calculation of Pe, you may use Table on the error function.", null, "", null, "Posted in Uncategorized" ]	[ null, "https://www.solutioninn.com/images2/19-E-T-E-C-S(339).PNG", null, "https://blueribbonwriters.com/wp-content/uploads/2020/01/order-supreme-essay.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.86802685,"math_prob":0.95889515,"size":602,"snap":"2021-43-2021-49","text_gpt3_token_len":143,"char_repetition_ratio":0.12374582,"word_repetition_ratio":0.0,"special_character_ratio":0.23421927,"punctuation_ratio":0.084033616,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9977606,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,1,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-12-08T00:41:06Z\",\"WARC-Record-ID\":\"<urn:uuid:5c669385-c2d4-48ac-bda1-16db1432772f>\",\"Content-Length\":\"34556\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:3d4711f4-fb80-4ec1-a1e4-ba15e548f29a>\",\"WARC-Concurrent-To\":\"<urn:uuid:b1c72cd8-faec-4f39-b7ed-9961bd39614e>\",\"WARC-IP-Address\":\"192.64.113.86\",\"WARC-Target-URI\":\"https://theuvocorp.com/a-using-the-two-upper-bounds-for-the-complementary-error-4/\",\"WARC-Payload-Digest\":\"sha1:YNWNJQ7OCP7DYK6RMWYEKBV63CLTBHWJ\",\"WARC-Block-Digest\":\"sha1:DCAJAXTUVYQKHNU3F4PCLX7PO43ZY7DQ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964363420.81_warc_CC-MAIN-20211207232140-20211208022140-00033.warc.gz\"}"}
https://www.arxiv-vanity.com/papers/1701.06118/	[ "arXiv Vanity renders academic papers from arXiv as responsive web pages so you don’t have to squint at a PDF. Read this paper on arXiv.org.\n\n# Numerical solution of space-fractional partial differential equations by a differential quadrature approach\n\nX. G. Zhu Y. F. Nie Department of Applied Mathematics, Northwestern Polytechnical University, Xi’ an 710129, P.R. China\n###### Abstract\n\nThis article aims to develop a direct numerical approach to solve the space-fractional partial differential equations (PDEs) based on a new differential quadrature (DQ) technique. The fractional derivatives are approximated by the weighted linear combinations of the function values at discrete grid points on problem domain with the weights calculated via using three types of radial basis functions (RBFs) as test functions. The method in presence is robust, straight forward to apply, and highly accurate under the condition that the shape parameters of RBFs are well chosen. Numerical tests are provided to illustrate its validity and capability.\n\n###### keywords:\nDQ method, RBFs, Fractional derivatives, Space-fractional PDEs.\njournal: Elsevier\n\n## 1 Introduction\n\nIn this study, we are mainly interested in an efficient method for numerically solving a class of space-fractional models in the following form\n\n ∂y(x,t)∂t−κ(x)∂αy(x,t)∂+xα−υ(x)∂αy(x,t)∂−xα=f(x,t), x∈Λ, 0\n\nsubjected to the initial and boundary conditions\n\n y(x,0)=ψ(x),x∈Λ, (1.2) y(a,t)=g1(t),y(b,t)=g2(t),0\n\nwhere , , , are non-negative but do not vanish altogether. only when and only when . In Eq. (1.1), the space-fractional derivatives are defined in Caputo sense, i.e.,\n\n ∂αy(x,t)∂+xα =1Γ(2−α)∫xa∂2y(ξ,t)∂ξ2dξ(x−ξ)α−1, ∂αy(x,t)∂−xα =1Γ(2−α)∫bx∂2y(ξ,t)∂ξ2dξ(ξ−x)α−1,\n\nwith the Euler’s Gamma function .\n\nThe space-fractional PDEs describe many physical phenomena such as anomalous transport, hereditary elasticity, and chaotic dynamics Metzler and Klafter (2000); Rossikhin and Shitikova (1997); Zaslavsky (2002), while compared favorably to the integer PDEs. Since they are frequently sufficing in the absence of exact closed-form solutions, various numerical algorithms have been designed to solve them, typically including general Padé approximation Ding (2016), finite difference methods Meerschaert and Tadjeran (2006); Tian et al. (2015); Wang et al. (2010), meshless point interpolation method Liu et al. (2015), finite element methods Ervin and Roop (2006); Zhang et al. (2010), discontinuous Galerkin method Xu and Hesthaven (2014), finite volume method Hejazi et al. (2014), spline approximation method (SAM) Sousa (2011), RBF Kansa method Pang et al. (2015), polynomial and fractional spectral collocation methods Tian et al. (2014); Zayernouri and Karniadakis (2015). In Chen et al. (2010); Doha et al. (2014); Ren et al. (2013); Saadatmandi and Dehghan (2011), a series of operational matrix methods are constructed via the approximate expansions using shifted Jacobi, Chebyshev, Legendre polynomials, and Haar wavelets functions, as elements, respectively. Some analytic techniques are referred to Pandey et al. (2012); Ray (2009); Ray and Sahoo (2015); Yıldırım and Koçak (2009) and references therein.\n\nDQ method is understood as a direct numerical approach for PDEs that evaluates the derivatives via representative weighted linear combinations of function values on problem domain Bellman and Casti (1971). A group of test functions to calculate these weights can be chosen as Lagrange basis functions, RBFs, and orthogonal polynomials Chen et al. (1997); Quan and Chang (1989); Shu and Richards (1992); Wu and Shu (2002). DQ method is characterized by a few advantages such as high accuracy, low occupancy cost, truly mesh-free and the ease of programming.\n\nDue to the non-locality of fractional derivatives, a great extra computational cost is usually incurred when a conventional algorithm is applied to a fractional PDE. In this work, we propose a new RBFs based DQ method (RBF-DQM) for Eqs. (1.1)-(1.3). Using three types of RBFs as test functions, the weights are successfully determined and with them, the equation under consideration degenerates to an ordinary differential system (ODS). A time-stepping RBF-DQM is derived by introducing a difference scheme in time. The presented technique inherits the features of classic DQ methods. More importantly, it is insensitive to dimensional change, so it serves as a good alternative for the high-dimensional or the other complex fractional models arising in actual applications.\n\nThe outline is as follows. In Section 2, we give a brief description of fractional derivatives. In Section 3, the weighted coefficients are calculated by commonly used RBFs, which are required to approximate the fractional derivatives. We propose a Crank-Nicolson RBF-DQM to discretize the model problem in Section 4 and test its codes on three illustrative examples in Section 5. A conclusion is drawn in the last section.\n\n## 2 Fractional derivatives\n\nAt first, some basic definitions are introduced for preliminaries. Let ; then the following formulas\n\ndefine the left and right -th Caputo derivatives, respectively, if , where for , for , and is the floor function.\n\nThe left and right Caputo derivatives have the properties\n\nand coincide with classic derivatives with exactness to a multiplier factor :\n\n 0Dsxf(x)=∂sf(x)∂xs,∗xDsbf(x)=(−1)s∂sf(x)∂xs,\n\nwhere and . We refer the readers to Kilbas et al. (2006); Podlubny (1999) for more properties.\n\n## 3 DQ formulations based on RBFs\n\nIn the sequel, DQ formulations for fractional derivatives based on RBFs are derived. Define a lattice on but not necessarily with equally spaced points, i.e., , . In general, we always approximate the exact solution of a PDE like Eqs. (1.1)-(1.3) in the form\n\n y(x,t)≅M∑k=0δk(t)ϕk(x), (3.4)\n\nwith a set of proper basis functions . However, if\n\n ∂αϕk(xi)∂+xα =M∑j=0a(α)ijϕk(xj), i,k=0,1,…M, (3.5) ∂αϕk(xi)∂−xα =M∑j=0b(α)ijϕk(xj), i,k=0,1,…M, (3.6)\n\nand on acting , on both sides of Eq. (3.4), we realize that\n\n ∂αy(xi,t)∂+xα≅M∑k=0δk(t)∂αϕk(xi)∂+xα=M∑k=0δk(t)M∑j=0a(α)ijϕk(xj)≅M∑j=0a(α)ijy(xj,t), (3.7) ∂αy(xi,t)∂−xα≅M∑k=0δk(t)∂αϕk(xi)∂−xα=M∑k=0δk(t)M∑j=0b(α)ijϕk(xj)≅M∑j=0b(α)ijy(xj,t), (3.8)\n\nthanks to the linearity of the fractional derivatives, namely, Eqs. (3.7)-(3.8) are valid as along as Eqs. (3.5)-(3.6) are satisfied. The idea behind this is referred to as DQ Bellman and Casti (1971); , , , are called the weighted coefficients of fractional derivatives and will be calculated by means of typical RBFs.\n\n### 3.1 Three typical RBFs\n\nRBFs are the functions of the distance from their centers. They are popular as an effective tool to set up numerical algorithms for PDEs since the superiority of potential spectral accuracy. Here, commonly used RBFs are involved, i.e.,\n\n• Gaussians (GA):\n\nwhere , and is the shape parameter. It is worthy to note that the value should be well prescribed in computation because it has a significant impact on the approximation power of a RBFs based method.\n\n### 3.2 Weighted coefficients for fractional derivatives\n\nIn order to obtain the weighted coefficients of the left and right fractional derivatives, we substitute the RBFs into Eqs. (3.5)-(3.6) to get\n\n ∂αφk(xi)∂+xα =M∑j=0a(α)ijφk(xj), i,k=0,1,…M, (3.9) ∂αφk(xi)∂−xα =M∑j=0b(α)ijφk(xj), i,k=0,1,…M. (3.10)\n\nRewriting Eqs. (3.9)-(3.10) in a matrix-vector form for each grid point yields\n\n ⎛⎜ ⎜ ⎜ ⎜ ⎜⎝φ0(x0)φ0(x1)⋯φ0(xM)φ1(x0)φ1(x1)⋯φ1(xM)⋮⋮⋱⋮φM(x0)φM(x1)⋯φM(xM)⎞⎟ ⎟ ⎟ ⎟ ⎟⎠M⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝ω(α)i0ω(α)i1⋮ω(α)iM⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝Dαφ0(xi)Dαφ1(xi)⋮DαφM(xi)⎞⎟ ⎟ ⎟ ⎟ ⎟⎠Dαφ(xi), (3.11)\n\nwhere if whereas if , . M is the interpolation matrix only related to the nodal distribution, being nonsingular for MQ and fully positive definite for IM, GA Cheng (2012). One has\n\n M=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝ϵ√(x1−x0)2+ϵ2⋯√(xM−x0)2+ϵ2√(x0−x1)2+ϵ2ϵ⋯√(xM−x1)2+ϵ2⋮⋮⋱⋮√(x0−xM)2+ϵ2√(x1−xM)2+ϵ2⋯ϵ⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠,\n\nin particular, when MQ RBFs are adopted.\n\nThere are clearly no explicit expressions for , ; fortunately, they can be approximated by numerical quadrature rules. Taking the transforms , of variables, respectively, reaches to\n\n ∂αφk(xi)∂+xα ∂αφk(xi)∂−xα\n\nA close examination reveals that both of the two formulas are the special cases of the following weakly singular integral, i.e.,\n\n ∫1−1(1−ζ)λ(1+ζ)μf(ζ)dζ,λ, μ>−1,\n\nwith , that can be handled by Gauss-Jacobi quadrature rules. , are then determined by solving Eqs. (3.11) for each and the fractional derivatives are removed from a fractional PDE by using Eqs. (3.7)-(3.8) as replacements, thus we obtain the required solution by solving a ODS instead.\n\n## 4 A time-stepping RBF-DQM for fractional PDEs\n\nIn this section, a RBF-DQM of fully discretization is derived for the space-fractional PDEs via the above direct approximations for fractional derivatives. Define a lattice on with equally spaced points , , . On substituting the weighted sums (3.7)-(3.8) into Eq. (1.1), we have\n\n ∂y(xi,t)∂t−κ(xi)M∑j=0a(α)ijy(xj,t)−υ(xi)M∑j=0b(α)ijy(xj,t)=f(xi,t), i=0,1,⋯,M,\n\nactually being a first-order ODS. Also, denote , , for brevity. Imposing the associated constraints (1.2)-(1.3) and rewriting the ODS in matrix-vector form, a time-stepping RBF-DQM is then derived by introducing a Crank-Nicolson scheme in time, given as\n\n (I−τκA+υB2)Yn=(I+τκA+υB2)Yn−1+τHn−1/2, (4.12)\n\nwhere I is an identity matrix, , , , and A, B, are as follows\n\nwith , . We perform the procedures on the nodal distribution , , . An detailed implementation of RBF-DQM is summarized in the following flowchart\n\n• Input , , , , and allocate , .\n\n• Form M, compute by Gauss-Jacobi quadrature rules, and solve Eqs. (3.11) for each so that the weighted coefficients are found.\n\n• Do a loop from to to solve Eqs. (4.12) for each by forming A, B first and output the desirable approximation at each time step.\n\n## 5 Illustrative examples\n\nIn this part, the proposed methods, termed as MQ-DQM, IM-DQM, and GA-DQM hereinafter, are studied on three numerical examples. The shape parameter should be adjusted with the grid number , so we select for MQ, for IM, and for GA, tentatively, by references to Fasshauer (2002); Franke (1982); Xiang and Wang (2009). The numerical errors are all defined in -norm and the fractional derivatives are computed with 16 quadrature points and weights, whose values corresponding to are given as reference in A.\n\nExample 5.1. Approximate by Eq. (3.7) on with the above , whose explicit expression is , where is the hypergeometric function. The numerical results are tabulated in Table 1. As observed, the approximation improves as increases, which implies that the DQ approximations for the fractional derivatives are valid. Moreover, under the given , GA-DQM seems to be more efficient than MQ-DQM and IM-DQM.\n\nExample 5.2. Let , and ; we solve Eqs. (1.1)-(1.3) on with zero initial-boundary conditions. Table 2 displays the numerical results at when and . From the table, we find that sufficiently small errors can be achieved by MQ-DQM, IM-DQM, and GA-DQM even if a few spatial grid points are utilized and all the methods are obviously convergent by taking their own , respectively.\n\nExample 5.3. Let , , and ; we solve Eqs. (1.1)-(1.3) on with , , and . The numerical errors of SAM Sousa (2011) and our methods at are reported in Table 3, when and . It is observed from the table that MQ-DQM, IM-DQM, and GA-DQM outperform SAM in term of computational accuracy.\n\n###### Remarks 5.1.\n\nWhen is fixed, the value is crucial to the accuracy of a RBFs based method, so is RBF-DQM. A general trade-off principle demanding attention is that one can adjust to decrease the approximate errors, but need to pay for this by increasing the condition number of the interpolation matrix which may cause an algorithm to be instable Schaback (1995), so a good that balances both the accuracy and stability is anticipated in practice. Nevertheless, how to select an optimal value is technical and is being an issue deserving to investigate.\n\n## 6 Conclusion\n\nIn this research, an efficient DQ method is proposed for the space-fractional PDEs of Caputo type based on commonly used RBFs as test functions, which enjoys some properties such as high accuracy, flexibility, truly meshless, and the simplicity in implementation. 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Comput. 217 (2010) 2534–2545." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8098322,"math_prob":0.96839243,"size":18625,"snap":"2020-24-2020-29","text_gpt3_token_len":5304,"char_repetition_ratio":0.14478277,"word_repetition_ratio":0.010909091,"special_character_ratio":0.3174765,"punctuation_ratio":0.21430436,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9965891,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-02T13:26:27Z\",\"WARC-Record-ID\":\"<urn:uuid:c7f1aa67-a0d4-4c8f-b408-491c1b228900>\",\"Content-Length\":\"557826\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:19cf11da-7bd1-4c65-9b94-b08532fc519e>\",\"WARC-Concurrent-To\":\"<urn:uuid:6f0bacf7-7ee2-42e8-ab21-cc9ee2f653ce>\",\"WARC-IP-Address\":\"104.28.20.249\",\"WARC-Target-URI\":\"https://www.arxiv-vanity.com/papers/1701.06118/\",\"WARC-Payload-Digest\":\"sha1:VJ3EYSKFPV5JIIW2YI7HM7EKNWWIH3EQ\",\"WARC-Block-Digest\":\"sha1:SOATB2SKE35WOFL5UIJ64A2XBP4NAWAY\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593655878753.12_warc_CC-MAIN-20200702111512-20200702141512-00181.warc.gz\"}"}
https://answers.everydaycalculation.com/divide-fractions/3-42-divided-by-2-6	[ "Solutions by everydaycalculation.com\n\n## Divide 3/42 with 2/6\n\n3/42 ÷ 2/6 is 3/14.\n\n#### Steps for dividing fractions\n\n1. Find the reciprocal of the divisor\nReciprocal of 2/6: 6/2\n2. Now, multiply it with the dividend\nSo, 3/42 ÷ 2/6 = 3/42 × 6/2\n3. = 3 × 6/42 × 2 = 18/84\n4. After reducing the fraction, the answer is 3/14\n\nMathStep (Works offline)", null, "Download our mobile app and learn to work with fractions in your own time:" ]	[ null, "https://answers.everydaycalculation.com/mathstep-app-icon.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.77565414,"math_prob":0.98030674,"size":332,"snap":"2022-27-2022-33","text_gpt3_token_len":166,"char_repetition_ratio":0.21646342,"word_repetition_ratio":0.0,"special_character_ratio":0.5060241,"punctuation_ratio":0.056818184,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9686116,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-07-05T08:37:03Z\",\"WARC-Record-ID\":\"<urn:uuid:d87317ef-2f28-4856-a765-a6d16e84a6c7>\",\"Content-Length\":\"7853\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2751c420-f4f3-468a-9137-c3c14e4b27a5>\",\"WARC-Concurrent-To\":\"<urn:uuid:dd489718-eba4-47f6-9fe3-95631423d432>\",\"WARC-IP-Address\":\"96.126.107.130\",\"WARC-Target-URI\":\"https://answers.everydaycalculation.com/divide-fractions/3-42-divided-by-2-6\",\"WARC-Payload-Digest\":\"sha1:QB7JRVOMP6T5HLIQE3K3HYI7OCWKJVXK\",\"WARC-Block-Digest\":\"sha1:T6W2BC5MNWLMBKHNWSBI25EGOHEJ6Y2W\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656104542759.82_warc_CC-MAIN-20220705083545-20220705113545-00007.warc.gz\"}"}
https://www.uregina.ca/science/mathstat/research/areas-expertise.html	[ "# Areas of Expertise\n\nA list of the general research areas of Department faculty is below. Further information about an individual faculty member may be found on that person's webpage.\n\n### Actuarial Mathematics\n\n• Dr. T. Bae (Financial mathematics, risk management)\n• Prof. P. Douglas (Applied stochastic modelling)\n• Prof. J.-P. Venter (Actuarial Science, Financial mathematics and investment theory, enterprise risk management, stochastic survival models)\n\n### Algebra, Discrete Mathematics, and Number Theory\n\n• Dr. S. Fallat (Linear algebra, combinatorial matrix theory)\n• Dr. A. Herman (Group theory, representation theory, algebraic combinatorics)\n• Dr. K. Meagher (Combinatorics and algebraic graph theory)\n• Dr. R. McIntosh (Computational number theory, q- hypergeometric series, mock theta functions)\n• Dr. S. A. Mojallal* (Spectral graph theory, matrix theory)\n• Dr. F. Szechtman (Group theory, Lie algebras, representation theory, linear algebra)\n• Dr. P. Vishwakarma* (Matrix analysis and positivity preservers, totally nonnegative matrices, and inverse eigenvalue problem for graphs)\n\n### Statistics and Probability Theory\n\n• Dr. D. Deng (Limit theory, generalized linear models)\n• Dr. M. Kozdron (Probability, stochastic processes, statistical physics)\n• Dr. A. Volodin (Statistical inference, limit theorems, distributions on abstract spaces, bootstrap)\n• Dr. Y. Zhao (Applied statistics and biostatistics)\n\n* Postdoctoral Fellow (PIMS) ** Postdoctoral Fellow" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.72627157,"math_prob":0.89669853,"size":2258,"snap":"2022-05-2022-21","text_gpt3_token_len":575,"char_repetition_ratio":0.14773735,"word_repetition_ratio":0.0,"special_character_ratio":0.22010629,"punctuation_ratio":0.24020888,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9862994,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-05-25T22:07:22Z\",\"WARC-Record-ID\":\"<urn:uuid:ecd5e98f-2da4-4d43-8e2a-2d006a7ed3c3>\",\"Content-Length\":\"13627\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0c688007-16d2-456b-9126-294d58c7d598>\",\"WARC-Concurrent-To\":\"<urn:uuid:5ac7a8dd-c445-472e-9738-7326e24f76ed>\",\"WARC-IP-Address\":\"142.3.152.9\",\"WARC-Target-URI\":\"https://www.uregina.ca/science/mathstat/research/areas-expertise.html\",\"WARC-Payload-Digest\":\"sha1:34HVIRIIXYVACNNEWY6YQKQMO6JRXWUO\",\"WARC-Block-Digest\":\"sha1:HDDZK4WIHNH6NJEPH3I4NIHTLFEZXQQ6\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-21/CC-MAIN-2022-21_segments_1652662594414.79_warc_CC-MAIN-20220525213545-20220526003545-00468.warc.gz\"}"}
https://www.mdpi.com/1996-1073/15/23/8924	[ "", null, "Font Type:\nArial Georgia Verdana\nFont Size:\nAa Aa Aa\nLine Spacing:\nColumn Width:\nBackground:\nArticle\n\n# Improved Thermal Performance of a Serpentine Cooling Channel by Topology Optimization Infilled with Triply Periodic Minimal Surfaces\n\n1\nInstitute of Turbomachinery, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China\n2\nDepartment of Mechanical Engineering and Science, Kyoto University, Kyoto 615-8540, Japan\n\nAuthor to whom correspondence should be addressed.\nEnergies 2022, 15(23), 8924; https://doi.org/10.3390/en15238924\nReceived: 6 October 2022 / Revised: 21 October 2022 / Accepted: 26 October 2022 / Published: 25 November 2022\n(This article belongs to the Special Issue New Insights of Gas Turbine Cooling Systems)\n\n## Abstract\n\n:\nThe present study utilizes a density-based topology optimization method to design a serpentine channel under turbulent flow, solving a high pressure loss issue and enhancing heat transfer capability. In the topology optimization, the kε turbulence model is modified by adding penalization terms to reveal turbulence effects. Heat transfer modeling is included by setting the modified energy equation with additional terms related to topology optimization. The main objective is to minimize pressure loss while restricting heat transfer. The 2D simplified model is topologically optimized. Then, the optimal solution with intermediate results is extruded in the 3D system and interpreted with triply periodic minimal surfaces (TPMS) to further enhance heat transfer performance. Compared to the baseline serpentine channel, the optimized model infilled with the diamond-TPMS structure lowers pressure loss by 30.8% and significantly enhances total heat transfer by up to 45.8%, yielding thermal performance of 64.8% superior to the baseline. The temperature uniformity is also improved. The simulation results show that the curvatures in the optimized model with diamond-TPMS structure eliminate the large recirculation flow and low heat transfer regions. This model diminishes the effect of Dean’s vortices but promotes high turbulent kinetic energy, leading to better uniform flow and heat transfer distributions.\n\n## 1. Introduction\n\nApplications of a multipass cooling channel under turbulent flow and heat transfer enhancement themes can be found in an internal cooling gas turbine blade, microelectronic cooling, etc. Generally, the serpentine channel is a standard internal cooling scheme in practical turbine blades, effectively utilizing the coolant in the midchord region [1,2]. However, complicated flow fields induced by sharp turns after bend regions cause a nonuniform flow distribution. Rao et al. have reported that the turning regions cause more than 90% of the total pressure drop of the multipass channel. These phenomena in the serpentine channels result in highly nonuniform-flow heat transfer and high pressure loss, decreasing overall thermal performance. Therefore, it is desirable to design multipass channels that generate a lower-pressure loss while maintaining the total heat transfer and achieving better uniformity.\nIn order to control the flow field and heat transfer distribution in serpentine channels, placing the turning guide vane at the bend region is a common method that has been examined in several previous studies. Schüler et al. and Guo et al. investigated experimentally and numerically different turning guide vane designs in the multipass channel. They found that the pressure loss was reduced by about 13.4%–48.4% when using the turning guide vanes, but the heat transfer deteriorated. Similarly, Lei et al. observed that the pressure loss was largely reduced when installing the turning vane, yet heat transfer decreased because the flow separation was suppressed. This design process was based on a trial-and-error method. To achieve low pressure loss while maintaining total heat transfer in a serpentine cooling channel, various geometrical parameters were varied, causing a cost- and time-consuming design process.\nAlternatively, optimization methods can speed up and ease the designs. Layout optimization is usually divided into size, shape and topology optimization (TO). Size and shape optimizations have been used to minimize pressure loss and enhance thermal performance in heat exchangers. The results reveal that the optimal design significantly reduces the pressure drop and achieves higher thermal efficiency than the original model . However, it is observed that the design process only adjusts the positions of points or boundaries of the channel to obtain the objective functions without changing the topology of the structure. Moreover, since additive manufacturing is rapidly developed, which can compete for limitations to produce relatively complex geometries, the size and shape optimizations cannot completely exploit this manufacturing scheme’s potential. Meanwhile, finding the optimal shapes and optimizing the topology of channels can be obtained simultaneously through topology optimization. Additionally, free-form and efficient geometries can be generated, satisfying solutions for additive manufacturing. Therefore, topology optimization is a more general design method than size and shape optimizations .\nTopology optimization has been employed in different design systems covered by various disciplines. It has been well developed and affirmed widely due to competitive designs in cost and functionality under various constraints [8,9,10,11,12,13,14]. Several methods, such as the homogenization approach, the evolutionary technique, the level-set and the density approach, have been developed for implementation.\nAmong topology optimization schemes, the density-based method has gained more popularity . In the comparisons of Manuel et al. , the density-based approach yields better outcomes for most cases in terms of iteration numbers. Additionally, the density method merits rapid convergence and weak dependence on the initial guess [8,16]. Furthermore, it could be highlighted that no prior research has considered turbulent flow systems except the density-based method. The key idea of the density-based method is the penalized interpolation function, which continuously forces fictitious porous cells, including fluid and solid, to be fluid or solid subdomains. A comprehensive review and its general formulation of topology optimization can be found in previous work . Here, a brief overview of the density-based topology optimization applied in recent research on thermofluid systems is shown in Table 1.\nAs highlighted in Table 1, most previous studies are 2D models with a steady-state laminar flow regime. The FEM is the most implemented discretization method since the topology optimization initiates from solid mechanics. Moreover, it is easier for the discrete adjoint scheme to manage the FEM .\nNevertheless, only limited works on turbulent flow topology optimization have been studied, which is primarily needed in most industrial flow applications, e.g., a turboshaft engine’s turbine blade serpentine cooling channels. Several works employed a frozen turbulence assumption to simplify sensitivity analysis for the topology optimization of turbulent flows [23,24]. However, as the frozen turbulence neglects the effect of porosity in the turbulent viscosity calculation, the optimized model negatively affects the accuracy. The shortcomings of this assumption have been presented by Dilgen et al. .\nRecently, study of the turbulence effects in the topology optimization for turbulent flow has appeared. Dilgen et al. optimized many 2D and 3D complex flow channels using Spalart–Allmaras and kω turbulence models, where k denotes turbulent kinetic energy and ω is the specific dissipation rate. The results showed that the generated structure in a U-bend channel significantly reduced the pressure drop across the channel. Later, Yoon proposed the topology optimization with the FEM using the kε turbulence model, where ε is the turbulent energy dissipation. However, it is noted that the energy equation for topology optimization has not yet been found, and convective heat transfer optimization has not yet been explored in those studies.\nEven though topology optimization is an efficient tool for generating complex structures, it still causes some problems. The results of density-based topology optimization may convert a number of intermediate densities. This type of conversion causes pressure maldistribution in the cooling channel between the design and its realization by additive manufacturing, which could not explore the full performance of the optimized structure. To overcome the issues, the solid domain of the optimized model, including the intermediate results, can be fulfilled with a graded lattice structure. This method has been observed to help manage the intermediate densities in the conventional topology optimization, reduce the structure’s weight, and preserve the original design structure ; however, this approach is rarely applied to cooling channel applications.\nAmong complex lattice structures, the repeating network, called triply periodic minimal surface (TPMS), has attracted much research interest due to its compact and significant thermal properties. Additionally, the TPMS is a self-supporting structure, which has been observed to present better mechanical performance than other strut-based lattice structures . The TPMS networks have also shown better heat transfer performance in cooling channels than the lattice-frame material due to a smoother surface and larger wetted area . The TPMS, with its complex and interconnected structures, provided consistent fluid streamlines, promoting permeation . Moreover, many previous works have found that the TPMS structures considerably increase heat transfer with reasonably increased pressure loss, improving the thermal performance of the cooling channel [31,32,33].\nThis work proposes an optimized design to minimize pressure and increase heat transfer, improving the thermal performance of a serpentine cooling channel. The density-based topology optimization for the thermofluid problem under turbulent flow is firstly utilized to minimize the pressure loss while maintaining heat transfer in the serpentine cooling channel of a turbine blade. The turbulence effects are revealed by adding penalization terms in the kε turbulence model programmed by COMSOL. The functions to activate/deactivate the convective and conduction terms in the modified energy equation are also included to account for the heat transfer modeling in the topology optimization. Three-dimensional (3D) serpentine models with the TPMS structures are obtained by infilling the TPMS structures in the solid domain and the intermediate density resulted from the conventional topology optimization. The final models are resimulated in ANSYS FLUENT to show elaborated flow structure, pressure loss and heat transfer advantages over the baseline and the channel generated from the conventional topology optimization. By infilling the TPMS structures in the conventional topology optimization, the intermediate result is eliminated, the original channel structure is preserved, and the heat transfer is further increased due to the complex structure and large wetted area of the TPMS.\n\n## 2. Serpentine Model\n\n#### 2.1. Model Description\n\nThe design model is inspired by an internal cooling channel of a turboshaft heat exchanger. The channel dimension of the numerical model in this work is consistent with the experimental serpentine cooling channel for a turbine blade with short passes, studied by Guo et al. . Figure 1 illustrates the three-short-pass serpentine channel for (a) a full-scale 3D model and (b) a simplified 2D design. Each channel pass has a cross-section of 120 mm × 60 mm, yielding a hydraulic diameter (Dh) of 80 mm and an aspect ratio (AR) of 2.0. The length of all the straight passages is 245 mm, resulting in a length-to-hydraulic diameter of 3.06. Before entering the heating section, entrance and exit sections extended with a length of 300 mm are created at the channel inlet and outlet, respectively. The first turning region is designed as a rectangular shape with an inner radius of 20 mm since this region near the tip of the turbine blade suffers from a high heat load, which needs a large heat transfer area to cool. Meanwhile, the second bend near the blade platform receives a lower heat load; hence it is designed as an arc shape with outer and inner diameters of 140 and 20 mm, to reduce the total pressure loss in the channel .\nTypically, the topology optimization requires hundreds of iterations to reach the convergence of the final optimal design. Hence, to save the cost of obtaining the optimized design, the 2D design simplified from the original 3D model is firstly used in this work [19,20,21], as demonstrated in Figure 1b. An inlet boundary condition (Γin) is set as uniform velocity (Uin) with a prescribed temperature (Tin). The inlet velocity is defined based on the experimental Reynolds number (Re), as follows:\n$R e = ρ U i n D h μ$\nwhere ρ is fluid density and μ is dynamic viscosity.\nAn outlet boundary condition (Γout) is set as a zero static pressure (p0). The boundary condition of the heating surfaces on the top and bottom is imposed with uniform heat flux (q0). The remaining walls are thus set as no-slip conditions (Γwall) to surround the serpentine channel. Figure 1b also shows the design domains, in which all the heating regions inside design domains (Ωid) are evolved. Meanwhile, the extended entrance and exit of the channel set as outside design domains (Ωod) are not considered for optimization.\n\n#### 2.2. Thermofluid Modeling\n\nPrevious works have considered an incompressible flow for thermofluid topology optimization to design turbine blade cooling channels . Moreover, it has been observed that when the Mach number is less than 0.3, the deviation of the compressible and incompressible flow can be neglected. Since the maximum bulk velocity of air in the channel is lower than 15 m/s, the Mach number is less than 0.1, then incompressible and steady-state flow are assumed throughout this work.\n\n#### 2.2.1. Fluid-Flow Modeling\n\nThe continuity equation and Navier–Stokes equation assuming the incompressible steady fluid are expressed as follows:\n$ρ ∇ ⋅ u = 0$\n$ρ ( u ⋅ ∇ ) u = ∇ ⋅ [ - p I + ( μ + μ T ) ( ∇ u + ( ∇ u ) ) ] + F$\nwhere u indicates the fluid velocity vector; $∇$ signifies the gradient operator; p denotes the pressure field; I is the identical tensor; μT is the turbulent dynamic viscosity; * is the transpose; and F represents the Brinkman friction term introduced by Borrvall and Petersson , calculated from Equation (4). Here, the body force is not exerted on the fluid outside the design domain (F = 0 in Ωod).\n$F = − [ ( α u , max − α u , min ) I u ( γ ) + α u , min ] u$\nwhere αu,max is the maximum inverse permeability of the porous medium, while αu,min is the minimum one. Iu(γ) denotes the inverse permeability interpolated function stated in Section 3.1.\nFrom Equation (5), αu,max can be defined depending on Darcy number (Da):\n$α u , max = μ D a ⋅ D h 2$\n\n#### 2.2.2. Turbulence Modeling\n\nAs the topology optimization progressively evolves the design domain to allow material distribution, modifying the turbulence model is also necessary. In the present work, the authors modify the standard kε model in COMSOL by adding penalty terms to reveal the effects of turbulence on the serpentine cooling channel for topology optimization, which is modified as follows:\n$ρ ( u ⋅ ∇ ) k = ∇ ⋅ [ ( μ + μ T σ k ) ∇ k ] + P k − ρ ε − α k ( γ ) ( k − k 0 )$\n$ρ ( u ⋅ ∇ ) ε = ∇ ⋅ [ ( μ + μ T σ ε ) ∇ ε ] + C ε 1 ε k P k − C ε 2 ρ ε 2 k − α ε ( γ ) ( ε − ε 0 )$\nThe closure coefficients and all parameters in Equations (6) and (7) are described as follows:\n$μ T = ρ C μ k 2 ε , P k = μ T [ ∇ u : ( ∇ u + ( ∇ u ) * ) ] , C ε 1 = 1.44 , C ε 2 = 1.92 , σ k = 1 , σ ε = 1$\nwhere Pk signifies the turbulent kinetic energy source term.\nThe last terms in Equations (6) and (7) are augmented to enforce k and ε to k0 and ε0 for the solid regions in the design domain. They are defined as follows :\n$α k ( γ ) ( k − k 0 ) = [ ( α k , max − α k , min ) I k ( γ ) + α k , min ] ( k − k 0 )$\n$α ε ( γ ) ( ε − ε 0 ) = [ ( α ε , max − α ε , min ) I ε ( γ ) + α ε , min ] ( ε − ε 0 )$\nwhere αk,max and αε,max are the maximum limits of the penalization for k and ε, respectively, while αk,min and αε,min are the minimum ones. Ik(γ) and Iε) are material interpolated functions in both Equations (8) and (9). Here, the values of αk,min, αε,min, k0 and ε0 are assigned to zero, while the value of αk,max and αε,max are set equal to the value of αu,max. The material interpolations are detailed in Section 3.1.\n\n#### 2.2.3. Heat Transfer Modeling\n\nThe thermal convection–diffusion equation outside the design domain is given as follows:\n$ρ c f u ∇ T - ∇ ⋅ ( k f ∇ T ) = 0$\nwhere cf denotes the fluid heat capacity, kf indicates the fluid thermal conductivity and T is the temperature field in the serpentine channel.\nIn the design domain, as the distribution of the fluid and solid domains is determined after the optimization, the authors build up the energy equation by adding the functions to control the convective and conduction terms as follows:\n$ρ c f I c ( γ ) u ∇ T - ∇ ⋅ ( k f I K ( γ ) ∇ T ) = q 0$\nwhere Ic(γ) is the function used to activate/deactivate the convective term for fluid (Ic(γ) = 1) and solid (Ic(γ) = 0). IK(γ) is used to interpolate between the thermal conductivity of the fluid and the thermal conductivity of the solid (ks). Lastly, the term q0 indicates the constant bottom wall heat flux.\nIt should be noted that q0 is the uniform heat flux applied to the heating boundaries in the 2D model. Additionally, the energy equation is weakly coupled to the fluid flow problem to obtain u; then, it substitutes the energy equation to solve T. Furthermore, the Reynolds number and heat flux applied in this work are fixed at Re = 10,000 and q0 = 1500 W/m2, respectively, because varying the velocity or the heat input in the topology optimization could change the topology of the structure in the final result [11,14].\n\n## 3. Optimization Procedures\n\nThe procedure for obtaining the optimized model with lattice structures in this study is depicted in Figure 2. Firstly, the 2D simplified model, shown in Figure 1b, is topologically optimized using the boundary conditions and governing equations described in Section 2. The optimal solid domain, including the intermediate densities, is then extruded into the 3D model and interpreted with the TPMS structures. Finally, the final designs are recalculated for comparison.\n\n#### 3.1. Topology Optimization\n\nIn this section, the details of the topology optimization process to obtain the 2D optimized structure are presented.\n\n#### 3.1.1. Material Distribution\n\nThe material distribution in the density-based method described by the design variable (γ) takes a value between 0 and 1 at any point in the design domain. Here, γ = 1 denotes free flow in the channel, while γ = 0 is solid regions. The discrete 0/1 problem is relaxed to use gradient-based methods. As the material distribution takes intermediate values (0 ≤ γ ≤ 1), this relaxation requires the interpolation of transition areas. The interpolations of the fluid flow equation in the design domain are expressed in Equations (12)–(14):\n$I u ( γ ) = q u 1 − γ q u + γ$\n$I k ( γ ) = q k 1 − γ q k + γ$\n$I ε ( γ ) = q ε 1 − γ q ε + γ$\nwhere qu, qk and qε are tuning parameters to control the function curvature.\nFor the thermal convection and conductivity, the interpolation terms Ic(γ) and IK(γ) are calculated using solid isotropic material penalization (SIMP) in Equations (15) and (16), respectively:\n$I c ( γ ) = γ n c + ( 1 − γ n c ) δ$\n$I K ( γ ) = γ n K + ( 1 − γ n K ) k s k f$\nwhere nc and nK are penalization power coefficients; δ is fixed at δ = 10−15 to avoid numerical problems when solving Equation (11) .\n\n#### 3.1.2. Problem Formulation\n\nIn this study, minimizing the pressure loss is set as the objective function (J), while the heat transfer across the serpentine channel is constrained. In minimizing the pressure loss for the cooling channel, the power dissipation within the volume is generally selected to proceed in the optimization process since it shows the flow fluctuation in the considered domain. It provides robust convergence, and better flow uniformity can be expected when minimizing this function [11,12]. The power dissipation is expressed as follows:\n$J = ∫ Ω [ ( μ + μ T ) ( ∇ u + ( ∇ u ) * ) : ∇ u + α u ⋅ u ] d Ω$\nIn prior works, the temperature through the boundary has been defined as another objective function to maximize heat transfer . This function represents the temperature gain of fluid across the channel. Since the majority of the previous work has observed that the use of the turning vanes deteriorated the heat transfer [4,5,6], the temperature constraint (C) is thus employed to restrict the heat transfer in the present work. It is written as follows:\n$C = ∫ Γ out T ( u ⋅ n ) d Γ out$\nHere, Γout is the outlet boundary, and n is the normal vector. With this expression, the heat transfer in the channel can be constrained (g) as follows:\n$g = C C r e f − 1 ≥ 0$\nwhere Cref is the reference value calculated from the conventional serpentine channel.\nHence, the mathematical optimization model is written in Equation (20):\n$g = ∫ Γ out T ( u ⋅ n ) d Γ o u t ≥ C r e f$\n$0 ≤ γ ( x ) ≤ 1 , ∀ x ∈ Ω i d$\nwhere S is the vectors with discrete state fields, R indicates the residual vector function evaluated from the discretization of the governing equations, g represents inequality constraints and x denotes the design variable vector.\n\n#### 3.1.3. Density Filter and Projection\n\nDuring the iterative process, the design variable changes independently in each cell, causing a large difference value of the design variables in the adjacent cells. In the present optimization, a partial differential equation filter is adopted to mitigate such phenomena, which is defined as follows:\n$− R min 2 ∇ 2 γ f + γ f = γ$\nwhere γf signifies a smoothed design variable. Rmin is a filter radius calculated by = 1.5 × maximum mesh size in the considered domain.\nAdditionally, a projection technique, which projects the design variable into either 0 or 1, is adopted to mitigate the intermediate results. Here, a smoothed Heaviside projection is applied as follows:\n$γ p = tanh ( β η ) + tanh ( β ( γ f − η ) ) tanh ( β η ) + tanh ( β ( 1 − η ) )$\nwhere γp is the projected design variable, β is a parameter used to control the steepness of the projection, and η denotes the projection threshold, fixed at η = 0.5 in this study.\n\n#### 3.1.4. Summary of Topology Optimization Process\n\nThe present study uses COMSOL Multiphysics to run the topology optimization process. The flow chart of the topology optimization process is demonstrated in Figure 3. After the design variables and the parameters are initialized, the governing equations are solved to attain physical variable fields. The objective and the constraint functions are then computed. If the convergence criterion is not achieved, the sensitivity is computed by the adjoint method to obtain the gradient information. The design variable field is then updated using GCMMA and regularized. When the constraint is achieved, and the relative change in the objective is less than 10−4, or after every 100 iterations, the optimization proceeds to the next continuation steps. The continuation approach is used to avoid the low-quality local minima result and ease the convergence in this optimization process . Eventually, the optimization is considered to be completed when all the continuation steps are computed.\n\n#### 3.2. Infilling Lattice Structures\n\nAfter obtaining the 2D optimized model, the optimal solid domain within the channel, including intermediate results generated by the conventional topology optimization, is extruded into the 3D model and replaced by TPMS structures. The TPMS networks are used to infill the extruded optimized model with a range of thickness according to the optimal density results. Here, the diamond and gyroid-sheet structures, as shown in Figure 4, are chosen to replace the solid domains and intermediate results of the extruded optimized model due to their excellent thermal performance [31,32,33]. The final designs are then recomputed.\n\n## 4. Results and Discussion\n\nIn this section, two main parts are presented. For the topology optimization, the problem setup and 2D simplified model validation are detailed in the first part. The optimal density distribution in the 2D optimized model is extruded into the 3D domain and fulfilled with the TPMS structures. The final designs are then recalculated and validated in the second part. The pressure drop, heat transfer and temperature uniformity are compared with the baseline serpentine channels.\n\n#### 4.1.1. Validation of the 2D Model\n\nSince the 3D channel is simplified into the 2D model to reduce the computational cost for topology optimization, the accuracy of the stated variable field obtained from 2D calculations should be satisfied . Air and copper are selected in this work for fluid and solid properties for the simulation. The boundary conditions and governing equations explained in Section 2 are used to compute the flow, turbulence effect and heat transfer. It should be marked that in COMSOL, the out-of-plane heat flux (q0) in the 2D model is equivalent to the uniform heat flux boundary in the 3D system.\nThe mesh system of the 2D and 3D models is demonstrated in Figure 5. It is suggested that the boundary layer mesh evolves during the optimization for the accurate turbulence calculation, or the mesh should be fine enough to obtain accurate and reliable results. The extra-fine mesh setting at 42,468 elements, containing mainly triangle element type, is thus generated for all 2D models, as shown in Figure 5a. Meanwhile, for the 3D model, the tetrahedral mesh is generated, and the boundary layer mesh with a prism shape is employed in the wall regions. The fine mesh system is selected at 2.38 million elements, as shown in Figure 5b.\nTo verify the discrepancy between the 2D and 3D models, averaged values of the pressure, velocity and temperature at 12 different positions are plotted in Figure 6. In all 2D cases, the results are averaged from 12-cut lines, whereas in all 3D cases, the outcomes are extracted from the planes, then averaged. Overall, the 2D results have similar trends to the 3D ones for all variables. The pressure values in Figure 6a and velocity magnitudes in Figure 6b for the 2D model are lower than in the 3D model, while the temperature values in Figure 6c are opposite. The 2D model overestimates the temperature by about 0.4%–2.5%, and the average error value is 1.6%. The temperature discrepancy between the 2D and 3D models can attribute to the limitation of a pseudo-3D model, which has also been observed in several investigations [13,19]. However, the temperature distributions that slightly decreased along the channel path are well estimated, providing similar trends among them. The variations can be acceptable for the optimization process. Therefore, comparing results with the 3D numerical model, the 2D model can be accurately employed to perform topology optimization in this work.\n\n#### 4.1.2. Optimized Structure\n\nThe initial design is a uniform density field of γ0 = 0.6. The model is evaluated through the continuation parameters, as given in Table 2, to find the optimal structure. After the simulation, the design converges to the optimized structure, as presented in Figure 7. It can be noticed that the model leaves the intermediate fields to satisfy the temperature constraint by distributing the material in the channel. The intermediate results are due to the high conductivity difference between the copper and air used in this work . For the 2D optimized design, the total pressure loss between inlet and outlet boundaries decreases by about 33.9% compared to the original 2D model, while the constraint temperature in the channel is maintained.\nThe optimal density distribution within the 2D optimized channel, including intermediate results generated by conventional topology optimization, is extracted with ranges of density filtering from 0.1 to 0.9 and extruded into the 3D model, as shown in Figure 8a. The TPMS structures are then used to infill with a range of thickness between 2–4 mm. The lowest thickness of 2 mm is specified at the density filtering of 0.9, and the thickness is increased accordingly, reaching the highest thickness of 4 mm at the density filtering of 0.1, as demonstrated in Figure 8b. The thickness range is selected based on the printable capability in an actual scale of the cooling channel for a gas turbine blade. Here, the single unit cell size of the diamond- and gyroid-sheet structures is 120 × 120 × 120 mm3.\n\n#### 4.2. 3D Model Comparisons\n\nIn this section, the optimized model extracted with a density filtering of 0.5 (TO model) is also included to compare with the baseline channel. All models and the boundary conditions are shown in Figure 9. These models are applicable at the Reynolds number Re = 10,000 and heat flux q0 = 1500 W/m2 according to the design settings in the conventional topology optimization, as mentioned in Section 2. For all cases, the channel dimensions are consistent with the baseline serpentine channel, as described in Figure 1a. The inlet and outlet boundaries are the same as the baseline channel. Here, the red zones are treated by the uniform heat flux, including the top, bottom and extruded areas from the optimization and TPMS structures. The remaining surfaces are set as no-slip walls.\nThe flow and heat transfer for all models in this section are simulated using the commercial CFD solver ANSYS FLUENT 18.0, which provides lower computational costs and consumes lesser computer memory for 3D complex cooling channels than COMSOL. The overview of mesh systems for the optimized model with the diamond-sheet structure is demonstrated as an example in Figure 10. The hybrid mesh involving tetrahedron and prism elements is created to discretize the entire domain. The tetrahedron meshes are finely dense in the small- and high-curvature regions, while the boundary layer meshes with a prism shape are used near the no-slip walls. The meshes near the wall are refined to ensure that y+ values are less than one.\n\n#### 4.2.1. 3D Model Validation\n\nFor the validation, the conventional three-short-pass smooth channel investigated by Guo et al. is used as the referee design. The SIMPLEC algorithm is utilized for the pressure–velocity coupling, and the second-order upwind method is adopted. The convergence criterion is less than 10−4 for the continuity, velocity and turbulence quantities and less than 10−6 for the energy equation .\nThree sets of the meshes at 4.95, 7.31 and 11.6 million elements are simulated to check mesh independence for the baseline channel. The numerical results are steady and converge to a certain value as the mesh elements increase. The grid convergence index (GCI) is evaluated using the spanwise-averaged surface temperature to estimate the numerical accuracy of the mesh at 7.31 million, as demonstrated in Figure 11. The line denotes the highest mesh solution, while the red bars represent the error. It is found that the maximum error of the samples is less than 0.5%, and the overall GCI value on the spanwise-averaged temperature is about 0.1%, which indicates that the numerical results are independent of the meshes. Therefore, the mesh element is selected at 7.31 million for the baseline serpentine channel, while this mesh setting is also used for the other models.\nIn order to select the turbulence model with the highest accuracy, several Reynolds-averaged Navier–Stokes equations (RANS), i.e., SST kω, realizable kε and RNG kε, are executed and compared with the baseline serpentine channel. SST means the shear stress transport turbulence model, and RNG denotes renormalization group methods. Here, the globally averaged Nusselt number of each pass ($N u ¯ ¯$) is used to compare the heat transfer with the experimental data from Guo et al. . It can be evaluated as follows:\n$N u ¯ ¯ = Q t o t A w Δ T D h k f$\nwhere Qtot denotes the total heat transfer, Aw is the wetted area, and ΔT signifies the log mean temperature difference of the heating wall, calculated as follows:\n$Δ T = ( T w − T i n ) − ( T w − T o u t ) ln ( T w − T i n T w − T o u t )$\nwhere Tw is the average wall temperature, while Tin and Tout are the inlet and outlet fluid temperatures.\nThe globally averaged Nusselt number simulated from different turbulence models for the baseline serpentine channel at the Reynolds number of 10,000 is shown in Figure 12. It should be noted that Guo et al. have found that the SST kω turbulence model could accurately simulate the heat transfer of the three-pass channel with ribs. This is because the SST kω turbulence model captures well the reattachment flow from the ribbed walls. However, in this work, SST kω and RNG kε considerably overestimates the results in the second pass by 25.9% and 21.4%, respectively. Meanwhile, the average Nusselt number simulated using realizable kε agrees well with the experimental results for all smooth passages due to correctly predicting the large recirculation flow. The mean error deviates by about 10.7% compared to the experimental data. Hence, the realizable kε turbulence model with enhanced wall treatment is selected to simulate all models in this study.\n\n#### 4.2.2. Pressure Loss\n\nThe relative pressure coefficient (Cp) is employed to characterize the pressure loss from different models. The relative pressure coefficient along the channel flow path is calculated as follows:\n$C p = p l o c − p r e f ρ U i n 2 / 2$\nwhere ploc is the local static pressure at measuring points and pref indicates the pressure value at the channel inlet.\nThe relative pressure coefficient at three different locations at the Reynolds number of 10,000 is demonstrated in Figure 13. In the first pass, as the structure for all models is the same without changing from the baseline channel, the values are comparable because the coolant flows smoothly with minor flow resistance. The losses are higher when the coolant enters the second and third passes. Here, the baseline serpentine channel causes the highest losses, followed by the TO model with the gyroid-TPMS structure. In contrast, the TO model provides the lowest values, which are lower than the baseline model by 14.5% and 44.3% for the second and third passages.\nThe friction factor calculated to compare the total pressure loss in the channel can be expressed as follows:\n$f = 2 Δ p D h ρ U i n 2 L$\nwhere Δp is calculated from the pressure at the inlet boundary minus the pressure at the outlet boundary. L is the total length of the serpentine cooling channel.\nThe friction factor for all models at the Reynolds number of 10,000 is compared in Figure 14. All cases cause lower friction loss than the baseline channel, and the TO model provides the lowest value. It can be observed that the TO model with diamond- and TO model with gyroid-TPMS structures cause slightly higher friction loss than the TO model attributed to their complex curvatures. Compared to the baseline channel, the TO model, TO model with diamond- and TO model with gyroid-TPMS structures show a friction loss reduction of 38.0%, 30.8% and 19.0%, respectively.\n\n#### 4.2.3. Heat Transfer\n\nThe globally average Nusselt number, including two turning regions, at the Reynolds number of 10,000 is presented in Figure 15. It is evident that all topology optimization models provide higher heat transfer than the baseline channel, particularly in the TO model with the diamond-TPMS structure. This result is because the heat transfer is constrained during the optimization, and the heat transfer is further increased when interpreting the intermediate results with the TPMS structures. The globally averaged Nusselt number in the TO model, TO model with diamond- and TO model with gyroid-TPMS structures are higher than in the baseline channel by about 9.6%, 21.1% and 17.6%, respectively.\nThe total Nusselt number ($N u ¯ ¯ t o t$) presents the total heat transfer performance based on the flat base heating area. The total Nusselt number includes the heat transfer contribution from all wetted surfaces, as the total heat transfer is experienced by the turbine blade’s suction and pressure walls, and it is calculated as follows:\n$N u ¯ ¯ t o t = N u ¯ ¯ ( A w A h )$\nwhere Ah is the flat heated area.\nIt is noted that the total wetted area of the TO model, TO model with diamond- and TO model with gyroid-TPMS structures are increased by 8.5%, 15.6% and 13.3% compared to the baseline channel. The total Nusselt number at the Reynolds number of 10,000 is shown in Figure 16. The total Nusselt number is higher than the globally averaged Nusselt number, particularly in the TO model with the diamond-TPMS structure. This substantial increase is because the diamond-sheet structure has a larger contact area than the TO model curvature and the gyroid-sheet structure. The total Nusselt number for the TO model, TO model with diamond- and TO model with gyroid-TPMS structures increases by 25.3%, 45.8% and 36.6%, respectively, compared to the baseline channel.\nTo show the heat transfer enhancements in the serpentine channel, the Dittus–Boelter equation is employed to normalize the globally averaged and total Nusselt numbers, defined as follows:\n$N u ¯ ¯ 0 = 0.023 R e 0.8 P r 0.4$\nwhere Pr is the Prandtl number, and the value of Pr for the air in this study is 0.7.\nThe globally averaged and total heat transfer enhancements at the Reynolds number of 10,000 are presented in Figure 17a,b, respectively. The enhanced heat transfer is observed for all models since the serpentine channels generate high heat transfer, particularly from the first turning region. Among the comparative models, the heat transfer enhancement is more evident for the TO model with diamond-TPMS structure since this model considerably increases total heat transfer in the channel.\n\n#### 4.2.4. Thermal Performance\n\nTo compare the thermal performance of the serpentine cooling channel, the heat transfer performance requires an assessment after involving friction loss. In this study, the thermal performance of the serpentine channels can be calculated as follows:\n$T P = N u ¯ ¯ t o t / N u ¯ ¯ 0 ( f / f 0 ) 1 / 3$\nwhere f0 is the Blasius correlation, defined in Equation (30):\n$f 0 = 0.316 R e − 0.25$\nThe thermal performance for all cases at the Reynolds number 10,000 is presented in Figure 18. It is evident that the baseline serpentine channel shows the lowest thermal performance due to causing the highest pressure loss and lowest heat transfer in the channel. Meanwhile, the TO model with the diamond-TPMS structure exhibits the highest value because the optimized curvatures help minimize the pressure loss in the serpentine channel, while the diamond-sheet structure enhances high heat transfer. The thermal performance for the TO model, TO model with diamond- and TO model with gyroid-TPMS structures improves by 47.0%, 64.8% and 46.5%, respectively, compared to the baseline channel.\n\n#### 4.2.5. Temperature Uniformity\n\nThe high temperature difference between leading and trailing surfaces generated in the gas turbine channel is observed when the channel rotates at high speed [1,2]. It is essential to be concerned about the temperature uniformity that affects the temperature difference on the surface. The uniformity index of temperature (θ) is introduced in this section to determine surface temperature uniformity. It can be calculated as follows:\n$θ = 1 − ∑ i = 1 n \| T i − T a v g \| 2 T a v g A h$\nwhere Ti denotes the local temperature, and Ai is the local area. Tavg is the average temperature of the considered region, calculated as follows:\n$T a v g = 1 A p ∑ i = 1 n T i A i$\nThe uniformity index of surface temperature for all serpentine channels at the Reynolds number of 10,000 is presented in Figure 19. In the first pass, all cases exhibit comparable values, while the uniformities drastically decrease in the second pass for the baseline model. These negative results are due to the strong turning effect at the first rectangular bend, causing nonuniform flow distribution throughout the second pass . However, the uniformities are improved in the TO model, TO model with diamond- and TO model with gyroid-TPMS structures owing to the curvatures that manipulate smoother flow distributions. For all optimized models, the temperature uniformity on the surface improves by about 4.5%–5.6% in the second passage and 0.4%–2.0% in the third passage, superior to the baseline channel.\nIn short, the optimized model infilled with the diamond-sheet structure shows the best thermal performance because it provides low pressure loss and enhances the highest heat transfer. This structure also improves the temperature uniformity throughout the channel compared to the baseline model. Therefore, this study selects the TO model with diamond-TPMS structure as the optimal serpentine channel.\n\n#### 4.3. Detailed Flow and Heat Transfer Characteristics of the Optimal Serpentine Channel\n\nAccording to the analysis of the pressure loss, heat transfer and temperature uniformity in Section 4.2, the optimized serpentine channel with diamond-sheet structure is selected to be the optimal model, and the detailed flow and heat transfer characteristics are demonstrated and compared to the baseline serpentine channel.\n\n#### 4.3.1. Flow Characteristics\n\nThe velocity distributions and streamlines on the middle section (Z = 30 mm) of the serpentine channel at the Reynolds number of 10,000 are shown in Figure 20 for (a) the baseline channel and (b) the TO model with diamond-TPMS structure. It is clearly seen that for the baseline serpentine channel, the large recirculation flow is generated at the second bend region and third passage. The high velocity can be observed at the outer wall of the third passage. These phenomena cause nonuniform distribution of the flow. Meanwhile, the recirculation flow in the TO model with the diamond-TPMS structure is eliminated, as seen in Figure 20b. The curvature in this model induces the coolant to flow near the inner wall of the second passage. The flow in the diamond-sheet structure is also distributed evenly, leading to a more uniform velocity distribution throughout the channel.\nThe pressure distribution on the middle section of the channel at the Reynolds number of 10,000 is demonstrated in Figure 21 for (a) the baseline model and (b) the TO model with diamond-TPMS structure. For the baseline channel, the high-pressure loss emerges in the two bend regions, which could be attributed to the generation of the Dean’s vortices . However, after inserting the diamond-sheet structure into the optimized serpentine channel, the pressure loss is significantly reduced. It can be observed that the pressure loss is reduced, particularly in the second passage inside the diamond-sheet structure, and the area of low pressure continues to the third passage. The TO model with the diamond-TPMS structure manipulates the fluid well, reducing the recirculation flow, which lowers the total pressure loss in the serpentine channel.\nThe flow mechanisms and turning effects can be discussed with the dimensionless turbulent kinetic energy (TKE/Uin2) and streamlines on different cross-sections of the serpentine channel, as presented in Figure 22. For both models, the turbulent kinetic energy and streamlines exhibit the same characteristics in the first passage. The higher turbulent kinetic energy values can be observed in the second and third passages.\nFor the baseline model, as shown in Figure 22a, the Dean’s vortices can be observed on the outer wall when the fluid enters the second passage on plane B3. These vortices are prominent in plane B2, while the fluid mixes at the end of the second passage, resulting in high turbulent kinetic energy in the middle region. The coolant turning into the third passage further increases the turbulent kinetic energy and continues until it leaves the channel, as seen on planes C1–C3. Even though the strong mixing flow could enhance high heat transfer, this effect causes substantial differences in the flow distribution, eventually leading to non-uniform heat transfer in the channel.\nFor the TO model with the diamond-TPMS structure, as shown in Figure 22b, the Dean’s vortices at the inner wall, as found in the baseline model, are diminished because the large solid volume of the diamond-sheet structure distributes the flow in the second passage, as demonstrated on planes B3–B1. However, the high turbulent kinetic energy areas can be observed. The main coolant is forced to flow along the curvature of the diamond-sheet structure, intensifying high turbulent kinetic energy. Moreover, the complex flow generated inside the diamond-sheet structure expands large regions of high turbulence. The turbulent kinetic energy in the third passage is also more uniform than in the baseline model. These phenomena in the TO model with the diamond-TPMS structure lead to higher heat transfer and better uniformity than in the baseline channel.\n\n#### 4.3.2. Heat Transfer Characteristics\n\nThe local Nusselt number (Nu) is employed to analyze the surface heat transfer. The local Nusselt number is evaluated from the local heat transfer coefficient based on the local wall heat flux and the local temperature difference between the wall and the corresponding fluid bulk temperatures. The Nusselt number contour at the Reynolds number of 10,000 is illustrated in Figure 23 for (a) the baseline model and (b) the TO model with diamond-TPMS structure. For the baseline channel, as presented in Figure 23a, it can be clearly seen that the low Nusselt number region near the inner wall of the second passage is considerably expanded due to the large recirculation flow. However, this low heat transfer disappears in the TO model with diamond-TPMS structure, particularly in the second passage, as shown in Figure 23b. The high-value areas can also be observed inside the diamond-sheet structure because the curved interconnected walls generate impingement flow on the surface, as shown in the enlarged view. The average heat transfer at the second passage increases by 69.8% compared to the baseline model. Moreover, as the recirculation flow is eliminated due to the curvature of the diamond-sheet structure in the second and third passages of the TO model with diamond-TPMS structure, more uniform distributions of high heat transfer can be observed. The uniform heat transfer distributions are beneficial to mitigate the local thermal stress in the gas turbine blade.\n\n## 5. Conclusions\n\nIn this work, the concept of density-based topology optimization is utilized to design the serpentine cooling channel under turbulent flow. In the topology optimization, the penalization terms are added to the kε turbulence model to reveal the turbulence effects. The functions to activate/deactivate the convective and conduction terms in the energy equation are also carried out. The main objective is to minimize the pressure drop while maintaining heat transfer. The 2D simplified models are topologically optimized. The optimized solution, including the intermediate results, is then extruded into the 3D system and infilled with TPMS structures. The final models are resimulated and validated to compare with the baseline serpentine channel. The main conclusions are drawn as follows:\n• The conventional topology optimization model achieves a lower pressure loss and provides higher heat transfer than the baseline channel. Infilling the diamond- and gyroid-sheet structures in the optimal solution, including the intermediate results of the conventional topology optimization model, further enhance high heat transfer and maintains lower pressure loss than the baseline channel.\n• The 3D optimized models with the TPMS structures provide lower pressure loss by about 19.0%–30.8% and higher total heat transfer by 36.6%–45.8%, compared to the 3D baseline channel. The optimized mode infilled with the diamond-TPMS structure is selected as the optimal serpentine channel in this study since it provides the best thermal performance, up to 64.8%, superior to the baseline channel. The temperature uniformity on the surface is also improved, particularly in the second passage.\n• The 3D optimized model with the diamond-TPMS structure significantly eliminates the recirculation flow and the low heat transfer regions at the second and third passages. This model also reduces the influence of Dean’s vortices but maintains high turbulence kinetic energy, leading to better uniform flow and heat transfer distributions.\nOverall, the method in this work provides an advanced tool to manage the intermediate results in the conventional topology optimization of turbulent flow and heat transfer systems. Complex structures generated by this approach can be fabricated with additive manufacturing. Even though the present optimization models could be employed at a specific Reynolds number and heat input, this method can be applied to design cooling channels at higher Reynolds numbers and different cooling channel shapes, achieving higher thermal performance for gas turbine blades, heat sinks for high-power electronics, etc.\n\n## Author Contributions\n\nConceptualization, K.Y. and Y.R.; methodology, K.Y., L.Y. and H.L.; software, K.Y. and H.L.; validation, K.Y., Y.R. and L.Y.; formal analysis, K.Y. and Y.R.; investigation, K.Y. and H.L.; resources, Y.R.; data curation, K.Y.; writing—original draft preparation, K.Y.; writing—review and editing, K.Y., Y.R., L.Y. and H.L.; visualization, K.Y., Y.R., L.Y. and H.L.; supervision, Y.R. and L.Y.; project administration, Y.R.; funding acquisition, Y.R. All authors have read and agreed to the published version of the manuscript.\n\n## Funding\n\nThis research was funded by the National Science and Technology Major Project, grant number 2017-III-0009-0035 and the National Natural Science Foundation of China, grant numbers 11972230 and 51676119.\n\n## Conflicts of Interest\n\nThe authors declare no conflict of interest.\n\n## Nomenclature\n\n Ah Flat heated area (m2) Ai Local area (m) Aw Wetted area (m2) AR Aspect ratio of the channel C Constraint function cf Fluid heat capacity (J/(kg K)) Cp Relative pressure coefficient Cref Reference value in constraint function Da Darcy number Dh Hydraulic diameter of the channel (m) F Fictitious body-force (N/m3) f Friction factor f0 Blasius correlation g Inequality constraints Iu, Ik, Iε, Ic, IK Material interpolated functions J Objective function k, k0 Turbulent kinetic energy of the fluid and solid domains (m2/s2) kf, ks Thermal conductivity of the fluid and solid (W/(m K)) L Total length of the serpentine channel (m) n Normal vector nc, nK Penalization power coefficients in material interpolated functions $N u ¯ ¯$, $N u ¯ ¯ t o t$ Globally and total Nusselt numbers $N u ¯ ¯ 0$ Dittus–Boelter equation Pk Turbulent kinetic energy source term (W/m3) p0 Zero static pressure (Pa) ploc Local static pressure at measuring points (Pa) pref Pressure value at the channel inlet (Pa) Pr Prandtl number Qtot Total heat transfer (W) q0 Uniform heat flux (W/m2) qu, qk, qε Tuning parameters in material interpolated functions R Residual vector function Re Reynolds number Rmin Filter radius (m) S Vectors with discrete state fields TP Thermal performance of cooling channels Tavg Average temperature of the considered surface (m2) Ti Local temperature at the consider area (K) Tin, Tout Inlet and outlet temperatures (K) Tw Average wall temperature (K) u Fluid velocity vector Uin Inlet velocity (m/s) x Design variable vector y+ Nondimensional distance from the endwall to the first element node Greek Letters αu,max, αu,min Maximum and minimum inverse permeabilities of the porous medium (Pa s/m) β Projection slope γ Design variable γf, γp Smoothed and projected design variables ε, ε0 Turbulent energy dissipation for fluid and solid regions (m2/s3) μ Dynamic viscosity (N s/m2) μT Turbulent dynamic viscosity (N s/m2) ρ Fluid density (kg/m3) η Projection threshold θ Uniformity index of temperature ω Specific rate of dissipation in the k–ω turbulence model (1/s) Γin, Γout Inlet and outlet boundary conditions Γwall No-slip boundary conditions Ωid, Ωod Inside and outside design domains\n\n## References\n\n1. 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Methods Eng. 2011, 86, 765–781. [Google Scholar] [CrossRef]\n36. Celik, I.B.; Ghia, U.; Roache, P.J.; Freitas, C.J.; Coleman, H.; Raad, P.E. Procedure for Estimation and Reporting of Uncertainty Due to Discretization in CFD Applications. J. Fluids Eng. Trans. ASME 2008, 130, 0780011–0780014. [Google Scholar] [CrossRef]\n37. Winterton, R.H.S. Where Did the Dittus and Boelter Equation Come From? Int. J. Heat Mass Transf. 1998, 41, 809–810. [Google Scholar] [CrossRef]\nFigure 1. Numerical model of a three-short-pass serpentine channel: (a) full 3D model; (b) 2D simplified design from the 3D model.\nFigure 1. Numerical model of a three-short-pass serpentine channel: (a) full 3D model; (b) 2D simplified design from the 3D model.\nFigure 2. Overview of the procedures in this study.\nFigure 2. Overview of the procedures in this study.\nFigure 3. Flow chart of the topology optimization process.\nFigure 3. Flow chart of the topology optimization process.\nFigure 4. A single unit cell of (a) diamond-sheet structure; (b) gyroid-sheet structure for fulfilling the optimal results in this study.\nFigure 4. A single unit cell of (a) diamond-sheet structure; (b) gyroid-sheet structure for fulfilling the optimal results in this study.\nFigure 5. (a) The mesh system for 2D model validation; (b) mesh system in the 3D model.\nFigure 5. (a) The mesh system for 2D model validation; (b) mesh system in the 3D model.\nFigure 6. Comparison of the average values of the state variables between 2D and 3D at Re = 10,000: (a) pressure values; (b) velocity magnitudes; (c) temperature values.\nFigure 6. Comparison of the average values of the state variables between 2D and 3D at Re = 10,000: (a) pressure values; (b) velocity magnitudes; (c) temperature values.\nFigure 7. The optimized 2D model with intermediate results.\nFigure 7. The optimized 2D model with intermediate results.\nFigure 8. (a) 3D optimal solution including intermediate results; (b) topology optimization model infilled with lattice structures.\nFigure 8. (a) 3D optimal solution including intermediate results; (b) topology optimization model infilled with lattice structures.\nFigure 9. Models for comparison: (a) optimized model extracted with density filtering of 0.5 (TO model); (b) optimized model infilled with diamond-sheet structure (TO model with diamond-TPMS); (c) optimized model infilled with diamond-sheet structure (TO model with gyroid-TPMS).\nFigure 9. Models for comparison: (a) optimized model extracted with density filtering of 0.5 (TO model); (b) optimized model infilled with diamond-sheet structure (TO model with diamond-TPMS); (c) optimized model infilled with diamond-sheet structure (TO model with gyroid-TPMS).\nFigure 10. An example of mesh systems for the 3D models.\nFigure 10. An example of mesh systems for the 3D models.\nFigure 11. The spanwise-averaged temperature and grid convergence index on the surface of the conventional serpentine channel at Re = 10,000.\nFigure 11. The spanwise-averaged temperature and grid convergence index on the surface of the conventional serpentine channel at Re = 10,000.\nFigure 12. Comparisons of globally averaged Nusselt number in each pass for different turbulence models at Re = 10,000.\nFigure 12. Comparisons of globally averaged Nusselt number in each pass for different turbulence models at Re = 10,000.\nFigure 13. Comparisons of relative pressure coefficient for all serpentine channels at Re = 10,000.\nFigure 13. Comparisons of relative pressure coefficient for all serpentine channels at Re = 10,000.\nFigure 14. Friction factors for all serpentine channels at Re = 10000.\nFigure 14. Friction factors for all serpentine channels at Re = 10000.\nFigure 15. Globally averaged Nusselt numbers for all serpentine channels at Re = 10,000.\nFigure 15. Globally averaged Nusselt numbers for all serpentine channels at Re = 10,000.\nFigure 16. Total Nusselt numbers for all serpentine channels at Re = 10,000.\nFigure 16. Total Nusselt numbers for all serpentine channels at Re = 10,000.\nFigure 17. Heat transfer enhancements for all serpentine models at Re = 10,000: (a) globally averaged Nusselt number ratio; (b) total Nusselt number ratio.\nFigure 17. Heat transfer enhancements for all serpentine models at Re = 10,000: (a) globally averaged Nusselt number ratio; (b) total Nusselt number ratio.\nFigure 18. Thermal performances for all serpentine models at Re = 10,000.\nFigure 18. Thermal performances for all serpentine models at Re = 10,000.\nFigure 19. Temperature uniformity on the surface for all serpentine models at Re = 10,000.\nFigure 19. Temperature uniformity on the surface for all serpentine models at Re = 10,000.\nFigure 20. Velocity distributions and streamlines in the middle section of the serpentine channel: (a) baseline model; (b) TO model with diamond-TPMS structure.\nFigure 20. Velocity distributions and streamlines in the middle section of the serpentine channel: (a) baseline model; (b) TO model with diamond-TPMS structure.\nFigure 21. Pressure distributions in the middle section of the serpentine channel: (a) baseline model; (b) TO model with diamond-TPMS structure.\nFigure 21. Pressure distributions in the middle section of the serpentine channel: (a) baseline model; (b) TO model with diamond-TPMS structure.\nFigure 22. Local dimensionless turbulent kinetic energy and velocity streamline on the cross-sections in the multipass channel: (a) baseline model; (b) TO model with diamond-TPMS structure.\nFigure 22. Local dimensionless turbulent kinetic energy and velocity streamline on the cross-sections in the multipass channel: (a) baseline model; (b) TO model with diamond-TPMS structure.\nFigure 23. Nusselt number contours of the serpentine channel: (a) baseline model; (b) TO model with diamond-TPMS structure.\nFigure 23. Nusselt number contours of the serpentine channel: (a) baseline model; (b) TO model with diamond-TPMS structure.\nTable 1. An overview of density-based topology optimization in recent studies on thermofluid systems.\nTable 1. An overview of density-based topology optimization in recent studies on thermofluid systems.\nAuthorFlow RegimeStructureDiscretizationOptimizer\nAlexandersen et al. Laminar2D and 3DFEM 1MMA 3\nDilgen et al. Turbulent2D and 3DFVM 2 MMA\nHaertel et al. Laminar2DFEMGCMMA 4\nLi et al. [20,21]Laminar2DFEMSNOPT 5\nZhang et al. Laminar2DFEMGCMMA\n1 finite element method. 2 finite volume method. 3 method of moving asymptotes. 4 globally convergent method of moving asymptotes. 5 sparse nonlinear optimizer.\nTable 2. Parameters used in the 2D topology optimization process.\nTable 2. Parameters used in the 2D topology optimization process.\nParameterDaβηquqkqεncnK\nValue10−4–10−61.5–16.00.50.1–100.0010.13–53–5\n Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.\n\n## Share and Cite\n\nMDPI and ACS Style\n\nYeranee, K.; Rao, Y.; Yang, L.; Li, H. Improved Thermal Performance of a Serpentine Cooling Channel by Topology Optimization Infilled with Triply Periodic Minimal Surfaces. Energies 2022, 15, 8924. https://doi.org/10.3390/en15238924\n\nAMA Style\n\nYeranee K, Rao Y, Yang L, Li H. Improved Thermal Performance of a Serpentine Cooling Channel by Topology Optimization Infilled with Triply Periodic Minimal Surfaces. Energies. 2022; 15(23):8924. https://doi.org/10.3390/en15238924\n\nChicago/Turabian Style\n\nYeranee, Kirttayoth, Yu Rao, Li Yang, and Hao Li. 2022. \"Improved Thermal Performance of a Serpentine Cooling Channel by Topology Optimization Infilled with Triply Periodic Minimal Surfaces\" Energies 15, no. 23: 8924. https://doi.org/10.3390/en15238924\n\nNote that from the first issue of 2016, this journal uses article numbers instead of page numbers. 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https://metanumbers.com/588552	[ "## 588552\n\n588,552 (five hundred eighty-eight thousand five hundred fifty-two) is an even six-digits composite number following 588551 and preceding 588553. In scientific notation, it is written as 5.88552 × 105. The sum of its digits is 33. It has a total of 6 prime factors and 32 positive divisors. There are 193,664 positive integers (up to 588552) that are relatively prime to 588552.\n\n## Basic properties\n\n• Is Prime? No\n• Number parity Even\n• Number length 6\n• Sum of Digits 33\n• Digital Root 6\n\n## Name\n\nShort name 588 thousand 552 five hundred eighty-eight thousand five hundred fifty-two\n\n## Notation\n\nScientific notation 5.88552 × 105 588.552 × 103\n\n## Prime Factorization of 588552\n\nPrime Factorization 23 × 3 × 137 × 179\n\nComposite number\nDistinct Factors Total Factors Radical ω(n) 4 Total number of distinct prime factors Ω(n) 6 Total number of prime factors rad(n) 147138 Product of the distinct prime numbers λ(n) 1 Returns the parity of Ω(n), such that λ(n) = (-1)Ω(n) μ(n) 0 Returns: 1, if n has an even number of prime factors (and is square free) −1, if n has an odd number of prime factors (and is square free) 0, if n has a squared prime factor Λ(n) 0 Returns log(p) if n is a power pk of any prime p (for any k >= 1), else returns 0\n\nThe prime factorization of 588,552 is 23 × 3 × 137 × 179. Since it has a total of 6 prime factors, 588,552 is a composite number.\n\n## Divisors of 588552\n\n32 divisors\n\n Even divisors 24 8 4 4\nTotal Divisors Sum of Divisors Aliquot Sum τ(n) 32 Total number of the positive divisors of n σ(n) 1.4904e+06 Sum of all the positive divisors of n s(n) 901848 Sum of the proper positive divisors of n A(n) 46575 Returns the sum of divisors (σ(n)) divided by the total number of divisors (τ(n)) G(n) 767.171 Returns the nth root of the product of n divisors H(n) 12.6367 Returns the total number of divisors (τ(n)) divided by the sum of the reciprocal of each divisors\n\nThe number 588,552 can be divided by 32 positive divisors (out of which 24 are even, and 8 are odd). The sum of these divisors (counting 588,552) is 1,490,400, the average is 46,575.\n\n## Other Arithmetic Functions (n = 588552)\n\n1 φ(n) n\nEuler Totient Carmichael Lambda Prime Pi φ(n) 193664 Total number of positive integers not greater than n that are coprime to n λ(n) 12104 Smallest positive number such that aλ(n) ≡ 1 (mod n) for all a coprime to n π(n) ≈ 48113 Total number of primes less than or equal to n r2(n) 0 The number of ways n can be represented as the sum of 2 squares\n\nThere are 193,664 positive integers (less than 588,552) that are coprime with 588,552. And there are approximately 48,113 prime numbers less than or equal to 588,552.\n\n## Divisibility of 588552\n\n m n mod m 2 3 4 5 6 7 8 9 0 0 0 2 0 6 0 6\n\nThe number 588,552 is divisible by 2, 3, 4, 6 and 8.\n\n• Arithmetic\n• Abundant\n\n• Polite\n\n## Base conversion (588552)\n\nBase System Value\n2 Binary 10001111101100001000\n3 Ternary 1002220100020\n4 Quaternary 2033230020\n5 Quinary 122313202\n6 Senary 20340440\n8 Octal 2175410\n10 Decimal 588552\n12 Duodecimal 244720\n16 Hexadecimal 8fb08\n20 Vigesimal 3db7c\n36 Base36 cm4o\n\n## Basic calculations (n = 588552)\n\n### Multiplication\n\nn×i\n n×2 1177104 1765656 2354208 2942760\n\n### Division\n\nni\n n⁄2 294276 196184 147138 117710\n\n### Exponentiation\n\nni\n n2 346393456704 203870561730052608 119988426847345922543616 70619428597859137404890284032\n\n### Nth Root\n\ni√n\n 2√n 767.171 83.8034 27.6979 14.2547\n\n## 588552 as geometric shapes\n\n### Circle\n\nRadius = n\n Diameter 1.1771e+06 3.69798e+06 1.08823e+12\n\n### Sphere\n\nRadius = n\n Volume 8.53971e+17 4.35291e+12 3.69798e+06\n\n### Square\n\nLength = n\n Perimeter 2.35421e+06 3.46393e+11 832338\n\n### Cube\n\nLength = n\n Surface area 2.07836e+12 2.03871e+17 1.0194e+06\n\n### Equilateral Triangle\n\nLength = n\n Perimeter 1.76566e+06 1.49993e+11 509701\n\n### Triangular Pyramid\n\nLength = n\n Surface area 5.99971e+11 2.40264e+16 480551\n\n## Cryptographic Hash Functions\n\nmd5 c271ddb95bc465647b080d8d1843b6c8 39731d286980d283d4d0a3a6626d3eaac256bed5 11da15afd31cd0ea89a6cac7dc253ea7722aa296ae2ca3859a9be642d435f9af 0df2137ad95f8b16a715d3c6228f59253dcd275516bd69cee1ad4c67cc93c5a70259a30a495413fb7bf645590f3caf7b415177ea2a5307a1194da97ddd424f42 f74609035e1559fd433ff13c631ade2f1a451c37" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.63130075,"math_prob":0.9862669,"size":4635,"snap":"2021-21-2021-25","text_gpt3_token_len":1624,"char_repetition_ratio":0.11876485,"word_repetition_ratio":0.033923306,"special_character_ratio":0.46752965,"punctuation_ratio":0.08025478,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9958108,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-16T07:21:37Z\",\"WARC-Record-ID\":\"<urn:uuid:2209f420-f71c-4cfb-a580-28c030e18295>\",\"Content-Length\":\"48850\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:18020814-704d-44af-ac37-21c323c36629>\",\"WARC-Concurrent-To\":\"<urn:uuid:f5416a25-8dfa-4616-8795-2542989a3da4>\",\"WARC-IP-Address\":\"46.105.53.190\",\"WARC-Target-URI\":\"https://metanumbers.com/588552\",\"WARC-Payload-Digest\":\"sha1:I7WB3CC77K4RI6QC26RWPRKGND4CSMUG\",\"WARC-Block-Digest\":\"sha1:AECQ2XTCKOFILNXGOPEMVVA3PCJL5VCL\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623487622234.42_warc_CC-MAIN-20210616063154-20210616093154-00388.warc.gz\"}"}
https://brandonrozek.com/ta/fall2017/cpsc220/dec6/	[ "", null, "", null, "# Final Review December 6\n\n## Classes\n\nHere is how you can create a class called “Employee” with a non-default constructor (a constructor that takes parameters) and a getter and setter\n\n``````public class Employee {\n// Our private variables\nprivate String name;\nprivate double salary;\n// Non-default constructor\npublic Employee(String name, double salary) {\nthis.name = name;\nthis.salarly = salary;\n}\n// This is a getter\npublic string getName() {\nreturn name;\n}\npublic double setSalarly(double salary) {\nthis.salary = salary;\n}\n}\n``````\n\n## For Loops + Arrays\n\nFor loops are constructed in the following way\n\n`for (initialization; condition to stop; increment to get closer to condition to stop)`\n\n``````//Assume an array with variable name array is declared before\nfor (int i = 0; i < array.length; i++) {\n// This code will loop through every entry in the array\n}\n``````\n\nNote that you don’t always have to start from zero, you can start anywhere from the array.\n\n## For Loops + Arrays + Methods\n\nThis is an example of how you can take in an array in a method\n\n``````public static boolean isEven(int[] array) { // <-- Note the int[] array\nfor (int i = 0; i < array.length; i++) { // Iterate through the entire array\n// If you find an odd number, return false\nif (array[i] % 2 == 1) {\nreturn false;\n}\n}\n// If you didn't find any odd numbers, return true\nreturn true;\n}\n``````\n\n## File I/O\n\nLet’s say that you have the following file\n\n``````4\nchicken\n3\n``````\n\nAnd you want to make it so that you take the number, and print the word after it a certain number of times. For this example we would want to see the following\n\n``````chicken chicken chicken chicken\n``````\n\nHere is the code to write it\n\n``````public static void printStrings() {\nFileInputStream file = new FileInputStream(\"stuff.txt\"); // File contents are in stuff.txt\nScanner scnr = new Scanner(file); // Scanner takes in a file to read\nwhile (scnr.hasNext()) { // While there are still stuff in the file to read\nint number = scnr.nextInt(); // Grab the number\nString word = scnr.next(); // Grab the word after the number\n// Print the word number times\nfor (int i = 0; i < number; i++) {\nSystem.out.print(word);\n}\n// Put a new line here\nSystem.out.println();\n}\n\n}\n``````\n\n## Recursion\n\nLook at handout and carefully trace recursion problems\n\n## 2D Arrays\n\nDeclare a 2D int array with 4 rows and 7 columns\n\n``````int[][] dataVals = new int;\n``````\n\nA 2D array with 4 rows and 7 columns has 7 * 4 = 28 entries.\n\nIf you want to sum up all the numbers in a 2 dimensional array, do the following\n\n``````// Assume numbers is declared beforehand\nint sum = 0;\nfor (int i = 0; i < numbers.length; i++) { // Loop through every row in the 2D array\nfor (int j = 0; j < numbers[i].length; j++) { // Loop through every column in a row\n// This code now looks at one entry in a 2D array\nsum += numbers[i][j];\n}\n}\n``````" ]	[ null, "https://brandonrozek.com//img/avatar.jpg", null, "https://brandonrozek.com//img/avatar-border.svg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7296745,"math_prob":0.8387951,"size":2742,"snap":"2021-04-2021-17","text_gpt3_token_len":709,"char_repetition_ratio":0.10518627,"word_repetition_ratio":0.055226825,"special_character_ratio":0.28118163,"punctuation_ratio":0.10285714,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98945916,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-01-17T08:26:40Z\",\"WARC-Record-ID\":\"<urn:uuid:3a4ff6ef-acd6-404b-b2ff-b8881dadf282>\",\"Content-Length\":\"17419\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c9260918-ecb4-4b49-96d6-780ddc192fe9>\",\"WARC-Concurrent-To\":\"<urn:uuid:1b96b55a-54e7-4870-b5d9-942881d2641e>\",\"WARC-IP-Address\":\"159.203.122.95\",\"WARC-Target-URI\":\"https://brandonrozek.com/ta/fall2017/cpsc220/dec6/\",\"WARC-Payload-Digest\":\"sha1:UI47RS4PUAGAU6YIMXNQVMXEIEXS7BKX\",\"WARC-Block-Digest\":\"sha1:NFRFPKHLR2ZLM7II5LP2TAUDN32PH2OY\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-04/CC-MAIN-2021-04_segments_1610703511903.11_warc_CC-MAIN-20210117081748-20210117111748-00290.warc.gz\"}"}
https://socratic.org/questions/your-piggy-bank-has-25-coins-in-it-some-are-quarters-and-some-are-nickels-you-ha	[ "# Your piggy bank has 25 coins in it; some are quarters and some are nickels. You have $3.45. How many nickels do you have? ##### 1 Answer Mar 4, 2018 See a solution process below: #### Explanation: First, let's call the number of quarters you have: $q$And, the number of nickels you have: $n$Using these variables and the information in the problem we can write two equations: • Equation 1: $q + n = 25$• Equation 2: $0.25q + $0.05n =$3.45\n\nStep 1) Solve the first equation for $q$:\n\n$q + n = 25$\n\n$q + n - \\textcolor{red}{n} = 25 - \\textcolor{red}{n}$\n\n$q + 0 = 25 - n$\n\n$q = 25 - n$\n\nStep 2) Substitute $\\left(25 - n\\right)$ for $q$ in the second equation and solve for $n$ to find the number of nickels you have:\n\n$0.25q +$0.05n = $3.45 becomes: $0.25(25 - n) + $0.05n =$3.45\n\n($0.25 xx 25) - ($0.25 xx n) + 0.05n = $3.45 $6.25 - $0.25n + 0.05n =$3.45\n\n$6.25 + (-$0.25 + 0.05)n = $3.45 $6.25 + (-$0.20)n =$3.45\n\n$6.25 -$0.20n = $3.45 $6.25 - color(red)($6.25) -$0.20n = $3.45 - color(red)($6.25)\n\n0 - $0.20n = -$2.80\n\n-$0.20n = -$2.80\n\n(-$0.20n)/(color(red)(-)color(red)($)color(red)(0.20)) = (-$2.80)/(color(red)(-)color(red)($)color(red)(0.20))\n\n(color(red)(cancel(color(black)(-$0.20)))n)/cancel(color(red)(-)color(red)($)color(red)(0.20)) = (color(red)(cancel(color(black)(-$)))2.80)/(cancel(color(red)(-)color(red)($))color(red)(0.20))\n\n$n = \\frac{2.80}{\\textcolor{red}{0.20}}$\n\n$n = 14$\n\nYou would have 14 nickels" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.70101273,"math_prob":1.0000085,"size":1142,"snap":"2019-51-2020-05","text_gpt3_token_len":484,"char_repetition_ratio":0.228471,"word_repetition_ratio":0.035714287,"special_character_ratio":0.5595447,"punctuation_ratio":0.18214285,"nsfw_num_words":2,"has_unicode_error":false,"math_prob_llama3":0.9999993,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-01-20T11:26:36Z\",\"WARC-Record-ID\":\"<urn:uuid:6037f922-273f-45b1-ac66-00fffd39d7db>\",\"Content-Length\":\"37377\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:6df4de06-4f6e-4eb0-a2fa-7fbf2258173c>\",\"WARC-Concurrent-To\":\"<urn:uuid:73433790-06d3-4d54-bfe6-f52c0ef40fc7>\",\"WARC-IP-Address\":\"54.221.217.175\",\"WARC-Target-URI\":\"https://socratic.org/questions/your-piggy-bank-has-25-coins-in-it-some-are-quarters-and-some-are-nickels-you-ha\",\"WARC-Payload-Digest\":\"sha1:PDN7VKTEGWH5X4TK6IRBA6N2GRVBYBDZ\",\"WARC-Block-Digest\":\"sha1:YWJBHEVL4DEZXCRVKHGY7UNK2T7IBFOD\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-05/CC-MAIN-2020-05_segments_1579250598726.39_warc_CC-MAIN-20200120110422-20200120134422-00276.warc.gz\"}"}
https://homework.cpm.org/category/ACC/textbook/gb8i/chapter/4%20Unit%205/lesson/INT1:%204.2.1/problem/4-56	[ "", null, "", null, "### Home > GB8I > Chapter 4 Unit 5 > Lesson INT1: 4.2.1 > Problem4-56\n\n4-56.\n\nIn problem 4-25 you looked at the data for a study conducted on a vitamin supplement that claims to shorten the length of the common cold. The data is shown in the table below: Homework Help ✎\n\n Number of months taking supplement $0.5$ $2.5$ $1$ $2$ $0.5$ $1$ $2$ $1$ $1.5$ $2.5$ Number of days cold lasted $4.5$ $1.6$ $3$ $1.8$ $5$ $4.2$ $2.4$ $3.6$ $3.3$ $1.4$\n1.", null, "You previously created a linear model for this data by sketching a line of best fit. Now create a model that is consistent with your classmates by determining the LSRL. Sketch the graph and the LSRL.\n\nReview the LSRL procedure in the previous lesson.\n\n2. Draw the upper and lower boundary lines on the graph following the process you used in problem 4-24. What is the equation of the upper boundary line? Of the lower boundary line?\n\nAnswer: $y = 6.16 − 1.58x$ and $y = 4.58 − 1.58x$, based on a maximum residual of $−0.79$\n\n3. Based on the upper and lower boundary lines of your model, what do you predict is the length of a cold for a person who has taken the supplement for three months? Consider the precision of the data and use an appropriate number of decimal places in your response.\n\nSince the number of days the cold lasted is decreasing as the number of months taking the supplement increases,\nit would seem the length of the cold should be between $1$ and $0.5$ days.\n\n4. How long do you predict a cold will last for a person who has taken no supplement? Interpret the y-intercept in context.\n\nSince the $y$-intercept is around $5.3$, the number of days a cold should last for a person who hasn't taken the supplement should be around $5.3$ days.\n\n5. How long do you predict a cold will last for a person who has taken six months of supplements?\n\nNegative values in this case do not make sense. Statistical models often cannot be extrapolated far beyond the edges of the data.\n\n6. If you have a cold, would you prefer a negative or positive residual?\n\n• If you have a cold, your goal would be to decrease the number of days the cold lasts. Would a positive or negative residual represent a decrease in the number of days that a cold lasts?" ]	[ null, "https://homework.cpm.org/dist/7d633b3a30200de4995665c02bdda1b8.png", null, 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https://controlentertainmentonline.com/ans/1-cos-x-identity.html	[ "# 1 Cos X Identity\n\n1 Cos X Identity. Because the two sides have been shown to be equivalent, the equation is an identity. The angle in the one plus cos double angle trigonometric identity can be represented by any symbol but it is popularly written in two different forms. 1 + cos ( 2 x) = 2 cos 2 x. 1 + cos ( 2 a) = 2 cos 2 a.", null, "Ze \|á¨dâåhz\\$`x. b7su%ðxü¡_ âàþ˜õæg ù´ øùœ vóæ~i© ¯a jb—€2+ bb/ü•bøp+& rx·8„© h:ê îb÷q‰‚{â. °ä¤·ª ! Ƒ¾¸ù© îy0ƒ üuõ µ \|è u ! `†æ. —ã½½òâ¸uoñ w¦mæüêýk úî h6’‡vn ë9 +mcxž±z‰“n þ¸i 4bq i. Learn how to solve trigonometric identities problems step by step online. The least common multiple (lcm) of a sum of algebraic fractions consists of the product of the common factors with the greatest exponent, and the uncommon. An easy way to show that this is not an identity is to plug in 0 for x. In fact, if we use the identity sin(2x) = 2sin(x)cos(x) we can show that there are no real values for which the questioned equality is true. Tan(x y) = (tan x tan y) / (1 tan x tan y). Tan(2x) = 2 tan(x) / (1.", null, "source: www.chegg.com Solved: Verify The Identity. Tan(x)/1- Cos(x) = Csc (x) (1... \| Chegg.com. Because the two sides have been shown to be equivalent, the equation is an identity.", null, "source: socratic.org How do you verify the identity (sin2x)/(sinx) = 2/(secx)? \| Socratic. The angle in the one plus cos double angle trigonometric identity can be represented by any symbol but it is popularly.", null, "source: socratic.org How do you verify (sin x/ (1 - cos x)) + ((1 - cos x)/ sin x) = 2csc x. 1 + cos ( 2 x) = 2 cos 2 x.", null, "source: socratic.org How do you verify this identity: (cosx)/(1+sinx) + (1+sinx)/(cosx. 1 + cos ( 2 a) = 2 cos 2 a.", null, "source: www.youtube.com Establishing a trig identity involving sin x and 1 - cos x - YouTube. Thus, the one plus cosine of double angle rule can be.", null, "source: socratic.org How do you verify the identity: (1-(cosx-sinx)^(2))/cosx=2sinx? \| Socratic. Ze \|á¨dâåhz\\$`x. b7su%ðxü¡_ âàþ˜õæg ù´ øùœ vóæ~i© ¯a jb—€2+ bb/ü•bøp+& rx·8„© h:ê îb÷q‰‚{â. °ä¤·ª !", null, "source: boingboing.net This trig problem kept me up too late last night / Boing Boing. Ƒ¾¸ù© îy0ƒ üuõ µ \|è u !", null, "source: socratic.org How do you simplify the identify sin^2x/(1-cosx) = 1+cosx? \| Socratic. `†æ. —ã½½òâ¸uoñ w¦mæüêýk úî h6’‡vn ë9 +mcxž±z‰“n þ¸i 4bq i.", null, "source: www.slideshare.net Unit 5.2. Learn how to solve trigonometric identities problems step by step online.", null, "source: ar.howtodoiteasy.com إذا كانت Sin x - cos x = 1/3 ، فما هي sin 3x + cos 3x؟ 2021. The least common multiple (lcm) of a sum of algebraic fractions consists of the product of the common factors with the.", null, "source: www.bartleby.com Answered: 1-cos2x/1+sinx=sinx. Prove the identity \| bartleby. An easy way to show that this is not an identity is to plug in 0 for x.", null, "source: sharedocnow.blogspot.com 1 Cos X 2 - sharedoc. In fact, if we use the identity sin(2x) = 2sin(x)cos(x) we can show that there are no real values for which the.", null, "source: socratic.org How do you verify the identity (1+ sinx)/(cos x) + (cos x)/(1+sin x. Tan(x y) = (tan x tan y) / (1 tan x tan y).", null, "source: socratic.org How do you verify the identity cos t/(1-sin t) = sec t+tan t? \| Socratic. Tan(2x) = 2 tan(x) / (1.", null, "source: www.chegg.com Solved: 1-8 Verify Each Identity 1. Tan ? Sin ? Cos ? = Se... \| Chegg.com. By using this website, you agree to our cookie policy.", null, "source: www.youtube.com Trigonometric Identity (1 + sin x)/(1 - sin x) = (sec x + tan x)^2 CBSE. Trigonometric identities are useful whenever trigonometric functions are involved in an expression or an equation.", null, "source: socratic.org How do you verify sin^2(x) = (1/2)(1-cos2x)? \| Socratic. Trigonometric identities are true for every value of variables occurring on both sides of an equation.", null, "source: socratic.org How do you prove 2sin x + sin 2 x = (2 sin^3 x) / (1 – cos x)? \| Socratic. Geometrically, these identities involve certain trigonometric functions (such as sine, cosine, tangent) of one or more.", null, "source: www.youtube.com tan x + cot x = 1/(sin x cos x) verify the trig identity - YouTube. Sine, cosine and tangent are.", null, "source: www.bartleby.com Answered: tan2x(1+cos2x)=1-cos2x. Verify the… \| bartleby. Is it is a open ended equation identity." ]	[ null, "https://useruploads.socratic.org/wHYm142SQyOFK0hbZhXK_mat1.jpg>", null, "https://tse4.mm.bing.net/th", null, "https://tse3.mm.bing.net/th", null, "https://tse2.mm.bing.net/th", null, "https://tse3.mm.bing.net/th", null, "https://tse3.mm.bing.net/th", null, "https://tse2.mm.bing.net/th", null, "https://tse2.mm.bing.net/th", null, "https://tse2.mm.bing.net/th", null, "https://tse1.mm.bing.net/th", null, "https://tse4.mm.bing.net/th", null, "https://tse4.mm.bing.net/th", null, "https://tse4.mm.bing.net/th", null, "https://tse4.mm.bing.net/th", null, "https://tse1.mm.bing.net/th", null, "https://tse4.mm.bing.net/th", null, "https://tse4.mm.bing.net/th", null, "https://tse2.mm.bing.net/th", null, "https://tse4.mm.bing.net/th", null, "https://tse2.mm.bing.net/th", null, "https://tse1.mm.bing.net/th", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7053314,"math_prob":0.9932627,"size":4153,"snap":"2022-40-2023-06","text_gpt3_token_len":1476,"char_repetition_ratio":0.14678235,"word_repetition_ratio":0.38731444,"special_character_ratio":0.3159162,"punctuation_ratio":0.15762004,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99761325,"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,1,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,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-09-28T12:03:14Z\",\"WARC-Record-ID\":\"<urn:uuid:34eb6b72-337d-4c01-b567-e04f05c47dd9>\",\"Content-Length\":\"41547\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7877c97b-9330-4792-8ac1-025fdd2939cd>\",\"WARC-Concurrent-To\":\"<urn:uuid:7e6ec118-27f5-4dce-bcd8-752d8f78723b>\",\"WARC-IP-Address\":\"172.67.206.153\",\"WARC-Target-URI\":\"https://controlentertainmentonline.com/ans/1-cos-x-identity.html\",\"WARC-Payload-Digest\":\"sha1:S73GJFYTLMCUHCWLZUVP4LFRJDMHKNZB\",\"WARC-Block-Digest\":\"sha1:BDFET47YJTLGCLF6EFLFSOCZS2R4LT7R\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030335254.72_warc_CC-MAIN-20220928113848-20220928143848-00205.warc.gz\"}"}
https://nylogic.github.io/set-theory-seminar/2020/08/28/normal-ultrapowers-with-many-sets-of-ordinals.html	[ "August 28\nMiha Habič, Bard College at Simon’s Rock\nNormal ultrapowers with many sets of ordinals\n\nAny ultrapower $M$ of the universe by a normal measure on a cardinal $\\kappa$ is quite far from $V$ in the sense that it computes $V_{\\kappa+2}$ incorrectly. If GCH holds, this amounts to saying that $M$ is missing a subset of $\\kappa^+$. Steel asked whether, even in the absence of GCH, normal ultrapowers at $\\kappa$ must miss a subset of $\\kappa^+$. In the early 90s Cummings gave a negative answer, building a model with a normal measure on $\\kappa$ whose ultrapower captures the entire powerset of $\\kappa^+$. I will present some joint work with Radek Honzík in which we improved Cummings’ result to get this capturing property to hold at the least measurable cardinal.\n\nVideo" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9020492,"math_prob":0.98904735,"size":777,"snap":"2020-34-2020-40","text_gpt3_token_len":205,"char_repetition_ratio":0.12807244,"word_repetition_ratio":0.0,"special_character_ratio":0.23680824,"punctuation_ratio":0.06944445,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9855111,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-09-26T08:46:45Z\",\"WARC-Record-ID\":\"<urn:uuid:be0a1359-577b-44c0-b007-db60ec311368>\",\"Content-Length\":\"5041\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:20bd3dd3-ca61-4f23-9039-b080bc4b60b8>\",\"WARC-Concurrent-To\":\"<urn:uuid:5721403e-eaaa-41cd-ab4f-5cd8a6ecdb3e>\",\"WARC-IP-Address\":\"185.199.109.153\",\"WARC-Target-URI\":\"https://nylogic.github.io/set-theory-seminar/2020/08/28/normal-ultrapowers-with-many-sets-of-ordinals.html\",\"WARC-Payload-Digest\":\"sha1:QKBMD2NDK2RZB6ZVXK3SLLRFSHPEYD3B\",\"WARC-Block-Digest\":\"sha1:XDO2KAGYTHWJJ7LVJYCOSBLKQQJ25XD5\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600400238038.76_warc_CC-MAIN-20200926071311-20200926101311-00288.warc.gz\"}"}
https://www.statmodel.com/HTML_UG/chapter15v8.htm	[ "CHAPTER 15\n\nTITLE, DATA, VARIABLE, AND DEFINE COMMANDS\n\nIn this chapter, the TITLE, DATA, VARIABLE, and DEFINE commands are discussed. The TITLE command is used to provide a title for the analysis. The DATA command is used to provide information about the data set to be analyzed. The VARIABLE command is used to provide information about the variables in the data set to be analyzed. The DEFINE command is used to transform existing variables and create new variables.\n\nThe TITLE command is used to provide a title for the analysis. Following is the general format for the TITLE command:\n\n TITLE: title for the analysis\n\nThe TITLE command is not a required command. Note that commands can be shortened to four or more letters.\n\nThe TITLE command can contain any letters and symbols except the words used as Mplus commands when they are followed by a colon. These words are: title, data, variable, define, analysis, model, output, savedata, montecarlo, and plot. These words can be included in the title if they are not followed by a colon. Colons can be used in the title as long as they do not follow words that are used as Mplus commands. Following is an example of how to specify a title:\n\nTITLE: confirmatory factor analysis of diagnostic criteria\n\nThe title is printed in the output just before the Summary of Analysis.\n\nThe DATA command is used to provide information about the data set to be analyzed. The DATA command has options for specifying the location of the data set to be analyzed, describing the format and type of data in the data set, specifying the number of observations and number of groups in the data set if the data are in summary form such as a correlation or covariance matrix, requesting listwise deletion of observations with missing data, and specifying whether the data should be checked for variances of zero.\n\nData must be numeric except for certain missing value flags and must reside in an external ASCII file. There is no limit on the number of variables or observations. The maximum record length is 10,000. Special features of the DATA command for multiple group analysis are discussed in Chapter 14. Monte Carlo data generation is discussed in Chapters 12 and 19. The estimator chosen for an analysis determines the type of data required for the analysis. Some estimators require a data set with information for each observation. Some estimators require only summary information.\n\nThere are six DATA transformation commands. They are used to rearrange data from a wide to long format, to rearrange data from a long to wide format, to create a binary and a continuous variable from a semicontinuous variable, to create a set of binary variables that are indicators of missing data, to create variables for discrete-time survival modeling, and to rearrange longitudinal data from a format where time points represent measurement occasions to a format where time points represent age or another time-related variable.\n\nFollowing are the options for the DATA and the DATA transformation commands:\n\n DATA: FILE IS file name; FORMAT IS format statement; FREE FREE; TYPE IS INDIVIDUAL; INDIVIDUAL COVARIANCE; CORRELATION; FULLCOV; FULLCORR; MEANS; STDEVIATIONS; MONTECARLO; IMPUTATION; NOBSERVATIONS ARE number of observations; NGROUPS = number of groups; 1 LISTWISE = ON; OFF; OFF SWMATRIX = file name; VARIANCES = CHECK; NOCHECK; CHECK DATA IMPUTATION: IMPUTE = NDATASETS = SAVE = FORMAT = MODEL = VALUES = ROUNDING = THIN = names of variables for which missing values will be imputed; number of imputed data sets; names of files in which imputed data sets are stored; format statement; COVARIANCE; SEQUENTIAL; REGRESSION; values imputed data can take; number of decimals for imputed continuous variables; k where every k-th imputation is saved; 5 F10.3 depends on analysis type no restrictions 3 100 DATA WIDETOLONG: WIDE = LONG = IDVARIABLE = REPETITION = names of old wide format variables; names of new long format variables; name of variable with ID information; name of variable with repetition information; ID REP\n\n DATA LONGTOWIDE: LONG = WIDE = IDVARIABLE = REPETITION = names of old long format variables; names of new wide format variables; name of variable with ID information; name of variable with repetition information (values); 0, 1, 2, etc. DATA TWOPART: NAMES = CUTPOINT = BINARY = CONTINUOUS = TRANSFORM = names of variables used to create a set of binary and continuous variables; value used to divide the original variables into a set of binary and continuous variables; names of new binary variables; names of new continuous variables; function to use to transform new continuous variables; 0 LOG DATA MISSING: NAMES = BINARY = names of variables used to create a set of binary variables; names of new binary variables; TYPE = MISSING; SDROPOUT; DDROPOUT; DESCRIPTIVE = sets of variables for additional descriptive statistics separated by the \| symbol; DATA SURVIVAL: NAMES = CUTPOINT = BINARY = names of variables used to create a set of binary event-history variables; value used to create a set of binary event-history variables from a set of original variables; names of new binary variables; DATA COHORT: COHORT IS COPATTERN IS COHRECODE = TIMEMEASURES = TNAMES = name of cohort variable (values); name of cohort/pattern variable (patterns); (old value = new value); list of sets of variables separated by the \| symbol; list of root names for the sets of variables in TIMEMEASURES separated by the \| symbol;\n\nThe DATA command is a required command. The FILE option is a required option. The NOBSERVATIONS option is required when summary data are analyzed. This option is not required when individual data are analyzed. Default settings are shown in the last column. If the default settings are appropriate for the options that are not required, nothing needs to be specified for these options.\n\nNote that commands and options can be shortened to four or more letters. Option settings can be referred to by either the complete word or the part of the word shown above in bold type.\n\nThe FILE option is used to specify the name and location of the ASCII file that contains the data to be analyzed. The FILE option is required for each analysis. It is specified for a single group analysis as follows:\n\nFILE IS c:\\analysis\\data.dat;\n\nwhere data.dat is the name of the ASCII file containing the data to be analyzed. In this example, the file data.dat is located in the directory c:\\analysis. If the full path name of the data set contains any blanks, the full path name must have quotes around it.\n\nIf the name of the data set is specified with a path, the directory specified by the path is checked. If the name of the data set is specified without a path, the local directory is checked. If the data set is not found in the local directory, the directory where the input file is located is checked.\n\nThe FORMAT option is used to describe the format of the data set to be analyzed. Individual data can be in fixed or free format. Free format is the default. Fixed format is recommended for large data sets because it is faster to read data using a fixed format. Summary data must be in free format.\n\nFor data in free format, each entry on a record must be delimited by a comma, space, or tab. When data are in free format, the use of blanks is not allowed. The number of variables in the data set is determined from information provided in the NAMES option of the VARIABLE command. Data are read until the number of pieces of information equal to the number of variables is found. The program then goes to the next record to begin reading information for the next observation.\n\nFor data in fixed format, each observation must have the same number of records. Information for a given variable must occupy the same position on the same record for each observation. A FORTRAN-like format statement describing the position of the variables in the data set is required. Following is an example of how to specify a format statement:\n\nFORMAT IS 5F4.0, 10x, 6F1.0;\n\nAlthough any FORTRAN format descriptor (i.e., F, I, G, E, x, t, /, etc.) is acceptable in a format statement, most format statements use only F, t, x, and /. Following is an explanation of how to create a FORTRAN-like format statement using these descriptors.\n\nThe F format describes the format for a real variable. F is followed by a number. It can be a whole number or a decimal, for example, F5.3. The number before the decimal point describes the number of columns reserved for the variable; the number after the decimal point specifies the number of decimal places. If the number 34234 is read with an F5.3 format, it is read as 34.234. If the data contain a decimal point, it is not necessary to specify information about the position of the decimal point. For example, the number 34.234 can be read with a F6 format as 34.234.\n\nThe F format can also be preceded by a number. This number represents the number of variables to be read using that format. The statement 5F5.3 is a shorthand way of saying F5.3, F5.3, F5.3, F5.3, F5.3.\n\nThere are three options for the format statement related to skipping columns or records when reading data: x, t, and /. The x option instructs the program to skip columns. The statement 10x says to skip 10 columns and begin reading in column 11. The t option instructs the program to go to a particular column and begin reading. For example, t130 says to go to column 130 and begin reading in column 130. The / option is used to instruct the program to go to the next record. Consider the following format statements:\n\n1. (20F4, 13F5, 3F2)\n\n2. (3F4.1,25x,5F5)\n\n3. (3F4.1,t38,5F5)\n\n4. (2F4/14F4.2//6F3.1)\n\n1. In the first statement, for each record the program reads 20 four-digit numbers followed by 13 five-digit numbers, then three two-digit numbers with a total record length of 151.\n\n2. In the second statement, for each record the program reads three four-digit numbers with one digit to the right of the decimal, skips 25 spaces, and then reads five five-digit numbers with a total record length of 62.\n\n3. The third statement is the same as the second but uses the t option instead of the x option. In the third statement, for each record the program reads three four-digit numbers with one digit to the right of the decimal, goes to column 38, and then reads five five-digit numbers.\n\n4. In the fourth statement, each observation has four records. For record one the program reads two four-digit numbers; for record two the program reads fourteen four-digit numbers with two digits to the right of the decimal; record three is skipped; and for record four the program reads six three-digit numbers with one number to the right of the decimal point.\n\nFollowing is an example of a data set with six one-digit numbers with no numbers to the right of the decimal point:\n\n123234\n\n342765\n\n348765\n\nThe format statement for the data set above is:\n\nFORMAT IS 6F1.0;\n\nor\n\nFORMAT IS 6F1;\n\nThe TYPE option is used in conjunction with the FILE option to describe the contents of the file named using the FILE option. It has the following settings:\n\nINDIVIDUAL Data matrix where rows represent observations and columns represent variables\n\nCOVARIANCE A lower triangular covariance matrix read row wise\n\nCORRELATION A lower triangular correlation matrix read row wise\n\nFULLCOV A full covariance matrix read row wise\n\nFULLCORR A full correlation matrix read row wise\n\nMEANS Means\n\nSTDEVIATIONS Standard deviations\n\nMONTECARLO A list of the names of the data sets to be analyzed\n\nIMPUTATION A list of the names of the imputed data sets to be analyzed\n\nINDIVIDUAL\n\nThe default for the TYPE option is INDIVIDUAL. The TYPE option is not required if individual data are being analyzed where rows represent observations and columns represent variables.\n\nSUMMARY DATA\n\nThe TYPE option is required when summary data such as a covariance matrix or a correlation matrix are analyzed. The TYPE option has six settings related to the analysis of summary data. They are: COVARIANCE, CORRELATION, FULLCOV, FULLCORR, MEANS, and STDEVIATIONS. Summary data must reside in a free format external ASCII file. The number of observations must be specified using the NOBSERVATIONS option of the DATA command.\n\nWhen summary data are analyzed and one or more dependent variables are binary or ordered categorical (ordinal), only a correlation matrix can be analyzed. When summary data are analyzed and all dependent\nvariables are continuous, a covariance matrix is usually analyzed. In some cases, a correlation matrix can be analyzed.\n\nA data set with all continuous dependent variables in the form of a correlation matrix, standard deviations, and means is specified as:\n\nTYPE IS CORRELATION MEANS STDEVIATIONS;\n\nThe program creates a covariance matrix using the correlations and standard deviations and then analyzes the means and covariance matrix.\n\nThe external ASCII file for the above example contains the means, standard deviations, and correlations in free format. Each type of data must begin on a separate record even if the data fits on less than one record. The means come first; the standard deviations begin on the record following the last mean; and the entries of the lower triangular correlation matrix begin on the record following the last standard deviation. The data set appears as follows:\n\n.4 .6 .3 .5 .5\n\n.2 .5 .4 .5 .6\n\n1.0\n\n.86 1.0\n\n.56 .76 1.0\n\n.78 .34 .48 1.0\n\n.65 .87 .32 .56 1.0\n\nor alternatively:\n\n.4 .6 .3 .5 .5\n\n.2 .5 .4 .5 .6\n\n1.0 .86 1.0 .56 .76 1.0 .78 .34 .48 1.0 .65 .87 .32 .56 1.0\n\nMONTECARLO\n\nThe MONTECARLO setting of the TYPE option is used when the data sets being analyzed have been generated and saved using either the REPSAVE option of the MONTECARLO command or by another computer program. The file named using the FILE option of the DATA command contains a list of the names of the data sets to be analyzed and summarized as in a Monte Carlo study. This ASCII file is created automatically when the data sets are generated and saved in a prior analysis using the REPSAVE option of the MONTECARLO command. This file must be created by the user when the data sets are generated and saved using another computer program. Each record of the file must contain one data set name. For example, if five data sets are being analyzed, the contents of the file would be:\n\ndata1.dat\n\ndata2.dat\n\ndata3.dat\n\ndata4.dat\n\ndata5.dat\n\nwhere data1.dat, data2.dat, data3.dat, data4.dat, and data5.dat are the names of the five data sets generated and saved using another computer program. All files must be in the same format. Files saved using the REPSAVE option are in free format.\n\nWhen the MONTECARLO option is used, the results are presented in a Monte Carlo summary format. The output includes the population value for each parameter, the average of the parameter estimates across replications, the standard deviation of the parameter estimates across replications, the average of the estimated standard errors across replications, the mean square error for each parameter (M.S.E.), 95 percent coverage, and the proportion of replications for which the null hypothesis that a parameter is equal to zero is rejected at the .05 level. In addition, the average fit statistics and the percentiles for the fit statistics are given if appropriate. A description of Monte Carlo output is given in Chapter 12.\n\nIMPUTATION\n\nThe IMPUTATION setting of the TYPE option is used when the data sets being analyzed have been generated using multiple imputation procedures. The file named using the FILE option of the DATA command must contain a list of the names of the multiple imputation data sets to be analyzed. Parameter estimates are averaged over the set of analyses. Standard errors are computed using the average of the squared standard errors over the set of analyses and the between analysis parameter estimate variation (Rubin, 1987; Schafer, 1997). A chi-square test of overall model fit is provided (Asparouhov & Muthén, 2008c; Enders, 2010). The ASCII file containing the names of the data sets must be created by the user. Each record of the file must contain one data set name. For example, if five data sets are being analyzed, the contents of the file would be:\n\nimp1.dat\n\nimp2.dat\n\nimp3.dat\n\nimp4.dat\n\nimp5.dat\n\nwhere imp1.dat, imp2.dat, imp3.dat, imp4.dat, and imp5.dat are the names of the five data sets created using multiple imputation.\n\nThe NOBSERVATIONS option is required when summary data are analyzed. When individual data are analyzed, the program counts the number of observations. The NOBSERVATIONS option can, however, be used with individual data to limit the number of records used in the analysis. For example, if a data set contains 20,000 observations, it is possible to analyze only the first 1,000 observations by specifying:\n\nNOBSERVATIONS = 1000;\n\nThe NGROUPS option is used for multiple group analysis when summary data are analyzed. It specifies the number of groups in the analysis. It is specified as follows:\n\nNGROUPS = 3;\n\nwhich indicates that the analysis is a three-group analysis. Multiple group analysis is discussed in Chapter 14.\n\nLISTWISE\n\nThe LISTWISE option is used to indicate that any observation with one or more missing values on the set of analysis variables not be used in the analysis. The default is to estimate the model under missing data theory using all available data. To turn on listwise deletion, specify:\n\nLISTWISE = ON;\n\nSWMATRIX\n\nThe SWMATRIX option is used with TYPE=TWOLEVEL and weighted least squares estimation to specify the name and location of the file that contains the within- and between-level sample statistics and their corresponding estimated asymptotic covariance matrix. The univariate and bivariate sample statistics are estimated using one- and two-dimensional numerical integration with a default of 7 integration points. The INTEGRATION option of the ANALYSIS command can be used to change the default. It is recommended to save this information and use it in subsequent analyses along with the raw data to reduce computational time during model estimation. Analyses using this information must have the same set of observed dependent and independent variables, the same DEFINE command, the same USEOBSERVATIONS statement, and the same USEVARIABLES statement as the analysis which was used to save the information. It is specified as follows:\n\nSWMATRIX = swmatrix.dat;\n\nwhere swmatrix.dat is the file that contains the within- and between-level sample statistics and their corresponding estimated asymptotic covariance matrix.\n\nFor TYPE=IMPUTATION, the file specified contains a list of file names. These files contain the within- and between-level sample statistics and their corresponding estimated asymptotic covariance matrix for a set of imputed data sets.\n\nVARIANCES\n\nThe VARIANCES option is used to check that the analysis variables do not have variances of zero in the sample used for the analysis. Checking for variances of zero is the default. To turn off this check, specify:\n\nVARIANCES = NOCHECK;\n\nThe DATA IMPUTATION command is used when a data set contains missing values to create a set of imputed data sets using multiple imputation methodology. Imputation refers to the estimation of missing values in a data set to create a data set without missing values. Multiple imputation refers to the creation of several data sets where missing values have been imputed. Multiple imputation is carried out using Bayesian estimation. The multiple imputations are random draws from the posterior distribution of the missing values (Rubin, 1987; Schafer, 1997). For an overview, see Enders (2010). The multiple imputation data sets can be used for subsequent model estimation using maximum likelihood or weighted least squares estimation of each data set where the parameter estimates are averaged over the data sets and the standard errors are computed using the Rubin formula (Rubin, 1987). A chi-square test of overall model fit is provided (Asparouhov & Muthén, 2008c; Enders, 2010). Plausible values for latent variables in the model can be saved by specifying SAVE=FSCORES in conjunction with the FACTORS option of the SAVEDATA command.", null, "The figure above shows three ways that data imputation can be done. The first path in the figure uses an unrestricted H1 imputation model and saves the imputed data sets for a subsequent analysis. In this case, TYPE=BASIC is specified in the ANALYSIS command. See Example 11.5. To use the data sets in a subsequent analysis, specify TYPE=IMPUTATION in the DATA command. See Example 13.13. The second path in the figure uses an unrestricted H1 imputation model with an estimator other than BAYES. In this case, the model is estimated immediately after the data are imputed. See Example 11.6. The third path in the figure uses an H0 imputation model and ESTIMATOR=BAYES. The H0 model specified in the MODEL command is used to impute the data. See Example 11.7.\n\nIMPUTE\n\nThe IMPUTE option is used to specify the analysis variables for which missing values will be imputed. Data can be imputed for all or a subset of the analysis variables. These variables can be continuous or categorical. If they are categorical a letter c in parentheses must be included after the variable name. If a variable is on the CATEGORICAL list in the VARIABLE command, it must have a c in parentheses following its name. A variable not on the CATEGORICAL list can have a c in parentheses following its name. Following is an example of how to specify the IMPUTE option:\n\nIMPUTE = y1-y4 u1-u4 (c) x1 x2;\n\nwhere values will be imputed for the continuous variables y1, y2, y3, y4, x1, and x2 and the categorical variables u1, u2, u3, and u4.\n\nThe IMPUTE option has an alternative specification that is convenient when there are several variables that cannot be specified using the list function. When c in parentheses follows the equal sign, it means that c applies to all of the variables that follow. For example, the following IMPUTE statement specifies that the variables x1, x3, x5, x7, and x9 are categorical:\n\nIMPUTE = (c) x1 x3 x5 x7 x9;\n\nThe keyword ALL can be used to indicate that values are to be imputed for all variables in the dataset. The ALL option can be used with the c setting, for example,\n\nIMPUTE = ALL (c);\n\nindicates that all of the variables in the data set are categorical.\n\nNDATASETS\n\nThe NDATASETS option is used to specify the number of imputed data sets to create. The default is five. Following is an example of how to specify the NDATASETS option:\n\nNDATASETS = 20;\n\nwhere 20 is the number of imputed data sets that will be created. The default for the NDATASETS option is 5.\n\nSAVE\n\nThe SAVE option is used to save the imputed data sets for subsequent analysis using TYPE=IMPUTATION in the DATA command. It is specified as follows:\n\nSAVE = impute.dat;\n\nwhere the asterisk () is replaced by the number of the imputed data set. A file is also produced that contains the names of all of the data sets. To name this file, the asterisk () is replaced by the word list.\n\nThe FORMAT option is used to specify the format in which the imputed data will be saved. All dependent and independent variables used in the analysis are saved. In addition, all other variables that are used in conjunction with the analysis are saved as well as any variables specified using the AUXILIARY option of the VARIABLE command. The names of the data sets along with the names of the variables saved and the format are printed in the output. The default is to save the analysis variables using a fixed format.\n\nFollowing is an example of how to specify the FORMAT option to save imputed data in a free format:\n\nFORMAT IS FREE;\n\nImputed data can also be saved in a fixed format specified by the user. The user has the choice of which F or E format the analysis variables are saved in with the format of other saved variables determined by the program. This option is specified as:\n\nFORMAT IS F2.0;\n\nwhich indicates that all analysis variables will be saved with an F2.0 format.\n\nMODEL\n\nThe MODEL option is used to specify the type of unrestricted H1 model to use for imputation (Asparouhov & Muthén, 2010b). The MODEL option has three settings: COVARIANCE, SEQUENTIAL, and REGRESSION. The default is COVARIANCE. The COVARIANCE setting uses a model of unrestricted means, variances, and covariances for a set of continuous variables. The SEQUENTIAL setting uses a sequential regression method also referred to as the chained equations algorithm in line with Raghunathan et al. (2001). The REGRESSION setting uses a model where variables with missing data are regressed on variables without missing data (Asparouhov & Muthén, 2010b). To request the sequential regression method, specify:\n\nMODEL = SEQUENTIAL;\n\nVALUES\n\nThe VALUES option is used to provide the values for continuous variables that the imputed data can take. The default is to put no restrictions on the values that the imputed data can take. The values must be integers. For example, four five-category variables not declared as categorical can be restricted to take on only the values of one through five by specifying:\n\nVALUES = y1-y4 (1-5);\n\nThe closest value to the imputed value is used. If the imputed value is 2.7, the value 3 will be used.\n\nROUNDING\n\nThe ROUNDING option is used to specify the number of decimals that imputed continuous variables will have. The default is three. To request that five decimals be used, specify:\n\nROUNDING = y1-y10 (5);\n\nThe value zero is used to specify no decimals, that is, integer values.\n\nTHIN\n\nThe THIN option is used to specify which intervals in the draws from the posterior distribution are used for imputed values. The default is to use every 100th iteration. To request that every 200th iteration be used, specify:\n\nTHIN = 200;\n\nThere are six DATA transformation commands. They are used to rearrange data from a wide to long format, to rearrange data from a long to wide format, to create a binary and a continuous variable from a semicontinuous variable, to create a set of binary variables that are indicators of missing data for another set of variables, to create variables for discrete-time survival modeling where a binary variable represents the occurrence of a single non-repeatable event, and to rearrange longitudinal data from a format where time points represent measurement occasions to a format where time points represent age or another time-related variable. The DATA transformation commands are executed after the statements in the DEFINE command that come before the CLUSTER_MEAN, CENTER, and STANDARDIZE options of the DEFINE command. The CLUSTER_MEAN, CENTER, and STANDARDIZE options are then executed in the order mentioned followed by the execution of any statements that follow them.\n\nTHE DATA WIDETOLONG COMMAND\n\nIn growth modeling an outcome measured at four time points can be represented in a data set in two ways. In the wide format, the outcome is represented as four variables on a single record. In the long format, the outcome is represented as a single variable using four records, one for each time point. The DATA WIDETOLONG command is used to rearrange data from a multivariate wide format to a univariate long format.\n\nWhen the data are rearranged, the set of outcomes is given a new variable name and ID and repetition variables are created. These new variable names must be placed on the USEVARIABLES statement of the VARIABLE command if they are used in the analysis. They must be placed after any original variables. If the ID variable is used as a cluster variable, this must be specified using the CLUSTER option of the VARIABLE command.\n\nThe creation of the new variables in the DATA WIDETOLONG command occurs after any transformations in the DEFINE command and any of the other DATA transformation commands. If listwise deletion is used, it occurs after the data have been rearranged. Following is a description of the options used in the DATA WIDETOLONG command.\n\nWIDE\n\nThe WIDE option is used to identify sets of variables in the wide format data set that will be converted into single variables in the long format data set. These variables must be variables from the NAMES statement of the VARIABLE command. The WIDE option is specified as follows:\n\nWIDE = y1-y4 \| x1-x4;\n\nwhere y1, y2, y3, and y4 represent one variable measured at four time points and x1, x2, x3, and x4 represent another variable measured at four time points.\n\nLONG\n\nThe LONG option is used to provide names for the new variables in the long format data set. There should be the same number of names as there are sets of variables in the WIDE statement. The LONG option is specified as follows:\n\nLONG = y \| x;\n\nwhere y is the name assigned to the set of variables y1-y4 on the WIDE statement and x is the name assigned to the set of variables x1-x4.\n\nIDVARIABLE\n\nThe IDVARIABLE option is used to provide a name for the variable that provides information about the unit to which the record belongs. In univariate growth modeling, this is the person identifier which is used as a cluster variable. The IDVARIABLE option is specified as follows:\n\nIDVARIABLE = subject;\n\nwhere subject is the name of the variable that contains information about the unit to which the record belongs. If an id variable is specified using the IDVARIABLE option of the VARIABLE command, the values of this variable are used for the variable specified using the IDVARIABLE option. This option is not required.\n\nREPETITION\n\nThe REPETITION option is used to provide a name for the variable that contains information on the order in which the variables were measured. The REPETITION option is specified as follows:\n\nREPETITION = time;\n\nwhere time is the variable that contains information on the order in which the variables were measured. This variable assigns consecutive values starting with zero to the repetitions. This variable can be used in a growth model as a time score variable. This option is not required.\n\nTHE DATA LONGTOWIDE COMMAND\n\nIn growth modeling an outcome measured at four time points can be represented in a data set in two ways. In the long format, the outcome is represented as a single variable using four records, one for each time point. In the wide format, the outcome is represented as four variables on a single record. The DATA LONGTOWIDE command is used to rearrange data from a univariate long format to a multivariate wide format.\n\nWhen the data are rearranged, the outcome is given a set of new variable names. These new variable names must be placed on the USEVARIABLES statement of the VARIABLE command if they are used in the analysis. They must be placed after any original variables.\n\nThe creation of the new variables in the DATA LONGTOWIDE command occurs after any transformations in the DEFINE command and any of the other DATA transformation commands. Following is a description of the options used in the DATA LONGTOWIDE command.\n\nLONG\n\nThe LONG option is used to identify the variables in the long format data set that will be used to create sets of variables in the wide format data set. These variables must be variables from the NAMES statement of the VARIABLE command. The LONG option is specified as follows:\n\nLONG = y \| x;\n\nwhere y and x are two variables that have been measured at multiple time points which are represented by multiple records.\n\nWIDE\n\nThe WIDE option is used to provide sets of names for the new variables in the wide format data set. There should be the same number of sets of names as there are variables in the LONG statement. The number of names in each set corresponds to the number of time points at which the variables in the long data set were measured. The WIDE option is specified as follows:\n\nWIDE = y1-y4 \| x1-x4;\n\nwhere y1, y2, y3, and y4 are the names for the variable y in the wide data set and x1, x2, x3, and x4 are the names for the variable x in the wide data set.\n\nIDVARIABLE\n\nThe IDVARIABLE option is used to identify the variable in the long data set that contains information about the unit to which each record belongs. The IDVARIABLE option is specified as follows:\n\nIDVARIABLE = subject;\n\nwhere subject is the name of the variable that contains information about the unit to which each record belongs. This variable becomes the identifier for each observation in the wide data set. The IDVARIABLE option of the VARIABLE command cannot be used to select a different identifier.\n\nREPETITION\n\nThe REPETITION option is used to identify the variable that contains information about the times at which the variables in the long data set were measured. The REPETITION option is specified as follows:\n\nREPETITION = time;\n\nwhere time is the variable that contains information about the time at which the variables in the long data set were measured. If the time variable does not contain consecutive integer values starting at zero, the time values must be given. For example,\n\nREPETITION = time (4 8 16);\n\nspecifies that the values 4, 8, and 16 are the values of the variable time. The number of values should be equal to the number of variables in the WIDE option and the order of the values should correspond to the order of the variables.\n\nTHE DATA TWOPART COMMAND\n\nThe DATA TWOPART command is used to create a binary and a continuous variable from a continuous variable with a floor effect for use in two-part (semicontinuous) modeling (Duan et al., 1983; Olsen & Schafer, 2001). One situation where this occurs is when variables have a preponderance of zeros.\n\nA set of binary and continuous variables are created using the value specified in the CUTPOINT option of the DATA TWOPART command or zero which is the default. The two variables are created using the following rules:\n\n1. If the value of the original variable is missing, both the new binary and the new continuous variable values are missing.\n\n2. If the value of the original variable is greater than the cutpoint value, the new binary variable value is one and the new continuous variable value is the log of the original variable as the default.\n\n3. If the value of the original variable is less than or equal to the cutpoint value, the new binary variable value is zero and the new continuous variable value is missing.\n\nThe new variables must be placed on the USEVARIABLES statement of the VARIABLE command if they are used in the analysis. These variables must come after any original variables. If the binary variables are used as dependent variables in the analysis, they must be declared as categorical using the CATEGORICAL option of the VARIABLE command.\n\nThe creation of the new variables in the DATA TWOPART command occurs after any transformations in the DEFINE command and before any transformations using the DATA MISSING command. Following is a description of the options used in the DATA TWOPART command.\n\nNAMES\n\nThe NAMES option identifies the variables that are used to create a set of binary and continuous variables. These variables must be variables from the NAMES statement of the VARIABLE command. The NAMES option is specified as follows:\n\nNAMES = smoke1-smoke4;\n\nwhere smoke1, smoke2, smoke3, and smoke4 are the semicontinuous variables that are used to create a set of binary and continuous variables.\n\nCUTPOINT\n\nThe CUTPOINT option is used to provide the value that is used to divide the original variables into a set of binary and continuous variables. The default value for the CUTPOINT option is zero. The CUTPOINT option is specified as follows:\n\nCUTPOINT = 1;\n\nwhere variables are created based on values being less than or equal to one or greater than one.\n\nBINARY\n\nThe BINARY option is used to assign names to the new set of binary variables. The BINARY option is specified as follows:\n\nBINARY = u1-u4;\n\nwhere u1, u2, u3, and u4 are the names of the new set of binary variables.\n\nCONTINUOUS\n\nThe CONTINUOUS option is used to assign names to the new set of continuous variables. The CONTINUOUS option is specified as follows:\n\nCONTINUOUS = y1-y4;\n\nwhere y1, y2, y3, and y4 are the names of the new set of continuous variables.\n\nTRANSFORM\n\nThe TRANSFORM option is used to transform the new continuous variables. The LOG function is the default. The following functions can be used with the TRANSFORM option:\n\nLOG base e log LOG (y);\n\nLOG10 base 10 log LOG10 (y);\n\nEXP exponential EXP (y);\n\nSQRT square root SQRT (y);\n\nABS absolute value ABS(y);\n\nSIN sine SIN (y);\n\nCOS cosine COS (y);\n\nTAN tangent TAN(y);\n\nASIN arcsine ASIN (y);\n\nACOS arccosine ACOS (y);\n\nATAN arctangent ATAN (y);\n\nNONE no transformation\n\nThe TRANSFORM option is specified as follows:\n\nTRANSFORM = NONE;\n\nwhere specifying NONE results in no transformation of the new continuous variables.\n\nTHE DATA MISSING COMMAND\n\nThe DATA MISSING command is used to create a set of binary variables that are indicators of missing data or dropout for another set of variables. Dropout indicators can be scored as discrete-time survival indicators or dropout dummy indicators. The new variables can be used to study non-ignorable missing data (Little & Rubin, 2002; Muthén et al., 2011).\n\nThe new variables must be placed on the USEVARIABLES statement of the VARIABLE command if they are used in the analysis. These variables must come after any original variables. If the binary variables are used as dependent variables in the analysis, they must be declared as categorical using the CATEGORICAL option of the VARIABLE command.\n\nThe creation of the new variables in the DATA MISSING command occurs after any transformations in the DEFINE command and after any transformations using the DATA TWOPART command. Following is a description of the options used in the DATA MISSING command.\n\nNAMES\n\nThe NAMES option identifies the set of variables that are used to create a set of binary variables that are indicators of missing data. These variables must be variables from the NAMES statement of the VARIABLE command. The NAMES option is specified as follows:\n\nNAMES = drink1-drink4;\n\nwhere drink1, drink2, drink3, and drink4 are the set of variables for which a set of binary indicators of missing data are created.\n\nBINARY\n\nThe BINARY option is used to assign names to the new set of binary variables. The BINARY option is specified as follows:\n\nBINARY = u1-u4;\n\nwhere u1, u2, u3, and u4 are the names of the new set of binary variables.\n\nFor TYPE=MISSING, the number of binary indicators is equal to the number of variables in the NAMES statement. For TYPE=SDROPOUT and TYPE=DDROPOUT, the number of binary indicators is one less than the number of variables in the NAMES statement because dropout cannot occur before the second time point an individual is observed.\n\nTYPE\n\nThe TYPE option is used to specify how missingness is coded. It has three settings: MISSING, SDROPOUT, and DDROPOUT. The default is MISSING. For the MISSING setting, a binary missing data indicator variable is created. For the SDROPOUT setting, which is used with selection missing data modeling, a binary discrete-time (event-history) survival dropout indicator is created. For the DDROPOUT setting, which is used with pattern-mixture missing data modeling, a binary dummy dropout indicator is created. The TYPE option is specified as follows:\n\nTYPE = SDROPOUT;\n\nFollowing are the rules for creating the set of binary variables for the MISSING setting:\n\n1. If the value of the original variable is missing, the new binary variable value is one.\n\n2. If the value of the original variable is not missing, the new binary variable value is zero.\n\nFor the SDROPOUT and DDROPOUT settings, the set of indicator variables is defined by the last time point an individual is observed.\n\nFollowing are the rules for creating the set of binary variables for the SDROPOUT setting:\n\n1. The value one is assigned to the time point after the last time point an individual is observed.\n\n2. The value missing is assigned to all time points after the value of one.\n\n3. The value zero is assigned to all time points before the value of one.\n\nFollowing are the rules for creating the set of binary variables for the DDROPOUT setting:\n\n1. The value one is assigned to the time point after the last time point an individual is observed.\n\n2. The value zero is assigned to all other time points.\n\nDESCRIPTIVE\n\nThe DESCRIPTIVE option is used in conjunction with TYPE=BASIC of the ANALYSIS command and the SDROPOUT and DDROPOUT settings of the TYPE option to specify the sets of variables for which additional descriptive statistics are computed. For each variable, the mean and standard deviation are computed using all observations without missing on the variable. Means and standard deviations are provided for the following sets of observations whose definitions are based on missing data patterns:\n\nDropouts after each time point – Individuals who drop out before the next time point and do not return to the study\n\nNon-dropouts after each time point – Individuals who do not drop out before the next time point\n\nTotal Dropouts – Individuals who are missing at the last time point\n\nDropouts no intermittent missing – Individuals who do not return to\n\nthe study once they have dropped out\n\nDropouts intermittent missing – Individuals who drop out and return\n\nto the study\n\nTotal Non-dropouts – Individuals who are present at the last time point\n\nNon-dropouts complete data – Individuals with complete data\n\nNon-dropouts intermittent missing – Individuals who have missing\n\ndata but are present at the last time point\n\nTotal sample\n\nThe first set of variables given in the DESCRIPTIVE statement is the outcome variable. This set of variables defines the number of time points in the model. If the other sets of variables do not have the same number of time points, the asterisk () is used as a placeholder. Sets of variables are separated by the \| symbol. Following is an example of how to specify the DESCRIPTIVE option:\n\nDESCRIPTIVE = y0-y5 \| x0-x5 \| * z1-z5;\n\nThe first set of variables, y0-y5 defines the number of time points as six. The last set of variables has only five measures. An asterisk () is used as a placeholder for the first time point.\n\nTHE DATA SURVIVAL COMMAND\n\nThe DATA SURVIVAL command is used to create variables for discrete-time survival modeling where a binary discrete-time survival (event-history) variable represents whether or not a single non-repeatable event has occurred in a specific time period.\n\nA set of binary discrete-time survival variables is created using the following rules:\n\n1. If the value of the original variable is missing, the new binary variable value is missing.\n\n2. If the value of the original variable is greater than the cutpoint value, the new binary variable value is one which represents that the event has occurred.\n\n3. If the value of the original variable is less than or equal to the cutpoint value, the new binary variable value is zero which represents that the event has not occurred.\n\n4. After a discrete-time survival variable for an observation is assigned the value one, subsequent discrete-time survival variables for that observation are assigned the value of the missing value flag.\n\nThe new variables must be placed on the USEVARIABLES statement of the VARIABLE command if they are used in the analysis. These variables must come after any original variables. If the binary variables are used as dependent variables in the analysis, they must be declared as categorical using the CATEGORICAL option of the VARIABLE command.\n\nThe creation of the new variables in the DATA SURVIVAL command occurs after any transformations in the DEFINE command, the DATA TWOPART command, and the DATA MISSING command. Following is a description of the options used in the DATA SURVIVAL command.\n\nNAMES\n\nThe NAMES option identifies the variables that are used to create a set of binary event-history variables. These variables must be variables from the NAMES statement of the VARIABLE command. The NAMES option is specified as follows:\n\nNAMES = dropout1-dropout4;\n\nwhere dropout1, dropout2, dropout3, and dropout4 are the variables that are used to create a set of binary event-history variables.\n\nCUTPOINT\n\nThe CUTPOINT option is used provide the value to use to create a set of binary event-history variables from a set of original variables. The default value for the CUTPOINT option is zero. The CUTPOINT option is specified as follows:\n\nCUTPOINT = 1;\n\nwhere variables are created based on values being less than or equal to one or greater than one.\n\nBINARY\n\nThe BINARY option is used to assign names to the new set of binary event-history variables. The BINARY option is specified as follows:\n\nBINARY = u1-u4;\n\nwhere u1, u2, u3, and u4 are the names of the new set of binary event-history variables.\n\nTHE DATA COHORT COMMAND\n\nThe DATA COHORT command is used to rearrange longitudinal data from a format where time points represent measurement occasions to a format where time points represent age or another time-related variable. It is available only for continuous outcomes. Multiple cohort analysis is described in Chapter 14.\n\nThe new variables must be placed on the USEVARIABLES statement of the VARIABLE command if they are used in the analysis.\n\nThese variables must come after any original variables. The creation of the new variables in the DATA COHORT command occurs after any transformations in the DEFINE command. Following is a description of the options used in the DATA COHORT command.\n\nThe COHORT option is used when data have been collected using a multiple cohort design. The COHORT option is used in conjunction with the TIMEMEASURES and TNAMES options that are described below. Variables used with the COHORT option must be variables from the NAMES statement of the VARIABLE command. Following is an example of how the COHORT option is specified:\n\nCOHORT IS birthyear (63 64 65);\n\nwhere birthyear is a variable in the data set to be analyzed, and the numbers in parentheses following the variable name are the values that the birthyear variable contains. Birth years of 1963, 1964, and 1965 are included in the example below. The cohort variable must contain only integer values.\n\nCOPATTERN\n\nThe COPATTERN option is used when data are both missing by design and have been collected using a multiple cohort design. Variables used with the COPATTERN option must be variables from the NAMES statement of the VARIABLE command. Following is an example of how the COPATTERN option is specified:\n\nCOPATTERN = cohort (67=y1 y2 y3 68=y4 y5 y6 69=y2 y3 y4);\n\nwhere cohort is a variable that provides information about both the cohorts included in the data set and the patterns of variables for each cohort. In the example above, individuals in cohort 67 should have information on y1, y2, and y3; individuals in cohort 68 should have information on y4, y5, and y6; and individuals in cohort 69 should have information on y2, y3, and y4. Individuals who have missing values on any variable for which they are expected to have information are eliminated from the analysis. The copattern variable must contain only integer values.\n\nCOHRECODE\n\nThe COHRECODE option is used in conjunction with either the COHORT or COPATTERN options to recode the values of the cohort or copattern variable. The COHRECODE option is specified as follows:\n\nCOHRECODE = (1=67 2=68 3=69 4=70);\n\nwhere the original values of 1, 2, 3, and 4 of the cohort or copattern variable are recoded to 67, 68, 69, and 70, respectively. If the COHRECODE option is used, all values of the original variable must be recoded to be included in the analysis. Observations with values that are not recoded will be eliminated from the analysis.\n\nThe TIMEMEASURES option is used with multiple cohort data to specify the years in which variables to be used in the analysis were measured. It is used in conjunction with the COHORT and COPATTERN options to determine the ages that are represented in the multiple cohort data set. Variables used with the TIMEMEASURES option must be variables from the NAMES statement of the VARIABLE command. Following is an example of how the TIMEMEASURES option is specified:\n\nTIMEMEASURES = y1 (82) y2 (84) y3 (85) y4 (88) y5 (94);\n\nwhere y1, y2, y3, y4, and y5 are original variables that are to be used in the analysis, and the numbers in parentheses following each of these variables represent the years in which they were measured. In this situation, y1, y2, y3, y4, and y5 are the same measure, for example, frequency of heavy drinking measured on multiple occasions.\n\nThe TIMEMEASURES option can be used to identify more than one measure that has been measured repeatedly as shown in the following example:\n\nTIMEMEASURES = y1 (82) y2 (84) y3 (85) y4 (88) y5 (94) \|\n\ny6 (82) y7 (85) y8 (90) y9 (95) \|\n\nx1 (83) x2(88) x3 (95);\n\nwhere each set of variables separated by the symbol \| represents repeated measures of that variable. For example, y1, y2, y3, y4, and y5 may represent repeated measures of heavy drinking; y6, y7, y8, and y9 may represent repeated measures of alcohol dependence; and x1, x2, and x3 may represent repeated measures of marital status.\n\nThe TNAMES option is used to generate variable names for the new multiple cohort analysis variables. A root name is specified for each set of variables mentioned using the TIMEMEASURES option. The age of the respondent at the time the variable was measured is attached to the root name. The age is determined by subtracting the cohort value from the year the variable was measured. Following is an example of how the TNAMES option is specified:\n\nTNAMES = hd;\n\nwhere hd is the root name for the new variables.\n\nFollowing is an example of how the TNAMES option is specified for the TIMEMEASURES and COHORT options when multiple outcomes are measured:\n\nTNAMES = hd \| dep \| marstat;\n\nFollowing are the variables that would be created:\n\nhd22, hd24, hd25, hd26, hd27, hd28, hd29, hd30,\n\nhd31, hd32, hd33, hd34, hd36, hd37, hd38, hd39,\n\ndep22, dep24, dep25, dep26, dep27, dep28, dep29, dep30,\n\ndep32, dep33, dep35, dep36, dep37, dep38, dep39, dep40,\n\nmarstat23, marstat25, marstat26, marstat27, marstat28\n\nmarstat30, marstat31, marstat32, marstat33, marstat35\n\nmarstat37, marstat38, marstat39, marstat40.\n\nThere is no hd variable for ages 23 and 35, no dep variable for ages 23, 31, and 34, and no marstat variable for ages 24, 29, 34, and 36 because these ages are not represented by the combination of cohort values and years of measurement.\n\nThe VARIABLE command is used to provide information about the variables in the data set to be analyzed. The VARIABLE command has options for naming and describing the variables in the data set to be analyzed, subsetting the data set on observations, subsetting the data set on variables, and specifying missing values for each variable.\n\nFollowing are the options for the VARIABLE command:\n\n VARIABLE: NAMES ARE names of variables in the data set; USEOBSERVATIONS ARE conditional statement to select observations; all observations in data set USEVARIABLES ARE names of analysis variables; all variables in NAMES MISSING ARE variable (#); . ; ; BLANK; CENSORED ARE names, censoring type, and inflation status for censored dependent variables; CATEGORICAL ARE names of binary and ordered categorical (ordinal) dependent variables (model); NOMINAL ARE names of unordered categorical (nominal) dependent variables; COUNT ARE names of count variables (model); DSURVIVAL ARE names of discrete-time survival variables; GROUPING IS name of grouping variable (labels); IDVARIABLE IS name of ID variable; _RECNUM; FREQWEIGHT IS name of frequency (case) weight variable; TSCORES ARE names of observed variables with information on individually-varying times of observation; AUXILIARY = names of auxiliary variables; names of auxiliary variables (M); names of auxiliary variables (R3STEP); names of auxiliary variables (R); names of auxiliary variables (BCH); names of auxiliary variables (DU3STEP); names of auxiliary variables (DCATEGORICAL); names of auxiliary variables (DE3STEP); names of auxiliary variables (DCONTINUOUS); names of auxiliary variables (E); CONSTRAINT = names of observed variables that can be used in the MODEL CONSTRAINT command; PATTERN IS name of pattern variable (patterns); STRATIFICATION IS name of stratification variable; CLUSTER IS name of cluster variables; WEIGHT IS name of sampling weight variable;\n\n WTSCALE IS UNSCALED; CLUSTER CLUSTER; ECLUSTER; BWEIGHT name of between-level sampling weight variable; B2WEIGHT IS name of the level 2 sampling weight variable; B3WEIGHT IS name of the level 3 sampling weight variable; BWTSCALE IS UNSCALED; SAMPLE; SAMPLE REPWEIGHTS ARE names of replicate weight variables; SUBPOPULATION IS conditional statement to select subpopulation; all observations in data set FINITE = name of variable; name of variable (FPC); name of variable (SFRACTION); name of variable (POPULATION); FPC CLASSES = names of categorical latent variables (number of latent classes); KNOWNCLASS = name of categorical latent variable with known class membership (labels); TRAINING = names of training variables; names of variables (MEMBERSHIP); names of variables (PROBABILITIES); names of variables (PRIORS); MEMBERSHIP WITHIN ARE WITHIN ARE (label) names of individual-level observed variables; names of individual-level observed variables; BETWEEN ARE BETWEEN ARE (label) names of cluster-level observed variables; names of cluster-level observed variables; SURVIVAL ARE names and time intervals for time-to-event variables; TIMECENSORED ARE LAGGED ARE TINTERVAL IS names and values of variables that contain right censoring information; names of lagged variables (lag); name of time variable (interval); (0 = NOT 1 = RIGHT)\n\nThe VARIABLE command is a required command. The NAMES option is a required option. Default settings are shown in the last column. If the default settings are appropriate for the analysis, nothing needs to be specified except the NAMES option.\n\nNote that commands and options can be shortened to four or more letters. Option settings can be referred to by either the complete word or the part of the word shown above in bold type.\n\nThe NAMES option is used to assign names to the variables in the data set named using the FILE option of the DATA command. This option is required. The variable names can be separated by blanks or commas and can be up to 8 characters in length. Variable names must begin with a letter. They can contain only letters, numbers, and the underscore symbol. The program makes no distinction between upper and lower case letters. Following is an example of how the NAMES option is specified:\n\nNAMES ARE gender ethnic income educatn drink_st agedrink;\n\nVariable names are generated if a list of variables is specified using the NAMES option. For example,\n\nNAMES ARE y1-y5 x1-x3;\n\ngenerates the variable names y1 y2 y3 y4 y5 x1 x2 x3.\n\nNAMES ARE itema-itemd;\n\ngenerates the variable names itema itemb itemc itemd.\n\nThere are options for selecting a subset of observations or variables from the data set named using the FILE option of the DATA command. The USEOBSERVATIONS option is used to select a subset of observations from the data set. The USEVARIABLES option is used to select a subset of variables from the data set.\n\nThe USEOBSERVATIONS option is used to select observations for an analysis from the data set named using the FILE option of the DATA command. This option is not available for summary data. The USEOBSERVATIONS option selects only those observations that satisfy the conditional statement specified after the equal sign. For example, the following statement selects observations with the variable ethnic equal to 1 and the variable gender equal to 2:\n\nUSEOBSERVATIONS = ethnic EQ 1 AND gender EQ 2;\n\nOnly variables from the NAMES statement of the VARIABLE command can be used in the conditional statement of the USEOBSERVATIONS option. Logical operators, not arithmetic operators, must be used in the conditional statement. Following are the logical operators that can be used in conditional statements to select observations for analysis:\n\nAND logical and\n\nOR logical or\n\nNOT logical not\n\nEQ equal ==\n\nNE not equal /=\n\nGE greater than or equal to >=\n\nLE less than or equal to <=\n\nGT greater than >\n\nLT less than <\n\nAs shown above, some of the logical operators can be referred to in two different ways. For example, equal can be referred to as EQ or ==.\n\nThe USEVARIABLES option is used to select variables for an analysis. It can be used with individual data or summary data. Variables included on the USEVARIABLES statement can be variables from the NAMES statement of the VARIABLE command and variables created using the DEFINE command and the DATA transformation commands. New variables created using the DEFINE command and the DATA transformation commands must be included on the USEVARIABLES statement.\n\nThe USEVARIABLES option identifies the observed dependent and independent variables that are used in an analysis. Variables with special functions such as grouping variables do not need to be included on the USEVARIABLES statement unless they are new variables created using the DEFINE command or the DATA transformation commands. Following is an example of how to specify the USEVARIABLES option:\n\nUSEVARIABLES ARE gender income agefirst;\n\nVariables on the USEVARIABLES statement must follow a particular order. The order of the variables is important because it determines the order of variables used with the list function. The set of original variables from the NAMES statement of the VARIABLE command must be listed before the set of new variables created using the DEFINE command or the DATA transformation commands. Within the two sets of original and new variables, any order is allowed.\n\nIf all of the original variables plus some of the new variables are used in the analysis, the keyword ALL can be used as the first entry in the USEVARIABLES statement. This indicates that all of the original variables from the NAMES statement of the VARIABLE command are used in the analysis. The keyword ALL is followed by the names of the new variables created using the DEFINE command or the DATA transformation commands that will be used in the analysis. Following is an example of how to specify the USEVARIABLES option for this situation:\n\nUSEVARIABLES = ALL hd1 hd2 hd3;\n\nwhere ALL refers to the total set of original variables and hd1, hd2, and hd3 are new variables created using the DEFINE command or the DATA transformation commands.\n\nThe MISSING option is used to identify the values or symbol in the analysis data set that are treated as missing or invalid. Any numeric value and the non-numeric symbols of the period, asterisk (), or blank can be used as missing value flags. There is no default missing value flag. Numeric and non-numeric missing value flags cannot be combined. The blank cannot be used as a missing value flag for data in free format. When a list of missing value flags contains a negative number, the entries must be separated by commas.\n\nThe period (.), the asterisk (), or the blank can be used as non-numeric missing value flags. Only one non-numeric missing value flag can be used for a particular data set. This missing value flag applies to all variables in the data set. The blank cannot be used with free format data. With fixed format data, blanks in the data not declared as missing value flags are treated as zeroes.\n\nThe following command indicates that the period is the missing value flag for all variables in the data set:\n\nMISSING ARE . ;\n\nThe blank can be a missing value flag only in fixed format data sets. The following command indicates that blanks are to be considered as missing value flags:\n\nMISSING = BLANK;\n\nAny number of numeric values can be specified as missing value flags for the complete data set or for individual variables. The keyword ALL can be used with the MISSING option if all variables have the same numeric value(s) as missing value flags.\n\nThe following statement specifies that the number 9 is the missing value flag for all variables in the data set:\n\nMISSING ARE ALL (9);\n\nThe following example specifies that for the variable ethnic, the numbers 9 and 99 are missing value flags, while for the variable y1, the number 1 is the missing value flag:\n\nMISSING ARE ethnic (9 99) y1 (1);\n\nThe list function can be used with the MISSING option to specify a list of missing value flags and/or a set of variables. The order of variables in the list is determined by the order of variables in the NAMES statement of the VARIABLE command. Values of 9, 99, 100, 101, and 102 can be declared as missing value flags for all variables in a data set by the following specification:\n\nMISSING ARE ALL (9 99-102);\n\nMissing values can be specified for a list of variables as follows:\n\nMISSING ARE gender-income (9 30 98-102);\n\nThe statement above specifies that the values of 9, 30, 98, 99, 100, 101, and 102 are missing value flags for the list of variables beginning with gender and ending with income.\n\nIf a single missing value flag is negative, it can be specified as described above. If there are several negative missing value flags, they must be separated by commas to distinguish between the list function and a negative value. The following example specifies that for the variable ethnic, the numbers -9 and -99 are missing value flags.\n\nMISSING ARE ethnic (-9, -99);\n\nThe list function can be used with negative missing value flags. It is specified as:\n\nMISSING = ALL (-778- -775);\n\nThe statement above specifies that the values of -778, -777, -776, and -775 are missing value flags for all variables in the data set.\n\nAll observed dependent variables are assumed to be measured on a continuous scale unless the CENSORED, CATEGORICAL, NOMINAL, and/or COUNT options are used. The specification of the scale of the dependent variables determines how the variables are treated in the model and its estimation. Independent variables can be binary or continuous. The scale of the independent variables has no influence on the model or its estimation. The distinction between dependent and independent variables is described in Chapter 17.\n\nVariables named using the CENSORED, CATEGORICAL, NOMINAL, and/or COUNT options can be variables from the NAMES statement of the VARIABLE command and variables created using the DEFINE command and the DATA transformation commands.\n\nCENSORED\n\nThe CENSORED option is used to specify which dependent variables are treated as censored variables in the model and its estimation, whether they are censored from above or below, and whether a censored or censored-inflated model will be estimated.\n\nThe CENSORED option is specified as follows for a censored model:\n\nCENSORED ARE y1 (a) y2 (b) y3 (a) y4 (b);\n\nwhere y1, y2, y3, y4 are censored dependent variables in the analysis. The letter a in parentheses following the variable name indicates that the variable is censored from above. The letter b in parentheses following the variable name indicates that the variable is censored from below. The lower and upper censoring limits are determined from the data.\n\nThe CENSORED option is specified as follows for a censored-inflated model:\n\nCENSORED ARE y1 (ai) y2 (bi) y3 (ai) y4 (bi);\n\nwhere y1, y2, y3, y4 are censored dependent variables in the analysis. The letters ai in parentheses following the variable name indicate that the variable is censored from above and that a censored-inflated model will be estimated. The letters bi in parentheses following the variable name indicate that the variable is censored from below and that a censored-inflated model will be estimated. The lower and upper censoring limits are determined from the data.\n\nWith a censored-inflated model, two variables are considered, a censored variable and an inflation variable. The censored variable takes on values for individuals who are able to assume values of the censoring point and beyond. The inflation variable is a binary latent variable for which the value one denotes that an individual is unable to assume any value except the censoring point. The inflation variable is referred to by adding to the name of the censored variable the number sign (#) followed by the number 1. In the example above, the censored variables available for use in the MODEL command are y1, y2, y3, and y4, and the inflation variables available for use in the MODEL command are y1#1, y2#1, y3#1, and y4#1.\n\nThe CATEGORICAL option is used to specify which dependent variables are treated as binary or ordered categorical (ordinal) variables in the model and its estimation and the type of model to be estimated. Both probit and logistic regression models can be estimated for categorical variables. For binary variables, the following IRT models can be estimated: two-parameter normal ogive, two-parameter logistic, three-parameter logistic, and four-parameter logistic. For ordered categorical (ordinal) variables, the following IRT models can be estimated: generalized partial credit with logistic and graded-response with probit (normal ogive) and logistic. For a nominal IRT model, use the NOMINAL option.\n\nFor categorical dependent variables, there are as many thresholds as there are categories minus one. The thresholds are referred to in the MODEL command by adding to the variable name the dollar sign (\\$) followed by a number. The threshold for a binary variable u1 is referred to as u1\\$1. The two thresholds for a three-category variable u2 are referred to as u2\\$1 and u2\\$2. Ordered categorical dependent variables cannot have more than 10 categories. The number of categories is determined from the data.\n\nThe CATEGORICAL option is specified as follows:\n\nCATEGORICAL ARE u2 u3 u7-u13;\n\nwhere u2, u3, u7, u8, u9, u10, u11, u12, and u13 are binary or ordered categorical dependent variables in the analysis. With weighted least squares and Bayes estimation, a probit model is estimated. For binary variables, this is a two-parameter normal ogive model. For ordered categorical (ordinal) variables, this is a graded response model. With maximum likelihood estimation, a logistic model is estimated as the default. For binary variables, this is a two-parameter logistic model. For ordered categorical (ordinal) variables, this is a proportional odds model which is the same as a graded response model. Probit models can also be estimated with maximum likelihood estimation using the LINK option of the ANALYSIS command.\n\nThe CATEGORICAL option for a generalized partial credit model is specified as follows:\n\nCATEGORICAL = u1 –u3 (gpcm) u10 (gpcm);\n\nwhere the variables u1, u2, u3, and u10 are ordered categorical (ordinal) variables for which a generalized partial credit model will be estimated. The partial credit model has c-1 step parameters for an item with c categories and one slope parameter (Asparouhov & Muthén, 2016). The step parameters are referred to in the same way as thresholds. The first step parameter for a three-category ordered categorical (ordinal) variable u1 is referred to as u1\\$1. The second step parameter is referred to as u1\\$2.\n\nThe CATEGORICAL option for a three-parameter logistic model is specified as follows:\n\nCATEGORICAL = u1 –u3 (3pl) u10 (3pl);\n\nwhere the variables u1, u2, u3, and u10 are binary variables for which a three-parameter logistic model will be estimated. The guessing parameter cannot be referred to directly. Instead a parameter related to the guessing parameter is referred to (Asparouhov & Muthén, 2016). This parameter is referred to as the second threshold. The first threshold for a binary variable u1 is referred to as u1\\$1. The second threshold is referred to as u1\\$2.\n\nThe CATEGORICAL option for a four-parameter logistic model is specified as follows:\n\nCATEGORICAL = u1 –u3 (4pl) u10 (4pl);\n\nwhere the variables u1, u2, u3, and u10 are binary variables for which a four-parameter logistic model will be estimated. The lower asymptote (guessing) and upper asymptote parameters cannot be referred to directly. Instead a parameter which is related to the lower asymptote (guessing) and a parameter which is related to the upper asymptote parameter are referred to (Asparouhov & Muthén, 2016). The parameter related to the lower asymptote (guessing) parameter is referred to as the second threshold. The parameter related to the upper asymptote parameter is referred to as the third threshold. The first threshold for a binary variable u1 is referred to as u1\\$1. The second threshold is referred to as u1\\$2. The third threshold is referred to as u1\\$3.\n\nRECODING OF DEPENDENT VARIABLES\n\nThe estimation of the model for binary or ordered categorical dependent variables uses zero to denote the lowest category, one to denote the second lowest category, two to denote the third lowest category, etc. If the variables are not coded this way in the data, they are automatically recoded as described below. When data are saved for subsequent analyses, the recoded categories are saved.\n\nFollowing are examples of situations in which data are recoded by the program:\n\nCategories in Original Data Categories in Recoded Data\n\n1 2 3 4 0 1 2 3\n\n2 3 4 5 0 1 2 3\n\n2 5 8 9 0 1 2 3\n\n0 1 no recode needed\n\n1 2 0 1\n\nIn most situations, the default recoding is appropriate. In multiple group analysis and growth modeling, the default recoding may not be appropriate because the categories observed in the data for a variable may not be the same across groups or time. For example, it is sometimes the case that individuals are observed in lower categories at earlier time points and higher categories at later time points. Several variations of the CATEGORICAL option are available for these situations. These are allowed only for maximum likelihood estimation.\n\nUsing the automatic recoding, each variable is recoded using the categories found in the data for that variable. Following is an example of how to specify the CATEGORICAL option so that each variable is recoded using the categories found in the data for a set of variables:\n\nCATEGORICAL u1-u3 ();\n\nwhere u1, u2, and u3 are a set of ordered categorical variables and the asterisk () in parentheses indicates that the categories of each variable are to be recoded using the categories found in the data for the set of variables not for each variable. Based on the original data shown in the table below, where the rows represent observations and the columns represent variables, the set of variables are found to have four possible categories: 1, 2, 3, and 4. The variable u1 has observed categories 1 and 2; u2 has observed categories 1, 2, and 3; and u3 has observed categories 2, 3, and 4. The recoded values are shown in the table below.\n\n Categories in the Original Data Set Categories in the Recoded Data Set u1 u2 u3 u1 u2 u3 1 2 3 0 1 2 1 1 2 0 0 1 2 2 2 1 1 1 2 3 4 1 2 3\n\nThe CATEGORICAL option can be used to give a set of categories that are allowed for a variable or set of variables rather than having these categories determined from the data. Following is an example of how to specify this:\n\nCATEGORICAL = u1-u3 (1-6);\n\nwhere the set of variables u1, u2, and u3 can have the categories of 1, 2, 3, 4, 5, and 6. In this example, 1 will be recoded as 0, 2 as 1, 3 as 2, 4 as 3, 5 as 4, and 6 as 5.\n\nAnother variation of this is:\n\nCATEGORICAL = u1-u3 (2 4 6);\n\nwhere the set of variables u1, u2, and u3 can have the categories 2, 4, and 6. In this example, 2 will be recoded as 0, 4 as 1, and 6 as 2.\n\nThe CATEGORICAL option can be used to specify that different sets of variables have different sets of categories by using the \| symbol. For example,\n\nCATEGORICAL = u1-u3 () \| u4-u6 (2-5) \| u7-u9;\n\nspecifies that for the variables u1, u2, and u3, the possible categories are taken from the data for the set of variables; for the variables u4, u5, and u6, the possible categories are 2, 3, 4, and 5; and for the variables u7, u8, and u9, the possible categories are the default, that is, the possible categories are taken from the data for each variable.\n\nNOMINAL\n\nThe NOMINAL option is used to specify which dependent variables are treated as unordered categorical (nominal) variables in the model and its estimation. Unordered categorical dependent variables cannot have more than 10 categories. The number of categories is determined from the data. The NOMINAL option is specified as follows:\n\nNOMINAL ARE u1 u2 u3 u4;\n\nwhere u1, u2, u3, u4 are unordered categorical dependent variables in the analysis.\n\nFor nominal dependent variables, all categories but the last category can be referred to. The last category is the reference category. The categories are referred to in the MODEL command by adding to the variable name the number sign (#) followed by a number. The three categories of a four-category nominal variable are referred to as u1#1, u1#2, and u1#3.\n\nThe estimation of the model for unordered categorical dependent variables uses zero to denote the lowest category, one to denote the second lowest category, two to denote the third lowest category, etc. If the variables are not coded this way in the data, they are automatically recoded as described below. When data are saved for subsequent analyses, the recoded categories are saved.\n\nFollowing are examples of situations in which data are recoded by the program:\n\nCategories in Original Data Categories in Recoded Data\n\n1 2 3 4 0 1 2 3\n\n2 3 4 5 0 1 2 3\n\n2 5 8 9 0 1 2 3\n\n0 1 no recode needed\n\n1 2 0 1\n\nCOUNT\n\nThe COUNT option is used to specify which dependent variables are treated as count variables in the model and its estimation and the type of model to be estimated. The following models can be estimated for count variables: Poisson, zero-inflated Poisson, negative binomial, zero-inflated negative binomial, zero-truncated negative binomial, and negative binomial hurdle (Long, 1997; Hilbe, 2011). The negative binomial models use the NB-2 variance representation (Hilbe, 2011, p. 63). Count variables may not have negative or non-integer values.\n\nThe COUNT option can be specified in two ways for a Poisson model:\n\nCOUNT = u1 u2 u3 u4;\n\nor\n\nCOUNT = u1 (p) u2 (p) u3 (p) u4 (p);\n\nor using the list function:\n\nCOUNT = u1-u4 (p);\n\nThe COUNT option can be specified in two ways for a zero-inflated Poisson model:\n\nCOUNT = u1-u4 (i);\n\nor\n\nCOUNT = u1-u4 (pi);\n\nwhere u1, u2, u3, and u4 are count dependent variables in the analysis. The letter i or pi in parentheses following the variable name indicates that a zero-inflated Poisson model will be estimated.\n\nWith a zero-inflated Poisson model, two variables are considered, a count variable and an inflation variable. The count variable takes on values for individuals who are able to assume values of zero and above following the Poisson model. The inflation variable is a binary latent variable with one denoting that an individual is unable to assume any value except zero. The inflation variable is referred to by adding to the name of the count variable the number sign (#) followed by the number 1. If the inflation parameter value is estimated at a large negative value corresponding to a probability of zero, the inflation part of the model is not needed.\n\nFollowing is the specification of the COUNT option for a negative binomial model:\n\nCOUNT = u1 (nb) u2 (nb) u3 (nb) u4 (nb);\n\nor using the list function:\n\nCOUNT = u1-u4 (nb);\n\nWith a negative binomial model, a dispersion parameter is estimated. The dispersion parameter is referred to by using the name of the count variable. If the dispersion parameter is estimated at zero, the model is a Poisson model.\n\nFollowing is the specification of the COUNT option for a zero-inflated negative binomial model:\n\nCOUNT = u1- u4 (nbi);\n\nWith a zero-inflated negative binomial model, two variables are considered, a count variable and an inflation variable. The count variable takes on values for individuals who are able to assume values of zero and above following the negative binomial model. The inflation variable is a binary latent variable with one denoting that an individual is unable to assume any value except zero. The inflation variable is referred to by adding to the name of the count variable the number sign (#) followed by the number 1. If the inflation parameter value is estimated at a large negative value corresponding to a probability of zero, the inflation part of the model is not needed.\n\nFollowing is the specification of the COUNT option for a zero-truncated negative binomial model:\n\nCOUNT = u1-u4 (nbt);\n\nCount variables for the zero-truncated negative binomial model must have values greater than zero.\n\nFollowing is the specification of the COUNT option for a negative binomial hurdle model:\n\nCOUNT = u1-u4 (nbh);\n\nWith a negative binomial hurdle model, two variables are considered, a count variable and a hurdle variable. The count variable takes on values for individuals who are able to assume values of one and above following the truncated negative binomial model. The hurdle variable is a binary latent variable with one denoting that an individual is unable to assume any value except zero. The hurdle variable is referred to by adding to the name of the count variable the number sign (#) followed by the number 1.\n\nDSURVIVAL\n\nThe DSURVIVAL option is used in conjunction with the PLOT command to identify the discrete-time survival variables so that survival curves are generated. The DSURVIVAL option is specified as follows:\n\nDSURVIVAL = u1-u4;\n\nwhere u1 to u4 are discrete-time survival variables.\n\nVARIABLES WITH SPECIAL FUNCTIONS\n\nThere are several options that are used to identify variables that have special functions. These variables can be variables from the NAMES statement of the VARIABLE command and variables created using the DEFINE command and the DATA transformation commands. Following is a description of these options and their specifications.\n\nThe GROUPING option is used to identify the variable in the data set that contains information on group membership when the data for all groups are stored in a single data set. Multiple group analysis is discussed in Chapter 14. A grouping variable must contain only integer values. Only one grouping variable can be used. If the groups to be analyzed are a combination of more than one variable, a single grouping variable can be created using the DEFINE command. Following is an example of how to specify the GROUPING option:\n\nGROUPING IS gender (1=male 2 = female);\n\nThe information in parentheses after the grouping variable name assigns labels to the values of the grouping variable found in the data set. In the example above, observations with gender equal to 1 are assigned the label male, and individuals with gender equal to 2 are assigned the label female. These labels are used in conjunction with the MODEL command to specify model statements specific to each group. Observations that have a value on the grouping variable that is not specified using the GROUPING option are not included in the analysis.\n\nIn situations where there are many groups, a shorthand notation can be used for the GROUPING option. It is specified as follows:\n\nGROUPING = country (101-200 225 350-360);\n\nwhere country is the grouping variable and 101 through 200, 225, and 350 through 360 are the values of country that will be used as groups. The values of the variable country are used as labels in group-specific MODEL commands.\n\nThe GROUPING option can be specified by mentioning only the number of groups, for example,\n\nGROUPING = country (34);\n\nwhere country is the grouping variable and the number 34 specifies that there are 34 groups. The group values are taken from the data. The reference group is the group with the lowest value. Default group labels are used. G1 is the label for the group with the lowest value, g2 is the label for the group with the next value, etc..\n\nIDVARIABLE\n\nThe IDVARIABLE option is used in conjunction with the SAVEDATA command to provide an identifier for each observation in the data set that is saved. This is useful for merging the data with another data set. The IDVARIABLE option is specified as follows:\n\nIDVARIABLE = id;\n\nwhere id is a variable that contains a unique numerical identifier for each observation. The length of this variable may not exceed 16.\n\nIf a data set does not contain an identifier variable, the _RECNUM setting can be used as follows to create one:\n\nIDVARIABLE = _RECNUM;\n\nThe unique numerical identifier corresponds to the record number of the data set specified using the FILE option of the DATA command.\n\nFREQWEIGHT\n\nThe FREQWEIGHT option is used to identify the variable that contains frequency (case) weight information. Frequency weights are used when a single record in the data set represents the responses of more than one individual. Frequency weight values must be integers. Frequency weights do not have to sum to the total number of observations in the analysis data set and are not rescaled in any way. Frequency weights are available for all analysis types except TYPE=COMPLEX, TYPE=TWOLEVEL, TYPE=THREELEVEL, TYPE=CROSSCLASSIFIED, and TYPE=EFA. With TYPE=RANDOM, frequency weights are available only with ALGORITHM=INTEGRATION. Following is an example of how the FREQWEIGHT option is specified:\n\nFREQWEIGHT IS casewgt;\n\nwhere casewgt is the variable that contains frequency weight information.\n\nThe TSCORES option is used in conjunction with TYPE=RANDOM to identify the variables in the data set that contain information about individually-varying times of observation for the outcome in a growth model. Variables listed in the TSCORES statement can be used only in AT statements in the MODEL command to define a growth model. For TYPE=TWOLEVEL, ALGORITHM=INTEGRATION must be specified in the ANALYSIS command for this type of analysis. The TSCORES option is specified as follows:\n\nTSCORES ARE a1 a2 a3 a4;\n\nwhere a1, a2, a3, and a4 are observed variables in the analysis data set that contain the individually-varying times of observation for an outcome at four time points.\n\nAuxiliary variables are variables that are not part of the analysis model. The AUXILIARY option has three uses. One is to identify a set of variables that is not used in the analysis but is saved for use in a subsequent analysis. A second is to identify a set of variables that will be used as missing data correlates in addition to the analysis variables. The third is to automatically carry out the 3-step approach of TYPE=MIXTURE.\n\nSAVED\n\nIn the first use of the AUXILIARY option, variables listed on the AUXILIARY statement are saved along with the analysis variables if the SAVEDATA command is used. These variables can be used in graphical displays if the PLOT command is used. If these variables are created using the DEFINE command or the DATA transformation commands, they must be listed on the USEVARIABLES statement of the VARIABLE command in addition to being listed on the AUXILIARY statement. The AUXILIARY option is specified as follows:\n\nAUXILIARY = gender race educ;\n\nwhere gender, race, and educ are variables that are not used in the analysis but that are saved in conjunction with the SAVEDATA and/or the PLOT commands.\n\nMISSING DATA CORRELATES\n\nIn the second use of the AUXILIARY option, it is used in conjunction with TYPE=GENERAL with continuous dependent variables and maximum likelihood estimation to identify a set of variables that will be used as missing data correlates in addition to the analysis variables (Collins, Schafer, & Kam, 2001; Graham, 2003; Asparouhov & Muthén, 2008b; Enders, 2010). This use is not available with MODINDICES, BOOTSTRAP, and models with a set of exploratory factor analysis (EFA) factors in the MODEL command. The setting M in parentheses is placed behind the variables on the AUXILIARY statement that will be used as missing data correlates. Following is an example of how to specify the M setting:\n\nAUXILIARY = z1-z4 (M);\n\nwhere z1, z2, z3, and z4 are variables that will be used as missing data correlates in addition to the analysis variables.\n\nAn alternative specification that is convenient when there are several variables that cannot be specified using the list function is:\n\nAUXILIARY = (M) x1 x3 x5 x7 x9;\n\nwhere all variables after (M) will be used as missing data correlates in addition to the analysis variables.\n\n3-STEP APPROACH\n\nIn the third use of the AUXILIARY option, the 3-step approach using TYPE=MIXTURE is automatically carried out. There are eight settings of the AUXILIARY option that automatically carry out the 3-step approach. Two of these settings are used to identify a set of variables not used in the first step of the analysis that are possible covariates in a multinomial logistic regression for a categorical latent variable. The multimonial logistic regression uses all covariates at the same time. Six of the settings are used to identify a set of variables not used in the first step of the analysis for which the equality of means across latent classes will be tested. The equality of means is tested one variable at a time. Only one of these eight settings can be used in an analysis at a time. Only one categorical latent variable is allowed with the 3-step approach. The manual 3-step approach in described in Asparouhov and Muthen (2014a, b).\n\nThe two settings that are used to identify a set of variables not used in the first step of the analysis that are possible covariates in a multinomial logistic regression for a categorical latent variable are R3STEP (Vermunt, 2010; Asparouhov & Muthén, 2012b) and R (Wang et al., 2005). R3STEP is preferred. R is superseded by R3STEP and should be used only for methods research.\n\nThe six settings that are used to identify a set of variables not used in the first step of the analysis for which the equality of means across latent classes will be tested are BCH (Bakk & Vermunt, 2015), DU3STEP (Asparouhov & Muthén, 2012b), DCATEGORICAL (Lanza et al., 2013), DE3STEP (Asparouhov & Muthén, 2012b), DCONTINUOUS (Lanza et al., 2013), and E (Asparouhov, 2007). BCH is preferred for continuous distal outcomes. DU3STEP should be used only when there are no class changes between the first and last steps. DCATEGORICAL is for categorical distal outcomes. The following settings for continuous distal outcomes, DE3STEP, DCONTINUOUS, and E, should be used only for methods research.\n\nAll of the settings are specified in the same way. The setting in parentheses is placed behind the variables on the AUXILIARY statement that will be used as covariates in the multinomial logistic regression or for which the equality of means will be tested. Following is an example of how to specify the R3STEP setting:\n\nAUXILIARY = race (R3STEP) ses (R3STEP) x1-x5 (R3STEP);\n\nwhere race, ses, x1, x2, x3, x4, and x5 will be used as covariates in a multinomial logistic regression in a mixture model.\n\nAn alternative specification for the eight settings that is convenient when there are several variables that cannot be specified using the list function is:\n\nAUXILIARY = (R3STEP) x1 x3 x5 x7 x9;\n\nwhere all variables after R3STEP) will be used as covariates in a multinomial logistic regression in a mixture model.\n\nFollowing is an example of how to specify more than one setting in the same AUXILIARY statement:\n\nAUXILIARY = gender age (BCH) educ ses (BCH) x1-x5 (BCH);\n\nwhere all of the variables on the AUXILIARY statement will be saved if the SAVEDATA command is used, will be available for plots if the PLOT command is used, and tests of equality of means across latent classes will be carried out for the variables age, ses, x1, x2, x3, x4, and x5.\n\nCONSTRAINT\n\nThe CONSTRAINT option is used to identify the variables that can be used in the MODEL CONSTRAINT command. These can be not only variables used in the MODEL command but also other variables. All variables on the CONSTRAINT list are treated as continuous variables in the analysis. Only variables used by the following options cannot be included: GROUPING, PATTERN, COHORT, COPATTERN, CLUSTER, STRATIFICATION, and AUXILIARY. Variables that are part of these options can be used in DEFINE to create new variables that can be used in the CONSTRAINT statement. The CONSTRAINT option is not available for TYPE=RANDOM, TYPE=TWOLEVEL, TYPE=THREELEVEL, TYPE=CROSSCLASSIFIED, TYPE=COMPLEX, and for estimators other than ML, MLR, and MLF. The CONSTRAINT option is specified as follows:\n\nCONSTRAINT = y1 u1;\n\nwhere y1 and u1 are variables that can be used in the MODEL CONSTRAINT command.\n\nPATTERN\n\nThe PATTERN option is used when data are missing by design. The typical use is in situations when, because of the design of the study, all variables are not measured on all individuals in the analysis. This can occur, for example, when individuals receive different forms of a measurement instrument that contain different sets of items. Following is an example of how the PATTERN option is specified:\n\nPATTERN IS design (1= y1 y3 y5 2= y2 y3 y4 3= y1 y4 y5);\n\nwhere design is a variable in the data set that has integer values of 1, 2, and 3. The variable names listed after each number and the equal sign are variables used in the analysis which should have no missing values for observations with that value on the pattern variable. For example, observations with the value of one on the variable design should have information for variables y1, y3, and y5 and have missing values for y2 and y4. Observations with the value of three on the variable design should have information for variables y1, y4, and y5 and have missing values for variables y2 and y3. The pattern variable must contain only integer values. Observations that have a value for the pattern variable that is not specified using the PATTERN option are not included in the analysis.\n\nThere are several options that are used for complex survey data. These include options for stratification, clustering, unequal probabilities of selection (sampling weights), and subpopulation analysis. The variables used with these options can be variables from the NAMES statement of the VARIABLE command and variables created using the DEFINE command and the DATA transformation commands. The exception is that variables used with the SUBPOPULATION option must be variables from the NAMES statement of the VARIABLE command. Following is a description of these options and their specifications.\n\nSTRATIFICATION\n\nThe STRATIFICATION option is used with TYPE=COMPLEX to identify the variable in the data set that contains information about the subpopulations from which independent probability samples are drawn. Following is an example of how the STRATIFICATION option is used:\n\nSTRATIFICATION IS region;\n\nwhere region is the variable that contains stratification information.\n\nThe CLUSTER option is used with TYPE=TWOLEVEL, TYPE=THREELEVEL, TYPE=CROSSCLASSIFIED, and TYPE=COMPLEX to identify the variables in the data set that contain clustering information. One cluster variable is used for TYPE=TWOLEVEL and TYPE=COMPLEX. Two cluster variables are used for TYPE=THREELEVEL, TYPE=CROSSCLASSIFIED, and TYPE=COMPLEX TWOLEVEL. Three cluster variables are used for TYPE=COMPLEX THREELEVEL.\n\nFollowing is an example of how the CLUSTER option is used with TYPE=TWOLEVEL and TYPE=COMPLEX:\n\nCLUSTER IS school;\n\nwhere school is the variable that contains clustering information.\n\nFollowing is an example of how the CLUSTER option is used with TYPE=THREELEVEL:\n\nCLUSTER IS school class;\n\nwhere school and class are the variables that contain clustering information. The cluster variable for the highest level must come first, that is, classrooms are nested in schools.\n\nFollowing is an example of how the CLUSTER option is used with TYPE=CROSSCLASSIFIED:\n\nCLUSTER = neighbor school;\n\nwhere neighbor and school are the variables that contain clustering information. Students are nested in schools crossed with neighborhoods.\n\nFollowing is an example of how the CLUSTER option is used with TYPE=CROSSCLASSIFIED and time series analysis:\n\nCLUSTER = subject time;\n\nwhere subject and time are the variables that contain clustering information. In cross-classified time series analysis, subject and time are crossed. There is no nesting because each subject is observed only once at any one time. The cluster variable for subject must precede the cluster variable for time. Within each cluster, data must be ordered by time.\n\nTYPE=COMPLEX TWOLEVEL can be used with either two cluster variables, one stratification and two cluster variables, or one stratification and one cluster variable. Following is an example of using two cluster variables:\n\nCLUSTER = school class;\n\nwhere school and class are the variables that contain clustering information. The clusters for TYPE=TWOLEVEL are classroom. The standard error and chi-square computations for TYPE=COMPLEX are based on school.\n\nFollowing is an example with stratification and clustering:\n\nSTRATIFICATION = region;\n\nCLUSTER = school;\n\nwhere the clusters for TYPE=TWOLEVEL are schools and the standard error and chi-square computations for TYPE=COMPLEX are based on region.\n\nTYPE=COMPLEX THREELEVEL can be used with either three cluster variables, one stratification and three cluster variables, or one stratification and two cluster variables. Following is an example of how the CLUSTER option is used with TYPE=COMPLEX THREELEVEL:\n\nCLUSTER IS psu school class;\n\nwhere psu, school, and class are the variables that contain clustering information. The cluster variable for the highest level must come first, that is, classrooms are nested in schools and schools are nested in psu’s. The clusters for TYPE=THREELEVEL are classroom and school. The standard error and chi-square computations for TYPE=COMPLEX are based on psu’s.\n\nWEIGHT\n\nThe WEIGHT option is used to identify the variable that contains sampling weight information for non-clustered data using TYPE=GENERAL, clustered data using TYPE=COMPLEX, and level 1 data using TYPE=TWOLEVEL and TYPE=THREELEVEL. Sampling weights are not available for TYPE=CROSSCLASSIFIED. Sampling weights are used when data have been collected with unequal probabilities of being selected. Sampling weight values must be non-negative real numbers. If the sum of the sampling weights is not equal to the total number of observations in the analysis data set, the weights are rescaled so that they sum to the total number of observations. Sampling weights are available for all analysis types. Sampling weights are available for ESTIMATOR=MLR, MLM, MLMV, WLS, WLSM, WLSMV, and ULS. There are two exceptions. They are not available for WLS when all dependent variables are continuous and are not available for MLM or MLMV for EFA. Following is an example of how the WEIGHT option is used to identify a sampling weight variable:\n\nWEIGHT IS sampwgt;\n\nwhere sampwgt is the variable that contains sampling weight information.\n\nWTSCALE\n\nThe WTSCALE option is used with TYPE=TWOLEVEL to adjust the weight variable named using the WEIGHT option. With TYPE=TWOLEVEL, the WEIGHT option is used to identify the variable that contains within-level sampling weight information.\n\nThe WTSCALE option has the following settings: UNSCALED, CLUSTER, and ECLUSTER. CLUSTER is the default.\n\nThe UNSCALED setting uses the within weights from the data set with no adjustment. The CLUSTER setting scales the within weights from the data set so that they sum to the sample size in each cluster. The ECLUSTER setting scales the within weights from the data so that they sum to the effective sample size (Pothoff, Woodbury, & Manton, 1992).\n\nThe WTSCALE option is specified as follows:\n\nWTSCALE = ECLUSTER;\n\nwhere scaling the within weights so that they sum to the effective sample size is chosen.\n\nBWEIGHT\n\nThe BWEIGHT option is used with TYPE=TWOLEVEL to identify the variable that contains between-level sampling weight information.\n\nThe BWEIGHT option is specified as follows:\n\nBWEIGHT = bweight;\n\nwhere bweight is the variable that contains between-level sampling weight information.\n\nB2WEIGHT\n\nThe B2WEIGHT option is used with TYPE=THREELEVEL to identify the variable that contains level 2 sampling weight information. Sampling weights are not available for TYPE=CROSSCLASSIFIED.\n\nThe B2WEIGHT option is specified as follows:\n\nB2WEIGHT = b2weight;\n\nwhere b2weight is the variable that contains level 2 sampling weight information.\n\nB3WEIGHT\n\nThe B3WEIGHT option is used with TYPE=THREELEVEL to identify the variable that contains level 3 sampling weight information. Sampling weights are not available for TYPE=CROSSCLASSIFIED.\n\nThe B3WEIGHT option is specified as follows:\n\nB3WEIGHT = b3weight;\n\nwhere b3weight is the variable that contains level 3 sampling weight information.\n\nBWTSCALE\n\nThe BWTSCALE option is used in with TYPE=TWOLEVEL to adjust the between-level sampling weight variable named using the BWEIGHT option. The BWTSCALE option is used in with TYPE=THREELEVEL to adjust the level 2 and level 3 sampling weight variables named using the B2WEIGHT and B3WEIGHT options.\n\nThe BWTSCALE option has the following settings: UNSCALED and SAMPLE. SAMPLE is the default.\n\nThe UNSCALED setting uses the between weights from the data set with no adjustment. The SAMPLE option adjusts the between weights so that the product of the between and the within weights sums to the total sample size.\n\nThe BWTSCALE option is specified as follows:\n\nBWTSCALE = UNSCALED;\n\nwhere no adjustment is made to the between weights.\n\nREPWEIGHTS\n\nReplicate weights summarize information about a complex sampling design (Korn & Graubard, 1999; Lohr, 1999). They are used to properly estimate standard errors of parameter estimates. Replicate weights are available for TYPE=COMPLEX for continuous variables using maximum likelihood estimation and for binary, ordered categorical, and censored variables using weighted least squares estimation. The SUBPOPULATION option is not available with replicate weights. Replicate weights can be used or generated. When they are generated, they can be used in the analysis and/or saved (Asparouhov & Muthén, 2009b).\n\nThe REPWEIGHTS option is used to identify the replicate weight variables. The STRATIFICATION and CLUSTER options may not be used in conjunction with the REPWEIGHTS option. The WEIGHT option must be used. Following is an example of how to specify the REPWEIGHTS option:\n\nREPWEIGHTS = rweight1-rweight80;\n\nwhere rweight1 through rweight80 are replicate weight variables.\n\nUSING REPLICATE WEIGHTS\n\nWhen existing replicate weights are used, the REPWEIGHTS option of the VARIABLE command is used in conjunction with the WEIGHT option of the VARIABLE command and the REPSE option of the ANALYSIS command. The sampling weights are used in the estimation of parameter estimates. The replicate weights are used in the estimation of standard errors of parameter estimates. The REPSE option specifies the resampling method that is used in the computation of the standard errors.\n\nGENERATING REPLICATE WEIGHTS\n\nWhen replicate weights are generated, the REPSE option of the ANALYSIS command and the WEIGHT option of the VARIABLE command along with the CLUSTER and/or the STRATIFICATION options of the VARIABLE command are used.\n\nSUBPOPULATION\n\nThe SUBPOPULATION option is used with TYPE=COMPLEX to select observations for an analysis when a subpopulation (domain) is analyzed. If the SUBPOPULATION option is used, the USEOBSERVATIONS option cannot be used. When the SUBPOPULATION option is used, all observations are included in the analysis although observations not in the subpopulation are assigned weights of zero (see Korn & Graubard, 1999, pp. 207-211). The SUBPOPULATION option is not available for multiple group analysis.\n\nThe SUBPOPULATION option identifies those observations for analysis that satisfy the conditional statement specified after the equal sign and assigns them non-zero weights. For example, the following statement identifies observations with the variable gender equal to 2:\n\nSUBPOPULATION = gender EQ 2;\n\nOnly variables from the NAMES statement of the VARIABLE command can be used in the conditional statement of the SUBPOPULATION option. Logical operators, not arithmetic operators, must be used in the conditional statement. Following are the logical operators that can be used in the conditional statement of the SUBPOPULATION option:\n\nAND logical and\n\nOR logical or\n\nNOT logical not\n\nEQ equal ==\n\nNE not equal /=\n\nGE greater than or equal to >=\n\nLE less than or equal to <=\n\nGT greater than >\n\nLT less than <\n\nAs shown above, some of the logical operators can be referred to in two different ways. For example, equal can be referred to as EQ or ==.\n\nFINITE\n\nFor TYPE=COMPLEX, the FINITE option is used to identify the variable that contains the finite population correction factor, the sampling fraction, or the population size for each stratum. If the sampling fraction or the population size for each stratum is provided, these are used to compute the finite population correction factor. The finite population correction factor is used to adjust standard errors when clusters have been sampled without replacement (WOR) from strata in a finite population. The finite population correction factor is equal to one minus the sampling fraction. The sampling fraction is equal to the number of sampled clusters in a stratum divided by the number of clusters in the population for that stratum. The population size for each stratum is the number of clusters in the population for that stratum. The FINITE option is not available with replicate weights.\n\nThe FINITE option has three settings: FPC, SFRACTION, and POPULATION. FPC is used when the finite population correction factor is provided. SFRACTION is used when the sampling fraction is provided. POPULATION is used when the population size for each stratum is provided. FPC is the default. Following is an example of how the FINITE option is used to identify a finite population correction variable:\n\nFINITE IS sampfrac (SFRACTION);\n\nwhere sampfrac is the variable that contains the sampling fraction for each stratum.\n\nMIXTURE MODELS\n\nThere are three options that are used specifically for mixture models. They are CLASSES, KNOWNCLASS, and TRAINING.\n\nThe CLASSES option is used to assign names to the categorical latent variables in the model and to specify the number of latent classes in the model for each categorical latent variable. This option is required for TYPE=MIXTURE. Between-level categorical latent variables must be identified as between-level variables using the BETWEEN option.\n\nFollowing is an example of how the CLASSES option is used:\n\nCLASSES = c1 (2) c2 (2) c3 (3);\n\nwhere c1, c2, and c3 are the names of the three categorical latent variables in the model. The numbers in parentheses specify the number of classes for each categorical latent variable in the model. The categorical latent variable c1 has two classes, c2 has two classes, and c3 has three classes.\n\nWhen there is more than one categorical latent variable in the model, there are rules related to the order of the categorical latent variables. The order is taken from the order of the categorical latent variables in the CLASSES statement. Because of the order in the CLASSES statement above, c1 is not allowed to be regressed on c2 in the model. It is only possible to regress c2 on c1 and c3 on c2 or c1. This order restriction does not apply to PARAMETERIZATION=LOGLINEAR.\n\nKNOWNCLASS\n\nThe KNOWNCLASS option is used for multiple group analysis with TYPE=MIXTURE. The KNOWNCLASS option is used to identify the categorical latent variable for which latent class membership is known and equal to observed groups in the sample. Only one KNOWNCLASS variable can be used. Following is an example of how to specify the KNOWNCLASS option:\n\nKNOWNCLASS = c1 (gender = 0 1);\n\nwhere c1 is a categorical latent variable named and defined using the CLASSES option. The information in parentheses following the categorical latent variable name defines the known classes using an observed variable. In this example, the observed variable gender is used to define the known classes. The first class consists of individuals with the value 0 on the variable gender. The second class consists of individuals with the value 1 on the variable gender.\n\nFollowing is an example with many known classes that uses the list function:\n\nKNOWNCLASS = c (country = 101-110 112 113-115);\n\nwhere c is a categorical latent variable named and defined using the CLASSES option. The information in parentheses following the categorical latent variable name defines the known classes using an observed variable. In this example, the observed variable country is used to define the known classes. The first class consists of individuals with the value 101 on the variable country. The last class consists of individuals with the value 115 on the variable country. There are a total of 14 classes.\n\nThe KNOWNCLASS option can be specified by mentioning only the name of the observed variable that defines the known classes without giving the values of the observed variable:\n\nKNOWNCLASS = c (country);\n\nwhere c is a categorical latent variable named and defined using the CLASSES option. In this example, the observed variable country is used to define the known classes. The number of known classes is taken from the CLASSES option. The values of the variable country are taken from the data. The first class consists of individuals with the lowest value on the variable country. The last class consists of individuals with the highest value on the variable country.\n\nThe TRAINING option is used to identify the variables that contain information about latent class membership and specify whether the information is about class membership, probability of class membership, or priors of class membership. The TRAINING option has three settings: MEMBERSHIP, PROBABILITIES, and PRIORS. MEMBERSHIP is the default. Training variables can be variables from the NAMES statement of the VARIABLE command and variables created using the DEFINE command and the DATA transformation commands. This option is available only for models with a single categorical latent variable.\n\nFollowing is an example of how the TRAINING option is used for an example with three latent classes where the training variables contain information about class membership:\n\nTRAINING = t1 t2 t3;\n\nwhere t1, t2, and t3 are variables that contain information about latent class membership. The variable t1 provides information about membership in class 1, t2 provides information about membership in class 2, and t3 provides information about membership in class 3. An individual is allowed to be in any class for which they have a value of one on a training variable. An individual who is known to be in class 2 would have values of 0, 1, and 0 on t1, t2, and t3, respectively. An individual with unknown class membership would have the value of 1 on t1, t2, and t3. An alternative specification is:\n\nTRAINING = t1 t2 t3 (MEMBERSHIP);\n\nFractional values can be used to provide information about the probability of class membership when class membership is not estimated but is fixed at fractional values for each individual. For example, an individual who has a probability of .9 for being in class 1, .05 for being in class 2, and .05 for being in class 3 would have t1=.9, t2=.05, and t3=.05. Fractional training data must sum to one for each individual.\n\nFollowing is an example of how the TRAINING option is used for an example with three latent classes where the training variables contain information about probabilities of class membership:\n\nTRAINING = t1 t2 t3 (PROBABILITIES);\n\nwhere t1, t2, and t3 are variables that contain information about the probability of latent class membership. The variable t1 provides information about the probability of membership in class 1, t2 provides information about the probability of membership in class 2, and t3 provides information about the probability of membership in class 3.\n\nPriors can be used when individual class membership is not known but when information is available on the probability of an individual being in a certain class. For example, an individual who has a probability of .9 for being in class 1, .05 for being in class 2, and .05 for being in class 3 would have t1=.9, t2=.05, and t3=.05. Prior values must sum to one for each individual.\n\nFollowing is an example of how the PRIORS option is used for an example with three latent classes where the training variables contain information about priors of class membership:\n\nTRAINING = t1 t2 t3 (PRIORS);\n\nwhere t1, t2, and t3 are variables that contain information about the probability of being in a certain class. The variable t1 provides information about the probability of membership in class 1, t2 provides information about the probability of membership in class 2, and t3 provides information about the probability of membership in class 3.\n\nMULTILEVEL MODELS\n\nThere are two options specific to multilevel models. They are WITHIN and BETWEEN. Variables identified using the WITHIN and BETWEEN options can be variables from the NAMES statement of the VARIABLE command and variables created using the DEFINE command and the DATA transformation commands.\n\nWITHIN\n\nThe WITHIN option is used with TYPE=TWOLEVEL, TYPE=THREELEVEL, and TYPE=CROSSCLASSIFIED to identify the variables in the data set that are measured on the individual level and to specify the levels on which they are modeled. All variables on the WITHIN list must be measured on the individual level. An individual-level variable can be modeled on all or some levels.\n\nFor TYPE=TWOLEVEL, an individual-level variable can be modeled on only the within level or on both the within and between levels. If a variable measured on the individual level is mentioned on the WITHIN list, it is modeled on only the within level. It has no variance in the between part of the model. If it is not mentioned on the WITHIN list, it is modeled on both the within and between levels. The WITHIN option is specified as follows:\n\nWITHIN = y1 y2 x1;\n\nwhere y1, y2, and x1 are variables measured on the individual level and modeled on only the within level.\n\nFor TYPE=THREELEVEL, an individual-level variable can be modeled on only level 1, on levels 1 and 2, levels 1 and 3, or on all levels. Consider a model where students are nested in classrooms and classrooms are nested in schools. Level 1 is student; level 2 is classroom; and level 3 is school. If a variable measured on the individual level is mentioned on the WITHIN list without a label, it is modeled on only level 1. It has no variance on levels 2 and 3. If it is mentioned on the WITHIN list with a level 2 cluster label, it is modeled on levels 1 and 2. It has no variance on level 3. If it is mentioned on the WITHIN list with a level 3 cluster label, it is modeled on levels 1 and 3. It has no variance on level 2. If it is not mentioned on the WITHIN list, it is modeled on all levels.\n\nFollowing is an example of how to specify the WITHIN option for TYPE=THREELEVEL:\n\nWITHIN = y1-y3 (class) y4-y6 (school) y7-y9;\n\nIn the example, y1, y2, and y3 are variables measured on the individual level and modeled on only level 1, student. Variables modeled on only level 1 must precede variables modeled on the other levels. Y4, y5, and y6 are variables measured on the individual level and modeled on level 1, student, and level 2, class, where class is the level 2 cluster variable. Y7, y8, and y9 are variables measured on the individual level and modeled on level 1, student, and level 3, school, where school is the level 3 cluster variable.\n\nAn alternative specification of the WITHIN option above reverses the order of the level 2 and level 3 variables:\n\nWITHIN = y1-y3 (school) y7-y9 (class) y4-y6;\n\nVariables modeled on only level 1 must precede variables modeled on the other levels. Another alternative specification is:\n\nWITHIN = y1-y3;\n\nWITHIN = (class) y4-y6;\n\nWITHIN = (school) y7-y9;\n\nIn this specification, the WITHIN statement for variables modeled on only level 1 must precede the other WITHIN statements. The order of the other WITHIN statements does not matter.\n\nFor TYPE=CROSSCLASSIFIED, an individual-level variable can be modeled on only level 1, on levels 1 and 2a, levels 1 and 2b, or on all levels. Consider a model where students are nested in schools crossed with neighborhoods. Level 1 is student; level 2a is school; and level 2b is neighborhood. If a variable measured on the individual level is mentioned on the WITHIN list without a label, it is modeled on only level 1. It has no variance on levels 2a and 2b. If it is mentioned on the WITHIN list with a level 2a cluster label, it is modeled on levels 1 and 2a. It has no variance on level 2b. If it is mentioned on the WITHIN list with a level 2b cluster label, it is modeled on levels 1 and 2b. It has no variance on level 2a. If it is not mentioned on the WITHIN list, it is modeled on all levels.\n\nFollowing is an example of how to specify the WITHIN option for TYPE=CROSSCLASSIFIED:\n\nWITHIN = y1-y3 (school) y4-y6 (neighbor) y7-y9;\n\nIn the example, y1, y2, and y3 are variables measured on the individual level and modeled on only level 1, student. Variables modeled on only level 1 must precede variables modeled on the other levels. Y4, y5, and y6 are variables measured on the individual level and modeled on level 1, student, and level 2a, school, where school is the level 2a cluster variable. Y7, y8, and y9 are variables measured on the individual level and modeled on level 1, student, and level 2b, neighborhood, where neighborhood is the level 2b cluster variable.\n\nBETWEEN\n\nThe BETWEEN option is used with TYPE=TWOLEVEL, TYPE=THREELEVEL, and TYPE=CROSSCLASSIFIED to identify the variables in the data set that are measured on the cluster level(s) and to specify the level(s) on which they are modeled. All variables on the BETWEEN list must be measured on a cluster level. A cluster-level variable can be modeled on all or some cluster levels.\n\nFor TYPE=TWOLEVEL, a cluster-level variable can be modeled on only the between level. The BETWEEN option is specified as follows:\n\nBETWEEN = z1 z2 x1;\n\nwhere z1, z2, and x1 are variables measured on the cluster level and modeled on the between level. The BETWEEN option is also used to identify between-level categorical latent variables with TYPE=TWOLEVEL MIXTURE.\n\nFor TYPE=THREELEVEL, a variable measured on level 2 can be modeled on only level 2 or on levels 2 and 3. A variable measured on level 3 can be modeled on only level 3. Consider a model where students are nested in classrooms and classrooms are nested in schools. Level 1 is student; level 2 is classroom; and level 3 is school. If a variable measured on level 2 is mentioned on the BETWEEN list without a label, it is modeled on levels 2 and 3. If a variable measured on level 2 is mentioned on the BETWEEN list with a level 2 cluster label, it is modeled on only level 2. It has no variance on level 3. A variable measured on level 3 must be mentioned on the BETWEEN list with a level 3 cluster label. Following is an example of how to specify the BETWEEN option for TYPE=THREELEVEL:\n\nBETWEEN = y1-y3 (class) y4-y6 (school) y7-y9;\n\nIn this example, y1, y2, and y3 are cluster-level variables measured on level 2, class, and modeled on both levels 2 and 3. Variables modeled on both levels 2 and 3 must precede variables modeled on only level 2 or level 3. Y4, y5, and y6 are cluster-level variables measured on level 2, class, and modeled on level 2. Y7, y8, and y9 are cluster-level variables measured on level 3 and modeled on level 3.\n\nAn alternative specification of the BETWEEN option above reverses the order of the level 2 and level 3 variables:\n\nBETWEEN = y1-y3 (school) y7-y9 (class) y4-y6;\n\nVariables modeled on both levels 2 and 3 must precede variables modeled on only level 2 or level 3. Another alternative specification is:\n\nBETWEEN = y1-y3;\n\nBETWEEN = (class) y4-y6;\n\nBETWEEN = (school) y7-y9;\n\nIn this specification, the BETWEEN statement for variables modeled on both levels 2 and 3 must precede the other BETWEEN statements. The order of the other BETWEEN statements does not matter.\n\nFor TYPE=CROSSCLASSIFIED, a variable measured on level 2a must be mentioned on the BETWEEN list with a level 2a cluster label. It can be modeled on only level 2a. A variable measured on level 2b must be mentioned on the BETWEEN list with a level 2b cluster label. It can be modeled on only level 2b. Consider a model where students are nested in schools crossed with neighborhoods. Level 1 is student; level 2a is school; and level 2b is neighborhood. Following is an example of how to specify the BETWEEN option for TYPE=CROSSCLASSIFIED:\n\nBETWEEN = (school) y1-y3 (neighbor) y4-y6;\n\nIn this example, y1, y2, and y3 are cluster-level variables measured on level 2a, school, and modeled on only level 2a. Y4, y5, and y6 are cluster-level variables measured on level 2b, neighborhood, and modeled on only level 2b.\n\nCONTINUOUS-TIME SURVIVAL MODELS\n\nThere are two options specific to continuous-time survival models. They are SURVIVAL and TIMECENSORED. Variables identified using the SURVIVAL and TIMECENSORED options can be variables from the NAMES statement of the VARIABLE command and variables created using the DEFINE command and the DATA transformation commands.\n\nSURVIVAL\n\nThe SURVIVAL option is used to identify the variables that contain information about time to event and to provide information about the number and lengths of the time intervals in the baseline hazard function to be used in the analysis. The SURVIVAL option must be used in conjunction with the TIMECENSORED option. The SURVIVAL option can be specified in five ways: the default baseline hazard function, a non-parametric baseline hazard function, a semi-parametric baseline hazard function, a parametric baseline hazard function, and a constant baseline hazard function.\n\nThe SURVIVAL option is specified as follows when using the default baseline hazard function:\n\nSURVIVAL = t;\n\nwhere t is the variable that contains time-to-event information. The default is either a semi-parametric baseline hazard function with ten time intervals or a non-parametric baseline hazard function. The default is a semi-parametric baseline hazard function with ten time intervals for models where t is regressed on a continuous latent variable, for multilevel models, and for models that require Monte Carlo numerical integration. In this case, the lengths of the time intervals are selected internally in a non-parametric fashion. For all other models, the default is a non-parametric baseline hazard function as in Cox regression where the number and lengths of the time intervals are taken from the data and the baseline hazard function is saturated.\n\nThe SURVIVAL option is specified as follows when using a non-parametric baseline hazard function as in Cox regression:\n\nSURVIVAL = t (ALL);\n\nwhere t is the variable that contains time-to-event information and ALL is a keyword that specifies that the number and lengths of the time intervals are taken from the data and the baseline hazard is saturated. It is not recommended to use the keyword ALL when the BASEHAZARD option of the ANALYSIS command is ON because it results in a large number of baseline hazard parameters.\n\nThe SURVIVAL option is specified as follows when using a semi-parametric baseline hazard:\n\nSURVIVAL = t (10);\n\nwhere t is the variable that contains time-to-event information. The number in parentheses specifies that 10 intervals are used in the analysis where the lengths of the time intervals are selected internally in a non-parametric fashion.\n\nThe SURVIVAL option is specified as follows when using a parametric baseline hazard function:\n\nSURVIVAL = t (45 110);\n\nwhere t is the variable that contains time-to-event information. The numbers in parentheses specify that four time intervals of length five and one time interval of length ten are used in the analysis.\n\nThe SURVIVAL option is specified as follows when using a constant baseline hazard function:\n\nSURVIVAL = t (CONSTANT);\n\nwhere t is the variable that contains time-to-event information and CONSTANT is the keyword that specifies a constant baseline hazard function.\n\nTIMECENSORED\n\nThe TIMECENSORED option is used in conjunction with the SURVIVAL option to identify the variables that contain information about right censoring, for example, when an individual leaves a study or when an individual has not experienced the event before the study ends. There must be the same number and order of variables in the TIMECENSORED option as there are in the SURVIVAL option. The variables that contain information about right censoring must be coded so that zero is not censored and one is right censored. If they are not, this can be specified as part of the TIMECENSORED option. The TIMECENSORED option is specified as follows when the variable is coded zero for not censored and one for right censored:\n\nTIMECENSORED = tc;\n\nThe TIMECENSORED option is specified as follows when the variable is not coded zero for not censored and one for right censored:\n\nTIMECENSORED = tc (1 = NOT 999 = RIGHT);\n\nThe value one is automatically recoded to zero and the value 999 is automatically recoded to one.\n\nLAGGED\n\nThe LAGGED option is used in time series analysis to specify the maximum lag to use for an observed variable during model estimation. Following is an example of how to specify the LAGGED option:\n\nLAGGED = y (1);\n\nwhere y is the variable in a time series analysis and the number 1 in parentheses is the maximum lag that will be used in model estimation. The lagged variable is referred to in the MODEL command by adding to the name of the variable an ampersand (&) and the number of the lag. The variable y at lag one is referred to as y&1.\n\nFollowing is an example of how to specify a maximum lag of 2 for a set of variables:\n\nLAGGED = y1-y3 (2);\n\nwhere y1, y2, and y3 are variables in a time series analysis and the number 2 in parentheses is the maximum lag that will be used in model estimation. The lagged variables are referred to in the MODEL command by adding to the name of the variable an ampersand (&) and the number of the lag. The variable y1 at lag one is referred to as y1&1. The variable y1 at lag two is referred to as y1&2.\n\nTINTERVAL\n\nThe TINTERVAL option is used in time series analysis to specify the time interval that is used to create a time variable when data are misaligned with respect to time due to missed measurement occasions that are not assigned a missing value flag and when measurement occasions are random. The data set must be sorted by the time interval variable.\n\nThe time interval value represents the difference in time between two consecutive measurement occasions. Using this value and the lowest value of the time interval variable, intervals are created that are used to create a time variable. The first interval is the lowest value of the time interval variable plus/minus half of the time interval value. The second interval adds the time interval value to the values of the first interval etc.. All values of the time interval variable that fall into the same interval are given the same value on the time variable. If an interval does not contain a value, a missing value flag is assigned. Following is an example of how to specify the TINTERVAL option:\n\nTINTERVAL = time (1);\n\nwhere time is the time interval variable and the value one is the time interval value. If the lowest time interval variable value is one, the first three intervals are .5 to 1.5, 1.5 to 2.5, and 2.5 to 3.5. The first three time variable values are 1, 2, and 3. For further details, see Asparouhov, Hamaker, and Muthén (2017).\n\nThe DEFINE command is used to transform existing variables and to create new variables. It includes several options for transforming variables including a do loop for repeating statements. The operations available in DEFINE can be executed for all observations or selectively using conditional statements. Transformations of existing variables do not affect the original data but only the data used for the analysis. If analysis data are saved using the SAVEDATA command, the transformed values rather than the original values are saved.\n\nThe statements in the DEFINE command are executed one observation at a time in the order specified with one exception. The CLUSTER_MEAN, CENTER, and STANDARDIZE options are executed in the order mentioned after all transformations specified before them in the DEFINE command and the DATA transformation commands are executed. All statements specified after these options are then executed one observation at a time in the order specified. These transformations use the new values from the CLUSTER_MEAN, CENTER, and STANDARDIZE options where applicable. For example, if two variables are centered and an interaction between them is specified after the CENTER option, the interaction uses the centered variables. Any variable listed in the NAMES option of the VARIABLE command or created in the DEFINE command can be transformed or used to create new variables. New variables created in the DEFINE command that will be used in an analysis must be listed on the USEVARIABLES list after the original variables. All statements specified after the CLUSTER_MEAN, CENTER and STANDARDIZE options must refer to variables used in the analysis that are listed on the USEVARIABLES list.\n\nFollowing are examples of the types of transformations available:\n\n DEFINE: variable = mathematical expression; IF (conditional statement) THEN transformation statements; _MISSING variable = MEAN (list of variables); variable = SUM (list of variables); CUT variable or list of variables (cutpoints); variable = CLUSTER_MEAN (variable); CENTER variable or list of variables (GRANDMEAN); CENTER variable or list of variables (GROUPMEAN); CENTER variable or list of variables (GROUPMEAN label); STANDARDIZE variable or list of variables; DO (number, number) expression; DO (\\$, number, number) DO (#, number, number) expression;\n\nDEFINE is not a required command.\n\nThe following logical operators can be used in DEFINE:\n\nAND logical and\n\nOR logical or\n\nNOT logical not\n\nEQ = = equal\n\nNE /= not equal\n\nGE >= greater than or equal to\n\nLE <= less than or equal to\n\nGT > greater than\n\nLT < less than\n\nAs shown above, some of the logical operators can be referred to in two different ways. For example, equal can be referred to as EQ or = =.\n\nThe following arithmetic operations can be used in DEFINE:\n\n+ addition y + x;\n\n- subtraction y - x;\n\n multiplication y * x;\n\n/ division y / x;\n\n exponentiation y2;\n\n% remainder remainder of y/x;\n\nThe following functions can be used in DEFINE:\n\nLOG base e log LOG (y);\n\nLOG10 base 10 log LOG10 (y);\n\nEXP exponential EXP (y);\n\nSQRT square root SQRT (y);\n\nABS absolute value ABS(y);\n\nSIN sine SIN (y);\n\nCOS cosine COS (y);\n\nTAN tangent TAN(y);\n\nASIN arcsine ASIN (y);\n\nACOS arccosine ACOS (y);\n\nATAN arctangent ATAN (y);\n\nPHI standard normal PHI (y); PHI (#);\n\ndistribution function\n\nWhen a non-conditional statement is used to transform existing variables or create new variables, the variable on the left-hand side of the equal sign is assigned the value of the expression on the right-hand side of the equal sign, for example,\n\ny = y/100;\n\ntransforms the original variable y by dividing it by 100. Non-conditional statements can be used to create new variables, for example,\n\nabuse = item1 + item2 + item8 + item9;\n\nAn individual with a missing value on any variable used on the right-hand side of the equal sign is assigned a missing value on the variable on the left-hand side of the equal sign.\n\nCONDITIONAL STATEMENTS\n\nConditional statements can also be used to transform existing variables and to create new variables. Conditional statements take the following form:\n\nIF (gender EQ 1 AND ses EQ 1) THEN group = 1;\n\nIF (gender EQ 1 AND ses EQ 2) THEN group = 2;\n\nIF (gender EQ 1 AND ses EQ 3) THEN group = 3;\n\nIF (gender EQ 2 AND ses EQ 1) THEN group = 4;\n\nIF (gender EQ 2 AND ses EQ 2) THEN group = 5;\n\nIF (gender EQ 2 AND ses EQ 3) THEN group = 6;\n\nAn individual with a missing value on any variable used in the conditional statement is assigned a missing value on the new variable. If no value is specified for a condition, individuals with that condition are assigned a missing value.\n\nThe _MISSING keyword can be used in DEFINE to refer to missing values. The _MISSING keyword can be used to assign a missing value to a variable, for example,\n\nIF (y EQ 0) THEN u = _MISSING;\n\nIt can also be used as part of the condition, for example,\n\nIF (y EQ _MISSING) THEN u = 1;\n\nOPTIONS FOR DATA TRANSFORMATION\n\nThe DEFINE command has six options for data transformation. The first option creates a variable that is the average of a set of variables. The second option creates a variable that is the sum of a set of variables. The third option categorizes one or several variables using the same set of cutpoints. The fourth option creates a variable that is the average for each cluster of an individual-level variable. The fifth option centers a variable by subtracting the grand mean or group mean from each value. The sixth option standardizes a variable to have a mean of zero and a standard deviation of one.\n\nMEAN\n\nThe MEAN option is used to create a variable that is the average of a set of variables. It is specified as follows:\n\nmean = MEAN (y1 y3 y5);\n\nwhere the variable mean is the average of variables y1, y3, and y5. Averages are based on the set of variables with no missing values. Any observation that has a missing value on all of the variables being averaged is assigned a missing value on the mean variable.\n\nThe list function can be used with the MEAN option as follows:\n\nymean = MEAN (y1-y10);\n\nwhere the variable ymean is the average of variables y1 through y10.\n\nVariables used with the MEAN option must be original variables from the NAMES statement of the VARIABLE command or temporary variables created using the DEFINE command. The order of the variables for the list function is taken from the NAMES statement not the USEVARIABLES statement.\n\nSUM\n\nThe SUM option is used to create a variable that is the sum of a set of variables. It is specified as follows:\n\nsum = SUM (y1 y3 y5);\n\nwhere the variable sum is the sum of variables y1, y3, and y5. Any observation that has a missing value on one or more of the variables being summed is assigned a missing value on the sum variable.\n\nThe list function can be used with the SUM option as follows:\n\nysum = SUM (y1-y10);\n\nwhere the variable ysum is the sum of variables y1 through y10.\n\nVariables used with the SUM option must be original variables from the NAMES statement of the VARIABLE command or temporary variables created using the DEFINE command. The order of the variables for the list function is taken from the NAMES statement not the USEVARIABLES statement.\n\nThe CUT option categorizes a variable or list of variables using the same set of cutpoints. More than one CUT statement can be included in the DEFINE command. Following is an example of how the CUT option is used:\n\nCUT y1 y5-y7 (30 40);\n\nThis statement results in the variables y1, y5, y6, and y7 having three categories: less than or equal to 30, greater than 30 and less than or equal to 40, and greater than 40, with values of 0, 1, and 2, respectively. Any observation that has a missing value on a variable that is being cut is assigned a missing value on the cut variable.\n\nVariables used with the CUT option must be original variables from the NAMES statement of the VARIABLE command or temporary variables created using the DEFINE command. The order of the variables for the list function is taken from the NAMES statement not the USEVARIABLES statement.\n\nCLUSTER_MEAN\n\nThe CLUSTER_MEAN option is used with TYPE=TWOLEVEL and TYPE=COMPLEX along with the CLUSTER option to create a variable that is the average of the values of an individual-level variable for each cluster. In multiple group analysis, each group’s means are used for creating cluster means in that group. It is specified as follows:\n\nclusmean = CLUSTER_MEAN (x);\n\nwhere the variable clusmean is the average of the values of x for each cluster. Averages are based on the set of non-missing values for the observations in each cluster. Any cluster for which all observations have missing values is assigned a missing value on the cluster mean variable.\n\nAll transformations specified in the DATA transformation commands or before the CLUSTER_MEAN option in the DEFINE command use the original values of the variables. All transformations specified in the DEFINE command after the CLUSTER_MEAN option use the cluster-mean values of the variables. To be used with the CLUSTER_MEAN option, any new variables created using the DEFINE command must be placed on the USEVARIABLES list after the original variables. The order of the variables for the list function is taken from the USEVARIALES list. If there is not USEVARIABLES list, the order of the variables is taken from the NAMES list.\n\nCENTER\n\nThe CENTER option is used to center continuous observed variables by subtracting the grand mean or group mean from each variable. In multiple group analysis, each group’s means are used for centering in that group. Grand-mean centering is available for all analyses. Group-mean centering is available with TYPE=TWOLEVEL, TYPE=THREELEVEL, TYPE=CROSSCLASSIFIED, and TYPE=COMPLEX in conjunction with the CLUSTER option. The CENTER option has two settings: GRANDMEAN and GROUPMEAN.\n\nAll transformations that are specified in the DATA transformation commands or before the CENTER option in the DEFINE command use the original values of the variables. All transformations that are specified in the DEFINE command after the CENTER option use the centered values of the variables. To be used with the CENTER option, any new variables created using the DEFINE command must be placed on the USEVARIABLES list after the original variables. The order of variables for the list function is taken from the USEVARIABLES list. If there is no USEVARIABLES list, the order of the variables is taken from the NAMES list. Any observation that has a missing value on the variable to be centered is assigned a missing value on the centered variable.\n\nGRANDMEAN\n\nGrand-mean centering subtracts the overall mean from a variable. Grand-mean centering is available for all continuous observed variables used in an analysis. Following is an example of how to specify grand-mean centering for TYPE=TWOLEVEL and TYPE=COMPLEX:\n\nCENTER x1-x4 (GRANDMEAN);\n\nwhere x1, x2, x3, and x4 are the variables to be centered using the overall means for these variables.\n\nFor TYPE=THREELEVEL, there are three types of grand-mean centering. Consider a model where students are nested in classrooms and classrooms are nested in schools. Level 1 is student; level 2 is classroom; and level 3 is school. For variables modeled on level 1 and higher, grand-mean centering subtracts the overall mean from a variable. For variables modeled on level 2 and higher, grand-mean centering subtracts the cluster 1, classroom, mean from a variable. For variables modeled on level 3, grand-mean centering subtracts the cluster 2, school, mean from a variable.\n\nFollowing is an example of how to specify grand-mean centering for TYPE=THREELEVEL:\n\nCENTER x1-x4 (GRANDMEAN);\n\nwhere x1, x2, x3, and x4 are the variables to be centered using the overall means for variables modeled on level 1 and higher, the cluster 1 means for variables modeled on level 2 and higher, and the cluster 2 means for variables modeled on level 3 and higher.\n\nFor TYPE=CROSSCLASSIFIED, there are three types of grand-mean centering. Consider a model where students are nested in schools crossed with neighborhoods. Level 1 is student; level 2a is school; and level 2b is neighborhood. For variables modeled on level 1 and higher, grand-mean centering subtracts the overall mean from a variable. For variables modeled on level 2a, grand-mean centering subtracts the cluster 2a, school, mean from a variable. For variables modeled on level 2b, grand-mean centering subtracts the cluster 2b, neighborhood, mean from a variable.\n\nFollowing is an example of how to specify grand-mean centering for TYPE=CROSSCLASSIFIED:\n\nCENTER x1-x4 (GRANDMEAN);\n\nwhere x1, x2, x3, and x4 are the variables to be centered using the overall means for variables modeled on level 1 and higher, the cluster 2a means for variables modeled on level 2a, and the cluster 2b means for variables modeled on level 2b.\n\nGROUPMEAN\n\nFor TYPE=TWOLEVEL and TYPE=COMPLEX, group-mean centering subtracts the cluster mean from a variable. For TYPE=COMPLEX, group-mean centering is available for all continuous observed variables in an analysis. For TYPE=TWOLEVEL, group-mean centering is available for all continuous observed variables used in an analysis that are not modeled at the highest level. Following is an example of how to specify group-mean centering:\n\nCENTER x1-x4 (GROUPMEAN);\n\nwhere x1, x2, x3, and x4 are the variables to be centered using the cluster means for these variables.\n\nFor TYPE=THREELEVEL, group-mean centering is available for any continuous observed variable that is not modeled at the highest level. Consider a model where students are nested in classrooms and classrooms are nested in schools. Level 1 is student; level 2 is classroom; and level 3 is school. For variables modeled on levels 1 and 2 and for variables modeled on only level 2, group-mean centering subtracts the cluster 2, school, mean from a variable where the cluster 2 mean is the average of the cluster 1, class, means for each level 3, school, cluster. Following is an example of how to specify cluster 2 group-mean centering:\n\nCENTER x1-x4 (GROUPMEAN school);\n\nwhere school is the cluster 2 variable and x1, x2, x3, and x4 are the variables to be centered using cluster 2 means for these variables.\n\nFor variables modeled on only level 1, both cluster 1, classroom, and cluster 2, school, group-mean centering are available. Following is an example of how to specify group-mean centering using cluster 1 and cluster 2 means:\n\nCENTER x1-x2 (GROUPMEAN class);\n\nCENTER x3 x4 (GROUPMEAN school);\n\nIn this example, class is the cluster 1 variable and x1 and x2 are the variables to be centered using cluster 1 means for these variables. School is the cluster 2 variable and x3 and x4 are the variables to be centered using cluster 2 means for these variables. A variable cannot be centered using both cluster 1 and cluster 2 means.\n\nFor TYPE=CROSSCLASSIFIED, group-mean centering is available for any continuous observed variable that is not modeled at the highest level. Consider a model where students are nested in schools crossed with neighborhoods. Level 1 is student; level 2a is school; and level 2b is neighborhood. For variables modeled on only level 1, group-mean centering subtracts the cluster 2a, school, mean or the cluster 2b, neighborhood mean from a variable. Following is an example of how to specify cluster 2a and 2b group-mean centering:\n\nCENTER x1-x4 (GROUPMEAN school);\n\nCENTER x5-x8 (GROUPMEAN neighbor);\n\nIn this example, school is the cluster 2a variable and x1, x2, x3, and x4 are the variables to be centered using cluster 2a means for these variables. Neighbor is the cluster 2b variable and x5, x6, x7, and x8 are the variables to be centered using the cluster 2b means for these variables.\n\nSTANDARDIZE\n\nThe STANDARDIZE option is used to standardize continuous variables by subtracting the mean from each value and dividing each value by the standard deviation. In multiple group analysis, each group’s means are used for standardizing in that group. Following is an example of how the STANDARDIZE option is used:\n\nSTANDARDIZE y1 y5-y10 y14;\n\nwhere the variables y1, y5 through y10, and y14 will be standardized. Any observation that has a missing value on the variable to be standardized is assigned a missing value on the standardized variable.\n\nAll transformations that are specified in the DATA transformation commands or before the STANDARDIZE option in the DEFINE command use the original values of the variables. All transformations that are specified in the DEFINE command after the STANDARDIZE option use the standardized values of the variables. To be used with the STANDARDIZE option, any new variables created using the DEFINE command must be placed on the USEVARIABLES list after the original variables. The order of variables for the list function is taken from the USEVARIABLES list. If there is no USEVARIABLES list, the order is taken from the NAMES list.\n\nDO\n\nThe DO option provides a do loop and a double do loop to facilitate specifying the same transformation for a set of variables. Following is an example of how to specify a do loop:\n\nDO (1, 5) diff# = y# - x#;\n\nwhere the numbers in parentheses give the range of values the do loop will use. The number sign (#) is replaced by these values during the execution of the do loop. Following are the transformations that are executed based on the DO option specified above:\n\ndiff1 = y1 - x1;\n\ndiff2 = y2 - x2;\n\ndiff3 = y3 - x3;\n\ndiff4 = y4 - x4;\n\ndiff5 = y5 - x5;\n\nFollowing is an example of how to specify a double do loop where x1 is equal to 0.1 and x2 is equal to 2:\n\nDO (\\$,1,2) DO (#,1,4) y\\$# = x\\$ * u#;\n\nwhere the numbers in parentheses give the range of values the double do loop will use. The numbers replace the symbol preceding them. Following are the transformations that are executed based on the DO option specified above:\n\ny11 = 0.1 * u1;\n\ny12 = 0.1 * u2;\n\ny13 = 0.1 * u3;\n\ny14 = 0.1 * u4;\n\ny21 = 2 * u1;\n\ny22 = 2 * u2;\n\ny23 = 2 * u3;\n\ny24 = 2 * u4;" ]	[ null, "https://www.statmodel.com/HTML_UG/chapter15v8_files/image001.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.83981645,"math_prob":0.9501582,"size":140844,"snap":"2019-13-2019-22","text_gpt3_token_len":31546,"char_repetition_ratio":0.22654076,"word_repetition_ratio":0.35511255,"special_character_ratio":0.21501803,"punctuation_ratio":0.10463662,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9822093,"pos_list":[0,1,2],"im_url_duplicate_count":[null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-03-19T01:31:26Z\",\"WARC-Record-ID\":\"<urn:uuid:68886035-4037-4694-a313-f938f76b89c0>\",\"Content-Length\":\"319249\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ef82385a-3ad4-4968-b3c0-a048f69d63bc>\",\"WARC-Concurrent-To\":\"<urn:uuid:d7115f90-1347-4bbf-be09-f338725428bc>\",\"WARC-IP-Address\":\"74.208.236.37\",\"WARC-Target-URI\":\"https://www.statmodel.com/HTML_UG/chapter15v8.htm\",\"WARC-Payload-Digest\":\"sha1:C7P3L6ELQ722BZAP3X3H2HGGMFAVYFV5\",\"WARC-Block-Digest\":\"sha1:Z2JSW7633QG7B3YKZF6YQRB2ZJI6PPKH\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-13/CC-MAIN-2019-13_segments_1552912201882.11_warc_CC-MAIN-20190319012213-20190319034213-00431.warc.gz\"}"}
https://math.stackexchange.com/questions/14160/continuous-curve-interpolating-a-list-of-points	[ "# Continuous curve interpolating a list of points\n\nI need a function (a curve -- preferably a simple one) that, given $n$ points of a 2D space ($R^2$) passes (interpolates) through all points in a smooth/continuous way.\n\nFound out that what I need is a spline, however cannot find one that behaves how I need. $n$-degree Bezier curves and B-splines don't pass through the $n$ points, just move from the first to the last one using the others as control points.\n\nA Bezier spline of $n-1$ Bezier curves (defined between each pair of adjacent points) should do the trick, but to have a smooth result I would need some control points which I don't have and don't know how to compute.\n\nAny hints?" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.96369904,"math_prob":0.9725629,"size":637,"snap":"2023-40-2023-50","text_gpt3_token_len":162,"char_repetition_ratio":0.12480253,"word_repetition_ratio":0.0,"special_character_ratio":0.24175824,"punctuation_ratio":0.0703125,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9977321,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-07T01:45:16Z\",\"WARC-Record-ID\":\"<urn:uuid:7715a4c0-052c-4a9f-af3e-fa084e74b2df>\",\"Content-Length\":\"144128\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7066b205-9786-4277-a712-1ac928ccdcc3>\",\"WARC-Concurrent-To\":\"<urn:uuid:78aff7c6-c883-4124-9880-c6e7cd429ee3>\",\"WARC-IP-Address\":\"172.64.144.30\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/14160/continuous-curve-interpolating-a-list-of-points\",\"WARC-Payload-Digest\":\"sha1:PLJZLESLFHAOZGQBTPNYDJJGFWTNKK7J\",\"WARC-Block-Digest\":\"sha1:HZNDAAP5U3KWQ4GOTFDIRB7T6FS4NZ5C\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100626.1_warc_CC-MAIN-20231206230347-20231207020347-00177.warc.gz\"}"}
https://web2.0calc.com/questions/linear-inequality_13	[ "+0\n\n# linear inequality\n\n0\n173\n1\n\nFind a linear inequality with the following solution set. Each grid line represents one unit.", null, "(Give your answer in the form Ax + By >= C or Ax + By > C where A, B, and C are integers with no common factor other than 1.)\n\nJul 4, 2022\n\n#1\n+1\n\nThe points (1,1) and (0,4) are on the graph\n\nThe slope is ( 4 - 1) /( 0 -1) = 3/-1 = -3\n\nWe either have\n\ny ≤ -3x + 4\n\nor\n\ny ≥ -3x + 4\n\n(0,0) makes the first true......but this point is not in the feasible region\n\nSo\n\ny ≥ -3x + 4 is true\n\nSo\n\n3x + y ≥ 4", null, "", null, "", null, "Jul 4, 2022" ]	[ null, "https://web2.0calc.com/api/ssl-img-proxy", null, "https://web2.0calc.com/img/emoticons/smiley-cool.gif", null, "https://web2.0calc.com/img/emoticons/smiley-cool.gif", null, "https://web2.0calc.com/img/emoticons/smiley-cool.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.85278255,"math_prob":0.9950238,"size":458,"snap":"2023-40-2023-50","text_gpt3_token_len":159,"char_repetition_ratio":0.1123348,"word_repetition_ratio":0.01923077,"special_character_ratio":0.39082968,"punctuation_ratio":0.12612613,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99701124,"pos_list":[0,1,2,3,4,5,6,7,8],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-30T03:21:05Z\",\"WARC-Record-ID\":\"<urn:uuid:92c91735-1be8-439b-bdc4-f2b66ade714b>\",\"Content-Length\":\"22280\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2767d87a-2515-4c84-b2c2-01f451a5d3f5>\",\"WARC-Concurrent-To\":\"<urn:uuid:ba33a52d-d7cc-45d7-896c-c55b99cd1f79>\",\"WARC-IP-Address\":\"168.119.251.111\",\"WARC-Target-URI\":\"https://web2.0calc.com/questions/linear-inequality_13\",\"WARC-Payload-Digest\":\"sha1:CSKMGSPKY37457IMTNKOBWPFYYSRPSE5\",\"WARC-Block-Digest\":\"sha1:WWXAAJXDJ5EVLDGM6J5JMF5IWUKDPQWW\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233510575.93_warc_CC-MAIN-20230930014147-20230930044147-00823.warc.gz\"}"}
https://www.gradesaver.com/textbooks/science/physics/college-physics-4th-edition/chapter-5-problems-page-185/12	[ "## College Physics (4th Edition)\n\nWe can rank the flywheels in order of radial acceleration of a point at the rim, from largest to smallest: $c \\gt d \\gt a \\gt b \\gt e$\nIn general: $\\omega = \\frac{2\\pi}{T}$ $v = \\omega~r$ $a_c = \\frac{v^2}{r}$ We can find the radial acceleration of a point at the rim of each flywheel: (a) $\\omega = \\frac{2\\pi}{T} = \\frac{2\\pi}{0.0040~s} = 1571~rad/s$ $v = (1571~rad/s)(0.080~m) = 125.7~m/s$ $a_c = \\frac{(125.7~m/s)^2}{0.080~m} = 197,000~m^2/s$ (b) $\\omega = \\frac{2\\pi}{T} = \\frac{2\\pi}{0.0040~s} = 1571~rad/s$ $v = (1571~rad/s)(0.020~m) = 31.4~m/s$ $a_c = \\frac{(31.4~m/s)^2}{0.020~m} = 49,000~m^2/s$ (c) $\\omega = \\frac{2\\pi}{T} = \\frac{2\\pi}{0.0010~s} = 6283~rad/s$ $v = (6283~rad/s)(0.080~m) = 502.6~m/s$ $a_c = \\frac{(502.6~m/s)^2}{0.080~m} = 3,158,000~m^2/s$ (d) $\\omega = \\frac{2\\pi}{T} = \\frac{2\\pi}{0.0010~s} = 6283~rad/s$ $v = (6283~rad/s)(0.020~m) = 125.7~m/s$ $a_c = \\frac{(125.7~m/s)^2}{0.020~m} = 790,000~m^2/s$ (e) $\\omega = \\frac{2\\pi}{T} = \\frac{2\\pi}{0.0040~s} = 1571~rad/s$ $v = (1571~rad/s)(0.010~m) = 15.7~m/s$ $a_c = \\frac{(15.7~m/s)^2}{0.010~m} = 25,000~m^2/s$ We can rank the flywheels in order of radial acceleration of a point at the rim, from largest to smallest: $c \\gt d \\gt a \\gt b \\gt e$" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.62036633,"math_prob":1.0000098,"size":1246,"snap":"2019-43-2019-47","text_gpt3_token_len":603,"char_repetition_ratio":0.18599033,"word_repetition_ratio":0.4457143,"special_character_ratio":0.5770466,"punctuation_ratio":0.110119045,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000094,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-12T23:31:10Z\",\"WARC-Record-ID\":\"<urn:uuid:10af1a52-71b0-4504-b01e-6082cea344e9>\",\"Content-Length\":\"69160\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:5bb0ef62-e1dd-417e-807c-00f8293983ba>\",\"WARC-Concurrent-To\":\"<urn:uuid:56c2510f-c056-48ae-8bea-e67528a31597>\",\"WARC-IP-Address\":\"3.90.134.5\",\"WARC-Target-URI\":\"https://www.gradesaver.com/textbooks/science/physics/college-physics-4th-edition/chapter-5-problems-page-185/12\",\"WARC-Payload-Digest\":\"sha1:4LKHPSKIOIZX52WPPXICOI3IFUIHA7ZW\",\"WARC-Block-Digest\":\"sha1:CYYP4YU73JN7V4VHR4NECBFFNBVLDCO7\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496665809.73_warc_CC-MAIN-20191112230002-20191113014002-00045.warc.gz\"}"}
https://rnmrtk.uchc.edu/rnmrtk/Analysis.html	[ "", null, "Analysis\n\nadump dim;l filename;l row1;i [row2;i ...]\nadump dim;l filename;l row1;i comp1;l [row2;i comp2;l ...]\n\nPerforms an ASCII dump of the specified rows along the specified dimension. The component indicators must be R or I if one of the dimensions is complex, or RR, RI, IR, or II if both are complex.\n\ninject [peaklist-file;l]\n\nAdds synthetic decaying sinusoids to the data set. All the dimensions must be in the time domain. If a peaklist file is specified, the program will record the peaks added to the data; each line of the file will contain the peak number, amplitude, and frequency in each dimension. If no peaklist file is specified, the peak list is just written to the screen.\n\nOnce invoked, the program accepts the set of commands listed below. In these commands, the symbol \"n\" stands for a dimension number (1 through 4).\n\nxpeakfit dim;l {IN or ANTI} row1;i row2;i [nignore;i] noise;f lphase;f freq;f linewidth;f coupling;f\n\nPerforms a nonlinear least-squares fit to J-modulated doublet (in- phase or anti-phase) in the time domain. Row1 is the row number to use for the fit. Row2 is a row to be co-added (subtracted for anti-phase), or 0 for none. nignore is the number of points at the start of the FID to leave out of the fit (default is 0). Noise is the RMS (real) noise level, used for tests of significance. Lphase is the linear phase correction needed. Freq is the cross peak central frequency in ppm. Linewidth is the initial estimated linewidth in Hz (0. will probably work okay). Coupling is the initial estimated coupling constant in Hz.\n\nHZ\n\nSpecify that all frequencies will be given in Hz.\n\nPPM\n\nSpecify that frequencies will be given in ppm (except for linewidths, which are always in Hz).\n\nAMP amp-low;f [amp-high;f]\n\nSet a range of amplitudes for the synthetic peaks. If amp-high is not given, it is taken to be the same as amp-low.\n\nLWn linewidth;f\n\nSet the synthetic peaks' linewidth in a dimension. The linewidth is given in Hz.\n\nFn freq-low;f [freq-high;f]\n\nSet a range for the peaks' frequencies in a dimension. If freq-high is not given, it is taken to be the same as freq-low. The frequencies can be given in Hz or ppm (see HZ and PPM).\n\nPHASEn {cphase;f [lphase;f] or cphase;i [lphase;i]}\n\nSpecify the phase correction to be used in processing a dimension. The synthetic peaks will be generated with a phase opposite to the value specified, so that when the correction is applied the peaks will be in phase.\n\nSEED seed-1;i ... seed-n;i\n\nSpecify a set of seeds (one in each dimension) for the subrandom number generator, used to produce the peaks' frequencies.\n\nNPEAKS num;i\n\nSpecify the number of peaks to generate. The maximum is 100. The amplitude range, linewidths, frequency ranges must have been set previously (see LWn, and Fn). Phase corrections (PHASEn) are optional. The program will generate this many peaks, with amplitudes evenly spaced in the given range and with frequencies distributed in the appropriate ranges according to a subrandom algorithm (see SEED). The signals will be added to the data set and printed in the peak list file.\n\nPEAK [ppm] amp;f freq-1;f ... freq-n;f\n\nSpecify an individual peak with a given amplitude and frequencies. The frequencies can be given in Hz or ppm (see the HZ and PPM commands) The linewidths must already have been set (see LWn). The peak will be added to the data set (and printed in the peak list file) immediately.\n\nSpecify the name of a sampling schedule file and the dimension(s) to which the schedule applies. The dimensions must be contiguous in memory; all of them should have length 1 except one, which has length equal to the number of entries in the sampling schedule.\n\nThe sampling schedule file contains a list of lines, each referring to one data point. The format of the lines is:\n\nindex-a;i index-b;i ... [nacq;i]\n\nwhere index-a, index-b, and so on give the time index for the data point in the a-, b-, etc. dimensions. Nacq is the number of \"acquisition units\" for the point; it is used to describe data in which different points are recorded using different numbers of acquisitions (phase cycles or transients). The default value for nacq is 1.\n\nEXIT\n\nExit inject." ]	[ null, "https://rnmrtk.uchc.edu/rnmrtk/Analysis_files/shapeimage_1.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8583234,"math_prob":0.95448947,"size":4198,"snap":"2023-14-2023-23","text_gpt3_token_len":1027,"char_repetition_ratio":0.13376252,"word_repetition_ratio":0.053295933,"special_character_ratio":0.23034778,"punctuation_ratio":0.15021929,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9634357,"pos_list":[0,1,2],"im_url_duplicate_count":[null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-03-24T06:42:54Z\",\"WARC-Record-ID\":\"<urn:uuid:7f5b60a1-04b5-46b8-b2e4-f246bdc58228>\",\"Content-Length\":\"17777\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a6ca22f4-4375-4eea-acb9-a6653e7853f3>\",\"WARC-Concurrent-To\":\"<urn:uuid:976da65e-e7c8-4ed0-84aa-ffe3abaaf1db>\",\"WARC-IP-Address\":\"155.37.200.50\",\"WARC-Target-URI\":\"https://rnmrtk.uchc.edu/rnmrtk/Analysis.html\",\"WARC-Payload-Digest\":\"sha1:FOZHU5PEFK5QHBQKCF56PT6MMTVZWE4N\",\"WARC-Block-Digest\":\"sha1:O2M3WSELG3CRDBXOHHVHDNTIFIIBSBTS\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-14/CC-MAIN-2023-14_segments_1679296945248.28_warc_CC-MAIN-20230324051147-20230324081147-00702.warc.gz\"}"}
https://raresql.com/tag/missing-column-in-table-or-view/	[ "Feeds:\nPosts\n\n## SQL Server – Compare two databases schema\n\nToday, I received an email from one of my blog readers, who would like to match test and production databases. Here is a basic script to compare two databases.\n\nGiven below are the objects, script will compare in source and target databases.\n\n• Tables\n• Views\n• Stored Procedures\n• User Defined Functions\n• Triggers\n• Primary Keys\n• Indexes\n• Missing column in table or view\n• Mismatch data type in table or view\n• Missing Parameter in the stored procedure\n```Create Procedure Sp_Compare_two_database_Schema\n@Source_DB_Name nvarchar(Max),\n@Target_DB_Name varchar(Max)\n\nAs\nBEGIN\nDECLARE @SQL nvarchar(MAX)\n\nSET @Source_DB_Name=QUOTENAME(@Source_DB_Name)\nSET @Target_DB_Name=QUOTENAME(@Target_DB_Name)\n\nSET @SQL=''\n\nCREATE TABLE #Result\n(\n[Main Object Type] nvarchar(max),\n[Main Object Name] nvarchar(max),\n[Type] varchar(50),\n[Sub Object Type] nvarchar(max),\n[Sub Object Name] nvarchar(max)\n)\n\n--Match Main Objects Like Tables, view, Stored Procedure , triggers.\nSET @SQL= 'Insert into #Result Select A.[type_desc] as [Main Object Type],A.[name],''Object Missing'' as [Type]\n,A.[type_desc] as [Sub Object Type]\n,A.[name] as [Sub Object Name]\nfrom ' + @Source_DB_Name + '.sys.objects A\nWhere [Parent_object_id]=0 And A.[name] NOT IN(\nSelect [name] from ' + @Target_DB_Name + '.sys.objects)\nOrder By A.[type_desc]'\nPrint @SQL\nEXEC (@SQL)\n\n--Match Sub Objects Like Foreign Keys.\nSET @SQL= 'Insert into #Result Select A.[type_desc] as [Main Object Type],A.[name],''Object Missing'' as [Type]\n,B.[type_desc] as [Sub Object Type]\n,B.[name] as [Sub Object Name]\nfrom ' + @Source_DB_Name + '.sys.objects A\nInner Join ' + @Source_DB_Name + '.sys.objects B\nOn A.[object_id]=B.[Parent_object_id]\nWhere B.[name] NOT IN(\nSelect [name] from ' + @Target_DB_Name + '.sys.objects)\nOrder By A.[type_desc]'\nPrint @SQL\nEXEC (@SQL)\n\n--Find if any column is missing in target database.\n\nSET @SQL= ';With CteA AS (Select A.[type_desc] as [Main Object Type],A.[Name] as [Main Object Name],''Column Missing'' as [Type],B.[Name] as [Column Name]\nfrom ' + @Source_DB_Name + '.sys.objects A\nInner Join ' + @Source_DB_Name + '.sys.columns B\nOn A.[object_id] =B.[object_id]\nWhere A.[Type] In (''U'',''V'')\n)\n,CteB AS (Select A.[type_desc] as [Main Object Type],A.[Name] as [Main Object Name],''Column Missing'' as [Type],B.[Name] as [Column Name]\nfrom ' + @Target_DB_Name + '.sys.objects A\nInner Join ' + @Target_DB_Name + '.sys.columns B\nOn A.[object_id] =B.[object_id]\nWhere A.[Type] In (''U'',''V'')\n)\nInsert into #Result\nSelect A.[Main Object Type],A.[Main Object Name],A.[Type], ''Column'' as [Sub Object Type],A.[Column Name] from CTEA A\nLeft Join CTEB B On\nA.[Main Object Type]=B.[Main Object Type]\nAnd A.[Main Object Name]=B.[Main Object Name]\nAnd A.[Column Name]=B.[Column Name]\nWhere (B.[Main Object Name] is NULL OR B.[Column Name] is NULL)\nAnd A.[Main Object Name] Not In (Select [Main Object Name] from #Result A\nWhere A.[Type]=''Object Missing'')\nOrder By A.[Main Object Type],A.[Main Object Name], A.[Column Name]'\nPrint @SQL\nEXEC (@SQL)\n\n--Find if any column data type is not sync with target database.\n\nSET @SQL= ';With CteA AS (Select A.[type_desc] as [Main Object Type],A.[Name] as [Main Object Name]\n,''Data Type Difference'' as [Type],B.[Name] as [Column Name]\n,B.[system_type_id]\nfrom ' + @Source_DB_Name + '.sys.objects A\nInner Join ' + @Source_DB_Name + '.sys.columns B\nOn A.[object_id] =B.[object_id]\nWhere A.[Type] In (''U'',''V'')\n)\n,CteB AS (Select A.[type_desc] as [Main Object Type],A.[Name] as [Main Object Name],''Data Type Difference'' as [Type],B.[Name] as [Column Name],B.[system_type_id]\nfrom ' + @Target_DB_Name + '.sys.objects A\nInner Join ' + @Target_DB_Name + '.sys.columns B\nOn A.[object_id] =B.[object_id]\nWhere A.[Type] In (''U'',''V'')\n)\n\nInsert into #Result Select A.[Main Object Type],A.[Main Object Name],A.[Type], ''Column'' as [Sub Object Type], A.[Column Name] from CTEA A\nInner Join CTEB B On\nA.[Main Object Type]=B.[Main Object Type]\nAnd A.[Main Object Name]=B.[Main Object Name]\nAnd A.[Column Name]=B.[Column Name]\nWhere A.[system_type_id]<>B.[system_type_id]\nAnd A.[Main Object Name] Not In (Select [Main Object Name] from #Result A\nWhere A.[Type]=''Object Missing'')\nOrder By A.[Main Object Type],A.[Main Object Name], A.[Column Name]'\n\nPrint @SQL\nEXEC (@SQL)\n\n--Find if any parameter of the procedure is missing in target database.\n\nSET @SQL= ';With CteA AS (Select A.[type_desc] as [Main Object Type]\n,A.[Name] as [Main Object Name],''Parameter Missing'' as [Type],B.[name] as [Parameter Name]\nfrom' + @Source_DB_Name + '.sys.objects A\nInner Join ' + @Source_DB_Name + '.sys.all_parameters B On A.[object_id] =B.[Object_id]\n)\n,CteB AS (\nSelect A.[type_desc] as [Main Object Type]\n,A.[Name] as [Main Object Name],''Parameter Missing'' as [Type],B.[name] as [Parameter Name]\nfrom ' + @Target_DB_Name + '.sys.objects A\nInner Join ' + @Target_DB_Name + '.sys.all_parameters B On A.[object_id] =B.[Object_id]\n)\n\nInsert into #Result Select A.[Main Object Type],A.[Main Object Name],A.[Type]\n, ''Parameter'' as [Sub Object Type]\n, A.[Parameter Name] from CTEA A\nLeft Join CTEB B On\nA.[Main Object Type]=B.[Main Object Type]\nAnd A.[Main Object Name]=B.[Main Object Name]\nAnd A.[Parameter Name]=B.[Parameter Name]\nWhere (B.[Main Object Name] IS NULL OR B.[Parameter Name] IS NULL)\nAnd A.[Main Object Name] Not In (Select [Main Object Name] from #Result A\nWhere A.[Type]=''Object Missing'')\nOrder By A.[Main Object Type],A.[Main Object Name], A.[Parameter Name]'\nPrint @SQL\nEXEC (@SQL\n)\nSelect * from #Result A Order By A.[Main Object Type] DESC, A.[Main Object Name] ASC ,[Type] DESC\nEND\nGO\n\n--Syntax\n--Sp_Compare_two_database_Schema 'Source DatabaseName','Target DatbaseName'\n\n--Example\nSp_Compare_two_database_Schema 'Source_DB','Target_DB'\n```", null, "" ]	[ null, "https://raresql.files.wordpress.com/2012/10/compare-two-databases-1-11.png", null 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https://brilliant.org/discussions/thread/logical-transformations-as-a-klein-4-group/	[ "# Logical Transformations as a Klein 4-Group\n\nAfter recently studying statements and their logical transformations, I was happily surprised to discover a resemblance between these transformations and a special type of group known in group theory as a Klein 4-group.\n\nFirst off, the logical transformations I was studying were the contrapositive, the converse, and the inverse. For example, if we take the statement \"If x then y\", we can transform it into:\n\n1. The contrapositive: \"If not y, then not x\",\n2. The converse: \"If y, then x\", and\n3. The inverse: \"If not x, then not y\"\n\nFor convenience, I will abbreviate these $C_P$, $C$, and $I$, and the original statement $S$.\n\nIn exploring these transformations further, I realized that instead of going from the original statement to one of the transformations, I could apply a transformation on top of a transformation.\n\nFor example, the contrapositive of the converse is the contrapositive of \"If y, then x\", which is \"If not x, then not y\".\n\nHowever, these double transformations all became other transformations! In the example above, for instance, the resulting statement is really just the inverse of the original statement. After trying out many of these double transformations, I realized that I was actually constructing a Cayley table.\n\nWithout delving to deeply into group theory, a Cayley table is a table used to show how a collection of items behave when combined in certain ways. In my case, I realized I could use a Cayley table to show how logic transformations behaved when applied more then once on a statement.\n\nI came up with the following table:\n\n $~$ $S~$ $C_P$ $C~$ $I~$ $S~$ $S~$ $C_P$ $C~$ $I~$ $C_P$ $C_P$ $S~$ $I~$ $C~$ $C~$ $C~$ $I~$ $S~$ $C_P$ $I~$ $I~$ $C~$ $C_P$ $S~$\n\nWhich has been classified, more generally, by group theorists as the Klein 4-group. Technically, this represents a transformation in any row first being applied, and then a transformation in any column being applied second with their intersection being the resulting statement. However, as is the case with this and many other types of groups, the transformations are commutative, meaning the order they are applied in does not matter.\n\nMaybe it is just me, but I thought it so fascinating that something as fundamental as logic could be described by something so abstract as group theory, and thus relating it to a host of other applications such as physics, chemistry, and Rubik's cubes!\n\nI would love any feedback anyone has on this topic. I have not been able to find anything online relating logic to group theory, so any extra information or ideas would be much appreciated!", null, "Note by David Stiff\n2 years ago\n\nThis discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.\n\nWhen posting on Brilliant:\n\n• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .\n• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting \"I don't understand!\" doesn't help anyone.\n• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.\n\nMarkdownAppears as\nitalics or _italics_ italics\nbold or __bold__ bold\n- bulleted- list\n• bulleted\n• list\n1. numbered2. list\n1. numbered\n2. list\nNote: you must add a full line of space before and after lists for them to show up correctly\nparagraph 1paragraph 2\n\nparagraph 1\n\nparagraph 2\n\n[example link](https://brilliant.org)example link\n> This is a quote\nThis is a quote\n # I indented these lines\n# 4 spaces, and now they show\n# up as a code block.\n\nprint \"hello world\"\n# I indented these lines\n# 4 spaces, and now they show\n# up as a code block.\n\nprint \"hello world\"\nMathAppears as\nRemember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.\n2 \\times 3 $2 \\times 3$\n2^{34} $2^{34}$\na_{i-1} $a_{i-1}$\n\\frac{2}{3} $\\frac{2}{3}$\n\\sqrt{2} $\\sqrt{2}$\n\\sum_{i=1}^3 $\\sum_{i=1}^3$\n\\sin \\theta $\\sin \\theta$\n\\boxed{123} $\\boxed{123}$\n\nSort by:\n\nI don't think this is a particularly surprising result, as every group of order 4 is either cyclic or isomorphic to the Vierergruppe... Obviously it's not cyclic (since each logical statement is idempotent); hence, the only thing to verify is closure and associativity of composition of \"logical transformations\" (whatever that means). There's also an application of the Vierergruppe to music composition too...\n\nI think it would be beneficial for you to add this to the Klein 4-group Wikipedia page, as well as this page, but you may want to make sure that your writing is perfect.\n\nThis is a result of the fact that inverse and converse are both C2 groups and combining both of them is equivalent to taking a direct sum on some level.\n\n- 7 months, 3 weeks ago\n\nThanks Sam. I must admit that I am not very knowledgeable in the field of group theory. When I posted this, I merely discovered a resemblance to something called the Klein-4 group which I had read about briefly.\n\n- 7 months, 3 weeks ago" ]	[ null, "https://ds055uzetaobb.cloudfront.net/brioche/avatars-2/resized/45/9f3853a5d95700c2d4b2fa03e4920f39.ecf9808c04-wyAuq3FHsk.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9430001,"math_prob":0.937412,"size":4049,"snap":"2020-34-2020-40","text_gpt3_token_len":870,"char_repetition_ratio":0.13003708,"word_repetition_ratio":0.0,"special_character_ratio":0.21289207,"punctuation_ratio":0.12387097,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98422235,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-09-27T11:12:35Z\",\"WARC-Record-ID\":\"<urn:uuid:5e35810c-d895-48d7-8b86-94cd92f8c06a>\",\"Content-Length\":\"80295\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:07c32c40-c8ec-4792-b667-283c222b0d36>\",\"WARC-Concurrent-To\":\"<urn:uuid:4c0968af-8720-4f0a-9239-c4f3fb545ce6>\",\"WARC-IP-Address\":\"104.20.34.242\",\"WARC-Target-URI\":\"https://brilliant.org/discussions/thread/logical-transformations-as-a-klein-4-group/\",\"WARC-Payload-Digest\":\"sha1:WQPYLAU6XXVA3PI653SML4T35J5O3NJ7\",\"WARC-Block-Digest\":\"sha1:TPYUYP35Q34GIVZPZM7QMGGN2IOAHIC4\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600400274441.60_warc_CC-MAIN-20200927085848-20200927115848-00382.warc.gz\"}"}
https://aspire.eecs.berkeley.edu/publication/probably-efficient-algorithms-for-numerical-tensor-algebra/	[ "# Provably Efficient Algorithms for Numerical Tensor Algebra\n\nThis thesis targets the design of parallelizable algorithms and communication-efficient parallel schedules for numerical linear algebra as well as computations with higher-order tensors. Communication is a growing bottleneck in the execution of most algorithms on parallel computers, which manifests itself as data movement both through the network connecting different processors and through the memory hierarchy of each processor as well as synchronization between processors. We provide a rigorous theoretical model of communication and derive lower bounds as well as algorithms in this model. Our analysis concerns two broad areas of linear algebra and of tensor contractions. We demonstrate the practical quality of the new theoretically-improved algorithms by presenting results which show that our implementations outperform standard libraries and traditional algorithms.\nWe model the costs associated with local computation, communication, and synchronization. We introduce a new technique for deriving lower bounds on tradeoffs between these costs and apply them to algorithms in both dense and sparse linear algebra as well as graph algorithms. These lower bounds are attained by what we refer to as 2.5D algorithms, which we give for matrix multiplication, Gaussian elimination, QR factorization, the symmetric eigenvalue problem, and the Floyd-Warshall all-pairs shortest-paths algorithm. 2.5D algorithms achieve lower interprocessor bandwidth cost by exploiting auxiliary memory. Algorithms employing this technique are well known for matrix multiplication, and have been derived in the BSP model for LU and QR factorization, as well as the Floyd-Warshall algorithm. We introduce alternate versions of LU and QR algorithms which have measurable performance improvements over their BSP counterparts, and we give the first evaluations of their performance. For the symmetric eigenvalue problem, we give the first 2.5D algorithms, additionally solving challenges with memory-bandwidth efficiency that arise for this problem. We also give a new memory-bandwidth efficient algorithm for Krylov subspace methods (repeated multiplication of a vector by a sparse-matrix).\nThe latter half of the thesis contains algorithms for higher-order tensors, in particular tensor contractions. We introduce Cyclops Tensor Framework, which provides an automated mechanism for network-topology-aware decomposition and redistribution of tensor data. It leverages 2.5D matrix multiplication to perform tensor contractions communication-efficiently. The framework is capable of exploiting symmetry and antisymmetry in tensors and utilizes a distributed packed-symmetric storage format. Finally, we consider a theoretically novel technique for exploiting tensor symmetry to lower the number of multiplications necessary to perform a contraction via computing some redundant terms that allow preservation of symmetry and then cancelling them out with low-order cost. We analyze the numerical stability and communication efficiency of this technique and give adaptations to antisymmetric and Hermitian matrices. This technique has promising potential for accelerating coupled-cluster (electronic structure) methods both in terms of computation and communication cost.\n\nPaper\n\nPublication Date: September 2014\nConference: PhD Dissertation" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9280559,"math_prob":0.8631759,"size":4191,"snap":"2020-10-2020-16","text_gpt3_token_len":695,"char_repetition_ratio":0.13494149,"word_repetition_ratio":0.4116608,"special_character_ratio":0.15437843,"punctuation_ratio":0.071875,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9837108,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-02-25T22:00:55Z\",\"WARC-Record-ID\":\"<urn:uuid:05dcb7c3-48e9-4b8f-b20d-081456e57946>\",\"Content-Length\":\"28950\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0ecfbf56-39b1-4414-bccd-d31b31be6638>\",\"WARC-Concurrent-To\":\"<urn:uuid:35ebd5b8-8b91-451b-8245-16916e63123d>\",\"WARC-IP-Address\":\"128.32.37.248\",\"WARC-Target-URI\":\"https://aspire.eecs.berkeley.edu/publication/probably-efficient-algorithms-for-numerical-tensor-algebra/\",\"WARC-Payload-Digest\":\"sha1:QGXQRJYDDO4LFRWTX5SLZBRMYE64GQSA\",\"WARC-Block-Digest\":\"sha1:XVPO2TLR5WCPZLRROXWTXILTUILO7WEB\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-10/CC-MAIN-2020-10_segments_1581875146160.21_warc_CC-MAIN-20200225202625-20200225232625-00314.warc.gz\"}"}
https://www.uoguelph.ca/registrar/calendars/undergraduate/2018-2019/c12/c12math.shtml	[ "# XII. Course Descriptions\n\n## Mathematics\n\nDepartment of Mathematics and Statistics\n\nSuggested initial course sequence:\n\n1. For students with 4U or OAC Calculus and expecting to pursue further studies in mathematics or the physical sciences: MATH1200, MATH1210.\n\n2. For students interested in applications to the biological sciences: MATH1080, MATH2080.\n\n3. For students not expecting to pursue further studies in mathematics: MATH1030, one STATXXXX course.\n\nMATH1030 Business Mathematics F,W (3-0) [0.50]\nPrimarily intended for business and economics students, this course is designed to introduce and reinforce the essential mathematical skills needed to understand, analyze, and solve mathematical problems related to business and economics. Topics covered include basic algebra; functions, including a review of exponential and logarithmic functions; sequences and series with financial applications; limits; continuity; and differential calculus including derivatives, higher order derivatives, and curve sketching.\nOffering(s): Also offered through Distance Education format.\nRestriction(s): MATH1080, MATH1200 Not available to students registered in the BSC program.\nDepartment(s): Department of Mathematics and Statistics\nMATH1080 Elements of Calculus I F,W (3-1) [0.50]\nThis course provides an introduction to the calculus of one variable with emphasis on mathematical modelling in the biological sciences. The topics covered include elementary functions, sequences and series, difference equations, differential calculus and integral calculus.\nPrerequisite(s): 1 of 4U Advanced Functions, 4U Advanced Functions and Calculus or equivalent\nRestriction(s): IPS1500, MATH1200\nDepartment(s): Department of Mathematics and Statistics\nMATH1160 Linear Algebra I F,W (3-0) [0.50]\nThis course provides an introduction to linear algebra in Euclidean space. Topics covered include: N-dimensional vectors, dot product, matrices and matrix operations, systems of linear equations and Gaussian elimination, linear independence, subspaces, basis and dimension, matrix inverse, matrix rank and determinant, eigenvalues, eigenvectors and diagonalization, orthogonalization and projections, linear transformations. Some fundamental proofs and applications of these topics will be included.\nPrerequisite(s): 4U Calculus and Vectors or 4U Advanced Functions\nRestriction(s): ENGG1500, MATH2150, MATH2160\nDepartment(s): Department of Mathematics and Statistics\nMATH1200 Calculus I F (3-1) [0.50]\nThis is a theoretical course intended primarily for students who expect to pursue further studies in mathematics and its applications. Topics include inequalities and absolute value; compound angle formulas for trigonometric functions; limits and continuity using rigorous definitions; the derivative and derivative formulas (including derivatives of trigonometric, exponential and logarithmic functions); Fermat's theorem; Rolle's theorem; the mean-value theorem; applications of the derivative; Riemann sums; the definite integral; the fundamental theorem of calculus; applications of the definite integral; the mean value theorem for integrals.\nPrerequisite(s): 1 of 4U Calculus and Vectors, 4U Advanced Functions and Calculus or Grade 12 Calculus\nRestriction(s): IPS1500, MATH1080\nDepartment(s): Department of Mathematics and Statistics\nMATH1210 Calculus II W (3-1) [0.50]\nThis course is a continuation of MATH1200. It is a theoretical course intended primarily for students who need or expect to pursue further studies in mathematics, physics, chemistry, engineering and computer science. Topics include inverse functions, inverse trigonometric functions, hyperbolic functions, indeterminate forms and l'Hopital's rule, techniques of integration, parametric equations, polar coordinates, Taylor and Maclaurin series; functions of two or more variables, partial derivatives, and if time permits, an introduction to multiple integration.\nPrerequisite(s): MATH1080 or MATH1200\nRestriction(s): IPS1510, MATH2080\nDepartment(s): Department of Mathematics and Statistics\nMATH2000 Proofs, Sets, and Numbers F (3-1) [0.50]\nThis course exposes the student to formal mathematical proof, and introduces the theory of sets and number systems. Topics include relations and functions, number systems including formal properties of the natural numbers, integers, and the real and complex numbers. Equivalence relations and partial and total orders are introduced. The geometry and topology of the real number line and Cartesian plane are introduced. Techniques of formal proof are introduced including well-ordering, mathematical induction, proof by contradiction, and proof by construction. These techniques will be applied to fundamental theorems from linear algebra.\nPrerequisite(s): 1 of IPS1500, MATH1080, MATH1160, MATH1200\nDepartment(s): Department of Mathematics and Statistics\nMATH2080 Elements of Calculus II W (3-1) [0.50]\nThis course will expand on integration techniques, and introduce students to difference and differential equations, vectors, vector functions, and elements of calculus of two or more variables such as partial differentiation and multiple integration. The course will emphasize content relevant to analyzing biological systems, and methods will be illustrated by application to biological systems.\nPrerequisite(s): 1 of IPS1500, MATH1080, MATH1200\nRestriction(s): IPS1510, MATH1210\nDepartment(s): Department of Mathematics and Statistics\nMATH2130 Numerical Methods W (3-1) [0.50]\nThis course provides a theoretical and practical introduction to numerical methods for approximating the solution(s) of linear and nonlinear problems in the applied sciences. The topics covered include: solution of a single nonlinear equation; polynomial interpolation; numerical differentiation and integration; solution of initial value and boundary value problems; and the solution of systems of linear and nonlinear algebraic equations.\nPrerequisite(s): 1 of IPS1510, MATH1210, MATH2080\nDepartment(s): Department of Mathematics and Statistics\nMATH2200 Advanced Calculus I F (3-0) [0.50]\nThe topics covered in this course include infinite sequences and series, power series, tests for convergence, Taylor's theorem and Taylor series for functions of one variable, planes and quadratric surfaces, limits, and continuity, differentiability of functions of two or more variables, partial differentiation, directional derivatives and gradients, tangent planes, linear approximation, Taylor's theorem for functions of two variables, critical points, extreme value problems, implicit function theorem, Jacobians, multiple integrals, and change of variables.\nPrerequisite(s): 1 of IPS1510, MATH1210, MATH2080\nDepartment(s): Department of Mathematics and Statistics\nMATH2210 Advanced Calculus II W (3-0) [0.50]\nThis course continues the study of multiple integrals, introducing spherical and cylindrical polar coordinates. The course also covers vector and scalar fields, including the gradient, divergence, curl and directional derivative, and their physical interpretation, as well as line integrals and the theorems of Green and Stokes.\nPrerequisite(s): MATH2200\nDepartment(s): Department of Mathematics and Statistics\nMATH2270 Applied Differential Equations F (3-1) [0.50]\nThis course covers the solution of differential equations that arise from problems in engineering and science. Topics include linear equations of first and higher order, systems of linear equations, Laplace transforms, series solutions of second-order equations, and an introduction to partial differential equations.\nPrerequisite(s): (ENGG1500 or MATH1160), (1 of IPS1510, MATH1210, MATH2080)\nRestriction(s): MATH2170\nDepartment(s): Department of Mathematics and Statistics\nMATH3100 Differential Equations II W (3-1) [0.50]\nThis course continues the study of differential equations. Power series solutions around regular singular points including Bessel equations are presented. First order linear systems and their general solution by matrix methods are thoroughly covered. Nonlinear systems are introduced along with the concepts of linearization, stability of equilibria, phase plane analysis, Lyapunov's method, periodic solutions and limit cycles. Two-point boundary value problems are discussed and an introduction to linear partial differential equations and their solution by separation of variables and Fourier series is given.\nPrerequisite(s): (1 of MATH1160, MATH2150, MATH2160), (MATH2170 or MATH2270)\nDepartment(s): Department of Mathematics and Statistics\nMATH3130 Abstract Algebra F (3-0) [0.50]\nThis course is an introduction to abstract algebra, covering both group theory and ring theory. Specific topics covered include an introduction to group theory, permutations, symmetric and dihedral groups, subgroups, normal subgroups and factor groups. Group theory continues through the fundamental homomorphism theorem. Ring theory material covered includes an introduction to ring theory, subrings, ideals, quotient rings, polynomial rings, and the fundamental ring homomorphism theorem.\nOffering(s): Offered in even-numbered years.\nPrerequisite(s): MATH2000, (1 of MATH1160, MATH2150, MATH2160)\nDepartment(s): Department of Mathematics and Statistics\nMATH3160 Linear Algebra II W (3-0) [0.50]\nThe topics in this course include complex vector spaces, direct sum decompositions of vector spaces, the Cayley-Hamilton theorem, the spectral theorem for normal operators and the Jordan canonical form.\nPrerequisite(s): (MATH1160 or MATH2160), 1.00 credits in MATH or STAT at the 2000 level or above\nDepartment(s): Department of Mathematics and Statistics\nMATH3200 Real Analysis F (3-0) [0.50]\nThis course provides a basic foundation for real analysis. The rigorous treatment of the subject in terms of theory and examples gives students the flavour of mathematical reasoning and intuition for other advanced topics in mathematics. Topics covered include the real number line and the supremum property; metric spaces; continuity and uniform continuity; completeness and compactness; the Banach fixed-point theorem and its applications to ODEs; uniform convergence and the rigorous treatment of the Riemann integral.\nPrerequisite(s): MATH2000, MATH2210, (MATH1160 or MATH2160)\nDepartment(s): Department of Mathematics and Statistics\nMATH3240 Operations Research F (3-0) [0.50]\nThis is a course in mathematical modelling which has applications to engineering, economics, business and logistics. Topics covered include linear programming and the simplex method, network models and the shortest path, maximum flow and minimal spanning tree problems as well as a selection of the following: non-linear programming, constrained optimization, deterministic and probabilistic dynamic programming, game theory and simulation.\nOffering(s): Offered in odd-numbered years.\nPrerequisite(s): (1 of MATH1160, MATH2150, MATH2160), 0.50 credits in statistics\nCo-requisite(s): MATH2200\nDepartment(s): Department of Mathematics and Statistics\nMATH3260 Complex Analysis W (3-0) [0.50]\nThis course extends calculus to cover functions of a complex variable; it introduces complex variable techniques which are very useful for mathematics, the physical sciences and engineering. Topics include complex differentiation, planar mappings, analytic and harmonic functions, contour integration, Taylor and Laurent series, the residue calculus and its application to the computation of trigonometric and improper integrals, conformal mapping and the Dirichlet problem.\nPrerequisite(s): MATH2200\nDepartment(s): Department of Mathematics and Statistics\nMATH3510 Biomathematics W (3-0) [0.50]\nThis course will convey the fundamentals of applying mathematical modelling techniques to understanding and predicting the dynamics of biological systems. Students will learn the development, analysis, and interpretation of biomathematical models based on discrete-time and continuous-time models. Applications may include examples from population biology, ecology, infectious diseases, microbiology, and genetics.\nPrerequisite(s): (1 of MATH1160, MATH2150, MATH2160), (MATH2170 or MATH2270)\nDepartment(s): Department of Mathematics and Statistics\nMATH4050 Topics in Mathematics I W (3-0) [0.50]\nIn this course students will discuss selected topics at an advanced level. It is intended mainly for mathematics students in the 6th to 8th semester. Content will vary from year to year. Sample topics include: probability theory, Fourier analysis, mathematical logic, operator algebras, number theory combinatorics, philosophy of mathematics, fractal geometry, chaos, stochastic differential equations.\nOffering(s): Offered in odd-numbered years.\nPrerequisite(s): MATH3200\nDepartment(s): Department of Mathematics and Statistics\nMATH4060 Topics in Mathematics II W (3-0) [0.50]\nIn this course students will discuss selected topics at an advanced level as in MATH4050, but with different choice of topic.\nOffering(s): Offered in even-numbered years.\nPrerequisite(s): MATH3200\nDepartment(s): Department of Mathematics and Statistics\nMATH4150 Topics in Mathematics III F,W (3-0) [0.50]\nIn this course students will discuss selected topics at an advanced level as in MATH4050, but with different choice of topics.\nPrerequisite(s): MATH3200\nDepartment(s): Department of Mathematics and Statistics\nMATH4200 Advanced Analysis F (3-0) [0.50]\nThis senior course in analysis will cover basic operator theory on Hilbert spaces, including self-adjoint operators and the spectral theorem. Other topics may include weak solutions, Sobolev spaces, inverse problems, measure theoretic probability or advanced topics from linear or nonlinear functional analysis.\nPrerequisite(s): MATH3200\nDepartment(s): Department of Mathematics and Statistics\nMATH4240 Advanced Topics in Modeling and Optimization W (3-0) [0.50]\nThis course is a study of advanced topics in the areas of optimization and modeling. Topics may include continuous and discrete models together with techniques for their analysis and design, and optimization topics such as game theory, networks, nonlinear problems, Markov chains, queuing theory, agent-based models, computational intelligence based techniques and computational optimization techniques.\nPrerequisite(s): 0.50 credits in Mathematics at the 3000 level.\nDepartment(s): Department of Mathematics and Statistics\nMATH4270 Partial Differential Equations F (3-0) [0.50]\nThis course focuses on first and second-order partial differential equations, with examples and applications from selected fields such as physics, engineering and biology. Topics may include the wave equation, the heat equation, Laplace's equation, linearity and separation of variables, solution by Fourier series, Bessel, Legendre and Green's functions, an introduction to the method of characteristics and Fourier transforms. The classification of linear second-order partial differential equations is discussed.\nPrerequisite(s): MATH3100\nDepartment(s): Department of Mathematics and Statistics\nMATH4440 Case Studies in Mathematics and Statistics W (3-0) [0.50]\nThis capstone course for the Mathematical Science major provides students with an opportunity to synthesize knowledge and utilize problem-solving skills accumulated over the course of their studies. The course will focus on case studies drawn from engineering, computer science, biology, life and physical sciences, medicine, and/or economics.\nPrerequisite(s): At least 3.0 mathematics and/or statistics credits at the 3000 level or above.\nRestriction(s): Restricted to students in the Mathematical Science major.\nDepartment(s): Department of Mathematics and Statistics\nMATH4600 Advanced Research Project in Mathematics F,W (0-6) [1.00]\nEach student in this course will undertake an individual research project in some area of mathematics, under the supervision of a faculty member. A written report and a public presentation of the project will be required.\nRestriction(s): Approval of a supervisor and the course coordinator.\nDepartment(s): Department of Mathematics and Statistics\nUniversity of Guelph" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8477388,"math_prob":0.91268426,"size":15281,"snap":"2020-45-2020-50","text_gpt3_token_len":3433,"char_repetition_ratio":0.17863455,"word_repetition_ratio":0.105525136,"special_character_ratio":0.22419998,"punctuation_ratio":0.14794521,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9969004,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-25T08:30:34Z\",\"WARC-Record-ID\":\"<urn:uuid:9a1e0038-1dda-429a-bd1e-bc36adefd44d>\",\"Content-Length\":\"58922\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:8b2b91e1-3049-4fc5-91cb-2b8af55ac353>\",\"WARC-Concurrent-To\":\"<urn:uuid:2c6b1977-7116-4008-bed6-23b3c0565a9c>\",\"WARC-IP-Address\":\"131.104.93.93\",\"WARC-Target-URI\":\"https://www.uoguelph.ca/registrar/calendars/undergraduate/2018-2019/c12/c12math.shtml\",\"WARC-Payload-Digest\":\"sha1:SGQAOO3OGYJ7EXCCWDGIEFV22ZGPX7OC\",\"WARC-Block-Digest\":\"sha1:B54VATX42JYEKPYE3TRJ255B3UJXPVEV\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-45/CC-MAIN-2020-45_segments_1603107888402.81_warc_CC-MAIN-20201025070924-20201025100924-00199.warc.gz\"}"}
http://www.nwchem-sw.org/index.php/Special:AWCforum/st/id1987/Problem_with_numerical_frequency....html	[ "", null, "Problem with numerical frequency calculation\n\nViewed 1720 times, With a total of 2 Posts\n\n Olanky Member Profile Send PM\n Clicked A Few Times", null, "Threads 2 Posts 5\n 2:31:25 PM PST - Wed, Mar 2nd 2016 Hi NWChem users, I am trying to calculate the vibration frequency of a configuration where adatom moving near the optimized position on CNT surface. I first optimized the whole stucture (221 atoms), and then restart the job with: restart cnt_5_10-p1vib set gen_hess:actlist 221 # 221st atom is the adatom set hessian:compress .true. vib ```reuse temp 1 300 animate ``` end task dft freq numerical I have two questions: 1. Even thought I put those two set directives in the restart job (as suggested in http://www.nwchem-sw.org/index.php/Special:AWCforum/st/id518/printing_hessian_with_greater...), I am still getting all atom frequencies with a 663x663 matrix. Is there any trick here to freeze the atoms in CNT and actually have only adatom moving? 2. I also tried to restart again but this time I changed the temperature to 2000 K. However, the output frequencies of normal modes are exactly the same with those at 300 K. The only difference in the output file is in this block: ```Temperature = 300.00K frequency scaling parameter = 1.0000 ``` ```Zero-Point correction to Energy = 1852.825 kcal/mol ( 2.952665 au) Thermal correction to Energy = 1856.996 kcal/mol ( 2.959312 au) Thermal correction to Enthalpy = 1857.592 kcal/mol ( 2.960261 au) ``` ```Total Entropy = 58.653 cal/mol-K - Translational = 49.102 cal/mol-K (mol. weight =2322.2348) - Rotational = 0.000 cal/mol-K (symmetry # = 1) - Vibrational = 9.551 cal/mol-K ``` ```Cv (constant volume heat capacity) = 54.633 cal/mol-K - Translational = 2.979 cal/mol-K - Rotational = 1.986 cal/mol-K - Vibrational = 49.668 cal/mol-K ``` ```Temperature = 2000.00K frequency scaling parameter = 1.0000 ``` ```Zero-Point correction to Energy = 1852.825 kcal/mol ( 2.952665 au) Thermal correction to Energy = 3162.591 kcal/mol ( 5.039911 au) Thermal correction to Enthalpy = 3166.564 kcal/mol ( 5.046242 au) ``` ```Total Entropy = 1191.330 cal/mol-K - Translational = 58.522 cal/mol-K (mol. weight =2322.2348) - Rotational = 0.000 cal/mol-K (symmetry # = 1) - Vibrational = 1132.808 cal/mol-K ``` ```Cv (constant volume heat capacity) = 1095.424 cal/mol-K - Translational = 2.979 cal/mol-K - Rotational = 1.986 cal/mol-K - Vibrational = 1090.458 cal/mol-K ``` Why are those frequencies the same at different temperature? Is there anything wrong in my procedure? Thank you so much! Longtao Edited On 2:32:01 PM PST - Wed, Mar 2nd 2016 by Olanky\n\n Sean bureaucrat Profile Send PM\n Forum Regular", null, "Threads 1 Posts 176\n 5:40:47 PM PST - Wed, Mar 2nd 2016 Please, try the following input: ```restart cnt_5_10-p1vib ``` ```set gen_hess:actlist 221 # 221st atom is the adatom set hessian:compress .true. task dft hessian numerical ``` ```geometry adatom adatom_x adatom_y adatom_z end ``` ```vib reuse temp 1 300 animate end ``` ```task dft freq ``` You need to first calculate the Hessian only for the active atom(s). Then if you want to do the frequency analysis for just that, you need to define a new geometry that contains just the active atom(s), otherwise you will get an end-of-file error. In regards to your second question, the frequencies are calculated in a harmonic approximation and there is no temperature dependence so the only change in output when the temperature is changed should be the thermochemistry analysis.\n\n Olanky Member Profile Send PM\n Clicked A Few Times", null, "Threads 2 Posts 5\n 2:56:19 PM PDT - Wed, Mar 16th 2016 Thank you for your reply, Sean. I just got chance to try today. I found if I define a new geometry with just active atom, the normal mode frequencies will all be zero. From the xyz file it's clear that the only atom in the geometry is not moving at all. The script in my first post can do the job, but it will generate thousands of xyz files. That's why it made me think if I did something wrong. I understand now to the second question, thank you! Edited On 2:57:05 PM PDT - Wed, Mar 16th 2016 by Olanky\n\n Who's here now Members 0 Guests 1 Bots/Crawler 0\n\nAWC's: 2.5.10 MediaWiki - Stand Alone Forum Extension\nForum theme style by: AWC", null, "" ]	[ null, "http://www.nwchem-sw.org/skins/gumax/blank.gif", null, "http://www.nwchem-sw.org/extensions/awc/forums/images/avatars/avatar_default.gif", null, "http://www.nwchem-sw.org/extensions/awc/forums/images/avatars/avatar_default.gif", null, "http://www.nwchem-sw.org/extensions/awc/forums/images/avatars/avatar_default.gif", null, "http://www.nwchem-sw.org/skins/gumax/blank.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8333849,"math_prob":0.7971458,"size":1598,"snap":"2019-26-2019-30","text_gpt3_token_len":439,"char_repetition_ratio":0.08782936,"word_repetition_ratio":0.06734007,"special_character_ratio":0.2753442,"punctuation_ratio":0.11246201,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9745041,"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\":\"2019-06-24T18:52:50Z\",\"WARC-Record-ID\":\"<urn:uuid:b334c5cc-aac5-4e0a-889a-7c27e124b80f>\",\"Content-Length\":\"47330\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:76fee81d-083e-43a1-9af0-d874a4237155>\",\"WARC-Concurrent-To\":\"<urn:uuid:b39a1295-2a4a-4877-b992-8e4f1eae0f85>\",\"WARC-IP-Address\":\"192.101.105.206\",\"WARC-Target-URI\":\"http://www.nwchem-sw.org/index.php/Special:AWCforum/st/id1987/Problem_with_numerical_frequency....html\",\"WARC-Payload-Digest\":\"sha1:BUWHDUFCNFCBDZ7T52F4KSV36VGI4P2N\",\"WARC-Block-Digest\":\"sha1:ABIDECHK7LZZWHYCBTUR2PLVNTPK3ARH\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-26/CC-MAIN-2019-26_segments_1560627999620.99_warc_CC-MAIN-20190624171058-20190624193058-00103.warc.gz\"}"}
https://www.pakainfo.com/get-table-name-from-model-using-laravel-5-7-example/	[ "# Get Table name From Model using Laravel 5.7 Example\n\nToday, We want to share with you Get Table name From Model using Laravel 5.7 Example.In this post we will show you get table name from model laravel 5.7, hear for laravel get table name from model, laravel eloquent table name we will give you demo and example for implement.In this post, we will learn about How to get table name and table column names from model in Laravel 5? with an example.\n\n## Get Table name From Model using Laravel 5.7 Example\n\nThere are the Following The simple About Get Table name From Model using Laravel 5.7 Example Full Information With Example and source code.\n\nRead Also: Get Current Url Parameter using Laravel\n\nAs I will cover this Post with live Working example to develop Get table column names as array from Eloquent model, so the get all column of a table using laravel eloquent for this example is following below.\n\n### How to return database table name in Laravel\n\n```\\$obj_item = new Item;\n\\$my_table_name = \\$obj_item->getTable();\n\nprint_r(\\$my_table_name);\n```\n\n### Get Table Name From Model in Laravel\n\n```\\$member = new Member;\n\\$table = \\$member->getTable();\nprint_r(\\$table);\n```\n\n### Get Table Column Name From Model in Laravel\n\ndefine a method in model class\n\n```<?php\nnamespace App;\nuse Illuminate\\Database\\Eloquent\\Model;\nclass Member extends Model\n{\npublic function getTableColumns() {\nreturn \\$this->getConnection()->getSchemaBuilder()->getColumnListing(\\$this->getTable());\n}\n}\n```\n\nget all columns of “members” table\n\n``` \\$member=new Member;\n\\$columns=\\$member->getTableColumns();\nprint_r(\\$columns);\n```" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7246123,"math_prob":0.65873736,"size":1837,"snap":"2021-43-2021-49","text_gpt3_token_len":412,"char_repetition_ratio":0.15657392,"word_repetition_ratio":0.07168459,"special_character_ratio":0.22482309,"punctuation_ratio":0.12536444,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9750062,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-26T11:39:53Z\",\"WARC-Record-ID\":\"<urn:uuid:d0574731-3c74-4985-b283-28812a0e7d14>\",\"Content-Length\":\"91921\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2c5a2869-104e-4964-a334-9479d27f11a5>\",\"WARC-Concurrent-To\":\"<urn:uuid:9aa7e875-6a92-4b70-8deb-339d2cf05972>\",\"WARC-IP-Address\":\"172.67.130.14\",\"WARC-Target-URI\":\"https://www.pakainfo.com/get-table-name-from-model-using-laravel-5-7-example/\",\"WARC-Payload-Digest\":\"sha1:P7S3T7RKIO425MKDLKNFR7QP3KDNOFPQ\",\"WARC-Block-Digest\":\"sha1:QRMMSRCGHZSZXYGVSIJPQEBMEZ7BIH45\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323587877.85_warc_CC-MAIN-20211026103840-20211026133840-00331.warc.gz\"}"}
https://tug.org/pipermail/metapost/2005-March/000329.html	[ "# [metapost] The strangest issues?\n\nDaniel H. Luecking luecking at uark.edu\nFri Mar 18 23:53:00 CET 2005\n\nOn Fri, 18 Mar 2005, Giuseppe Bilotta wrote:\n\n> Thursday, March 17, 2005 Daniel H. Luecking wrote:\n>\n> > (\\gamma(t) -\\omega(\\tau)).\\gamma'(t) = 0\n> > (\\gamma(t) -\\omega(\\tau)).\\omega'(\\tau) = 0\n>\n> > will be a minimum (or a saddle point). And minimums happen when the\n\nActually, I fogrgot to include local maxima.\n\n> > curves cross. Minimizing over the lambdas will now likely give you the\n> > crossing points. Of course, if \\gamma is a straight line, the\n> > alternative is that \\omega is a parallel line and (P1-P0) x (P3-P2) = 0.\n>\n> Well, t and \\tau are restricted to the [0,1] range and since\n> I'm for local, not global, extremal points I really don't\n> care if they are minima or maxima until I get down to\n> analyzing them. Which I cannot do if I cannot find them, of\n> course :)\n>\n> Plus, the curves \\omega and \\gamma are not entirely\n> arbitrary. While I'm trying not to put any constraints on\n> \\gamma, the endpoints of \\omega and its initial and final\n> tangents are in very well fixed relations to \\gamma.\n> Constraints on the lambdas can be also easily created so\n> that the two curves don't cross in the [0,1]^2 box, unless\n> the original curve is knotted itself or the radius is higher\n> than a critical value.\n>\n> Does this mean I only have to look at the values for which\n> t and \\tau are not differentiable?\n\nSince the two equations to be solved for t and \\tau are both cubic\nin one variable and quintic in the other, if they could be solved\nfor t and \\tau in terms of the lambdas they would be multiple-valued.\nThere would therefore be the possibility (even likelyhood) of \"branch\npoints\" where these different values come together and differential\nanalysis is impossible.\n\nAnd here we have rapidly converged to the limits of my knowledge of\nthese things.\n\n--\nDan Luecking\nDept. of Mathematical Sciences\nUniversity of Arkansas\nFayetteville, AR 72101" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9083364,"math_prob":0.79565066,"size":2014,"snap":"2019-35-2019-39","text_gpt3_token_len":562,"char_repetition_ratio":0.0920398,"word_repetition_ratio":0.010989011,"special_character_ratio":0.27507448,"punctuation_ratio":0.11025641,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9755914,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-09-15T14:33:47Z\",\"WARC-Record-ID\":\"<urn:uuid:b2dba9a3-5254-430f-9fce-a7e71fc2b4d5>\",\"Content-Length\":\"4570\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c3cfaf71-6470-4244-9778-175206107db6>\",\"WARC-Concurrent-To\":\"<urn:uuid:54767d4d-5ca3-4924-83c2-6e6336bf0266>\",\"WARC-IP-Address\":\"91.121.174.77\",\"WARC-Target-URI\":\"https://tug.org/pipermail/metapost/2005-March/000329.html\",\"WARC-Payload-Digest\":\"sha1:VIS7FRTLVLIRQIYSIO2BUPLHDKCULOFG\",\"WARC-Block-Digest\":\"sha1:FDDTXVXWCZRAHTGLVWAIWFJ3GL472F7P\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-39/CC-MAIN-2019-39_segments_1568514571506.61_warc_CC-MAIN-20190915134729-20190915160729-00380.warc.gz\"}"}
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西双版纳傣族自治州\n• 玉溪市\n• 昭通市\n• 新疆\n• 阿克苏地区\n• 阿勒泰地区\n• 巴音郭楞蒙古自治州\n• 博尔塔拉蒙古自治州\n• 昌吉回族自治州\n• 哈密地区\n• 和田地区\n• 喀什地区\n• 克拉玛依市\n• 克孜勒苏柯尔克孜自治州\n• 直辖行政单位\n• 塔城地区\n• 吐鲁番地区\n• 乌鲁木齐市\n• 伊犁哈萨克自治州\n• 黑龙江\n• 大庆市\n• 大兴安岭地区\n• 哈尔滨市\n• 鹤岗市\n• 黑河市\n• 鸡西市\n• 佳木斯市\n• 牡丹江市\n• 七台河市\n• 齐齐哈尔市\n• 双鸭山市\n• 绥化市\n• 伊春市\n• 香港\n• 香港\n• 九龙\n• 新界\n• 澳门\n• 澳门\n• 其它地区\n• 台湾\n• 台中市\n• 台南市\n• 高雄市\n• 台北市\n• 基隆市\n• 嘉义市\n•", null, "10吨吊车价格\|开封品质好的10吨吊车推荐\n\n品牌:远东重工,,\n\n出厂地:昭平县(昭平镇)\n\n报价：面议\n\n河南远东重工机械制造有限公司\n\n黄金会员：", null, "主营：吊车,随车吊,起重机,随车起重机,流动式汽车起重机\n\n•", null, "蒙古包厂家_内蒙古高质量的蒙古包生产制作工艺\n\n品牌:诺瑜玛,,\n\n出厂地:融安县(长安镇)\n\n报价：面议\n\n内蒙古诺瑜玛民族用品有限公司\n\n黄金会员：", null, "主营：蒙古包,工艺品,木质工艺家具,蒙古桌,防寒门帘\n\n•", null, "哈尔滨有口碑的哈尔滨水处理\|哈尔滨过滤设备\n\n品牌:金源,金源水处理,\n\n出厂地:鱼峰区\n\n报价：面议\n\n哈尔滨金源水处理有限公司\n\n黄金会员：", null, "主营：水处理设备,过滤设备,纯水设备,水处理材料,水处理安装\n\n•", null, "优良的治疗慢性胃炎推荐，开封中医治疗慢性胃炎怎么样\n\n品牌:济生堂,,\n\n出厂地:昭平县(昭平镇)\n\n报价：面议\n\n支国顺诊所\n\n黄金会员：", null, "主营：新乡肠胃科疾病,耳鼻喉疾病,肿瘤科疾病,治疗肠胃病\|耳鼻喉疾病,看肠胃病哪家好\n\n•", null, 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https://de.scribd.com/document/338645568/COE10703-Designing-Cathodic-Protection-Systems-pdf	[ "Sie sind auf Seite 1von 99\n\n# Engineering Encyclopedia\n\n## Designing Cathodic Protection Systems\n\nNote: The source of the technical material in this volume is the Professional Engineering\nDevelopment Program (PEDP) of Engineering Services.\nWarning: The material contained in this document was developed for Saudi Aramco and is\nintended for the exclusive use of Saudi Aramcos employees. Any material contained\nin this document which is not already in the public domain may not be copied,\nreproduced, sold, given, or disclosed to third parties, or otherwise used in whole, or in\npart, without the written permission of the Vice President, Engineering Services, Saudi\nAramco.\n\n## Chapter : Cathodic Protection For additional information on this subject, contact\n\nFile Reference: COE10703 D.R. Catte on 873-0153\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nCONTENTS PAGES\n\n## DESIGNING CATHODIC PROTECTION SYSTEMS FOR BURIED PIPELINES 1\n\nGalvanic Anode System Design for Road and Camel Crossings 2\nSaudi Aramco Engineering Standards and Drawings 2\nNumber of Galvanic Anodes Required 3\nCircuit Resistance 4\nGalvanic Anode Current Output 7\nGalvanic Anode Life 7\nExample 8\nNumber of Anodes 8\nCircuit Resistance 8\nGalvanic Anode Current Output 8\nGalvanic Anode Life 9\nImpressed Current System Design for Buried Pipelines 9\nSaudi Aramco Engineering Standards and Drawings 9\nMinimum Number of Impressed Current Anodes 12\nAnode Bed Resistance 13\nAmount of Coke Breeze Required 15\nExample 15\nMinimum Number of Impressed Current Anodes 15\nAnode Bed Resistance 16\nAmount of Coke Breeze Required 18\nDESIGNING CATHODIC PROTECTION SYSTEMS FOR ONSHORE WELL CASINGS 19\nSaudi Aramco Engineering Standards and Drawings 20\nCathodic Protection Current Requirements 23\nSurface Anode Bed Design 25\nDeep Anode Bed Design 26\nLength of the Coke Breeze Column 26\nCircuit Resistance 27\nAmount of Coke Breeze Required 28\nExample 29\nLength of the Coke Breeze Column 29\nCircuit Resistance 31\nAmount of Coke Breeze Required 31\n\n## Saudi Aramco DeskTop Standards\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## DESIGNING CATHODIC PROTECTION SYSTEMS FOR VESSEL AND TANK INTERIORS 32\n\nSaudi Aramco Engineering Standards and Drawings 33\nGalvanic Anode System Design for Vessel and Tank Interiors 36\nCurrent Output Per Anode 36\nNumber of Galvanic Anodes Required 37\nGalvanic Anode Life 37\nExample 38\nCurrent Output Per Anode 38\nNumber of Galvanic Anodes Required 38\nGalvanic Anode Life 38\nImpressed Current System Design for Vessel and Tank Interiors 40\nNumber of Impressed Current Anodes Required 40\nCircuit Resistance 41\nExample 42\nNumber of Impressed Current Anodes 42\nCircuit Resistance 43\nDESIGNING CATHODIC PROTECTION SYSTEMS FOR IN-PLANT FACILITIES 44\nSaudi Aramco Engineering Standards and Drawings 45\nNumber and Placement of Anodes in Distributed Anode Beds 47\nCircuit Resistance 50\nExample 52\nNumber and Placement of Impressed Current Anodes 52\nDESIGNING CATHODIC PROTECTION SYSTEMS FOR MARINE STRUCTURES 54\nSaudi Aramco Engineering Standards and Drawings 56\nGalvanic Anode System Design for Marine Structures 59\nNumber of Galvanic Anodes Required 59\nCircuit Resistance 60\nGalvanic Anode Life 60\nNumber and Spacing of Galvanic Anode Bracelets 61\nExample 62\nNumber of Anodes 62\nGalvanic Anode Life 63\nNumber and Spacing of Galvanic Anode Bracelets 63\nImpressed Current System Design for Marine Structures 64\nCorrected Current Requirement 64\nNumber of Impressed Current Anodes Required 64\nRectifier Voltage Requirement 65\n\n## Saudi Aramco DeskTop Standards\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nExample 66\nCorrected Current Requirement 66\nNumber of Anodes Required 66\nRectifier Voltage Requirement 67\nWORK AID 1: DATA BASE, FORMULAS, AND PROCEDURES TO DESIGN CATHODIC PROTECTION\nSYSTEMS FOR BURIED PIPELINES 68\nWork Aid 1A: Data Base, Formulas, and Procedure to Design Galvanic Anode Systems for Road and Camel\nCrossings 68\nWork Aid 1B: Formulas and Procedure to Design Impressed Current Systems for Buried Pipelines 71\nWORK AID 2: FORMULAS AND PROCEDURE TO DESIGN CATHODIC PROTECTION SYSTEMS FOR\nONSHORE WELL CASINGS 75\nWORK AID 3: FORMULAS AND PROCEDURES TO DESIGN CATHODIC PROTECTION SYSTEMS\nFOR VESSEL & TANK INTERIORS 78\nWork Aid 3A: Formulas and Procedure for the Design of Galvanic Anode Systems for Vessel & Tank\nInteriors 78\nWork Aid 3B: Formulas and Procedure for the Design of Impressed Current Systems for Vessel & Tank\nInteriors 81\nFormulas 81\nWORK AID 4: FORMULAS AND PROCEDURE TO DESIGN CATHODIC PROTECTION SYSTEMS FOR\nIN-PLANT FACILITIES 83\nWORK AID 5: FORMULAS AND PROCEDURES TO DESIGN CATHODIC PROTECTION SYSTEMS\nFOR MARINE STRUCTURES 85\nWork Aid 5A: Data Base, Formulas, and Procedure for the Design of Galvanic Anode Systems for Marine\nStructures 85\nWork Aid 5B: Formulas and Procedure for the Design of Impressed Current Systems for Marine Structures\n89\nGLOSSARY 92\nAPPENDIX 1 94\nSaudi Aramco Engineering Standards 94\nSaudi Aramco Standard Drawings 94\nSaudi Aramco Material System Specifications 95\n\n## Saudi Aramco DeskTop Standards\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Designing Cathodic Protection Systems for Buried Pipelines\n\nThis section is divided into two parts. The first part covers galvanic anode system designs for short pipeline\nsegments such as road and camel crossings. Galvanic anodes are used if the cathodic protection current\nrequirement is small and the soil resistivity is low. The second part will cover impressed current systems for\nburied pipelines which require much more cathodic protection current. Normally, Saudi Aramco protects\nonshore pipelines with impressed current systems.\n\nDesigns for galvanic anode and impressed current systems designs are prepared after the following has been\naccomplished:\n\n## the cathodic protection current requirements have been calculated\n\nthe effective resistivity of the soil has been determined\nthe anode bed location has been selected\nthe allowable anode bed resistance has been calculated\n\nIn Module 107.01, you calculated the current requirements for various structures. In Module 107.02, you\nselected an anode bed site based on soil resistivity, current distribution, and available utilities. You also\nrepresented proposed CP systems as equivalent electrical circuits and calculated their allowable anode bed\nresistance. In this section, you will be given the above information and other criteria that will allow you to\ndesign cathodic protection systems for buried pipelines.\n\n## Saudi Aramco DeskTop Standards 1\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Galvanic Anode System Design for Road and Camel Crossings\n\nDesign standards and practices for galvanic anode systems for road and camel crossings are presented below.\nThe design of galvanic anode systems for pipelines involves determining the following:\n\n## design requirements using Saudi Aramco standards and drawings\n\nthe number of galvanic anodes required\ncircuit resistance\ngalvanic anode current output\ngalvanic anode life\n\nAfter describing these requirements and calculations, an example is provided which demonstrates the design of\na galvanic anode system for a section of pipeline.\n\n## Saudi Aramco Engineering Standards and Drawings\n\nSaudi Aramco Engineering Standard SAES-X-400 provides minimum design requirements that govern CP\nsystems for buried onshore pipelines. CP systems inside plant facilities are not included. SAES-X-400 requires\ngalvanic anodes at the following sites:\n\n## buried pipelines at paved road crossings\n\nburied pipelines at camel crossings\nshort segments of pipelines that are not part of an impressed current system\n\nSaudi Aramco uses either pre-packaged or bare magnesium anodes to protect short pipeline segments. Bare\nanodes are used only in Subkha areas. The design calculations in this module are based on construction\nstandards in Standard Drawing AA-036352 - Galvanic Anodes for Road & Camel P/L Crossings, P/L Repair\nLocations. Figures 1A, 1B, and 1C show typical galvanic anode installations from Standard Drawing AA-\n036352.\n\nBonding station\nmarker plate\n\nmin.\nThermite weld\n600 mm min.\n\n1500 mm min.\nMagnesium anodes Cross section\nTypical Galvanic Anode Installation for a Road Crossing\nFigure 1A\n\n## Saudi Aramco DeskTop Standards 2\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nBonding station\n3600 mm marker plate\nmin.\n\nThermite weld\n\nMagnesium anodes\nCross section\n\nFigure 1B\n\nJunction box\n\nThermite weld\n\nburied\nvalve\n\nmagnesium anodes\n\nFigure 1C\n\n## the size (weight) of the anodes\n\nthe length of the pipe\nthe diameter of the pipe\n\nAt least two anodes are required for any installation. Work Aid1A provides a table from Standard Drawing\nAA-036352 and a procedure for determining the number of magnesium anodes required.\n\n## Saudi Aramco DeskTop Standards 3\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nCircuit Resistance\n\nThe circuit resistance of the galvanic anode system, RC, is represented by the electrical circuit in Figure 2.\n\nBonding\nstation I\n\nI1 I2\n\nED\nRA1 RA2\n\nGalvanic Anodes I\nRS\n\n## Galvanic Anodes at a Road Crossing and an Equivalent Electrical Circuit\n\nFigure 2\n\nThe structure-to-electrolyte resistance is represented by RS in the electrical circuit. The anode resistances are\nRA1 and RA2. Because the anodes are connected in parallel, their equivalent resistance is obtained from the\nfollowing formula:\n\n1 1 1 1\n= + + +\nR eq R A 1 R A 2 R AN\n\nIf the anodes resistances are equal, the equivalent resistance is given by the following formula.\n\n1 RA\n= 1 + 1 + + 1 = N R eq =\nR eq R A R A R AN RA N\n\nRA = RLW + RV,\n\nwhere -\n\n## RLW = the average anode lead wire resistance in ohms\n\nRV = the anode-to-electrolyte resistance of one vertical anode in ohms\n\n## Saudi Aramco DeskTop Standards 4\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Therefore, the circuit resistance is determined by the following equation:\n\nRA R + RV\nRc = R s + = R s LW\nN N\n\nFor an anode buried in chemical backfill as shown in Figure 3, the total resistance between the anode and\nelectrolyte includes (1) the resistance from the anode to the outer edge of the backfill package and (2) the\nresistance between the backfill package and the soil. The resistance from the anode to the outer edge of the\nbackfill is called the anode internal resistance. The resistance between the backfill and the soil is commonly\ncalled the anode-to-earth resistance.\n\nBag\n\n## Soil Chemical Anode\n\nbackfill\n\nAnode- Anode\nto-earth internal\nresistance resistance\n\nFigure 3\n\n## Saudi Aramco DeskTop Standards 5\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nBecause the contribution of the anode internal resistance is very small, Saudi Aramco only considers the anode-\nto-earth resistance. The anode-to-earth resistance of a single vertical anode is calculated using the Dwight\nEquation as follows:\n\n0.159\n1\n8L\nRV = ln\nL d\n\nwhere -\n\n## RV = resistance of one vertical anode to earth in ohms\n\nr = resistivity of backfill material (or soil) in ohm-cm\nL = length of anode (or backfill column) in centimeters\nd = diameter of anode (or backfill column) in centimeters\n\nPrepackaged magnesium anodes are used in most soil installations. Therefore, L and d above will be the\nnominal length and diameter of the anode backfill package.\n\nYou can calculate the anode bed resistance of two or more vertical anodes in parallel by using the Sunde\nEquation as follows:\n\n0.159\n1 +\n8L 2L\nR=\nNL\nln\nd S (\nln 0.656 N\n\n)\nwhere -\n\nR = resistance, in ohms, of N vertical anodes in parallel and spaced S centimeters apart along a\nstraight line.\nr = soil resistivity in ohm-cm\nN = number of anodes\nL = length of anode (or backfill column) in centimeters\nd = diameter of anode (or backfill column) in centimeters\nS = anode spacing in centimeters\n\nAnodes are usually cast in the shape of a trapezoid rather than a cylinder. If an anode is installed in Subkha\nwithout a backfill package, its effective diameter must be calculated. For example, a trapezoidal anode with\nnominal 7.5 cm sides has a circumference of 4 x 7.5 cm = 30 cm. The effective diameter is 30 cm/, or 9.5\ncm.\n\n## Saudi Aramco DeskTop Standards 6\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Galvanic Anode Current Output\n\nSAES-X-400 and SADP-X-100 require a calculation of the anode current output. The current output of a\ngalvanic anode system is a function of its driving potential and circuit resistance, as shown in the following\nformula:\n\nIA = ED/RC\nwhere -\n\n## IA = anode current output\n\nED = the anode driving potential\nRC = the circuit resistance\n\nThe driving potential, ED, is the difference between the anodes solution potential and the protected potential of\nthe pipeline.\n\n## Galvanic Anode Life\n\nThe life of a galvanic anode can be estimated if its weight and current output are known. The expected life of a\ngalvanic anode is given by the following equation from SADP-X-100, section 4.2, Eqn. 23.\n\nW UF\nY=\nC IA\n\nwhere -\n\n## Y = anode life in years\n\nC = actual consumption rate in kg/A-yr\nW = anode mass in kg\nIA = anode current output in amperes\nUF = utilization factor\n\nThe actual consumption rate, C, of standard and high potential magnesium anodes is 7.1 kg per ampere-year.\nAn anode needs to be replaced when there is not enough of it remaining to produce the required current. The\nutilization factor, UF, is the percentage of the anode that is consumed before it needs to be replaced. For\nmagnesium anodes, the utilization factor is 85%.\n\n## Saudi Aramco DeskTop Standards 7\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nExample\n\nWe will use the following data to determine the number and current output of pre-packaged 27.3 kg (60 lb.)\nmagnesium anodes required to protect a 15-meter section of 12\" pipe. Use the following engineering data:\n\n## Driving potential: 0.50 V versus Cu-CuSO4\n\nStructure-to-electrolyte resistance: 2.67 ohms\nBackfill package dimensions: 8\" dia. x 84\" (20.33 cm dia. x 213.36 cm)\nSoil resistivity: 1,000 ohm-cm\nNumber of Anodes\n\nAccording to the table in Work Aid 1A, two anodes are required for 15 meters of 12\" pipe.\nCircuit Resistance\n\nThe anode-to-earth resistance of one anode is given by the Sunde Equation as shown below:\n\n0.159\nln 8L 1 + ln 0.656 N )\n2L\nRV =\nNL d S\n(\n0.159( ohm cm) 8(213.36 cm) 2(213.36)\n=\n2(213.36 cm )\nln\n20.33 cm\n1 +\n1, 500\n(ln1.312 )\nR V = 1.307 ohm\n\n## RC = 2.67 + 0.025 + 1.307 = 4.00 ohms.\n\nGalvanic Anode Current Output\n\n## I = ED/RC = 0.50/4.00 = 0.13 A. (or 0.065 A for each anode)\n\nSaudi Aramco normally uses magnesium anodes in areas where soil resistivity is less than 5,000 ohm-cm. In\n5,000 ohm-cm soil, the anode-to-earth resistance in the example above would be 6.53 ohms (five times as much\nas in 1,000 ohm-cm soil). The circuit resistance would increase to 9.21 ohms and the current output would\ndecrease as follows:\n\n## Saudi Aramco DeskTop Standards 8\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Galvanic Anode Life\n\nThe expected lifetime of one 27.3 kg anode with a current output of 0.065 A in 1,000 ohm-cm soil is shown\nbelow:\n\n27.3 kg 0.85\nY=\n7.1 kg / amp yr 0.065 amp\nY = 50 years\n\nThe anode requirements, formulas, and procedure needed to design galvanic anode systems for short sections of\nburied pipelines are provided in Work Aid 1A.\n\n## Impressed Current System Design for Buried Pipelines\n\nDesign standards and practices for impressed current systems for buried pipelines are presented below. These\nstandards and practices include the following determinations:\n\n## design requirements using Saudi Aramco standards and drawings\n\nthe minimum number of impressed current anodes\nanode bed resistance (based on number of anodes and anode spacing)\nthe amount of coke breeze required\n\nAfter a discussion of the above information, an example is provided that includes a more efficient method,\nusing an anode design chart for designing impressed current anode beds.\n\n## Saudi Aramco Engineering Standard SAES-X-400 states the following:\n\nTotal circuit resistance for a rectifier CP system shall not exceed 1.0 ohm.\nTotal circuit resistance for a solar CP system shall not exceed 0.5 ohm.\nImpressed current systems shall provide a minimum negative pipe-to-soil potential of 1.2 volts\nand a maximum of 3.0 volts versus a Cu-CuSO4 half-cell.\nImpressed current anode beds shall be sized to discharge 120% of the rated current output of\nthe dc power source.\nImpressed current systems shall have a design life of 20 years.\n\nSaudi Aramco Design Practice SADP-X-100 states that surface anode beds less than 15 meters deep should\nalways be used unless they are uneconomical. Surface anode beds with watering facilities are usually more\neconomical than deep anode beds. Deep anode beds are much more expensive to install than surface anode\nbeds.\n\n## Saudi Aramco DeskTop Standards 9\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nAnode bed design calculations are based on construction standards set by Saudi Aramco in Standard Drawing\nAA-036346, Surface Anode Bed Details. AA-036346 contains diagrams of vertical and horizontal anode\ninstallations as shown in Figure 4.\n\n## Dual vertical anodes Vertical anode\n\nin coke breeze in Subkha\n\nGravel\n600 mm\n900 mm\n\nWatering\n50 mm hole pipe\n\n4000 mm\nAnode Anode\n2100 mm\n\nCoke\nbreeze\nNative clean\nbackfill\n8000 mm\n\n150 mm\n1000 mm\nmin. dia.\n250\nmm\n\nNo. 6 AWG\n\n2100 mm\n\nFigure 4\n\n## Saudi Aramco DeskTop Standards 10\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nImpressed current anode beds should be remote from the protected structure to provide uniform current\ndistribution. Figure 5 gives the minimum distances allowed between anode beds and buried structures. These\ncriteria cover both surface and deep anode beds.\n\n## Minimum Distance from\n\nAnode Bed Capacity Underground Structures\n35 amperes 35 meters\n50 amperes 75 meters\n100 amperes 150 meters\n150 amperes 225 meters\n\n## Minimum Anode Bed Distance from Underground Structures in SAES-X-400\n\nFigure 5\n\nSAES-X-400 states that remote surface anode beds shall be used where soil resistivity is compatible with\nsystem design requirements and economic considerations. Figure 6 shows a typical anode bed of 10 vertical\nanodes from Standard Drawing AA-036346. Additional groups of 10 anodes can be installed as required to\nrectifiers, must be separated by at least 50 meters. If the output capacity of either anode bed is greater than 50\namperes, they must be separated by at least 100 meters.\n\n## Typical group of 10 anodes Additional group of 10 as required\n\nNo . 6 AWG\nJunction\nBox\n\nTo rectifier or\nd-c power source\n\n10 anodes as required\n\nFigure 6\n\n## Saudi Aramco DeskTop Standards 11\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Minimum Number of Impressed Current Anodes\n\nThere are two ways to calculate the minimum number of impressed current anodes required. One method\nconsiders the anodes maximum current output in the electrolyte and the other method considers the anodes\nconsumption rate. It is best to use the method that gives the more conservative value (the greatest number of\nanodes).\n\nTo calculate the minimum number of anodes based on the anodes maximum current density, the following\nformula is used:\n\nN = I ( dL A )\n\nwhere -\n\n## N = number of impressed current anodes\n\nI = total current required in milliamperes\nd = anode diameter in centimeters\nL = anode length in centimeters\nA = anode maximum current density in mA/cm2 (Appendix I of SAES-X-400)\n\nTo calculate the minimum number of anodes based on the anodes consumption rate, the following formula is\nused:\n\nN = Y I C\nW\n\nwhere -\n\n## N = number of impressed current anodes\n\nY = the impressed current system design life in years\nI = total current required in amperes\nC = anode consumption rate in kg/A-yr (Appendix I of SAES-X-400)\nW = weight of a single anode in kg\n\n* The total current required is usually multiplied by 120% to adequately size the anode bed.\n\n## Saudi Aramco DeskTop Standards 12\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Anode Bed Resistance\n\nThe current output of an impressed current system is a function of the dc power source driving voltage and the\ncircuit resistance. The current output, I, is given by the following formula:\n\nI = ED/RC\nwhere -\nED = the rated voltage of the dc power source (minus 2 volts if the anodes are installed in coke\nbreeze)\nRC = the circuit resistance\n\nIn Module 107.02, we used the following formula to calculate circuit resistance, RC, of an impressed current\nsystem circuit.\n\nRC = RS + RLW + Rgb\n\nwhere -\nRS = structure-to-electrolyte resistance\nRLW = total lead wire resistance\nRgb = the anode bed resistance\n\nThe anode bed resistance, Rgb, is the total resistance of all the anodes in the anode bed. If the anodes are\nsurrounded by a coke breeze column as shown in Figure 7, the resistance between each anode and electrolyte\nincludes the anode internal resistance and the anode-to-earth resistance.\n\nGravel\n\nAnode-\nto-earth\nresistance\n\nAnode\ninternal Coke breeze\nresistance\n\nSoil Coke\nbreeze\n\nFigure 7\n\n## Saudi Aramco DeskTop Standards 13\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nAs with galvanic anodes, the internal resistance does not add significantly to the anodes total resistance.\nTherefore, Saudi Aramco only considers the anode-to-earth resistance. You can calculate the anode-to-earth\nresistance of a single vertical impressed current anode by using the Dwight Equation as follows:\n\n0.159\n1\n8L\nRV =\nl\nn\nL d\n\nwhere -\nRV = resistance of one vertical anode to earth in ohms\nr = resistivity of soil in ohm-cm\nL = length of anode (or backfill column) in centimeters\nd = effective diameter of anode (or backfill column) in centimeters\n\nYou can calculate the anode bed resistance of two or more vertical anodes in parallel by using the Sunde\nEquation as follows:\n\n0.159\n1 + ( )\n8L 2L\nR= ln ln 0.656 N\nNL d S\n\nwhere -\nR = resistance, in ohms, of N vertical anodes in parallel and spaced S centimeters apart along a\nstraight line.\nr = soil resistivity in ohm-cm\nN = number of anodes\nL = length of anode (or backfill column) in centimeters\nd = diameter of anode (or backfill column) in centimeters\nS = anode spacing in centimeters\n\nAccording to the Sunde Equation, the anode bed resistance decreases with an increase in the number of anodes\nand/or an increase in the anode spacings. By adjusting the number and spacing of anodes, you can achieve a\ndesired anode bed resistance. The desired anode bed resistance should be less than the allowable anode bed\nresistance given by the following formula:\n\n## Ragb = Rmax - (RS + RLW )\n\nwhere -\nRagb = the allowable anode bed resistance\nRmax = the maximum allowable circuit resistance (the rectifiers rated voltage minus 2 volts,\ndivided by its rated current output)\nRS = structure-to-electrolyte resistance\nRLW = total lead wire resistance\n\n## Saudi Aramco DeskTop Standards 14\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Amount of Coke Breeze Required\n\nTo calculate the net volume of coke breeze in each backfill column, the anode volume is subtracted from the\nvolume of the backfill column. This net volume is multiplied by the number of anodes and the coke breeze\ndensity to obtain the weight of coke breeze required. An extra 20% is added to cover spills and other waste.\n\nExample\n\nThe following example assumes that the structure-to-electrolyte resistance and the lead wire resistance are\nknown and the maximum allowable anode bed resistance has been determined. We will determine the number\nand spacing of anodes needed so that the anode bed resistance does not exceed the allowable anode bed\nresistance. Use the following engineering data.\n\n## CP current required: 16.5 amperes\n\nAnode material: Silicon iron\nAnode dimensions: 7.6 cm dia. x 152 cm length\nAnode consumption rate: 1 kg/A-yr\nMax. anode current density: 1 mA/cm2\nAnode weight: 50 kg\nBackfill dimensions: 20 cm dia. x 300 cm\nSoil resistivity: 5,000 ohm-cm\nAllowable anode bed resistance: 0.84 ohm\nCoke breeze density: 730 kg/m3\nMinimum Number of Impressed Current Anodes\n\nWe will design the anode bed so that it can discharge 20 amperes 120% of the 16.5 amperes required. To\nestimate the number of anodes required, multiply the total current requirement by the design life and\nconsumption rate of the anode material as follows.\n\n(\n\n)\nN = Y I C = (20 years )(20 A)(1 kg/A yr )/50 kg = 8 anodes\nW\n\nWe will use 10 anodes for the first calculation. (Using the current density method to calculate the minimum\nnumber of anodes would result in 6 anodes.)\n\n## Saudi Aramco DeskTop Standards 15\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Anode Bed Resistance\n\nSubstitute 10 anodes for N, 305 cm (10 ft.) spacing for S, and the backfill dimensions into the Sunde Equation\nas follows.\n\nR=\n0.159 8L\nNL (\nln\nd\n1 + )\n2L\nS\n(ln 0.656 N)\n0.159 ( 5,000 ) 8(300) 2(300 )\n= ln 1 + ln(0.656)(10 )\n(10 )( 300 ) 20 (305 )\nR = 1.984 ohms\n\nThis anode bed resistance exceeds the maximum allowable anode bed resistance of 0.84 ohms. However,\naccording to the Sunde Equation, increasing the number of anodes can lower the resistance. If we substitute\nvalues of 20, 30, and 40 anodes for N at the 305 cm spacing, we obtain the following values.\n\n## No. of Anode Bed Resistance\n\nAnodes at 305 cm Spacing\n10 1.984\n20 1.173\n30 0.852\n40 0.677\n\nThe calculated anode bed resistance of 40 anodes installed with 305 cm spacings is less than the allowable\nresistance of 0.84 ohm. However, remember that increasing the anode spacing also decreases the anode bed\nresistance. If we repeat the calculations for spacings of 457, 610, 762, and 914 cm, (15, 20, 25, and 30 ft.) we\nobtain the following table.\n\n## No. of Anode Spacing in Centimeters\n\nAnodes 305 457 610 762 914\n10 1.984 1.658 1.494 1.396 1.331\n20 1.173 0.950 0.837 0.770 0.726\n30 0.852 0.680 0.593 0.542 0.507\n40 0.677 0.535 0.464 0.421 0.393\n\nBased on the allowable anode bed resistance of 0.84 ohms, one option appears to be 20 anodes with 610 cm\nspacings. Another option30 anodes with 457 cm spacings-would result in an anode bed resistance of 0.68\nohm. We can graph the values in the table to create a design chart as shown in Figure 8.\n\n## Saudi Aramco DeskTop Standards 16\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n10.0\n\n305 cm spacing\n457 cm spacing\n610 cm spacing\n762 cm spacing\n914 cm spacing\nRaab\n1.0\n0.84\n\n0.5\n\n0.1\n2 10 20 30 40\nNUMBER OF ANODES\n\n## Vertical Anode Design Chart for an Impressed Current Anode Bed\n\nin Soil with a Resistivity of 5,000 ohm-cm\nFigure 8\n\nDesign charts are an efficient alternative to making several calculations for each anode bed design. The design\nchart in Figure 8 is based on a soil resistivity of 5,000 ohm-cm. To use this chart for other soil resistivities, the\nallowable anode bed resistance, Ragb, must be converted to a value that corresponds to a soil resistivity of\n5,000 ohm-cm. The Sunde Equation can be used to show that anode bed resistance is directly proportional to\nsoil resistivity as follows:\n\nR ohm cm ohm cm\n=\nR 5,000 ohm cm 5,000 ohm cm\n\nTherefore,\n\n## Saudi Aramco DeskTop Standards 17\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nIn summary, the allowable anode bed resistance is determined for 5,000 ohm-cm soil. Then the design chart in\nFigure 8 is used to select the optimum number and spacing of anodes to achieve an anode bed resistance less\nthan or equal to the allowable anode bed resistance. Work Aid 1B provides a procedure for using a design chart\nto determine the optimum number and spacing of impressed current anodes.\n\n## Amount of Coke Breeze Required\n\nNext, we will calculate the amount of coke breeze required. Assume that the anode dimensions are 7.6 cm dia.\nx 152 cm and the coke breeze column dimensions are 20 cm. dia. x 300 cm length. First, the anode volume is\nsubtracted from the volume of the anode backfill column.\n\n## 0.09 - 0.007 = 0.083 m3.\n\nTo obtain the weight of coke breeze required, this net volume is multiplied by the number of anodes and the\ncoke breeze density. An extra 20% is added to cover spills.\n\n## (0.083 m3)(20 anodes)(730 kg/m3)(120%) = 1,454 kg\n\nThe formulas and procedure to design impressed current anode beds are provided in Work Aid 1B.\n\n## Saudi Aramco DeskTop Standards 18\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Designing Cathodic Protection Systems for Onshore Well Casings\n\nSaudi Aramco cathodically protects all onshore well casings with impressed current systems. Saudi Aramcos\ngoal is to protect both well casings and associated flowlines and pipelines as an integrated system. This is\naccomplished by minimizing the use of pipeline insulating devices. If an insulation device is installed, a\nbonding box is used in case it becomes necessary to short circuit the insulator. Saudi Aramco normally uses an\nindividual impressed current system to protect each well. However, multiple wells are sometimes protected by\na single impressed current system.\n\nSaudi Aramco uses both surface and deep anode beds to protect onshore well casings. The type of anode bed\nand its location are determined by the following:\n\n## its current output capacity\n\nthe surface soil resistivity\nthe number of well casings to be protected\nthe physical layout of the wells and facilities\neconomics\n\nSaudi Aramco uses remote surface anode beds where soil resistivity is low enough for adequate current\ndistribution. Where surface soil resistivity is high, deep anode beds are used. Deep anode beds are also used in\ncongested areas such as pipeline corridors and in-plant areas to provide better current distribution.\n\nBoth surface and deep anode bed designs involve the following determinations:\n\n## design requirements using Saudi Aramco Engineering Standards and Drawings\n\ncathodic protection current requirements\n\nDescriptions of both requirements are provided in this section. After the information on cathodic protection\ncurrent requirement is presented, surface and deep anode bed designs are discussed separately. Surface anode\nbed design for a well casing is similar to surface anode bed design for a buried pipeline, which was covered in\nthe first section of this module. Therefore, this section focuses mainly on the design of deep anode beds.\n\n## Saudi Aramco DeskTop Standards 19\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Saudi Aramco Engineering Standards and Drawings\n\nThe design of cathodic protection systems for onshore well casings is governed by Saudi Aramco Engineering\nStandard SAES-X-700. SAES-X-700 states the following:\n\nthe design capacity of impressed current systems shall be 50 amperes per well with uncoated\ncasings and 10 amperes per well with coated casings. The Consulting Services Department\nmay approve designs for lower capacity systems if adequate protection is verifiable.\n\na single impressed current system may be used to protect more than one well if the wells are\nless than 200 meters apart.\n\nimpressed current anode beds shall be sized to discharge 120% of the rated current output of\nthe dc power source.\n\n## impressed current systems shall have a design life of 20 years.\n\nAccording to G.I. 428.003, a minimum negative casing-to-soil potential of 1.0 volt (current off) versus Cu-\nCuSO4 is required.\n\nA minimum distance of 150 meters is required between a deep anode bed and the well casing it is to protect. A\nminimum distance of 150 meters is also required from the anode bed to plant (GOSP, etc.) perimeter fencing.\nIn addition, SAES-X-700 requires that deep anode beds are located remote from other buried structures. A\ndistance of 50 meters is required for deep anode beds with a design current output of less than 30 amperes. A\ndistance of 100 meters is required for anode beds with capacities between 30 and 50 amperes.\n\nSurface anode beds should be designed in accordance with Standard Drawing AA-036346. Scrap steel surface\nanode beds should be designed in accordance with Standard Drawing AA-036278.\n\n## Saudi Aramco DeskTop Standards 20\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## There are two types of deep anode beds: aquifer\n\npenetrating and non-aquifer penetrating. An aquifer Anode\npenetrating deep anode bed is shown in Figure 9. junction\nImpressed current anodes and a PVC vent pipe are box\nstrapped to 2-3/4\" steel tubing and surrounded by\ncoke breeze inside 9-5/8\" casing. A water and coke PVC vent\npipe\nbreeze slurry is pumped in the hole from the bottom\nup through the steel tubing. An individual lead wire\nPositive\nconnects each anode to the junction box. cable\nSurface from d-c\nAnode reactions with water or brine generate chlorine casing power\nsource\ngas and oxygen. If these gases cannot escape, they\nwill surround the anodes and increase the anode bed Lead wires\nresistance. The anodes are mounted on a perforated\nPVC pipe so that the gas can escape freely. Saudi\nAramco rarely uses aquifer penetrating deep anode\nbeds. Aquifer penetrating deep anode installations\nFormation\nmust be approved by Saudi Aramcos Hydrology interface\nDepartment. The Hydrology Department regulates the Pea gravel\ndrilling depth to minimize the chances of Top of coke\ncommunication between subsurface aquifers. breeze column\n2-3/4\" steel at least 6 m\ntubing above anodes\n\n9.625\" O.D.\ncasing\nCoke breeze\n\nAnode Anode\ncentralizer\n\nBottom of coke\nbreeze column\napprox. 1.5 m\nBottom of tubing below anodes\nslotted\nAA-036356\n\n## Aquifer Penetrating Deep Anode Bed from Standard\n\nDrawing AA-036356\nFigure 9\n\n## Saudi Aramco DeskTop Standards 21\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Non-aquifer penetrating deep anode beds contain\n\nanodes and coke breeze without a full length of casing Anode\n(Figure 10). Saudi Aramco installs a PVC vent pipe junction\nbox\nto allow gases formed by anodic reactions to escape.\nPVC vent\nA separate loading pipe is run to the bottom of the pipe\nhole and used to pump a water slurry of coke breeze\nfrom the hole as it is filled with coke breeze. This\nprocedure allows the slurry to be pumped upward\nfrom the bottom of the well until the anodes are\nCasing\ncompletely surrounded.\n\n## The Hydrology Department regulates the acceptable Lead\n\nwires Positive\ndepth of the deep anode bed. The location of the cable\nanode bed is approved in writing. from d-c\npower\nsource\nFormation\ninterface\nPea gravel\n\nCoke\nbreeze\n\nAnode\n\nPerforated PVC\nvent pipe AA-036385\n\n## Non-Aquifer Penetrating Deep Anode Bed from\n\nStandard Drawing AA-036385.\nFigure 10\n\n## Saudi Aramco DeskTop Standards 22\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Cathodic Protection Current Requirements\n\nThe current required to protect an onshore well casing depends on its environment. The operating environment\ncan be very complex. Environmental considerations include the following:\n\nwell spacing\n\nthe size, area, and depth of well casings, cementing information, and coatings (if used)\n\nprocess plants\n\nstorage tanks\n\n## hazardous or unique requirements at proposed sites\n\nCurrent requirements can be determined for a particular producing area since formation conditions and well\ncompletion methods are usually similar. Saudi Aramco uses casing potential profile techniques to determine\ncurrent requirements. Casing profiles are similar to line current surveys for buried pipelines. These tests are\nexpensive so they are not performed on every well. The tubing must be pulled so that the potential profile tool\ncan contact the internal casing wall. Saudi Aramco now uses a new logging tool which does not require the\nwell bore to be filled with a non-conducting fluid.\n\nBasically, a downhole logging tool measures the voltage (IR drop) at regular intervals in the casing. The\nlogging tool contains spring-loaded knife blades or hydraulically-activated contacts that are located several feet\napart.\n\nOnce the well bore has been prepared, the logging tool is lowered into the well. The voltage between the blades\nor contacts is measured by using a sensitive voltmeter. Readings are usually taken from the bottom to the top\nof the casing. The tool also measures casing resistance so an accurate current flow can be calculated (I=V/R).\n\n## Saudi Aramco DeskTop Standards 23\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nCurrent that flows onto the casing is assumed to be cathodic protection current. Current that flows away from\nthe casing is assumed to be corrosion current. Current must flow onto the entire casing for it to be adequately\nprotected. Figure 11 shows how the readings are plotted and interpreted.\n\nMicrovolts\n-400 -200 0 +200 +400\n0\nBottom of\nWell surface pipe\ncasing\n\nNegative\nindicate\ncurrent\nflow down\ncasing\n\nNegative slope\nindicates\ncurrent is\nindicate current casing\nflow up casing\nPositive\nslope indicates\ncurrent is entering\nthe casing\n900\n\n1200\n\nFigure 11\n\n## Saudi Aramco DeskTop Standards 24\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Surface Anode Bed Design\n\nSurface anode beds that protect well casings are designed similarly to anode beds that protect buried pipelines.\nThe number and spacing of anodes can be adjusted so that the total circuit resistance is less than the maximum\nallowable circuit resistance. As with anode beds for buried pipelines, Saudi Aramco only considers the anode-\nto-earth resistance. The resistance of a surface anode bed is given by the Sunde Equation.\n\n0.159\n1 +\n8L 2L\nR=\nNL\nln\nd S (ln 0.656 N)\n\nwhere -\n\nR = resistance, in ohms, of N vertical anodes in parallel and spaced S centimeters apart along a\nstraight line.\nr = soil resistivity in ohm-cm\nN = number of anodes\nL = length of anode (or backfill column) in centimeters\nd = diameter of anode (or backfill column) in centimeters\nS = anode spacing in centimeters\n\nThe formulas and procedure used to design surface anode beds for onshore well casings are similar to those\nused for buried pipelines, which are provided in Work Aid 1B.\n\n## Saudi Aramco DeskTop Standards 25\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Deep Anode Bed Design\n\nlength of the coke breeze column (based on the number of anodes required)\ncircuit resistance\namount of coke breeze required\n\nAfter describing how the above information is determined, an example, which demonstrates the design of a\ndeep anode bed, is provided.\n\n## Length of the Coke Breeze Column\n\nThe length of the coke breeze column depends on the number and spacing of anodes in the deep anode bed.\nThe anode spacing is determined in the field. Anodes are usually vertically spaced on 5 meter centers. As with\nsurface anode beds, the number of anodes needed can be calculated by using the anodes maximum current\noutput in the electrolyte or the anodes consumption rate. It is best to use the method that gives the more\nconservative value or the greater number of anodes.\n\nTo calculate the minimum number of anodes based on the anodes maximum current density, the following\nformula is used:\nN = I/(dL x A)\n\nwhere -\nN = number of impressed current anodes\nI = total current required in milliamperes times 120%\nd = anode diameter in centimeters\nL = anode length in centimeters\nA = anode maximum current density in mA/cm2\n\nTo calculate the minimum number of anodes based on the anodes consumption rate, the following formula is\nused:\n\n(\n\nN= Y I C\nW )\nwhere -\nN = number of impressed current anodes\nY = the impressed current system design life in years\nI = total current required in amperes times 120%\nC = anode consumption rate in kg/A-yr\nW = weight of a single anode\n\n## Saudi Aramco DeskTop Standards 26\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nCircuit Resistance\n\nThe total current output of a deep anode impressed current system is given by the formula:\n\nI = ED/RC\nwhere -\nED = the voltage capacity of the dc power source minus 2 volts\nRC = circuit resistance of the deep anode impressed current system\n\nThe circuit resistance, RC, is represented by the equivalent electrical circuit in Figure 12. For design purposes,\na deep anode bed is treated as if it were a single vertical anode.\n\nRRPL\n\nWell I\ncasing\n\nRLW\n\nED I\n\nRV\nRRNL\n\nRS\n\nFigure 12\n\n## RC = RRPL + RLW + RV + RS + RRNL\n\nwhere -\nRRPL = the resistance in the positive lead wire from the rectifier to the junction box\nRLW = the equivalent resistance of the anode lead wires in parallel\nRV = the resistance of the anode bed column as a single vertical anode\nRS = structure-to-electrolyte resistance\nRRNL = the resistance in the negative lead wire from the well casing to the rectifier\n\n## Saudi Aramco DeskTop Standards 27\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nBecause the anode bed is treated as a single vertical anode, the anode bed resistance can be calculated by using\nthe Dwight Equation as follows:\n\n0.159 eff 8L\nRV =\nln d\n1\nL\n\nwhere -\nRV = resistance of vertical anode to earth in ohms\neff = effective soil resistivity of the interval in ohm-cm\nL = length of coke breeze column in centimeters\nd = diameter of deep anode hole in centimeters\n\nThe effective soil resistivity, eff, is the average resistivity over the interval where the anodes will be placed.\nThe soil resistivity is measured by using Geonics instruments.\n\nThe circuit resistance, RC, must be less than the maximum allowable circuit resistance. The maximum circuit\nresistance, Rmax, is given by the following formula:\n\nRmax = ED/I\nwhere -\n\n## ED = the driving voltage of the dc power source\n\nI = the current output rating of the dc power source\n\n## Amount of Coke Breeze Required\n\nNormally, the amount or weight of coke breeze required is calculated by multiplying the net volume of coke\nbreeze (plus an extra 20% because of spillage) by the coke breeze density. The net volume of coke breeze\nrequired is calculated by subtracting the volumes of the anodes and vent pipe from the total volume of the\nbackfill column. However, for our purposes, we will use the total volume of the backfill column to calculate\nthe weight of coke breeze required.\n\n## Saudi Aramco DeskTop Standards 28\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nExample\n\nThis example will demonstrate the design of a deep anode bed to protect an onshore well casing in accordance\nwith Saudi Aramco standards and practices. Using the following data, we will design the anode bed:\n\n## Current required: 50 amperes\n\nWell casing-to-soil resistance: 0.08 ohm\nAnode material: High silicon chromium cast iron\nAnode consumption rate: 0.45 kg/A-yr\nWeight per anode: 50 kg\nAnode dimensions: 7.6 cm dia. x 152 cm length\nRectifier output rating: 50 V, 50 A\nLead wire resistance: No. 4 AWG - 0.85 x 10-3 ohm/m (rectifier to junction box and well)\nNo. 6 AWG - 1.35 x 10-3 ohm/m (anodes)\nCoke breeze density: 730 kg/m3\nDistance from rectifier to junction box: 5 meters\nDistance from rectifier to well casing: 150 meters\nDepth at top of coke breeze column: 69 meters\nDiameter of coke breeze column: 30 cm\n\n## Length of the Coke Breeze Column\n\nEight amperes of current are required to protect the well casing. According to SAES-X-700, we will design the\nsystem for 50 amperes. To estimate the number of anodes, the current required is multiplied by the design life\nand the anode consumption rate. Then the total weight is divided by the mass per anode as follows:\n\n## (20 years)(50 A)(120%)(0.45 kg/A-yr)/50 kg per anode = 11 anodes\n\nIf we use the current density formula for calculating the number of anodes needed, we get:\n\nN = I / ( dL A )\n\n=\n(50, 000 mA )(1.2)\n(7,6 cm)(152 cm )(1 mA / cm2 )\n= 16.5 anodes round up to 17anodes\n\nSince 17 anodes is the larger calculated by the two methods, we will design our anode bed with 17 anodes.\n\n## Saudi Aramco DeskTop Standards 29\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nSeventeen high silicon chromium cast iron anodes (1.52 meters long) spaced on 5 meter centers require an\ninterval of 81.5 meters (Figure 13). Standard Drawing AA-036356 requires at least 6 m of coke breeze above\nthe anodes and a minimum of 1.5 m below the anodes. Therefore, the minimum length of this particular coke\nbreeze column is 81.5 m + 6 m + 1.5 m = 89 m.\n\nPea gravel\n\n6 m minimum\n\n0.76 m\n1\n\nCoke breeze 5m\n124 m\n2\n\n15\n\n5m\n\n16\n\n5m\n\n17\n0.76 m\n\n1.5 m minimum\n\nFigure 13\n\n## Saudi Aramco DeskTop Standards 30\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nCircuit Resistance\n\nAssume that the Geonics instrument measured an effective soil resistivity of 2482 ohm-cm. By using eff and\ntreating the anode bed as a single anode, we can calculate the deep anode bed resistance. The anode bed is 30\ncm in diameter and 8,900 cm long. Therefore, the anode bed resistance is as follows:\n\nRV =\n( ) (\n0.159 2, 482 8 8, 900\nln )\n1 = 0.300 ohm\n8, 900 30\n\nNext, we must ensure that the total circuit resistance is less than the maximum allowable circuit resistance and\ncalculate the amount of coke breeze required. The resistance in the rectifiers negative and positive lead wires is\ncalculated as follows:\nRNLW + RPLW = (150m + 5m)(110%)(0.85 x 10-3 ohm/m) = 0.145 ohm\n\nThe following is the equivalent resistance of the lead wires from the junction box to the anodes:\n\n16\n(17 )(75 ) + i (5) meters\n\nR LW = i =0\n17 ( ( )\n120% ) 1.35 10 3 ohm m = 0.186 ohm\n\nIncluding the well casing-to-soil resistance of 0.08 ohm, the total circuit resistance is calculated as follows:\nRC = 0.300 + 0.145 + 0.186 + 0.08 = 0.711 ohm.\n\nThe total circuit resistance is less than the maximum allowable circuit resistance, Rmax.\n\n## Rmax = (50V 2V)/50 A = 0.96 ohm.\n\nAmount of Coke Breeze Required\n\n## (6.291 m3)(120%) (730kg/m3) = 5,510 kg.\n\nThe formulas and procedure to design deep anode beds are provided in Work Aid 2.\n\n## Saudi Aramco DeskTop Standards 31\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Designing Cathodic Protection Systems for Vessel and Tank Interiors\n\nProduction vessels and storage tanks contain fluids that range from very corrosive hot, sour brines to\ndemineralized water or steam condensate. Sometimes, coatings alone can adequately protect vessels. In most\ncases, both coatings and cathodic protection are required to prevent corrosion.\n\nGalvanic anodes are usually the most economical choice except in very large tanks. In drinking water systems,\nwhere contamination from anode corrosion products is a concern, Saudi Aramco uses indium activated\naluminum galvanic anodes. Saudi Aramco normally uses high silicon chromium cast iron impressed current\nanodes to protect the interiors of large tanks. Whenever impressed current systems are considered, an economic\nanalysis should be performed.\n\nThis section is divided into two parts. The first part covers galvanic anode system designs for vessel and tank\ninteriors. The second part covers impressed current system designs for tank interiors. The designs for both\ntypes of CP systems include determining the following:\n\n## cathodic protection current requirement\n\ndesign requirements in accordance with Saudi Aramco Engineering Standards and Drawings\n\nIn Module 107.01, we calculated the total current requirement by multiplying the required current density from\nSAES-X-500 by the water-wetted surface area. Therefore, the designs in this section assume that the total\ncurrent requirement has been calculated. After the following description of design requirements from Saudi\nAramcos standards and drawings, methods and examples for designing galvanic and impressed current\nsystems are presented.\n\n## Saudi Aramco DeskTop Standards 32\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Saudi Aramco Engineering Standards and Drawings\n\nThe design of cathodic protection systems for vessel and tank interiors is governed by Saudi Aramco\nEngineering Standard SAES-X-500. SAES-X-500 states the following:\n\nSection 4.1.1 - Cathodic protection is mandatory if the resistivity of the contents is expected to\nbe 1500 ohm-centimeter or less during the life of the tank or vessel.\n\nSection 4.3.1 - The design life of galvanic or impressed current anode systems shall be 5 years\nor the testing and inspection (T&I) period, whichever is greater.\n\nSection 4.3.2 - Galvanic anodes in dehydrator vessels shall be designed using a 20%\nefficiency factor. Designs for other wet crude handling vessels shall use an efficiency factor of\n50%.\n\nSection 4.5.1 - The steel-to-water potential shall be more negative than -0.90 V (current on)\nversus a Ag-AgCl reference electrode, or +0.15 V (current on) versus a zinc electrode.\n\nSection 4.6.3 - Aluminum and zinc anodes shall not be used if the water resistivity is more\nthan 1000 ohm-centimeters.\n\nSection 4.6.4 - Magnesium anodes shall not be used if the water resistivity is less than 500\nohm-centimeters.\n\nSection 4.6.5 - Zinc anodes shall not be used in environments where the temperature exceeds\n49 C.\n\nCathodic protection designs for tanks are based on construction standards set in the following Standard\nDrawings: AA-036354 (Water Storage Tanks Galvanic Anodes) and AA-036353 (Water Storage Tanks\nImpressed Current). The number, depth, and location of galvanic and impressed current anodes are based on\ntank size, water level variation, and water resistivity. Some diagrams from AA-036354 and AA-036353 are\nshown in Figures 14 and 15.\n\n## Saudi Aramco DeskTop Standards 33\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nJunction box\n0.01 ohm shunt\nWeld Access\nhatch\n\nCa ble\nPoly-\npropylene\nrope\nTop View\nRe ference electrode Anode Installation Detail\naccess hole\n\nAccess\nhatch\n\nAnode\nPoly- wire\npropylene\nrope\n\n## See Anode Ca ble tie\n\nInstallation Detail\n\nSee Anode\nString Detail\n\n1.5 m\nAnode String Detail\n\nDiagrams from Standard Drawing AA-036354, Water Storage Tanks Galvanic Anodes\nFigure 14\n\n## Saudi Aramco DeskTop Standards 34\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Anode Assembly Detail\n\nReference\nelectrode\n\ncable\n\nAnode\nassembly\n\nJunction\nbox\n\nTop View\n\nSee Anode\nAssembly Detail\n\nJunction box\n\nReference\nelectrode Center of h\nTank\n1/\n2h\n\nDiagrams from Standard Drawing AA-036353, Water Storage Tanks Impressed Current\nFigure 15\n\n## Saudi Aramco DeskTop Standards 35\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Galvanic Anode System Design for Vessel and Tank Interiors\n\nThe design of galvanic anode systems for vessel and tank interiors includes determining the following:\n\n## the current output per anode\n\nthe number of galvanic anodes required\ngalvanic anode life\n\nAfter describing these calculations, an example, which demonstrates the design of galvanic anode systems, is\nprovided.\nCurrent Output Per Anode\n\nThe current output of a single galvanic anode in a vessel or tank is given by the following formula\n\nIA = ED/RC\nwhere -\nIA = current output of a single anode\nED = anode driving potential\nRC = circuit resistance\n\nThe circuit resistance of a single anode, RC, is represented in Figure 16 in the equivalent electrical circuit.\n\nIA\n\nRLW\nED\n\nRV\n\nGalvanic anode RS\n\nTank Galvanic Anode System and Equivalent Electrical Circuit for Each Anode\nFigure 16\n\n## Saudi Aramco DeskTop Standards 36\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## The circuit resistance is given by the following formula:\n\nRC = RS + RLW + RV\nwhere -\nRS = structure-to-electrolyte resistance in ohms\nRLW = the anode lead wire resistance in ohms\nRV = the anode-to-electrolyte resistance in ohms\n\nThe anode-to-electrolyte resistance of a single vertical anode, RV, is given by the Dwight Equation.\n0.159 8L\nRV = 1\nl d\nn\nL\n\nwhere -\nRV = resistance of one vertical anode to the electrolyte in ohms\nr = resistivity of the electrolyte in ohm-cm\nL = length of the anode in centimeters\nd = diameter of the anode in centimeters\nNumber of Galvanic Anodes Required\n\nThe number of galvanic anodes required is calculated by dividing the total current requirement by the current\noutput of a single galvanic anode as shown in the following equation:\n\nN = I/IA\nwhere -\nN = the number of anodes\nI = the total current required to protect the structure\nIA = the current output of a single anode\nGalvanic Anode Life\n\nThe life of a galvanic anode can be estimated if its weight and current output are known. The expected life of a\ngalvanic anode is given by the following formula:\n\nW UF\nY=\nC IA\nwhere -\nY = anode life in years\nW = anode mass in kg\nC = actual consumption rate in kg/A-yr\nIA = anode current output in amperes\nUF = Utilization factor\n\n## Saudi Aramco DeskTop Standards 37\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nExample\n\nGiven the following engineering data, we will calculate the current output, number, and life of galvanic anodes\nrequired to protect the interior of a water storage tank.\n\n## Current required: 3.6 amperes\n\nStructure-to-electrolyte resistance: 0.042 ohms\nWater resistivity: 15 ohm-cm\nAnode: Hydral 2B\nAnode dimensions: 22 cm dia. x 22 cm\nAnode actual consumption: 4.11 kg/A-yr\nAnode weight: 22 kg\nAnode solution potential: -1.05 V versus Ag-AgCl\nRequired structure-to-electrolyte potential: -0.90 V versus Ag-AgCl\nCurrent Output Per Anode\n\n## I = ED/RC = (EA-ES)/(RS + RLW + RV)\n\nIf we calculate RV by using the Dwight Equation and insert the known values for EA, RS, and RLW, we can\ndetermine the anode current output of a single anode as a function of the structures potential as follows.\n0.159 8L\n1 =\n( )\n0.159 15 8 22\nl n ( )\n1 = 0.12 ohm\nRV = ln\nL d 22 22\n(\nI = 1.05 E S ) (0.042 + 0.024 + 0.12) = (1.05 E S) 0.186\n\n## At a negative structure potential of 0.90 volt, the anodes current output is\n\nI = (1.05-0.90)/0.186 = 0.81 A.\nNumber of Galvanic Anodes Required\n\nThe number of anodes required is 3.6 A/0.81 amperes per anode, or at least 5 anodes.\nGalvanic Anode Life\n\n## W UF 22 kg 0.85 = 5.6 years\n\nY= =\nC I A 4.11 kg / A yr 0.81 A\n\n## Saudi Aramco DeskTop Standards 38\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nWe can develop similar performance data for this particular Hydral 2B anode in electrolytes with different\nresistivities. For example, the current output of the Hydral 2B anode in a\n10 ohm-cm electrolyte is calculated as follows.\n\nI = (1.05 E S) 0.042 + 0.024 + 10 (0.12) = (1.05 E S ) 0.15\n15\n\nBy plotting the formulas at water resistivities of 5, 10, 15 and 20 ohm-cm, we obtain the performance chart\nshown in Figure 17. The anode life is shown on the right side of the performance chart.\n\n10.0\n8.0 0.6\n\n## 6.0 De sign Parameters 0.8\n\nAnode dimensions: 22 cm dia. x 22 cm\nAnode efficiency: 96% Wt: 22 kg\n4.0 Consum. rate: 3.95 kg/amp-yr UF: 85% 1.1\nR S: 0.042 ohm RLW : 0.024 ohm\nAnode solution potential: -1.05 V vs. Ag-AgCl\n\n2.0 2.3\n\n1.0 4.5\n0.8 5.7\n\n0.6 7.6\n\n0.4 11.4\n\n0.2 22.7\n\n0.1\n0.80 0.85 0.90 0.95 1.0\nStructure Potential (volts vs. Ag-AgCl)\n\n## Performance Chart of a Hydral 2B Anode\n\nFigure 17\n\nThe formulas and procedure used to design galvanic anode systems for vessel and tank interiors are provided\nin Work Aid 3A.\n\n## Saudi Aramco DeskTop Standards 39\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Impressed Current System Design for Vessel and Tank Interiors\n\nThe design of impressed current systems for vessel and tank interiors includes determining the following:\n\n## the number of impressed current anodes required\n\nthe circuit resistance\n\nAfter describing these calculations, an example, which demonstrates the design of an impressed current system\nfor a tank interior, is provided.\nNumber of Impressed Current Anodes Required\n\nThe number of anodes can be calculated based on the anodes maximum current output in the electrolyte or the\nanodes consumption rate. It is best to use the method that gives the more conservative value; that is, the\nmethod that results in the greatest number of anodes.\n\nTo calculate the minimum number of anodes based on the anodes maximum current density, the following\nformula is used:\n\nN = I/(dL x A)\nwhere -\nN = number of impressed current anodes\nI = total current required in milliamperes\nd = anode diameter in centimeters\nL = anode length in centimeters\nA = anode maximum current density in mA/cm2\n\nTo calculate the minimum number of anodes based on the anodes consumption rate, the following formula is\nused:\n\nY I C\nN=\nW\n\nwhere -\nN = number of impressed current anodes\nY = the impressed current system design life in years\nI = total current required in amperes\nC = anode consumption rate in kg/A-yr\nW = weight of a single anode\n\n## Saudi Aramco DeskTop Standards 40\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nCircuit Resistance\n\nImpressed current anodes in vessels or tanks are connected in parallel as shown in Figure 18. The circuit\nresistance includes the anode resistances in parallel and the resistances in the negative and positive lead wires of\nthe rectifier.\n\nRRPL\n\nI\nI1 I2\nED\n\nRA1 RA2\nRRNL\n\nI\n\nRS\n\n## Tank Impressed Current System and Equivalent Electrical Circuit\n\nFigure 18\n\nThe equivalent resistance of N resistances in parallel is obtained from the following formula:\n\n1 1 1 1\n= + +\nR eq R A 1 R A 2 R AN\n\nIf the resistances are equal, the equivalent resistance is given by the following formula:\n\n1 = 1 + 1 + 1 = N R eq =\nRA\nR eq R A 1 R A 2 R AN RA N\n\n## Therefore, the circuit resistance is given by the formula shown below\n\nRA\nR c = R RPL + + Rs + R RNL\nN\n\nwhere -\n\n## RC = the circuit resistance of the entire impressed current system in ohms\n\nRRPL = the resistance in the positive lead wire from the rectifier to the junction box\nN = the number of impressed current anodes\nRA = the resistance of a single impressed current anode\nRS = structure-to-electrolyte resistance\nRRNL = the resistance in the negative lead wire from the structure to the rectifier\n\n## Saudi Aramco DeskTop Standards 41\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nThe circuit resistance, RC, must be less than the maximum allowable circuit resistance given by the formula:\nRmax = ED/I\n\nwhere -\n\n## ED = the rated voltage of the dc power source\n\nI = the current output rating of the dc power source\n\nExample\n\nWe will design an impressed current system to protect a large, coated storage tank by using the following\ninformation:\n\n## Current required: 4.95 amperes\n\nStructure-to-electrolyte resistance: 0.06 ohms\nAnode lead wire resistance: 0.038 ohms\nRectifier negative lead resistance: 0.04 ohm\nRectifier positive lead resistance: 0.05 ohm\nWater resistivity: 15 ohm-cm\nAnode material: High silicon chromium cast iron\nAnode dimensions: 5.08 cm dia. x 152.4 cm (2\" dia. x 60\")\nAnode weight: 27.3 kg\nAnode maximum current density: 0.5 mA/cm2\nAnode consumption rate: 1 kg/A-yr\nRequired structure-to-electrolyte potential: -0.90 V versus Ag-AgCl\nRectifier output rating: 50 V, 50 A\n\n## Saudi Aramco DeskTop Standards 42\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nCircuit Resistance\n\n## The resistance of the 5 anodes in parallel is given by the following formula:\n\nR A R LW + R V\n=\nN N\n\nWe can solve for RV by using the Dwight Equation for a single anode as follows.\n\nRV =\n0.159 8L\n1 =\n( ) ln (\n0.159 15 8 152.4 )\n1 = 0.07 ohm\nl\nn\nL d 152.4 5.08\n\nSubstituting all resistance values into the circuit resistance formula we obtain the following circuit resistance:\n\nR LW + R V\nR c = R RNL + + Rs + R RPL\nN\n0.038 + 0.07\nR c = 0.04 + + 0.06 + 0.05\n5\nR c = 0.17 ohm\n\nThe calculated circuit resistance is less than the maximum allowable circuit resistance, which is\n\n## Rmax = 50 V/50 A = 1.0 ohm.\n\nThe formulas and procedure used to design an impressed current system to protect the interior of a vessel or\ntank are provided in Work Aid 3B.\n\n## Saudi Aramco DeskTop Standards 43\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Designing Cathodic Protection Systems For In-Plant Facilities\n\nThere are a particular set of problems involved when cathodically protecting structures within a plant area.\nHydrocarbon lines, firewater piping, buried valves, and tank bottoms are examples of critical systems, which\nrequire cathodic protection in plant areas. Some external corrosion problems are caused by the buried copper\ngrounding grid, which is designed to protect personnel in case of an electrical ground fault. Without cathodic\nprotection, buried steel piping corrodes faster because it becomes anodic to the copper grid.\n\nTank bottoms in contact with the earth are susceptible to corrosion due to moisture in the soil. Saudi Aramco\noften bonds tanks and buried structures together and cathodically protects them as a single unit. Cathodic\nprotection current is supplied by surface distributed impressed current or galvanic anode systems near tanks or\nbetween parallel pipes. This installation ensures uniform current distribution and prevents shielding.\n\nPrevious sections of this module have addressed the design of CP systems for piping and vessel and tank\ninteriors; therefore, this section focuses on CP system design for external tank bottoms. Saudi Aramco protects\nabove-ground storage tanks with close, or distributed, impressed current systems. This type of design is\napplicable in congested areas such as plants because (1) remote anode beds are electrically shielded by other\nburied structures, and (2) some buried metal in the plant does not require cathodic protection (e.g., a bare\ncopper grounding grid or rebar in foundations).\n\nThe design of impressed current systems that protect external tank bottoms involve determination of the\nfollowing:\n\n## design requirements using Saudi Aramco standards and drawings\n\nthe current required to shift the potential of the earth under the tank bottom\nthe number of impressed current anodes required\n\nAfter the following information about Saudi Aramcos standards and drawings is presented, a method and\nexample are given to demonstrate the design of impressed current systems to protect tank bottoms.\n\n## Saudi Aramco DeskTop Standards 44\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Saudi Aramco Engineering Standards and Drawings\n\nThe design of cathodic protection systems for in-plant facilities is governed by Saudi Aramco Engineering\nStandard SAES-X-600. Structures which are cathodically protected include the following:\n\n## pressurized steel hydrocarbon pipelines\n\nbottoms or soil side of above ground storage tanks\nburied tanks containing hydrocarbons\nsea walls and associated anchors\nburied steel bodied valves\n\n## The design life of impressed current anode systems shall be 20 years.\n\nAnode beds shall be sized to discharge 100% of the rated current capacity of the d-c power\nsource.\nThe maximum system operating voltage shall be 100 volts with a maximum circuit resistance\nof 1 ohm or less.\nDesigns for systems connected to plant ground, rebar in concrete, and other underground\nstructures shall provide distributed anodes.\n\nThe minimum structure-to-soil potentials of in-plant structures are listed in Figure 19.\n\n## Structure Required Potential\n\nCurrent On\nBuried plant piping -0.85 volt or more negative versus CuSO4 electrode\n\nTank bottom external -1.00 volt or more negative versus C uSO4 at periphery\n-0.85 volt or more negative versus permanent CuS O4\n+0.20 volt or less positive versus permanent Zn\n-0.35 volt change in structure potential vs CuS O4\nSea walls (water side) -0.90 volt or more negative versus AgCl electrode\nSea walls (soil side) -0.85 volt or more negative versus CuSO4 electrode\n\nFigure 19\n\n## Saudi Aramco DeskTop Standards 45\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nCathodic protection designs for tanks are based on construction standards set in Standard Drawing AA-036355-\nTank Bottom Impressed Current Details. AA-036355 requires a distance between the anodes and the tank of\nabout one-quarter of the tanks radius. The minimum distance is 3 meters and the maximum distance is 10\nmeters. Also, the maximum separation between distributed anodes is 20 meters. Some diagrams from AA-\n\nV+\n036355 are shown in\nFigure 20.\n\nRRLW\n=R\nR\nC +R\nRPL+\nRNL\nN\nDiagrams from Standard Drawing AA-036355, Tank Bottom Impressed Current Details\nFigure 20\n\n## Saudi Aramco DeskTop Standards 46\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Number and Placement of Anodes in Distributed Anode Beds\n\nSaudi Aramco uses distributed anode beds in congested areas where electrical shielding prevents the use of\nremote anode bed installations. Normally, high silicon chromium cast iron anodes are used. Distributed anode\nsystems are designed so that the structure to be protected is within the area of influence that surrounds each\nanode (Figure 21). The idea of this type of design is to change the potential of the earth around the structure.\nThe earth within the area of influence of each current-discharging anode will be positive with respect to remote\nearth. There is a limited area of the tank bottom where the net potential difference between the tank bottom and\nadjacent soil will be sufficient to attain cathodic protection. Note in the figure that although a single anode\nmay cathodically protect the tank periphery closest to it, the anode cannot adequately protect the rest of the\ntank.\n\n## Assume tank-to-soil Anode\n\npotential is -0.5 V Protected area header\nbefore energizing of tank bottom cable\nanode.\n\nEarth potential\nchange after anode\nis energized\n\n-1.0\n\n## Protected potential of tank center\n\n-0.85\nto tank-to-earth potential\nbefore anode is energized.\n\nTank Tank\nwall\ncenter\n\n-0.5\n8 6 4 2 0 2 4 6 8\nDistance from Tank Periphery to Tank Center (Meters)\n\nFigure 21\n\n## Saudi Aramco DeskTop Standards 47\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nIt must be remembered that the earth potential change is additive for all the anodes that cause a change (see\nFigure 22). Hence, the earth potential shift at a given point on the tank bottom must include the potential shift\ncaused by neighboring anodes. For example, if the earth potential shift at a given point is 0.2 volt from one\nanode and 0.1 volt from a neighboring anode, then the total earth potential change would be 0.3 volt.\n\n## Earth potential shift\n\ncaused by anode Impressed\ncurrent anode\n\nStorage tank\n\nJunction box\n\n## Additive Effect of Distributed Anodes\n\nFigure 22\n\nTo determine the spacing between anodes, there will be some geometry involved to be sure that an adequate\npotential shift is achieved at all points along the protected structure. Since the separation between anodes\ncannot exceed 20 meters, divide the circumference of the distributed anode system by 20 meters to determine\nthe total number of anodes. Round up to the nearest number of anodes.\n\n## Saudi Aramco DeskTop Standards 48\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nThe amount of earth potential change depends on (1) the size and shape of each anode, (2) the anodes position\nrelative to the structure to be protected, (3) the current flow, and (4) the soil resistivity. According to SADP-X-\n100, Section 18.3.7, the earth potential shift is given by the following formulas:\n\n## (1) For a single vertical anode\n\n0.5 I L2 + X 2 + L\nVx = ln , (see Figure 23).\nL X\n\n## (2) For a single horizontal anode\n\nVx =\nI\nl n\n(0.5L )2 + X 2 + h 2 + 0.5L\nL X 2 + h2\n\nwhere -\nVX = earth potential change at the center of the tank in volts\nI = current flow in amperes\nr = soil resistivity in ohm-cm\nL = anode length in cm\nX = horizontal distance from the anode to the center of the tank in cm (Figure 23)\nh = depth of burial to centerline of anode in cm\n\nsource\n\nh Anode\n\nX L\nTank\ncenter\n\nFigure 23\n\n## Saudi Aramco DeskTop Standards 49\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nCircuit Resistance\n\nImpressed current anodes around a tank are connected in parallel as shown in Figure 24. Saudi Aramco\nnormally uses high silicon chromium cast iron anodes.\n\ntank wall\n\nAnode\njunction box\n\nRectifier\n\npower source cable ring\n\nRRPL RCBL\n\nI\nI1 I2 I3 IN\nED\n\nRRNL\nRA1 RA2 RA3 ... RAN\n\nRS\n\nFigure 24\n\n## Saudi Aramco DeskTop Standards 50\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nThe circuit resistance of the impressed current system is given by the following formula:\n\nRA\nR C = R RPL + RCBL + + R S + R RNL\nN\n\nwhere -\nRC = the circuit resistance of the entire impressed current system\nRRPL = the resistance in the positive lead wire from the rectifier to the junction box\nRCBL = the resistance in the header cable\nN = the number of impressed current anodes\nRA = the resistance of a single impressed current anode\nRS = structure-to-electrolyte resistance\nRRNL = the resistance in the negative lead wire from the structure to the rectifier\n\n## The resistance, RA, is given by the following formula:\n\nRA = RLW + RV,\nwhere -\nRLW = the anode lead wire resistance in ohms\nRV = the anode-to-electrolyte resistance in ohms\n\nThe anode lead wire resistance, RLW, is very small and can be ignored. Therefore, RA is equal to the anode-\nto-electrolyte resistance of a single vertical anode, which is given by the Dwight Equation.\n\n0.159\n1\n8L\nRA = R V =\nl d\nn\nL\n\nwhere -\nRV = resistance of one vertical anode to the electrolyte in ohms\nr = resistivity of the electrolyte in ohm-cm\nL = length of the backfill in centimeters\nd = diameter of the backfill in centimeters\n\nFor high resistivity soils like those found in Saudi Arabia, RV is much greater than the sum of the other\nresistances. Therefore, RRPL, RRNL, RCBL, and RS, can be ignored.\n\n## Saudi Aramco DeskTop Standards 51\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nExample\n\nGiven the following engineering data, we will design an impressed current system to protect a bare tank\nbottom.\n\n## Anode material: High silicon chromium cast iron\n\nAnode dimensions: 7.6 cm dia. x 152 cm (backfill, 20 cm dia. x 180 cm)\nTank dimensions: 30 m diameter\nTank native potential: -0.5 V vs. CuSO4 electrode\nSoil resistivity: 2,000 ohm-cm\nRectifier output rating: 50 V, 35 A\nNumber and Placement of Impressed Current Anodes\n\nAccording to Standard Drawing AA-036355, the distance from the anodes to the tank wall should be\napproximately one-quarter of the tank radius. In the case of a 30 m dia. tank (15 m radius), the anodes will be\nplaced at a distance of 0.25 x 15 or 3.75 meters from the tank wall (see\nFigure 25). The radius of the system is, therefore, 15 + 3.75 or 18.75 m. The circumference of the circle at\nwhich the anodes will be located can be calculated as follows:\n\nC = 2r = 2(18.75) = 118 m\n\nAllowing a maximum separation of 20 m between each anode, we will need 118/20 = 5.9 or 6 anodes as a\nminimum number of anodes.\n\nCable Ring\n\n15 m Vertical Anode\nr\n\n## Anode junction box\n\nNe gative return lead to rectifier\n\nFigure 25\n\n## Saudi Aramco DeskTop Standards 52\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nUsing the equation for earth potential shift for a single vertical anode, calculate the current needed to give a\ntotal of shift of 0.35 volts at the center of the tank from all six anodes.\n\n0.5 I 2000 2 2\nVx = 0.35 V = l n 180 + 1875 + 180\n180 1875\n\n1000 I 2064\n0.35 V = ln\n180 1875\n( )(\n= I 1.768 l n 1.107 )\n( )( )\n0.35 V = I 1.768 0.1014 I = 1.95 amperes\n\nThis is the current that will shift the potential by 0.35 volts at the center of the tank. The formulas and\nprocedure that are used to calculate current required to shift earth potential are provided in Work Aid 4.\n\nTo complete the design, it is necessary to determine the total current requirement for the tank bottom and use\nsufficient anodes to assure a 20 year design life.\n\n## Current needed for tank bottom:\n\n(30)\n2\nd 2 2\nI= 0.02 A / m = 0.02 = 14.1 amperes\n4 4\n\nSAES-X-600 requires sufficient anodes to discharge the rectifier amperage rating without exceeding the\nmaximum anode current density. The current output for a single anode should not exceed:\n\n## I = dL x 1 mA/cm2 = (7.6)(152) x 1.0\n\nI = 3629 mA or 3.6 amperes\n\nThe rectifier output is 35 amperes. Therefore, the minimum number of anodes needed is\n35 3.6 = 9.7 anodes. Use 10 anodes.\n\n## Saudi Aramco DeskTop Standards 53\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Designing Cathodic Protection Systems For Marine Structures\n\nSaudi Aramco cathodically protects the entire submerged surface area of marine structures (see Figure 26).\nThis submerged surface area extends from the base of the structure to the Indian Spring Mean High Tide Level.\nTo calculate the current required to protect the structure, you must know the following:\n\n## the area of steel which is immersed in sea water\n\nthe area of steel which is immersed below the mud line\nthe actual or anticipated number of well casings\nany insulated or unprotected foreign structures\nand the required current density for the specific environment\n\nSplash zone\nWater line\n\nImmersed zone\n\nMud line\n\nOffshore Platform\nFigure 26\n\nThe immersed surface areas can be calculated from drawings and specifications of the structure or obtained\nfrom the structure designer.\n\n## Saudi Aramco DeskTop Standards 54\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nThis section is divided into two parts. The first part covers galvanic anode system designs for marine\nstructures. Saudi Aramco cathodically protects all marine structures and pipelines with galvanic anodes. The\nsecond part covers impressed current systems. Impressed current systems are used when ac power is available.\nWhen used with a galvanic anode system, an impressed current system is intended as the primary system. The\ngalvanic anode system is used as a backup for the following two reasons:\n\n## 1) To protect the structure until the impressed current system is energized.\n\n2) To protect the structure when electrical power is interrupted. Power can be interrupted during break\ndowns or during scheduled shutdowns.\n\nThe designs for both types of CP systems involve determination of design requirements by using Saudi Aramco\nEngineering Standards and Drawings. Therefore, after the following information about Saudi Aramcos\nstandards and drawings, methods and examples for designing galvanic and impressed current systems are\ndescribed separately.\n\n## Saudi Aramco DeskTop Standards 55\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## The design of cathodic protection systems for marine structures is governed by\n\nSAES-X-300. SAES-X-300 states the following:\n\nGalvanic anode systems, when used alone, shall have a design life of 25 years.\nGalvanic anode systems accompanied by impressed current systems shall have a design life of\n10 years and the impressed current system shall have a design life of 15 years.\nThe cathodic protection system shall achieve a minimum structure-to-electrolyte potential of -\n0.90 volt versus Ag-AgCl over the entire structure.\n\nSaudi Aramco requires the following current densities in the immersed surface areas.\n\nCoated Uncoated\n\n## Seawater structures 10.0* 50.0\n\nStructures in mud or soil 10.0 20.0\nMarine pipelines (coated) 2.5 --\n\n## Higher current density may be required depending on turbulence and/or velocity.\n\nCathodic protection designs for offshore structures are based on construction standards set in the following\nStandard Drawings: AA-036348 (Galvanic and Impressed Current Anodes on Offshore Structures), AA-036409\n(Replacement of Galvanic Anodes on Offshore Structures and Risers), and AA-036335 (Half Shell Bracelet\nType Anode for Pipe Sizes 4\" Through 60\"). Standard Drawing AA-036335 states that the maximum spacing\nfor all sizes of anode bracelets shall be 150 meters. Some diagrams from AA-036348, AA-036409, and AA-\n036335 are shown in Figures 27 and 28.\n\n## Saudi Aramco DeskTop Standards 56\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\ncoating\nremoved\n\nwelded to pipe\nAA-036335\n\n## Mean Sea Level Galvanic Anode Bracelet\n\nfor Submarine Pipelines\n\nPipeline Riser\n\nAnodes laid on\nsea bed under\npile structure\n\nAA-036409\n\nAA-036409\n\nFigure 27\n\n## Saudi Aramco DeskTop Standards 57\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nNylon\nStrapping\nGalvanic\nanodes\nImpressed\ncurrent anode\n\nDielectric\nImpressed shield\ncurrent anodes\n\nJunction Box.\n\n## 2\" PVC Coated\n\nConduit\n1-1/2\" Conduit\n\nMain Deck\nJunction Box Mounting for\nImpressed Current Anode Cables AA-036348\n\nFigure 28\n\n## Saudi Aramco DeskTop Standards 58\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Galvanic Anode System Design for Marine Structures\n\nSaudi Aramco uses indium-doped aluminum alloy or zinc-tin-doped aluminum alloy galvanic anodes to protect\nmarine structures. Galvanic anodes are usually installed at least 30 cm (1 ft.) from the structure. A calcareous\nbuild-up forms on the structure as it polarizes. This build-up increases the current distribution of the anodes.\nGalvanic anode bracelets are used to protect marine pipelines.\n\nThe design of galvanic anode systems for marine structures (such as platforms, mooring buoys, etc.) involves\ndetermining the following:\n\n## the number of galvanic anodes required\n\ngalvanic anode life\n\nThe design of galvanic anode systems for marine pipelines involves determining the following:\n\n## the number of galvanic anode bracelets required\n\nthe spacing of the bracelets\n\nAfter describing these calculations, an example, which demonstrates the design of a galvanic anode system for\na marine platform and pipeline, is provided.\nNumber of Galvanic Anodes Required\n\nThe number of anodes needed to protect a marine structure depends on the total current required and the current\noutput per anode. In Module 107.01, we calculated the total current requirement by multiplying the required\ncurrent density from SAES-X-300 by the immersed surface area of the marine structure. The total number of\nanodes is calculated by using the following equation:\n\nN = I/IA\nwhere -\nN = the number of anodes\nI = the total current required to protect the structure\nIA = the current output of a single anode\n\n## Saudi Aramco DeskTop Standards 59\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nAccording to SADP-X-100, Eqn. 20, the current output from a single anode, IA, can be found using the\nfollowing equation:\n\nIA = ED/RC,\nwhere -\nIA = anode current output in amperes\nED = the anode driving potential in volts versus Ag-AgCl\nRC = the circuit resistance in ohms\nCircuit Resistance\n\n## The circuit resistance, RC , is given by the following equation:\n\nRC = RS + RV\nwhere -\nRS = the structure-to-electrolyte resistance (for offshore structures, this is negligible)\nRV = the anode-to-electrolyte resistance\n\nFor galvanic anodes on marine structures, the Dwight Equation is used to calculate RV.\n\n0.159\n1\n8L\nRV =\nl\nn\nL d\n\nwhere -\nr = the electrolyte (seawater) resistivity in ohm-cm\nL = the length of the anode in centimeters\nd = the diameter of the anode in centimeters or the circumference divided by for non-\ncylindrical shapes\nGalvanic Anode Life\n\nThe anodes must last over the design life of the system. The anode life is given by the following equation.\nW UF\nY=\nC IA\n\nwhere -\nY = anode life in years\nW = mass of one anode in kg\nUF = utilization factor\nC = actual consumption rate in kg/A-yr\nIA = current output of one anode in amperes\n\n## Saudi Aramco DeskTop Standards 60\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Number and Spacing of Galvanic Anode Bracelets\n\nThe number of anode bracelets required to protect a marine pipeline is calculated as follows.\n\nN = L/150 m\nwhere -\nN = the number of anode bracelets\nL = length of the pipeline\n\nThe anode bracelets must last over the design life of the pipeline. The anode life is given by the following\nequation.\n\nY = W UF\nC IA\nwhere -\nY = anode life in years\nW = net weight of one anode bracelet in kg\nUF = utilization factor\nC = actual consumption rate in kg/A-yr\nIA = current output of one anode in amperes\n\nThe net weight per bracelet, W, can be obtained from Standard Drawing AA-036335 (see also Work Aid 5A).\nThe current requirement for one anode bracelet, IA, can be calculated by diving the total current requirement by\nthe number of anode bracelets.\n\nAn alternative method involves calculating the current output of a single anode bracelet by dividing the driving\npotential of the galvanic anode material by the circuit resistance. As shown previously, the circuit resistance is\nequivalent to the anode-to-electrolyte resistance because the structure-to-electrolyte resistance is negligible. For\nbracelet type anodes, the following equation from Design Practice SADP-X-100 (Eqn. 22, p. 33) is used to\ncalculate the anode-to-electrolyte resistance.\n\n0.315\nRA =\nA\nwhere -\nRA = the anode-to-electrolyte resistance for bracelet type anodes\nr = the electrolyte resistivity in ohm-cm\nA = the exposed surface area of the anode in cm2\n\nThen, the number of anodes can be calculated by dividing the total current requirement by the current output of\na single anode bracelet.\n\n## Saudi Aramco DeskTop Standards 61\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nExample\n\nWe will calculate the number of Galvalum III anodes needed to protect an offshore platform and a coated\nmarine pipeline. Assume that an impressed current system will also be installed to protect the platform. We\nwill use the following information to design the platforms galvanic anode system.\n\n## Current required: 250 amperes\n\nGalvalum III solution potential: -1.09 V versus Ag-AgCl\nGalvalum III anode dimensions: 28 cm x 28 cm x 304.8 cm (11\" x 11\" x 120\")\nGalvalum III anode weight: 566 kg (1,245 lbs.)\nGalvalum III consumption rate: 3.46 kg/A-yr\nWater resistivity: 15 ohm-cm\nRequired structure potential: -0.90 V versus Ag-AgCl\nNumber of Anodes\n\nThe current output of each anode is given by the equation I = ED/RA. The driving potential of the Galvalum III\nanode is\n\n## ED = 1.09 V - 0.90 V = 0.19 V versus Ag-AgCl.\n\nTo calculate the anode-to-electrolyte resistance of the anode, we must insert its dimensions and the water\nresistivity into the Dwight Equation. The effective diameter of the anode is\n\n## Therefore, the anode-to-electrolyte resistance is\n\nRV =\n0.159 8L\nl n 1 =\n( ) ( )\n0.159 15 ln 8 304.8 1 = 0.025 ohm\nL d 304.8 35.7\n\nand the current output of a single Galvalum III anode on the platform is\n\n## Saudi Aramco DeskTop Standards 62\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## The lifetime of one anode is\n\nW UF (\n566 kg .85 )( )\nY= = = 18 years\nCIA ( )(\n3.46 kg amp yr 7.6 amp )\nThis is greater than the design lifetime of 10 years.\n\nNow, using the following information, we will calculate the current requirement and number of Galvalum III\nanodes needed to protect the coated marine pipeline:\n\n## Length of pipeline: 4.5 km\n\nPipe diameter: 45.7 cm\nCurrent required: 14 amperes\nGalvalum III consumption rate: 3.46 kg/A-yr\n\n## N = 4500 m/150 m = 30 bracelets.\n\nNow we will make sure that the anodes will last over the design lifetime of 10 years. According to Standard\nDrawing AA-036335 (see table in Work Aid 5A), the net anode material weight of a bracelet for a 45.7 cm\ndiameter pipeline is 61 kg. Therefore, the lifetime of one anode bracelet is calculated as follows:\n\nY=\nW UF (61 kg )(0.85)\nC I (3.46 kg amp yr )(14 amps 30 bracelets) =\n= 32 years\n\nThe formulas and procedure used to design galvanic anode systems for marine structures and offshore pipelines\nare provided in Work Aid 5A.\n\n## Saudi Aramco DeskTop Standards 63\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Impressed Current System Design for Marine Structures\n\nThe driving potentials of impressed current anodes are much greater than galvanic anodes. Therefore, fewer\nimpressed current anodes are required to provide the same amount of current. However, their placement is\nmore critical to achieve adequate current distribution. An impressed current anode will tend to over-protect\nareas close to it and under-protect more remote areas. To improve the current distribution of impressed current\nanodes, the following methods are sometimes used:\nAn insulating shield is installed on the structure near impressed current anodes.\nImpressed current anodes are separated from the structure by at least 1.5 m.\n\nThe design of impressed current systems for marine structures involves determining:\nthe corrected current required\nthe number of impressed current anodes required\nthe rectifier voltage requirement\n\nAfter describing these calculations, an example, which demonstrates the design of an impressed current system\nto protect a marine platform, is provided.\nCorrected Current Requirement\n\nImpressed current anodes are considered 67-80% as effective as galvanic anodes. In the Arabian Gulf, 75%\neffectiveness is used in most design calculations. Therefore, we must modify the current requirement as\nfollows:\n\n## ICorr = I(1 + (100% %Efficiency)/100)\n\nwhere -\nICorr = corrected total current requirement for an impressed current system\nI = total current requirement for galvanic anode systems\nEfficiency = efficiency of the impressed current anodes\nNumber of Impressed Current Anodes Required\n\nThe number of impressed current anodes is calculated based on the maximum anode current output as follows:\nN = ICorr/IA\nwhere -\nICorr = corrected total current requirement for an impressed current system\nIA = the maximum current output of one impressed current anode\n\nThe maximum current output is the maximum current density of the anode material multiplied by the anode\nsurface area.\n\n## Saudi Aramco DeskTop Standards 64\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Rectifier Voltage Requirement\n\nSaudi Aramco sizes the rectifier to meet the total current requirement of the anodes based on a rectifier\nefficiency of 67%. The rectifier output voltage is given by the following formula:\n\nE = ICorrRC/Efficiency\n\n## The total circuit resistance, RC, is given by the following formula:\n\nR V + R LW\nR C = R RPL + R RNL +\nN\n\nwhere -\nRC = the circuit resistance of the entire impressed current system\nRRPL = the resistance in the positive lead wire from the rectifier to the junction box\nRRNL = the resistance in the negative lead wire from the structure to the rectifier\nN = the number of impressed current anodes\nRV = the resistance of a single impressed current anode (Dwight Equation)\nRLW = anode lead wire resistance\n\nNote that the structure-to-electrolyte resistance, RS, is omitted from the formula for RC. This is because RS is\nnegligible in seawater.\n\n## Saudi Aramco DeskTop Standards 65\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nExample\n\nWe will design an impressed current system to protect the previous offshore platform for which we designed a\ngalvanic anode system. However, assume that the platform is also electrically bonded to four conductor pipes.\n\n## Current required for platform: 251 amperes\n\nAnode material: Platinized niobium\nAnode dimensions: 7.6 dia x 76.2 cm (3\" dia. x 30\")\nAnode max. current output density: 40 mA/cm2\nWater resistivity: 15 ohm-cm\nAnode lead wire: No. 2 AWG, 50 meters long\nLead wire resistance: 0.531 x 10-3 ohm/m\nTotal resistance in both rectifier lead wires: 0.02 ohm\nCurrent requirement for conductor pipes: 3 amperes each\n\n## Corrected Current Requirement\n\nThe total current requirement for the platform and conductor pipes is\n\n## I = 251 A + (4)(3 A) = 263 A.\n\nThe corrected current required for an impressed current system is calculated as follows:\n\n## Saudi Aramco DeskTop Standards 66\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Rectifier Voltage Requirement\n\nThe output voltage is given by the equation E = ICorrRC. The total circuit resistance, RC, is calculated as\nfollows: (Remember, RS is negligible in seawater)\n\nR V + R LW\nR C = R RPL + R RNL +\nN\n\nThe anode-to-electrolyte resistance, RV, is calculated using the Dwight Equation as follows:\n\nRV =\n0.159 8L\n1 =\n( ) ln (\n0.159 15 8 76.2 )\n\n1 = 0.11 ohm\nl\nn\nL d 76.2 7.6\n\n## RLW = (50 m)(0.531 x 10-3 ohm/m) = 0.03 ohm.\n\nThe total resistance in the rectifier lead wires, RRPL + RRNL, is 0.02 ohm. Therefore, the circuit resistance is\n\n## RC = 0.02 + (0.11 + 0.03)/5 = 0.05 ohm.\n\nAllowing for a rectifier efficiency of 67%, the voltage requirement of the rectifier is\n\n## E = ICorrRC/Eff = (329 A)(0.05 ohms)/0.67 = 25 volts.\n\nFormulas and procedures used to design impressed current systems for marine structures are provided in Work\nAid 5B.\n\n## Saudi Aramco DeskTop Standards 67\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Work Aid 1: Data Base, Formulas, and Procedures to Design Cathodic\n\nProtection Systems for Buried Pipelines\n\nThis Work Aid provides formulas, and procedures to design galvanic and impressed current systems for buried\npipelines.\n\nWork Aid 1A: Data Base, Formulas, and Procedure to Design Galvanic Anode Systems for\n\nThis Work Aid provides requirements from Standard Drawing AA-036352, formulas, and a procedure for\ndetermining the number, circuit resistance, current output, and design life of galvanic anodes used to protect\nburied pipelines.\n\n## Dia. of Pipe (inches)\n\nPipe Length (meters) Up to 6\" Up to 12\" Up to 24\" Up to 36\" Over 36\"\n15 2 2 2 2 4\n30 2 2 4 4 6\n45 2 4 4 6 8\n60 2 4 6 8 10\n75 4 6 8 10 10\n90 4 6 10 10 12\n\nNOTES:\n\n## 1. Minimum number of anodes shall always be 2, regardless of pipe length or diameter.\n\n2. 100 lb. anodes are to be used only in Subkha areas. When substituting 100 lb. anodes for 60 lb.\nanodes, reduce anode quantity by one-half from that noted in table.\n\n3. One-half of the anodes shall be located on either side of crossing where practical on existing pipelines.\n\n## Saudi Aramco DeskTop Standards 68\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nFormulas\n\n## Galvanic Anode Current Output\n\nIA = ED/RC\nwhere -\nIA = anode current output (amperes)\nED = driving potential of the galvanic anode (volts)\nRC = circuit resistance (ohms)\n\nCircuit Resistance\n\nR LW + R V\nRC = R S +\nN\nwhere -\nRC = circuit resistance (ohms)\nRS = the structure-to-soil resistance (ohms)\nRLW = the lead wire resistance (ohms)\nRV = the resistance of a single vertical anode to earth (ohms)\nN = the number of anodes\n\n## Dwight Equation (for a single vertical anode)\n\n0.159\n1\n8L\nRS =\nl\nn\nL d\nwhere -\nRV = resistance of vertical anode to earth in ohms\nr = resistivity of soil in ohm-cm\nL = length of anode (or backfill column) in centimeters\nd = diameter of anode (or backfill column) in centimeters\n\n## Galvanic Anode Life\n\nW UF\nY=\nC IA\nwhere -\nY = life in years\nW = anode mass in kg\nUF = utilization factor\nC = actual consumption rate in kg/A-yr\nIA = anode current output in amperes\n\n## Saudi Aramco DeskTop Standards 69\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nProcedure\n\n## 1.1 Obtain the dimensions of buried pipe section.\n\n1.2 If using 60 lb. anodes, find number of anodes for pipe diameter and length in the Table at the\nbeginning of this Work Aid.\n\n## 2.1 Obtain the following information:\n\nanode dimensions (in centimeters)\nchemical backfill package dimensions (in centimeters)\nsoil resistivity\n\n2.2 If the anode is bare, determine the working diameter of the galvanic anode.\nIf anode is cylindrical, use its diameter (in centimeters)\nIf anode is not cylindrical, calculate its effective diameter (circumference/3.14).\n\n2.3 Calculate the anode-to-earth resistance by inserting the values for soil resistivity and the\nbackfill dimensions into the Dwight Equation. In Subkha, where no backfill package is used,\ninsert the anode dimensions.\n\n2.4 Divide the sum of the lead wire resistance and anode-to-earth resistance by the number of\nanodes. Add this resistance to the structure-to-electrolyte resistance to calculate the circuit\nresistance.\n\n## 3.0 Calculate the anode current output.\n\n3.1 Divide the anode driving potential by the circuit resistance calculated in Step 2.4.\n\n## 4.1 Obtain the following information:\n\nanode mass in kg\nanode utilization factor\nactual anode consumption rate in kg/A-yr\n\n4.2 Substitute the anode current output from Step 3.1 and the values from Step 4.1 into the\nGalvanic Anode Life formula and calculate the anode life.\n\n## Saudi Aramco DeskTop Standards 70\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nWork Aid 1B: Formulas and Procedure to Design Impressed Current Systems for Buried\nPipelines\n\nThis Work Aid provides formulas and procedures to calculate the number and spacing of impressed current\nanodes and the volume of coke breeze needed for the anode bed. This procedure assumes that you have\ndetermined the current requirement and allowable anode bed resistance.\n\nFormulas\n\n## Minimum Number of Anodes Based on Anode Maximum Current Density\n\nN = I/(dL x A)\n\nwhere -\nN = number of impressed current anodes\nI = total current required in milliamperes times 120%\nd = anode diameter in centimeters\nL = anode length in centimeters\nA = anode maximum current density in mA/cm2\n\n## Minimum Number of Anodes Based on Anode Consumption Rate\n\nY I C\nN=\nW\nwhere -\nN = number of impressed current anodes\nY = the impressed current system design life in years\nI = total current required in amperes times 120%\nC = anode consumption rate in kg/A-yr\nW = weight of a single anode in kg\n\n## Ragb = Rmax - (RS + RLW)\n\nwhere -\nRagb = the allowable anode bed resistance\nRmax = the maximum allowable circuit resistance (the rectifiers rated voltage minus\n2 volts, divided by its rated current output)\nRS = structure-to-electrolyte resistance\nRLW = total lead wire cable resistance\n\n## Saudi Aramco DeskTop Standards 71\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Sunde Equation (for multiple vertical anodes in parallel)\n\n0.159\n1 + (l n 0.656N\n)\n8L 2L\nR= l n\nNL d S\n\nwhere -\nR = resistance, in ohms, of N anodes in parallel and spaced S centimeters apart along a straight\nline.\n= soil resistivity in ohm-cm\nN = number of anodes\nL = length of anode (or backfill column) in centimeters\nd = diameter of anode (or backfill column) in centimeters\nS = anode spacing in centimeters\n\nCorrected Allowable Anode Bed Resistance (for use with Design Chart A in this Work Aid)\n\nR5000 = R(5,000/)\nwhere -\n\n## R5000 = allowable anode bed resistance corresponding to 5,000 ohm-cm soil\n\nR = allowable anode bed resistance of soil with resistivity of ohm-cm\n= soil resistivity in ohm-cm\n\n## Saudi Aramco DeskTop Standards 72\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nProcedure\n\n## 1.1 Obtain the following information:\n\nanode material\nanode weight (in kg)\nanode consumption rate\ncoke breeze backfill column dimensions (in centimeters)\nsoil resistivity (in ohm-cm)\ncurrent required\nallowable anode bed resistance\nstructure-to-electrolyte resistance\n\n1.2 Calculate the minimum number of anodes required by using the anode current density formula\nand anode consumption rate formula. Use the largest number of anodes calculated from the\ntwo formulas. Round up to the nearest multiple of 10.\n\n## 2.0 Determine the anode bed resistance.\n\n2.1 If the allowable anode bed resistance (Ragb) is not available, calculate Ragb by using the\nAllowable Anode Bed Resistance Formula.\n\n2.2 Correct the allowable anode bed resistance, Ragb, for soil with resistivity other than 5000\nohm-cm by using the Corrected Allowable Anode Bed Resistance formula.\n\n2.3 Use Design Chart A in Figure 30 to determine the optimum number and spacing of anodes so\nthat Rgb is less than the corrected value of Ragb. Ensure that the number of anodes is greater\nthan the minimum number from Step 1.2.\n\n3.0 Calculate the weight of coke breeze needed for the anode bed.\n\n## 3.1 Obtain the following information:\n\nanode diameter and length (in centimeters)\ncoke breeze column dimensions\ncoke breeze density\n\n3.2 Subtract the volume of one anode from the volume of the backfill column to obtain the net\nvolume of coke breeze.\n\n## Saudi Aramco DeskTop Standards 73\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n3.3 Multiply the net volume of coke breeze by 1.2 (for spillage) and by the number of anodes\nfrom Step 3.2.\n\n3.4 Multiply the total volume of backfill by the density of the coke breeze.\n\n10.0\nBackfill Column:\n7.0 L = 300 cm\nd = 20 cm\n5.0 = 5,000 ohm-cm\n\n## 3.0 305 cm spacing\n\n457 cm spacing\n2.0 610 cm spacing\n762 cm spacing\n914 cm spacing\n1.0\n\n0.7\n0.5\n\n0.3\n\n0.1\n2 10 20 30 40\nNumber of Anodes\n\nDesign Chart A\nFigure 30\n\n## Saudi Aramco DeskTop Standards 74\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Work Aid 2: Formulas and Procedure to Design Cathodic Protection\n\nSystems for Onshore Well Casings\n\nThis Work Aid provides formulas and procedures to design impressed current deep anode beds to protect\nonshore well casings. This procedure assumes that you have determined the current requirement and allowable\nanode bed resistance.\n\nFormulas\n\n## Minimum Number of Anodes Based on Anode Maximum Current Density\n\nN = I/(dL x A)\nwhere -\nN = number of impressed current anodes\nI = total current required in milliamperes times 120%\nd = anode diameter in centimeters\nL = anode length in centimeters\nA = anode maximum current density in mA/cm2\n\n## Minimum Number of Anodes Based on Anode Consumption Rate\n\nY I C\nN=\nW\nwhere -\nN = number of impressed current anodes\nY = the impressed current system design life in years\nI = total current required in amperes times 120%\nC = anode consumption rate in kg/A-yr\nW = weight of a single anode\n\nCircuit Resistance\n\n## RC = RRPL + RLW + RV + RS + RRNL\n\nwhere -\nRC = circuit resistance\nRRPL = the resistance in the positive lead wire from the rectifier to the junction box\nRLW = the equivalent resistance of the anode lead wires (the sum of the individual lead wire\nresistances divided by the number of lead wires)\nRV = the resistance of the anode bed as a single vertical anode\nRS = structure-to-electrolyte resistance\nRRNL the resistance in the negative lead wire from the well casing to the rectifier\n\n## Saudi Aramco DeskTop Standards 75\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n0.159 eff 8L\nRV = 1\nl d\nn\nL\n\nwhere -\n\n## RV = resistance of vertical anode to earth in ohms\n\neff = effective soil resistivity of the interval in ohm-cm\nL = length of the coke breeze column in centimeters\nd = diameter of deep anode hole in centimeters\n\nVC = (d2/4)H\n\nwhere -\n\n## d = diameter of the coke breeze column in meters\n\nH = height of the coke breeze column in meters\n\nProcedure\n\n## 1.1 Obtain the following information:\n\nanode material\nanode diameter and length (in centimeters) and weight (in kg)\nanode consumption rate\ncurrent required\nanode spacing\n\n1.2 Calculate the minimum number of anodes required by using the anode current density formula\nand anode consumption rate formula. Use the largest number of anodes calculated from the\ntwo formulas.\n\n1.3 Calculate the length of the coke breeze column. Allow at least 6 meters above the top anode\nand at least 1.5 meters below the bottom anode for the coke breeze backfill.\n\n## Saudi Aramco DeskTop Standards 76\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## 2.1 Obtain the following information:\n\neffective soil resistivity from Geonics measurement\nlength of coke breeze column (from Step 1.3)\ndiameter of coke breeze column\nmaximum allowable circuit resistance\nstructure-to-electrolyte resistance\n\n2.2 Calculate the deep anode bed resistance by inserting the effective soil resistivity and the\ndimensions of the coke breeze column into the Dwight Equation.\n\n2.3 Multiply the total length of the rectifier lead wires by both the lead wire resistance (in ohm/m)\nand 110%.\n\n2.4 Divide the total length of the anode lead wires by the number of lead wires. Multiply this\namount by the lead wire resistance (in ohm/m) and 120%.\n\n2.5 Add the resistances from Steps 2.2, 2.3, and 2.4 to the well casing-to-soil resistance. Make\nsure that this total circuit resistance is less than the maximum allowable circuit resistance,\nRmax. Rmax = (rectifier rated voltage - 2 volts)/ rectifier rated current output.\n\n## 3.1 Obtain the following information:\n\ncoke breeze density\ncoke breeze column dimensions\n\n3.2 Calculate the volume of coke breeze using the provided formula. Multiply the volume of coke\nbreeze by 120% (for spillage).\n\n3.3 Multiply the volume of coke breeze by the coke breeze density to obtain the weight of coke\nbreeze required.\n\n## Saudi Aramco DeskTop Standards 77\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Work Aid 3: Formulas and Procedures to Design Cathodic Protection\n\nSystems for Vessel & Tank Interiors\n\nThis Work Aid provides formulas and procedures to design galvanic and impressed current systems for the\ninterior of tanks and vessels.\n\nWork Aid 3A: Formulas and Procedure for the Design of Galvanic Anode Systems for\nVessel & Tank Interiors\nFormulas\n\n## Current Output of a Galvanic Anode in a Vessel or Tank\n\n1 1\nI = ED = ED\nRC R S + R LW + R V\nwhere -\nI = current output of the anode(s)\nED = anode driving potential\nRC = circuit resistance\nRS = structure-to-electrolyte resistance\nRLW = resistance of a single anode lead wire\nRV = the anode-to-electrolyte resistance of a single anode\n\n## Dwight Equation (for a single vertical anode)\n\n0.159\n1\n8L\nRV =\nl\nn\nL d\nwhere -\nRV = anode-to-electrolyte resistance of a single anode in ohms\n= electrolyte resistivity\nL = anode length in centimeters\nd = anode diameter in centimeters\n\n## Anode Life (galvanic anode)\n\nY W\nUF\nC I A\nwhere -\nY = life in years\nW = anode mass in kg\nUF = utilization factor\nC = actual consumption rate in kg/A-yr\nIA = anode current output in amperes\n\n## Saudi Aramco DeskTop Standards 78\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nProcedure\n\n## 1.0 Calculate the current output per anode.\n\n1.1 If you have the manufacturers performance chart for the anode, locate the protected potential\nof the structure on the horizontal or X axis. Move vertically up the chart until you intersect\nthe curve for the water resistivity of interest. Move horizontally along the chart and read the\nvalue of the anodes current output on the vertical or Y axis. Go to Step 2.1.\n\nCAUTION: Performance charts are developed based on specific design parameters. You must be sure\nthat the performance chart you use was developed for your particular situation.\n\n1.2 If you do not have the manufacturers performance chart, obtain the following information:\ntotal current required to protect the tank or vessel\nelectrolyte resistivity\nanode material\nanode diameter and length (in centimeters)\nmaximum allowable circuit resistance\nstructure-to-electrolyte resistance\n\n1.3 Insert the anode dimensions and water resistivity into the Dwight Equation to\ncalculate the anode-to-electrolyte resistance.\n\n1.4 Add the structure-to-electrolyte resistance, anode lead wire resistance, and the anode-to-\nelectrolyte resistance from Step 1.3 to calculate the circuit resistance.\n\n1.5 Subtract the required potential of the structure from the solution potential of the galvanic\nanode to calculate the driving potential of the anode.\n\n1.6 Divide the driving potential from Step 1.5 by the circuit resistance from Step 1.4 to calculate\nthe current output of a single galvanic anode.\n\n## 2.0 Determine the number of galvanic anodes.\n\n2.1 Divide the total current required by the anode current output from Step 1.6 to calculate the\nnumber of anodes required. Round up to the nearest integer.\n\n## Saudi Aramco DeskTop Standards 79\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## 3.1 Obtain the following information:\n\nanode mass in kg\nanode utilization factor\nanode actual consumption rate\n\n3.2 Divide the product of the anode mass and utilization factor by the product of the anode\nconsumption rate and anode current output calculated in Step 1.6.\n\n## Saudi Aramco DeskTop Standards 80\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nWork Aid 3B: Formulas and Procedure for the Design of Impressed Current Systems for\nVessel & Tank Interiors\n\nFormulas\n\n## Minimum Number of Anodes Based on Anode Maximum Current Density\n\nN = I/(dL x A)\n\nwhere -\nN = number of impressed current anodes\nI = total current required in milliamperes times 120%\nd = anode diameter in centimeters\nL = anode length in centimeters\nA = anode maximum current density in mA/cm2\n\n## Minimum Number of Anodes Based on Anode Consumption Rate\n\nY I C\nN=\nW\nwhere -\nN = number of impressed current anodes\nY = the impressed current system design life in years\nI = total current required in amperes times 120%\nC = anode consumption rate in kg/A-yr\nW = weight of a single anode\n\nCircuit Resistance\n\nR LW + R V\nR C = R RPL + + R S + R RNL\nN\nwhere -\nRC = the circuit resistance of the entire impressed current system\nRRPL = the resistance in the positive lead wire from the rectifier to the junction box\nN = the number of impressed current anodes\nRLW = anode lead wire resistance\nRV = the anode-to-electrolyte resistance of a single anode\nRS = structure-to-electrolyte resistance\nRRNL = the resistance in the negative lead wire from the structure to the rectifier\n\n## Saudi Aramco DeskTop Standards 81\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Dwight Equation (for a single vertical anode)\n\n0.159\n1\n8L\nRV =\nl\nn\nL d\nwhere -\nRV = anode-to-electrolyte resistance of a single anode in ohms\n= electrolyte resistivity\nL = anode length in centimeters\nd = anode diameter in centimeters\n\nProcedure\n\n## 1.1 Obtain the following information:\n\ntotal current required to protect the tank or vessel\nanode material and dimensions\nmaximum current density of the anode\n\n1.2 Calculate the minimum number of anodes required by using the anode current density formula\nand anode consumption rate formula. Use the largest number of anodes calculated from the\ntwo formulas. Round up to the nearest integer.\n\n## 2.1 Obtain the following information:\n\nstructure-to-electrolyte resistance\nrectifier to junction box lead wire resistance\nresistance in the lead wire from the tank or vessel to the rectifier\nwater resistivity\nrectifier voltage and current output ratings\n2.2 Calculate the anode-to-electrolyte resistance of a single anode by inserting the anode\ndimensions and the water resistivity into the Dwight Equation.\n2.3 Divide the sum of the lead wire resistance and the anode-to-electrolyte resistance by the\nnumber of anodes calculated in Step 1.2. To this resistance, add the structure-to-electrolyte\nresistance and the resistances in the positive and negative lead wires of the rectifier. This will\ngive you the total circuit resistance of the impressed current system.\n2.4 Divide the rated voltage of the rectifier by its output current rating to calculate the maximum\nallowable circuit resistance. Ensure that the circuit resistance you calculated in Step 2.3 is less\nthan the maximum allowable circuit resistance.\n\n## Saudi Aramco DeskTop Standards 82\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Work Aid 4: Formulas and Procedure to Design Cathodic Protection\n\nSystems for In-Plant Facilities\n\nThis Work Aid provides formulas and procedures to design impressed current systems to protect the bottom\nexterior of storage tanks using the earth potential shift formula.\n\nFormulas\n\n## For a single vertical anode\n\n0.5 I L2 + X 2 + L\nVx = ln\nL X\n\n## For a single horizontal anode\n\nI\nl n\n(0.5L )2 + X 2 + h 2 + 0.5L\nVx =\nL X 2 + h2\n\nwhere -\nVX = earth potential change at the tank center (volts)\nI = current flow (amperes)\n= soil resistivity (ohm-cm)\nL = anode backfill length (cm)\nX = horizontal distance from the anode to the center of the tank (cm)\nh = depth of burial to centerline of anode (cm)\n\n## Saudi Aramco DeskTop Standards 83\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nProcedure\n\n## 1.0 Determine the number and location of impressed current anodes.\n\n1.1 Select the location of the anodes within one-quarter of the tank radius from the tank wall\naccording to Standard Drawing AA-036355.\n\n1.2 Add the distance between one anode and the tank to the tank radius to obtain the radius of the\n\n1.3 Divide the anode header cable length by 20 m to obtain the minimum number of anodes\nrequired.\n\n## 2.1 Obtain the following information:\n\naverage tank native potential\nsoil resistivity\nanode and anode backfill dimensions\ndistance between the anodes and tank center\n\n2.2 Substitute the soil resistivity, anode distance, anode backfill length, and required earth\npotential shift (0.35 volts according to Saudi Aramco Standards) into the earth potential shift\nformula for a single vertical anode and solve for the current I, required.\n\n2.3 Divide the current flow by the number of anodes to obtain the estimated current required from\neach anode.\n\n3.0 Calculate the current required to protect the tank based on surface area and required current density.\n\n## Saudi Aramco DeskTop Standards 84\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n## Work Aid 5: Formulas and Procedures to Design Cathodic Protection\n\nSystems for Marine Structures\n\nThis Work Aid provides formulas and procedures to design galvanic anode and impressed current systems to\nprotect offshore platforms and submerged pipelines.\n\nWork Aid 5A: Data Base, Formulas, and Procedure for the Design of Galvanic Anode\nSystems for Marine Structures\n\nThis Work Aid provides requirements from Standard Drawing AA-036335, formulas, and a procedure for\ndetermining the number, circuit resistance, current output, and design life of galvanic anodes used to protect\nmarine platforms and pipelines.\n\nHALF SHELL ANODE BRACELET TYPE ANODE FOR PIPE SIZES 4\" THROUGH 60\"\n\n## Pipe Size Net Weight Nominal Weight\n\n10.2 cm (4\") NB 16 kg 24 kg\n15.2 cm (6\") NB 23 kg 31 kg\n20.3 cm (8\") NB 30 kg 39 kg\n25.4 cm (10\") NB 36 kg 46 kg\n30.5 cm (12\") NB 41 kg 51 kg\n35.6 cm (14\") OD 50 kg 61 kg\n40.6 cm (16\") OD 54 kg 66 kg\n45.7 cm (18\") OD 61 kg 74 kg\n50.8 cm (20\") OD 68 kg 82 kg\n55.9 cm (22\") OD 75 kg 89 kg\n61.0 cm (24\") OD 82 kg 96 kg\n66.0 cm (26\") OD 86 kg 109 kg\n71.1 cm (28\") OD 91 kg 116 kg\n76.2 cm (30\") OD 95 kg 120 kg\n81.3 cm (32\") OD 100 kg 127 kg\n86.4 cm (34\") OD 104 kg 132 kg\n91.4 cm (36\") OD 109 kg 138 kg\n106.7 cm (42\") OD 129 kg 161 kg\n116.8 cm (46\") OD 143 kg 177 kg\n121.9 cm (48\") OD 167 kg 184 kg\n132.1 cm (52\") OD 161 kg 204 kg\n152.4 cm (60\") OD 186 kg 230 kg\n\n## Saudi Aramco DeskTop Standards 85\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nFormulas\n\n## Current Output of a Galvanic Anode\n\nIA = ED/RC\nwhere -\nIA = anode current output in amperes\nED = the anode driving potential in volts versus Ag-AgCl\nRC = the circuit resistance in ohms\n\n## Circuit Resistance of a Galvanic Anode\n\nRC = RS + RA = RA\nwhere -\nRC = Circuit resistance in ohms\nRS = the structure-to-electrolyte resistance (approximately zero)\nRA = the anode-to-electrolyte resistance\n\nDwight Equation\n0.159\n1\n8L\nRA = R V =\nl d\nn\nL\nwhere -\n= the electrolyte resistivity in ohm-cm\nL = the length of the anode in centimeters\nd = the diameter of the anode in centimeters or the circumference\ndivided by p for non-cylindrical shapes\n\n## Number of Galvanic Anodes Required\n\nN = I/IA\nwhere -\nN = the number of anodes\nI = the total current required to protect the structure\nIA = the current output of a single anode\n\nY = W UF\nC IA\nwhere -\nY = anode life in years\nW = anode mass in kg\nUF = Utilization factor\nC = actual consumption rate in kg/A-yr\nIA = current output of one anode in amperes\n\n## Saudi Aramco DeskTop Standards 86\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nProcedure\n\n## 1.1 Obtain the following information:\n\nplatform surface area in seawater in m2\ncurrent density required in seawater in mA/m2\nplatform surface area below mud line in m2\ncurrent density required in mud in mA/m2\n\n1.2 To calculate the total current requirement, multiply the immersed surface area of the structure\nin seawater by Saudi Aramcos current density requirement. Multiply the surface area of the\nstructure below the mud line by Saudi Aramcos current density requirement. Add the two\ncurrent requirements together.\n\n## 2.1 Obtain the following information:\n\nanode solution potential in volts versus Ag-AgCl\nanode dimensions in centimeters\nanode weight in kg\nseawater resistivity in ohm-cm\nanode consumption rate in kg/A-yr\nanode utilization factor\ngalvanic anode design life in years\n\n2.2 If the anode is not cylindrical, determine its effective diameter by dividing its circumference\nby . Calculate the anode-to-electrolyte resistance of the anode by inserting its effective\ndiameter, length, and the electrolyte resistivity into the Dwight Equation.\n\n2.3 Subtract the required potential of the structure from the solution potential of the anode to\ncalculate the anode driving potential. Divide the anode driving potential by the anode-to-\nelectrolyte resistance from Step 2.2 to determine the current output of a single anode.\n\n2.4 Divide the total current required by the anode current output from Step 2.3 to calculate the\nnumber of anodes required. Round up to the nearest integer.\n\n## Saudi Aramco DeskTop Standards 87\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n2.5 Insert the weight of a single anode, utilization factor, consumption rate, and current output\nfrom Step 2.3 into the Galvanic Anode Lifetime formula. Ensure that the anode life is greater\nthan the required design life. If the anode life is less than the required design life, multiply\nthe number of anodes from Step 2.4 by the ratio of the design lifetime and calculated lifetime.\nThe result is the proper number of anodes required for the design life of the cathodic\nprotection system.\n\n3.0 Calculate the number of galvanic anode bracelets for marine pipelines.\n\n## 3.1 Obtain the following information:\n\npipeline surface area in seawater in m2\npipeline length in meters\npipeline diameter in cm\nanode consumption rate in kg/A-yr\nanode utilization factor\nanode design life in years\n\n3.2 To calculate the pipelines current requirement, multiply its surface area by Saudi Aramcos\nrequired current density of 2.5 mA/m2.\n\n3.3 Divide the length of the pipeline by 150 meters to calculate the number of anode bracelets\nrequired.\n\n3.4 Divide the total current requirement by the number of anode bracelets to calculate the current\noutput per anode bracelet. Locate the net weight anode weight per bracelet in the table\nprovided in this Work Aid.\n\n3.5 Verify that the anode bracelet will last over the required design life. Substitute the anode\nconsumption rate, current output, utilization factor, and net weight of anode material into the\ngalvanic anode life formula and solve for Y.\n\n## Saudi Aramco DeskTop Standards 88\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nWork Aid 5B: Formulas and Procedure for the Design of Impressed Current Systems for\nMarine Structures\nFormulas\n\n## ICorr = I(1 + (100% - %Efficiency)/100)\n\nwhere -\nICorr = corrected total current requirement for an impressed current system\nI = total current requirement (multiply total surface area by Saudi Aramcos current\ndensity requirement)\nEfficiency = efficiency of the impressed current anodes\n\n## Minimum Number of Anodes Based on Anode Maximum Current Density\n\nN = ICorr/(dL x A)\nwhere -\nN = number of impressed current anodes\nICorr = corrected total current requirement for an impressed current system in mA\nd = anode diameter in centimeters\nL = anode length in centimeters\nA = anode maximum current density in mA/cm2\n\nCircuit Resistance\n\nR V + R LW\nR C = R RPL + R RNL +\nN\nwhere -\nRC = the circuit resistance of the entire impressed current system\nRRPL = the resistance in the positive lead wire from the rectifier to the junction box\nRRNL = the resistance in the negative lead wire from the structure to the rectifier\nN = the number of impressed current anodes\nRV = the resistance of a single impressed current anode (Dwight Equation)\n\n## Saudi Aramco DeskTop Standards 89\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nDwight Equation\n0.159\n1\n8L\nRA = R V =\nl d\nn\nL\nwhere -\nRA = The anode-to-electrolyte resistance\n= the electrolyte resistivity in ohm-cm\nL = the length of the anode in centimeters\nd = the diameter of the anode in centimeters or the circumference divided by for non-\ncylindrical shapes\n\nProcedure\n\n## 1.0 Calculate the corrected current requirement.\n\n1.1 Add the current required to protect any conductor pipe and unprotected pipelines to the current\nrequired to protect the structure.\n\n1.2 Use the Current Requirement for Impressed Current Systems formula to calculate the corrected\ncurrent requirement.\n\n## 2.1 Obtain the following information:\n\nanode dimensions in centimeters\nanode maximum current density\n\n2.2 Calculate the minimum number of anodes required by using the anode current density\nformula. Round up to the nearest integer.\n\n## 3.1 Obtain the following information:\n\nanode dimensions in centimeters\nseawater resistivity in ohm-cm\n\n## Saudi Aramco DeskTop Standards 90\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\n3.2 Calculate the anode-to-electrolyte resistance of a single anode by inserting the anode\ndimensions and the seawater resistivity into the Dwight Equation.\n\n3.3 Divide the sum of the lead wire resistance and the anode-to-electrolyte resistance by the\nnumber of anodes calculated in Step 2.2. To this resistance, add the resistances in the positive\nand negative lead wires of the rectifier. This will give you the total circuit resistance of the\nimpressed current system.\n\n3.4 To calculate the voltage requirement of the rectifier, multiply the corrected current by the\ncircuit resistance. Divide this result by the rectifier efficiency to determine the actual voltage\nrequirement.\n\n## Saudi Aramco DeskTop Standards 91\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nGLOSSARY\n\nanode internal resistance The resistance from the anode to the outer edge of the backfill.\n\nanode-to-earth resistance The resistance between the anode, or backfill, and the soil.\n\n## backfill A low resistance, moisture-retaining material immediately surrounding a\n\nburied impressed current anode for the purpose of increasing the effective\narea of contact with the soil and thus reducing the resistance to earth.\nCalcined petroleum coke backfill is commonly used as backfill for deep and\nsurface anode beds in Saudi Aramco.\n\nconductor pipe Tubular members through which oil or gas wells are drilled and then through\nwhich casing and tubing are inserted and often grouted into place.\n\ncurrent density The direct current per unit are generally expressed as amperes per square\nmeter or milliamperes per square meter. Current density to achieve cathodic\nprotection varies depending on the environment and metal being protected.\n\ndeep anode bed A type of anode bed that uses a drilled vertical hole to contain\nimpressed current anodes.\n\ninsulated flange A flanged joint used to electrically isolate pipelines and systems. The flange\nfaces and securing bolts are electrically insulated from each other by\n\npolarization The change of potential of a metal surface resulting from the passage of\ncurrent to or from an electrolyte.\n\nprotective potential A term used in cathodic protection to define the minimum potential required\nto suppress corrosion. Protective potential depends on the structure metal\nand the environment.\n\nremote earth The area(s) in which the structure-to-earth potential change is negligible with\nchange in reference electrode position away from the structure.\n\n## Saudi Aramco DeskTop Standards 92\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nshielding The act of preventing or diverting cathodic protection current from reaching a\nstructure. Shielding may be caused by a non-metallic barrier or by metallic\nstructures that surround the structure to be protected.\n\n## structure-to- electrolyte The potential difference between a buried or immersed metallic\n\npotential structure and the electrolyte surrounding it, measured with a\nreference electrode in contact with the electrolyte.\n\nsurface anode bed A type of anode bed that uses vertically or horizontally placed impressed\ncurrent or galvanic anodes.\n\nutilization factor The factor determined by the amount of anode material consumed when the\nanode can no longer deliver the current required.\n\n## Saudi Aramco DeskTop Standards 93\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nAPPENDIX 1\n\n## SAES-B-068 Electrical Area Classification\n\nSAES-P-100 Basic Electrical Design Criteria\nSAES-P-107 Overhead Power Distribution (SCECO Standard)\nSAES-P-111 Grounding\nSAES-Q-001 Criteria for Design and Construction of Concrete Structures\nSAES-X-300 Cathodic Protection Marine Structures\nSAES-X-400 Cathodic Protection of Buried Pipelines\nSAES-X-500 Cathodic Protection Vessel and Tank Internals\nSAES-X-600 Cathodic Protection In-Plant Facilities\nSAES-X-700 Cathodic Protection of Onshore Well Casings\nGI 482.002 Commissioning Procedures for Cathodic Protection Installations\n\n## AB-036008 Lidan anode - Pile Mounted\n\nAA-036069 Galvanic Anodes at Thrust Anchors\nAA-036108 Offshore Negative Terminal Box\nAB-036272 Deep Anode Bed Steel Cased Hole\nAB-036274 Junction Box 5-Terminal\nAB-036275 Junction Box 12-Terminal\nAA-036276 Splice Box; Multi-Purpose Details\nAA-036277 Bond Box 5-Terminal\nAA-036278 Deep Anode Bed Scrap Steel\nAA-036280 Photovoltaic Power System\nAA-036304 Pile Mounted Anodes for Offshore\nAA-036335 Half Shell Bracelet Type Anode, for Pipe Sizes 4\" through 60\"\nAA-036336 Half Shell Bracelet Type Anode, for Pipe Sizes 26\" through 48\"\nAA-036346 Surface Anode Bed Details Horizontal and Vertical Anodes\nAA-036347 Junction Box 20-Terminal\nAA-036348 Anode Installation Details Galvanic and Impressed, Offshore Structures\nAA-036349 Bond Box 3-Terminal\nAA-036350 Bond Box 2-Terminal\nAA-036351 Marker Plate Details\n\n## Saudi Aramco DeskTop Standards 94\n\nEngineering Encyclopedia Cathodic Protection\nDesigning Cathodic Protection Systems\n\nAA-036352 Galvanic Anodes for Road and Camel P/L Crossings, P/L Repair Locations, Installations\nand Details\nAA-036353 Water Storage Tanks Impressed Current\nAA-036354 Water Storage Tanks Galvanic Anodes\nAA-036355 Tank Bottom Impressed Current Details\nAA-036356 Deep Anode Bed Details, Aquifer Penetrating\nAA-036378 Rectifier Installation Details\nAB-036381 Thermite Welding of Cables to Pipelines & Structures\nAA-036384 Junction Box, Offshore Anode\nAA-036385 Deep Anode Bed Details, Non-Aquifer Penetrating\nAA-036409 Replacement Galvanic Anodes for Offshore Structures & P/Ls\nAB-036478 Magnesium Anode Installation at P/L Repair Locations Layout & Details\nAC-036524 Galvanic Anode Details Submarine Pipelines\nAB-036540 Mounting Support Details for Junction Boxes\nAB-036558 Standard Insulating Assemblies for Ring Joint Flanges with Gask-O-Seal Filler Gaskets\nAA-036674 Bonding Methods for Onshore Pipelines and Flow Lines\nAA-036675 Direct Buried Electric D-C Cathodic Protection Positive or Negative Cable\nAA-036761 Lead Silver Anode Seabed Installation Details\nAC-036762 Crude and Product Tank Internal Galvanic Anode Installation\nAA-036782 Bond Box, 2-Terminal for Insulating Devices\nAE-036785 Symbols for Cathodic Protection\nAB-036787 Road Crossings Installation In Plant (Plastic Envelope)\nAB-036907 Test Stations For Buried Pipelines, Pipeline Kilometer Markers\n\n## 02-AMSS-008 Insulating Spools and Joints\n\n17-AMSS-004 Constant Voltage Rectifiers\n17-AMSS-005 Phase Controlled Rectifiers\n17-AMSS-006 Galvanic Anodes\n17-AMSS-007 Impressed Current Anodes\n17-AMSS-008 Cathodic Protection Junction Boxes\n17-AMSS-012 Photovoltaic Power Supply\n17-AMSS-017 Cathodic Protection Cables" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8475086,"math_prob":0.93914264,"size":81395,"snap":"2019-51-2020-05","text_gpt3_token_len":19606,"char_repetition_ratio":0.21282451,"word_repetition_ratio":0.23085444,"special_character_ratio":0.22600897,"punctuation_ratio":0.08695362,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96101457,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-12-11T09:30:00Z\",\"WARC-Record-ID\":\"<urn:uuid:ef7d26d1-30aa-48a4-ab8d-bf3f31d90845>\",\"Content-Length\":\"607493\",\"Content-Type\":\"application/http; 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https://artofproblemsolving.com/wiki/index.php?title=2021_AMC_10A_Problems/Problem_2&diff=cur&oldid=152470	[ "# Difference between revisions of \"2021 AMC 10A Problems/Problem 2\"\n\n## Problem\n\nPortia's high school has", null, "$3$ times as many students as Lara's high school. The two high schools have a total of", null, "$2600$ students. How many students does Portia's high school have?", null, "$\\textbf{(A)} ~600 \\qquad\\textbf{(B)} ~650 \\qquad\\textbf{(C)} ~1950 \\qquad\\textbf{(D)} ~2000\\qquad\\textbf{(E)} ~2050$\n\n## Solution 1 (Two Variables)\n\nThe following system of equations can be formed with", null, "$p$ representing the number of students in Portia's high school and", null, "$l$ representing the number of students in Lara's high school.", null, "$$p=3l$$", null, "$$p+l=2600$$ Substituting", null, "$p$ with", null, "$3l$ we get", null, "$4l=2600$. Solving for", null, "$l$, we get", null, "$l=650$. Since we need to find", null, "$p$ we multiply", null, "$650$ by 3 to get", null, "$p=1950$, which is", null, "$\\boxed{\\text{C}}$\n\n-happykeeper\n\n## Solution 2 (One Variable)\n\nSuppose Lara's high school has", null, "$x$ students, so Portia's high school has", null, "$3x$ students. We have", null, "$x+3x=2600,$ or", null, "$4x=2600.$ The answer is", null, "$$3x=2600\\cdot\\frac 34=650\\cdot3=\\boxed{\\textbf{(C)} ~1950}.$$\n\n~MRENTHUSIASM\n\n## Solution 3 (Arithmetic)\n\nClearly,", null, "$2600$ is", null, "$4$ times the number of students in Lara's high school. Therefore, Lara's high school has", null, "$2600\\div4=650$ students, and Portia's high school has", null, "$650\\cdot3=\\boxed{\\textbf{(C)} ~1950}$ students.\n\n~MRENTHUSIASM\n\n## Solution 4 (Answer Choices)\n\n### Solution 4.1 (Quick Inspection)\n\nThe number of students in Portia's high school must be a multiple of", null, "$3.$ This eliminates", null, "$\\textbf{(B)},\\textbf{(D)},$ and", null, "$\\textbf{(E)}.$ Since", null, "$\\textbf{(A)}$ is too small (as it is clear that", null, "$600+\\frac{600}{3}<2600$), we are left with", null, "$\\boxed{\\textbf{(C)} ~1950}.$\n\n~MRENTHUSIASM\n\n### Solution 4.2 (Plug in the Answer Choices)\n\nFor", null, "$\\textbf{(A)},$ we have", null, "$600+\\frac{600}{3}=800\\neq2600.$ So,", null, "$\\textbf{(A)}$ is incorrect.\n\nFor", null, "$\\textbf{(B)},$ we have", null, "$650+\\frac{650}{3}=866\\frac{2}{3}\\neq2600.$ So,", null, "$\\textbf{(B)}$ is incorrect.\n\nFor", null, "$\\textbf{(C)},$ we have", null, "$1950+\\frac{1950}{3}=2600.$ So,", null, "$\\boxed{\\textbf{(C)} ~1950}$ is correct.\n\nFor completeness, we will check answer choices", null, "$\\textbf{(D)}$ and", null, "$\\textbf{(E)}:$\n\nFor", null, "$\\textbf{(D)},$ we have", null, "$2000+\\frac{2000}{3}=2666\\frac{2}{3}\\neq2600.$ So,", null, "$\\textbf{(D)}$ is incorrect.\n\nFor", null, "$\\textbf{(E)},$ we have", null, "$2050+\\frac{2050}{3}=2733\\frac{1}{3}\\neq2600.$ So,", null, "$\\textbf{(E)}$ is incorrect.\n\n~MRENTHUSIASM\n\n## Video Solution 1 (Very Fast & Simple)\n\n~ Education, the Study of Everything\n\n## Video Solution 2 (Setting Variables)\n\nhttps://youtu.be/qNf6SiIpIsk?t=119 ~ThePuzzlr\n\n## Video Solution 3 (Solving by Equation)\n\nhttps://www.youtube.com/watch?v=aOpgeMfvUpE&list=PLexHyfQ8DMuKqltG3cHT7Di4jhVl6L4YJ&index=1 ~North America Math Contest Go Go Go\n\n- pi_is_3.14\n\n~savannahsolver\n\n~IceMatrix\n\n~MathWithPi\n\n## See Also\n\n 2021 AMC 10A (Problems • Answer Key • Resources) Preceded byProblem 1 Followed byProblem 3 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 All AMC 10 Problems and Solutions\n\nThe problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.", null, "Invalid username\nLogin to AoPS" ]	[ null, "https://latex.artofproblemsolving.com/7/c/d/7cde695f2e4542fd01f860a89189f47a27143b66.png ", null, "https://latex.artofproblemsolving.com/5/9/2/592c1d4eacb3ec769d6ac9542733c10632610d91.png ", null, "https://latex.artofproblemsolving.com/8/8/9/889914608a80f5ba7125667dde3bfbf29ec8465b.png ", null, "https://latex.artofproblemsolving.com/3/6/f/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png ", null, "https://latex.artofproblemsolving.com/9/b/2/9b25f8e64b484493fda944d25cad453423041fe6.png ", null, "https://latex.artofproblemsolving.com/8/c/2/8c2224957ef77b34e819f72306e49697a7add896.png ", null, "https://latex.artofproblemsolving.com/7/6/c/76ca5c7bf28e15fd26b38ca7a83517cc5dafd198.png ", 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https://www.lcaoutreach.org/habitability-of-planets-2/sidebar-integrating-the-planck-equation-over-a-selected-waveband/	[ "# Sidebar. Integrating the Planck equation over a selected waveband\n\nSidebar. Integrating the Planck equation over a selected waveband.\n\nStars act essentially as blackbodies, in the distribution of their electromagnetic radiation into various wavelengths. So do most parts of planetary surfaces – soils, rocks, vegetation (if any), inhabitants (if any). It’s important to know how much energy goes into different wavebands:\n\n• For sunlight / starlight to drive photosynthesis by organisms that are the primary producers, that is, the organisms that capture energy usable by all other organisms by making high-energy compounds such as sugars:\n• Only a certain range of wavelengths is usable by these organisms. On Earth the dominant green plants use radiation with wavelengths in the range from 400 nanometers (nm) to 700 nm. Some bacteria can use light down as far as 850 nm. There are very strong reasons from the photophysics of molecules for this restriction, as noted in the text in the discussion of the wonder molecule, chlorophyll. Longer wavelengths don’t have enough energy content per photon to move electrons among molecules. Shorter wavelengths in the ultraviolet can be damaging to DNA; plants synthesize flavonoids and other protectants.\n• So, we want to calculate what fraction of the star’s energy at the planet is in the good photosynthetically active region. Really, it’s the number of photons or the photon flux density, which I explain shortly.\n• We also want to calculate the flux density of photons in the damaging UV region. Together with the knowledge of the UV-shielding compounds in the planet’s atmosphere, we can estimate a safety margin.\n• For thermal infrared radiation, TIR, which is the means by which the planet surface (and all the organisms on it) gets rid of absorbed energy to balance the energy budget:\n• There is a total radiant energy emitted, with a flux density proportional to the fourth power of the absolute temperature, I = σT4, and more accurately with an emissivity factor less than 1.0, εσT4. Often, the emissivity is very close to 1.0 so we can just apply a total factor – say, ε = 0.96 for most stuff on Earth, including our own skin.\n• Sometimes the distribution among wavebands within the TIR is important. This is critical in calculating the greenhouse effect because the greenhouse gases in the atmosphere all absorb in fairly narrow but important bands. Knowing the emissivities in various TIR bands can also be important in usingTIR fluxes measured remotely to infer the surface temperature accurately. This is readily done on Earth, less so on Mars.\n\nFor energy balance, it’s the energy in a waveband that can be absorbed that counts – photons are “weighted” in significance by their energy content, proportional to their frequency or the inverse of their wavelength. A photon of blue light at, say, 440 nm, has 50% more energy than a photon of red light at 660 nm. For photosynthesis, all absorbed photons count the same. A photon of blue light creates an electronically excited state of chlorophyll that within a picosecond or so relaxes to another excited state of lower energy, the same state created by absorbing a photon of red light. Thus, in photosynthesis, it’s the number of photons, not their energy content, that counts. The common measure of radiation for studies of photosynthesis is the photosynthetic photon flux density, or PPFD, quoted in (micro) moles of photons (Avogadro’s number of photons) per square meter per second. For other biophysical reactions such as skin tanning or reddening or mutagenesis for skin cancer or similar effects in plants, the case is similar – the number counts within each narrow waveband, with one or more wavebands for each target molecule in us or other organisms, such as DNA or flavonoid protectants that can be overwhelmed.\n\nWe may examine two cases – radiation from the Sun reaching the Earth, and radiation for the much cooler star, Proxima Centauri, our nearest neighbor star, reaching its planet b, or Proxima Centauri b (PCb). This small planet was discovered in 2016 and touted as possibly habitable because the final intensity of radiation at its surface is not too far off what the intensity of solar energy is on Earth.\n\nWe may calculate (1) the total energy flux density and (2) the flux density as number of photons per area per second, in the\n\n• UV waveband, especially UVC at 200-280 nanometers (nm), to evaluate the threat to organisms;\n• visible waveband, 400-700 nm, which is also the main waveband for photosynthesis on Earth; there is also an extension to 850 nm for minor photosynthesizers, some bacteria; and\n• selected thermal infrared wavebands where greenhouse gases absorb to trap outgoing thermal infrared radiation, thus keeping our planet warm.\n\nWe need the expression for the number of photons coming off the surface of the star’s surface.\n\n• If you wish to skip to the results, they’re at the very end here.\n\nThat law, for a blackbody such as a star approximates well, was discovered by Max Planck. There’s a lot of physics behind the expression for the number of photons per area per solid angle that a blackbody emits, within a small range of wavelengths. It inherently involves the nature of light as quantized into the discrete particles, the photons, where at a given wavelength of radiation all photons have the same discrete amount of energy, (“aitch new”) = hc/ λ. Here, h is the universal Planck’s constant, discovered, of course, by Max Planck, ν is the frequency of the light, c is the speed of light, and λ is the wavelength of light. Planck derived an expression that fit observations for the energy emitted per area per solid angle within the range of wavelengths from λ to λ+dλ. Here, is a small increment in wavelength:", null, "If you haven’t had calculus, this formula might not mean too much to you. Here, T is, of course, the absolute temperature as discussed earlier. Basically, L(λ) is the increment in energy flow from considering a small range of wavelength, . There are several universal physical constants, h, which is Planck’s constant (connected to the finding that all energy is quantized!), c, the velocity of light, and k, Boltzmann’s constant (connected to how temperature is related to energy). These all have fascinating stories of their own, which I won’t go into here. In any case, you can see that a hot star such as our Sun puts out much more energy per area than a cool star such as Proxima Centauri; the total energy output is proportional to the area under the curve for each star.\n\nWe can also express the rate in terms of the energy emitted in a range of frequencies between ν and ν+dν,", null, "For computing rates of photosynthesis (or for DNA damage from UV wavebands) we really want the number of photons, not the amount of energy. Bacteria or plants do photosynthesis with each photochemical reaction driven by a photon, irrespective of its energy as long as that energy is above a minimum (not strictly true, given different degrees of absorption at different wavelengths, but this is a start). The rate expressed in number of photons is this last equation divided by the energy of a photon,", null, "and, back to wavelength terms,", null, "If we count all photons emitted, over all frequencies (all wavelengths, all energies), we get two expressions familiar in physics. The total amount of energy emitted is", null, "The factor in front of T4 is the Stefan-Boltzmann constant, 5.67×10-8 W m-2K-4. The total number of photons emitted is", null, "Here, ζ(3) is a mathematical constant, the Riemann zeta function of 3, with a value of about 1.202 as a simple (pure) number.\n\nLet’s focus on the photon count, not the energy, for those three wavebands. We need to integrate the expression for LP(λ) over the range of wavelengths for each case. There are two basic ways to do this. The first is numerical: Divide the waveband into equal small parts – e.g., 400-410 nm, 410-420 nm, and so on – calculate LP in each small part, add them up, and multiply by the interval Δλ. This is clearly an approximation and it works well only if the interval is short enough that LP(λ) is not changing fast within the interval. It’s easy to program in any language (Python, Fortran, …) or Microsoft Excel. I have two versions of the latter, currently set up to isolate the UV, PPFD (= the visible waveband, also), and infrared. The second version uses finer steps at short wavelengths.\n\nThe second route is analytical, using exact mathematical forms derived with integral calculus. The formulas are infinite series, that is, expressions with an infinite number of terms, but the terms converge to a very accurate answer so that only a moderate number is needed. This calculation is presented nicely on one website put up by a commercial corporation, GATS, Inc., of Newport News, VA. They start by converting from wavelength to a new variable, called wavenumber, the number of wavelengths in a meter, 1/λ, given the symbol σ (not the same as the Stefan-Boltzmann constant!). They actually use σ in older units of wavelengths per cm, not per m, but I’ll stay really metric here (really “SI”).\n\nWe want to do the integration (this is calculus) over all wavenumbers from a minimum value of σ, which is 1/(850 nm) = 1/(8.5×10-7 m) = 1.176×106 m-1, to infinity. Well, that’s too far, but the number of photons of very high energy (very high wavenumber, very short wavelength) is so small that doing the integral to infinity makes very little difference. After all, we are making a guess about the lower limit that bacteria can use, anyway. The math is shown nicely one of the company’s deeper webpages. The result is", null, "Here, they defined a new variable of integration, x=100 hcσ/(kT). We can do this sum readily because only the first several terms are big enough to matter.\n\nThe results\n\nThe fluxes reported here in the 3 shortwave radiation bands are, first, in photons per square meter per second at the surface of the star. The next line converts that to moles of photons, with a mole being Avogadro’s number of them, just as for molecules expressed as moles. The line after that is again in moles but projected to the distance of the planet. For this we use the law that the intensity (flux density) falls off as 1/r2 – that is the flux density at the planet is lower than that at the star’s surface by the factor (rstar/rorbit)2. Here, rstar is the radius of the star and rorbit is the radius of the planet’s orbit around the star, taken as closely circular.", null, "There are notable comparisons.\n\n• The UV flux at Proxima Centauri b is trivially low, only 1% as high as on Earth; a cool star has a tiny fraction of its output at short wavelengths. You would not get sunburned there. You would only get fried by stellar flares and the stellar wind, which also have taken away the atmosphere. Any remanent atmosphere would also condense on the permanently cold side of the tidally-locked planet.\n• A 10% increase in stellar temperature from that of the Sun doubles the UV flux (this calculation is approximate). Hot stars are anathema for life, for this as well as for short lifetimes for evolution to occur.\n• The visible light on PCb is 1/5th as strong as on Earth, while the 10% hotter star at the same distance from Earth as our Sun is give 52% more visible light. Wear sunglasses, though not as dense as the peril-sensitive glasses in The Hitchhiker’s Guide to the Galaxy.\n• If we allow that photosynthetic organisms might use starlight to the longest wavelengths that Earth’s organisms do, then PCb catches up somewhat, getting about ½ as much useful flux density as we do on Earth. On Earth the organisms using the range 700-850 nm are in minor niches, shaded out in general by green plants.\n• Going to integration over all wavelengths to get total energy interception, we wee that PCb is not terribly cold, better than Mars (see also the main text about planetary temperatures). a 10% hotter star, alas, would cook us if the Earth were to absorb the same 71% fraction of stellar energy as it does now.\n\nI repeat here a figure from the main text, showing where the energy lies for the 3 stars. Note that the plots are all relative to the peak in the energy. The peak energy for Proxima Centauri is only 5% that for our Sun, and our Sun has a peak that’s only 60% that of the star that’s 10% hotter. If we draw the figure on the absolute energy scale, Proxima Centauri almost gets “lost in the noise.” That’s the second figure, below." ]	[ null, "https://www.lcaoutreach.org/wp-content/uploads/2021/02/word-image-30.png", null, "https://www.lcaoutreach.org/wp-content/uploads/2021/02/word-image-31.png", null, "https://www.lcaoutreach.org/wp-content/uploads/2021/02/word-image-33.png", null, "https://www.lcaoutreach.org/wp-content/uploads/2021/02/word-image-35.png", null, "https://www.lcaoutreach.org/wp-content/uploads/2021/02/word-image-37.png", null, "https://www.lcaoutreach.org/wp-content/uploads/2021/02/word-image-38.png", null, "https://www.lcaoutreach.org/wp-content/uploads/2021/02/word-image-39.png", null, "https://www.lcaoutreach.org/wp-content/uploads/2021/02/word-image-17.jpeg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.93172663,"math_prob":0.97110635,"size":12371,"snap":"2022-27-2022-33","text_gpt3_token_len":2779,"char_repetition_ratio":0.13293442,"word_repetition_ratio":0.008438818,"special_character_ratio":0.2185757,"punctuation_ratio":0.10851761,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9705471,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"im_url_duplicate_count":[null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-08-17T22:33:09Z\",\"WARC-Record-ID\":\"<urn:uuid:c9b08023-3cee-42cf-86f1-6d5bbccc8f24>\",\"Content-Length\":\"59384\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b91a0150-d289-440d-ab7f-d197dee8255c>\",\"WARC-Concurrent-To\":\"<urn:uuid:06236444-630c-4cdf-8613-4ad6276ae733>\",\"WARC-IP-Address\":\"67.205.13.137\",\"WARC-Target-URI\":\"https://www.lcaoutreach.org/habitability-of-planets-2/sidebar-integrating-the-planck-equation-over-a-selected-waveband/\",\"WARC-Payload-Digest\":\"sha1:IE62IOE3EKXKMK5NFWPS6OSJEX4VBPYE\",\"WARC-Block-Digest\":\"sha1:RTNDG5WHJXEBH4X6BCYQUQM23XE2TK7E\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-33/CC-MAIN-2022-33_segments_1659882573118.26_warc_CC-MAIN-20220817213446-20220818003446-00580.warc.gz\"}"}